Project-1-6372- Life Expectancy Analysis
Jessica McPhaul, Ivan Chavez, Rafia Mirza
2024-02-11
##RPubs RPubs
Introduction and Objective
This project focuses on the analysis of life expectancy using WHO data to model country life expectancy based on economic and health factors using regression models. The primary aim is to explore key relationships using regression analysis and interpret these in a structured manner.
Data Description and Processing
Data from WHO covering health and economic indicators were used. The dataset includes 2,848 records, each representing annual health and economic indicators for various countries. Variables such as Life Expectancy, Adult Mortality, Alcohol Consumption, GDP per Capita, and others were included. Preprocessing involved handling missing values, eliminating columns with a substantial number of missing values, and standardizing column names.
Exploratory Data Analysis (EDA)
EDA revealed patterns and correlations. Key visualizations include box plots showing life expectancy variation by economy status and time, and correlation matrices. Scatter plots of selected variables against life expectancy were also used. These analyses provided foundational insights for model building.
Objective 1: Regression Model Development and Analysis
- Model Fitting Approach: Enhanced linear model with iterative refinement process, emphasizing statistically significant variables.
- Variable Selection: Based on statistical significance and EDA, with notable predictors including Adult Mortality, BMI, and GDP per Capita.
- Feature Selection Summary: Manual selection and statistical techniques ensured inclusion of significant predictors.
- Final Regression Model Definition: Incorporated predictors like Adult Mortality, Alcohol Consumption, GDP per Capita, etc.
- Coefficients Summary Table: Provided estimates, standard errors, and significance levels for each predictor.
- Regression Coefficient Interpretation: Interpretations focused on magnitude and significance of key coefficients.
- Model Evaluation and Comparison: Employed visualizations and statistical analyses like ANOVA for evaluation.
Objective 2: Further Model Evaluation and Comparative Analysis
Evaluation and comparison of different regression models for life expectancy prediction, with a focus on predictive accuracy and suitability.
- Advanced Model Development and Iterative EDA: Increased complexity of the model with sophisticated techniques and ongoing EDA.
- In-Depth Model Refinement: Continuous integration of complex terms into the model based on EDA insights.
Methodology and Decision Making
- Comprehensive Model Comparison: Included enhanced linear models and nonparametric models like Random Forest or KNN.
- Extended Model Evaluation Metrics: Employed MSE, R-squared/Adjusted R-squared, AIC, BIC, etc.
- Detailed Comparative Analysis and Final Recommendations: Focused on performance, strengths, and limitations of models in the context of life expectancy prediction.
- Key Focuses of Our Approach: Included iterative process, balance and flexibility, and data-driven decision-making.
Conclusion
Model_1 was identified as the best model for our analysis. RFE, VIF, AIC/BIC/MSE/RMSE/MAE/ANOVA/MANOVA processes were used to train, test, and validate the models.
Part A: Initial Exploration & Introduction to Dataset
Dataset Composition
- Observations: 2,848 entries spanning from 2000 to 2015.
- Variables: 23, including both health-related and economic indicators.
- Data Types: Combination of categorical (Country, Region) and numerical (health indicators, economic measures).
- Year Coverage: Data spans from 2000 to 2015, median and mean year of 2008.
- Country and Region: 2,848 entries for each, showcasing geographical diversity.
- Metrics: Include economy status (developed: 1, undeveloped: 0), GDP per capita, population, income composition, and schooling years.
Key Data Findings
Health Variables
- Life Expectancy: Ranges from 39.4 to 83.8 years (Median: 71.45 years).
- Adult Mortality: Median rate of 144 per 1000, range: 49.38 to 719.36.
- Alcohol Consumption: Ranges from 0 to 17.87 liters/person/year (Average: 4.83 liters).
- Infant Deaths: Minimum 1.8, maximum 138.1 per 1000 live births (Mean: 30.08).
- Vaccination Rates (Polio, Hepatitis B): Range from 0% to 99%.
- Measles Cases: 0 to 212,183 cases, indicating high variance.
- BMI: Range from 0 to 79.3.
- Under-Five Deaths: 0 to 2500 deaths.
- Diphtheria Vaccination Rate: Range from 0% to 99%.
- HIV Incidence: 0 to 43.5 cases per 1000 live births.
- Thinness in Teens/Children: Maximum values at 27.7% and 28.6%.
Economic Indicators
- GDP Per Capita: $255 to $118,514.
- Healthcare Expenditure: Wide variation, correlated with GDP.
- Schooling Years: 0 to 20 years (Average: 12 years).
- Health Expenditure (% of GDP): 0% to 17.6%.
- GDP per Capita: 0 to 119,172.7 USD.
- Population: 0 to ~1.29 billion.
- Income Composition/Schooling: 0 to 0.948 and 1.1 to 14.1 years, respectively.
- Economy Status (Developed): 20.79% of entries from developed economies.
Data Cleaning Process
- Missing Data: Imputed with median for numerical (e.g., median GDP $5,962) and mode for categorical.
- Data Completeness: All 23 variables retained post-cleaning.
Overview
- The dataset comprises a diverse range of health indicators and socio-economic factors across various countries and regions from 2000 to 2015.
- Numerical variables cover a wide array of data points including year, economic status, life expectancy, adult mortality rates, and more.
Key Statistics
- Life Expectancy: Median of 71.45 years, with a broad range.
- Adult Mortality Rates: Median of 163.28, maximum of 719.36.
- Alcohol Consumption: Average of 4.83 liters, demonstrating wide variability.
- Economic Diversity: Reflected in disparities in healthcare expenditure and GDP per capita.
- Nutritional Status: Indicated by a broad range of BMI values.
- Health Outcomes: Including infant/under-five deaths and disease incidences.
- Socio-Economic Factors: Like income composition and schooling, crucial for assessing development impacts on health.
Data Readiness
- Insight Potential: The dataset is well-suited for trend analysis and predictive modeling, given its diversity in health outcomes and economic statuses.
In Summary
- The dataset provides a comprehensive view of health and economic indicators across a diverse set of countries and regions, highlighting variations in life expectancy, mortality rates, disease prevalence, vaccination rates, health expenditures, and socio-economic factors over a 15-year period.
1. Library Optimization
library(readr)
library(dplyr)##
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## intersect, setdiff, setequal, unionlibrary(ggplot2)
library(plotly)##
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library(caret)## Loading required package: latticelibrary(randomForest)## randomForest 4.7-1.1## Type rfNews() to see new features/changes/bug fixes.##
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library(openxlsx)
library(jsonlite)
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## recodelibrary(broom)
library(tidyr)
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## ℹ Use the conflicted package (<http://conflicted.r-lib.org/>) to force all conflicts to become errorslibrary(webshot)
library(webshot2)## Registered S3 method overwritten by 'webshot2':
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## print.webshot webshot
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## appshot, resize, rmdshot, shrink, webshotlibrary(lattice)
library(car)
library(nnet) # for ANN
library(class) # for KNN
library(e1071) # for SVM
library(randomForest) # for RF
library(glmnet)## Loading required package: Matrix
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## Attaching package: 'Matrix'
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## expand, pack, unpack
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## Loaded glmnet 4.1-8library(name)2. Data Loading
data <- read_csv("lifev3.csv")## Rows: 2848 Columns: 23
## ── Column specification ────────────────────────────────────────────────────────
## Delimiter: ","
## chr (2): Country, Region
## dbl (21): Year, Economy_status_Developed, Life_expectancy, Adult_mortality, ...
##
## ℹ Use `spec()` to retrieve the full column specification for this data.
## ℹ Specify the column types or set `show_col_types = FALSE` to quiet this message.# Define a mode function for categorical imputation
mode <- function(x) {
ux <- unique(x)
ux[which.max(tabulate(match(x, ux)))]
} # Closing bracket added here
# Impute missing values
data <- data %>%
mutate_if(is.numeric, ~ifelse(is.na(.), median(., na.rm = TRUE), .)) %>% # Impute numerical columns with median
mutate_if(is.character, ~ifelse(is.na(.), mode(.), .)) # Impute categorical columns with mode
# 5. Drop columns with many missing values if needed
data <- data[ , !(names(data) %in% c("ColumnsWithTooManyNAs"))]
# 6. Check result
summary(data)## Country Region Year Economy_status_Developed
## Length:2848 Length:2848 Min. :2000 Min. :0.0000
## Class :character Class :character 1st Qu.:2004 1st Qu.:0.0000
## Mode :character Mode :character Median :2008 Median :0.0000
## Mean :2008 Mean :0.2079
## 3rd Qu.:2011 3rd Qu.:0.0000
## Max. :2015 Max. :1.0000
## Life_expectancy Adult_mortality Infant_deaths Alcohol_consumption
## Min. :39.40 Min. : 49.38 Min. : 1.80 Min. : 0.000
## 1st Qu.:62.90 1st Qu.:106.68 1st Qu.: 8.10 1st Qu.: 1.220
## Median :71.45 Median :163.28 Median : 19.40 Median : 4.050
## Mean :68.95 Mean :190.90 Mean : 30.08 Mean : 4.837
## 3rd Qu.:75.50 3rd Qu.:245.32 3rd Qu.: 47.00 3rd Qu.: 7.810
## Max. :83.80 Max. :719.36 Max. :138.10 Max. :17.870
## percentage expenditure Hepatitis_B Measles BMI
## Min. : 0.000 Min. : 0.00 Min. : 0 Min. : 0.00
## 1st Qu.: 6.915 1st Qu.: 9.00 1st Qu.: 0 1st Qu.:18.80
## Median : 71.851 Median :86.00 Median : 15 Median :43.10
## Mean : 785.160 Mean :64.65 Mean : 2129 Mean :37.76
## 3rd Qu.: 470.529 3rd Qu.:96.00 3rd Qu.: 309 3rd Qu.:56.30
## Max. :19479.912 Max. :99.00 Max. :212183 Max. :79.30
## Under_five_deaths Polio Total expenditure Diphtheria
## Min. : 0.00 Min. : 0.00 Min. : 0.000 Min. : 0.00
## 1st Qu.: 0.00 1st Qu.:76.75 1st Qu.: 3.640 1st Qu.:78.00
## Median : 3.00 Median :93.00 Median : 5.480 Median :92.00
## Mean : 40.05 Mean :80.96 Mean : 5.398 Mean :81.04
## 3rd Qu.: 24.25 3rd Qu.:97.00 3rd Qu.: 7.320 3rd Qu.:97.00
## Max. :2500.00 Max. :99.00 Max. :17.600 Max. :99.00
## Incidents_HIV GDP_per_capita Population_millions
## Min. : 0.000 Min. : 0.0 Min. :0.000e+00
## 1st Qu.: 0.100 1st Qu.: 370.9 1st Qu.:3.264e+04
## Median : 0.100 Median : 1479.4 Median :7.237e+05
## Mean : 1.634 Mean : 6772.3 Mean :1.016e+07
## 3rd Qu.: 0.600 3rd Qu.: 5194.7 3rd Qu.:4.678e+06
## Max. :43.500 Max. :119172.7 Max. :1.294e+09
## Thinness_ten_nineteen_years Thinness_five_nine_years
## Min. : 0.000 Min. : 0.000
## 1st Qu.: 1.400 1st Qu.: 1.400
## Median : 3.100 Median : 3.200
## Mean : 4.709 Mean : 4.745
## 3rd Qu.: 7.100 3rd Qu.: 7.100
## Max. :27.700 Max. :28.600
## Income composition of resources Schooling
## Min. :0.0000 Min. : 1.100
## 1st Qu.:0.4660 1st Qu.: 5.100
## Median :0.6655 Median : 7.900
## Mean :0.5954 Mean : 7.653
## 3rd Qu.:0.7790 3rd Qu.:10.300
## Max. :0.9480 Max. :14.100# 7. List all column names to verify the existence of 'Status'
colnames(data)## [1] "Country" "Region"
## [3] "Year" "Economy_status_Developed"
## [5] "Life_expectancy" "Adult_mortality"
## [7] "Infant_deaths" "Alcohol_consumption"
## [9] "percentage expenditure" "Hepatitis_B"
## [11] "Measles" "BMI"
## [13] "Under_five_deaths" "Polio"
## [15] "Total expenditure" "Diphtheria"
## [17] "Incidents_HIV" "GDP_per_capita"
## [19] "Population_millions" "Thinness_ten_nineteen_years"
## [21] "Thinness_five_nine_years" "Income composition of resources"
## [23] "Schooling"Part B: Data Visualization
Overview Data Preparation, Cleaning & Visualization in Part B:
Methodology: Imputed missing values; ensured robust dataset ready for analysis. Visualization Techniques:
Strategies Used: Utilized ggplot2 and plotly for rich, interactive data visualizations. Key Visualizations:
Types and Insights:
Economic Status and Life Expectancy Analyzed life expectancy differences between developed and undeveloped economies. Boxplot Findings: Developed countries (label ‘1’) show higher and less variable life expectancy than developing ones (label ‘0’). Revealed disparities in life expectancy between developed and undeveloped economies.
Correlation Analysis Strong Positive Correlations: GDP per capita with schooling. Strong Negative Correlations: Adult mortality with life expectancy. Weaker Correlations: Indicated by lighter colors in the correlation matrix. Showed strong positive and negative correlations between various health and economic factors.
Life Expectancy Trends and Relationships Trends Over Time: General upward trend in life expectancy over years. Bivariate Relationships: Explored through scatterplots for key factors like adult mortality and GDP per capita.
Interactive Visualization Utilized plotly for dynamic exploration of data, especially in regional life expectancy trends.
Common Value Determination: A function was developed to find the most common value (mode) in a column, aiding in imputing missing categorical data.
Imputation of Missing Values: Missing numerical values were imputed with the median of their respective columns, avoiding bias from mean values affected by outliers.
Removal of Excessively Incomplete Columns: To maintain analysis quality, columns with too many missing values were removed.
Data Structure Verification: Post-cleaning, the dataset’s structure and key columns were confirmed as analysis-ready.
Data Visualization and Analysis
- Status Column Conversion and Boxplot: The ‘Economy_status_Developed’ column was converted to a factor for categorical plotting. A boxplot was created to visualize life expectancy distribution by economic status, revealing significant differences.
- Correlation Matrix: A static and then interactive correlation matrix were plotted to investigate relationships between all numeric variables, highlighting those strongly associated with life expectancy, like adult mortality and schooling.
- Life Expectancy Over Time: A plot of life expectancy trends over time by country was created, likely showing a general upward trend.
- Pairwise Scatterplots: These plots examined relationships between select variables like life expectancy, adult mortality, and GDP per capita.
- Interactive Boxplot with Faceting: Faceting life expectancy data by region and plotting over time enabled comparison of regional trends and disparities in life expectancy.
Model Fitting and Evaluation
- Regression Model Construction: Multiple regression models were built and compared for predicting life expectancy.
- Model Evaluation: Models were evaluated using criteria like AIC and BIC, with lower values indicating better fit.
- Residual Analysis: Residuals were analyzed for patterns suggesting model fit issues, like non-linearity or heteroscedasticity.
Interactive Visualizations
- Enhancement to Interactivity: Static ggplot objects were transformed into interactive Plotly visualizations for an engaging data exploration experience.
- Transformation of Static Plots: Static ggplot objects were converted into interactive Plotly visualizations. This enhanced user engagement and information accessibility through features like hover-over details, allowing for a more dynamic exploration of the data.
Model Summaries
- Detailed Reporting: Summaries of regression models were provided, including coefficients, significance, and diagnostics, crucial for interpreting which variables significantly predict life expectancy.
- Regression Model Insights: Detailed summaries of regression models were provided. These included coefficients, statistical significance, and model diagnostics, essential for understanding which variables significantly impact life expectancy.
Specific Visualizations
Boxplot of Life Expectancy: Displayed differences in life expectancy between two groups, with the group labeled ‘1’ showing higher and less variable life expectancy than ‘0’. This plot differentiated between two categories (‘0’ and ‘1’). The ‘0’ category showed a broader range of life expectancy (40 to 80 years) with a median around 65 years and some outliers indicating very low life expectancy. In contrast, the ‘1’ category had a narrower and higher range of life expectancy (55 to just below 80 years) with a median around 75 years and no outliers, suggesting higher and more consistent life expectancy in this group.
Multiple Linear Regression Plots: A grid of scatter plots showing linear regressions between various variables. These plots consisted of a grid of scatter plots, each showing a linear regression between two variables. Different traces in the plots indicated variations in subsets or categories within the data. The plots effectively displayed pairwise linear relationships among selected variables.
Life Expectancy Trends by Region: Line graphs for different global regions, showing life expectancy changes over time. Line graphs for various global regions depicted life expectancy trends from 2000 to around 2019. Each line represented a different country within a region, illustrating both individual country trends and overall regional trends in life expectancy.
Correlation Heatmap: A heatmap displaying correlations between various health and socioeconomic indicators. This heatmap displayed correlations between various health and socioeconomic indicators, using color intensity to represent the strength of these correlations. The range of colors from blue (negative correlation) to red (positive correlation) allowed for quick visual identification of the nature and strength of the relationships.
Life Expectancy Over Time by Country: A combined plot of all countries showing life expectancy trends. Combining data from all countries, this plot showed life expectancy trends from 2000 to around 2015. Each line represented a different country, highlighting the variability and overall trajectories of life expectancy across nations.
Correlation Dot Matrix: A dot matrix emphasizing significant relationships with the size and color of dots. Similar to the heatmap, this dot matrix also showed correlation coefficients between various indicators. The size and color of the dots (blue for positive, red for negative) emphasized the strength and nature of the correlations, making the most significant relationships more visually prominent.
# 1. Status Column Conversion
data$Economy_status_Developed <- as.factor(data$Economy_status_Developed)
# 2. Plot the boxplot with the corrected column name
gg_boxplot <- ggplot(data, aes(x = Economy_status_Developed, y = Life_expectancy)) +
geom_boxplot() +
labs(title = "Boxplot of Life Expectancy by Economy Status",
x = "Economy Status", y = "Life Expectancy") +
theme_minimal()
ggsave("life_expectancy_by_status.png", plot = gg_boxplot, width = 10, height = 6)
# Convert to Plotly for interactive visualization
p_plotly <- ggplotly(gg_boxplot)
htmlwidgets::saveWidget(p_plotly, "Life_Expectancy_by_Status_Plotly.html")
webshot::webshot("Life_Expectancy_by_Status_Plotly.html", file = "Life_Expectancy_by_Status_Plotly.png")
# Convert to Plotly for interactive visualization
p_plotly <- ggplotly(gg_boxplot)
p_plotly# Plot static
gg_boxplot <- ggplot(data, aes(x = Economy_status_Developed, y = Life_expectancy)) +
geom_boxplot() +
labs(title = "Boxplot of Life Expectancy by Economy Status",
x = "Economy Status", y = "Life Expectancy") +
theme_minimal()
ggsave("life_expectancy_by_status.png", plot = gg_boxplot, width = 10, height = 6)
# Convert to Plotly for interactive visualization
p_plotly <- ggplotly(gg_boxplot)
htmlwidgets::saveWidget(p_plotly, "Life_Expectancy_by_Status_Plotly.html")
#### 3. Correlation Matrix
# Static Correlation Matrix
cor_matrix_static <- cor(data[, sapply(data, is.numeric)], use = "complete.obs")
png("correlation_matrix.png", width = 800, height = 800, res = 150)
corrplot::corrplot(cor_matrix_static, method = "circle")
dev.off()## png
## 2# Interactive Correlation Matrix with Plotly
cor_matrix_interactive <- cor(data %>% select_if(is.numeric), use = "complete.obs")
p_corr <- plot_ly(x = colnames(cor_matrix_interactive), y = colnames(cor_matrix_interactive), z = cor_matrix_interactive,
type = "heatmap", colors = colorRamp(c("blue", "white", "red")))
htmlwidgets::saveWidget(p_corr, "Correlation_Matrix_Plotly.html")
webshot::webshot("Correlation_Matrix_Plotly.html", file = "Correlation_Matrix_Plotly.png")
# Static Correlation Matrix
cor_matrix_static <- cor(data[, sapply(data, is.numeric)], use = "complete.obs")
corrplot(cor_matrix_static, method = "circle")
# Interactive Correlation Matrix with Plotly
cor_matrix_interactive <- cor(data %>% dplyr::select_if(is.numeric), use = "complete.obs")
p_corr <- plot_ly(x = colnames(cor_matrix_interactive), y = colnames(cor_matrix_interactive), z = cor_matrix_interactive,
type = "heatmap", colors = colorRamp(c("blue", "white", "red")))
# Embed Plotly plot directly
p_corr# Life Expectancy Over Time
gg_life_expectancy <- ggplot(data, aes(x = Year, y = Life_expectancy, group = Country, color = Country)) +
geom_line() +
labs(title = "Life Expectancy Over Time",
x = "Year", y = "Life Expectancy") +
theme_minimal()
ggsave("life_expectancy_over_time.png", plot = gg_life_expectancy, width = 10, height = 6)
# Convert to Plotly for interactive visualization
p_plotly <- ggplotly(gg_life_expectancy)
htmlwidgets::saveWidget(p_plotly, "Life_Expectancy_Over_Time_Plotly.html")
webshot::webshot("Life_Expectancy_Over_Time_Plotly.html", file = "Life_Expectancy_Over_Time_Plotly.png")
# Convert to Plotly for interactive visualization
p_life_expectancy <- ggplotly(gg_life_expectancy)
p_life_expectancy#### 5. Pairwise Scatterplots
select_vars <- c("Life_expectancy", "Adult_mortality", "Alcohol_consumption", "GDP_per_capita", "Schooling")
png("Pairwise_Scatterplots.png", width = 1200, height = 800)
pairs(data[select_vars])
dev.off()## png
## 2# Convert to interactive Plotly scatter plot matrix
p_pairwise <- plot_ly(data = data, type = "scatter", mode = "markers")
for(i in 1:length(select_vars)) {
for(j in 1:length(select_vars)) {
if(i != j) {
p_pairwise <- add_trace(p_pairwise, x = ~data[[select_vars[i]]], y = ~data[[select_vars[j]]],
xaxis = paste0("x", i), yaxis = paste0("y", j))
}
}
}
p_pairwise <- p_pairwise %>%
layout(grid = list(rows = length(select_vars), columns = length(select_vars)))
p_pairwise#### 6. Interactive Boxplot with Faceting
# Create a ggplot object with faceting
gg_facet_plot <- ggplot(data, aes(x = Year, y = Life_expectancy, color = Country)) +
geom_line() +
facet_wrap(~Region) + # Replace 'Region' with your actual faceting variable
theme_minimal() +
theme(legend.position = "none")
# Convert to a Plotly object for interactivity
p_facet_plotly <- ggplotly(gg_facet_plot)
htmlwidgets::saveWidget(p_facet_plotly, "Life_Expectancy_Faceted_Plot.html")
webshot::webshot("Life_Expectancy_Faceted_Plot.html", file = "Life_Expectancy_Faceted_Plot.png")
# Convert to a Plotly object for interactivity
p_facet_plotly <- ggplotly(gg_facet_plot)
# Display the interactive plot
p_facet_plotlyPart C: Model Training
Data Analysis & Findings From Part C:
- Enhanced Linear Model (LM) Analysis:
- The
enhanced_linear_modelshows a very high R-squared value of 0.9971, indicating that the model explains almost all the variability in life expectancy. - The low residual standard error (0.5275) suggests good accuracy.
- However, with 197 predictors, the model could be overfitting.
- The
- Variance Inflation Factor (VIF) Analysis:
- High VIFs in
vif_model_1(e.g., Alcohol_consumption: 12.666473) suggest multicollinearity issues, which can distort the interpretation of coefficients. vif_model_2presents lower VIFs, indicating reduced multicollinearity for a simpler model.
- High VIFs in
- Model 2 (Simpler LM) Analysis:
- This model shows a lower but still strong R-squared of 0.9285.
- The RMSE for cross-validation (2.494402) and test data (2.644088) are quite consistent, indicating the model generalizes well.
- Generalized Linear Model (GLM) Analysis:
- The GLM with a logarithmic link function also performs well (AIC: 7849.5).
- Some coefficients (e.g.,
Income composition of resources: 6.039e-03) are significant, showing their impact on life expectancy.
- Artificial Neural Network (ANN) Analysis:
- The ANN model with 205 input features shows complexity in the weights, requiring careful interpretation.
- K-Nearest Neighbors (KNN) Analysis:
- The KNN model summary doesn’t provide much detail, but it’s important to choose the right ‘k’ value for optimal performance.
- Support Vector Machine (SVM) Analysis:
- The SVM model is another complex model, effective for capturing non-linear relationships.
- Random Forest (RF) Analysis:
- The RF model includes 500 trees and shows good performance in terms of mean squared error (MSE) and the coefficient of determination (R-squared).
- Comparative Analysis:
- Comparing actual and predicted values across models, the RMSE values vary, indicating the effectiveness of each model.
- Choosing the best model depends on the balance between complexity and predictive accuracy.
- Overall Insights:
- The high R-squared values in several models suggest good predictive capabilities.
- Care should be taken to avoid overfitting, especially in complex models like ANN and RF.
- Models like the simpler LM and RF can provide a good balance between accuracy and interpretability.
trainData <- read_csv("train_data.csv")## Rows: 1996 Columns: 24
## ── Column specification ────────────────────────────────────────────────────────
## Delimiter: ","
## chr (2): Country, Region
## dbl (22): Column1, Year, Economy_status_Developed, Life_expectancy, Adult_mo...
##
## ℹ Use `spec()` to retrieve the full column specification for this data.
## ℹ Specify the column types or set `show_col_types = FALSE` to quiet this message.enhanced_linear_model <- lm(Life_expectancy ~ ., data = trainData)
summary(enhanced_linear_model)##
## Call:
## lm(formula = Life_expectancy ~ ., data = trainData)
##
## Residuals:
## Min 1Q Median 3Q Max
## -2.0695 -0.2235 -0.0037 0.1979 3.2671
##
## Coefficients: (9 not defined because of singularities)
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) -2.365e+02 2.919e+01 -8.104 9.75e-16
## Column1 -6.173e-02 1.771e-02 -3.487 0.000501
## CountryAlbania 5.188e+00 4.578e-01 11.333 < 2e-16
## CountryAlgeria 5.725e+00 5.247e-01 10.911 < 2e-16
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## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 0.5275 on 1798 degrees of freedom
## Multiple R-squared: 0.9971, Adjusted R-squared: 0.9968
## F-statistic: 3186 on 197 and 1798 DF, p-value: < 2.2e-16# Calculate Standardized Residuals
standardized_resids <- rstandard(enhanced_linear_model)
# Create an interactive plot of Standardized Residuals using plotly
p <- plot_ly(x = seq_along(standardized_resids), y = standardized_resids, type = 'scatter', mode = 'markers') %>%
layout(title = "Standardized Residuals",
xaxis = list(title = "Index"),
yaxis = list(title = "Standardized Residuals"),
shapes = list(type = "line", line = list(color = "red", width = 2), x0 = 0, x1 = 1, xref = "paper", y0 = 0, y1 = 0))
# Display the plot
p# Cook's Distance
cooksd <- cooks.distance(enhanced_linear_model)
# Load plotly library
# Determine the threshold for influential points
influential_threshold <- 4 / (nrow(trainData) - length(coef(enhanced_linear_model)))
# Create an interactive plot of Cook's Distance using plotly
p <- plot_ly(x = seq_along(cooksd), y = cooksd, type = 'scatter', mode = 'markers') %>%
layout(title = "Cook's Distance",
xaxis = list(title = "Index"),
yaxis = list(title = "Cook's Distance"),
shapes = list(type = "line", line = list(color = "red", width = 2), x0 = 0, x1 = 1, xref = "paper", y0 = influential_threshold, y1 = influential_threshold))
# Display the plot
p# Load the training data
trainData <- read_csv("train_data.csv")## Rows: 1996 Columns: 24
## ── Column specification ────────────────────────────────────────────────────────
## Delimiter: ","
## chr (2): Country, Region
## dbl (22): Column1, Year, Economy_status_Developed, Life_expectancy, Adult_mo...
##
## ℹ Use `spec()` to retrieve the full column specification for this data.
## ℹ Specify the column types or set `show_col_types = FALSE` to quiet this message.# Build the enhanced linear model
enhanced_linear_model <- lm(Life_expectancy ~ ., data = trainData)
# Predict values using the model
trainData$Predicted <- predict(enhanced_linear_model, trainData)## Warning in predict.lm(enhanced_linear_model, trainData): prediction from
## rank-deficient fit; attr(*, "non-estim") has doubtful cases# Create a ggplot object for Actual vs Predicted values
actual_vs_predicted_plot <- ggplot(trainData, aes(x = Life_expectancy, y = Predicted)) +
geom_point() +
geom_abline(intercept = 0, slope = 1, color = "red") +
labs(title = "Actual vs Predicted Life Expectancy", x = "Actual", y = "Predicted")
# Convert to a Plotly object for interactivity
actual_vs_predicted_plotly <- ggplotly(actual_vs_predicted_plot)
# Display the interactive plot
actual_vs_predicted_plotly## Residuals
# Calculate residuals
trainData$Residuals <- residuals(enhanced_linear_model)
# Create a ggplot object for Residuals vs Predicted values
residuals_plot <- ggplot(trainData, aes(x = Predicted, y = Residuals)) +
geom_point() +
geom_hline(yintercept = 0, linetype = "dashed", color = "red") +
labs(title = "Residuals vs Predicted", x = "Predicted", y = "Residuals")
# Convert to a Plotly object for interactivity
residuals_plotly <- ggplotly(residuals_plot)
# Display the interactive plot
residuals_plotly### Part C2: Additional Model Training with Train-Test Split
# Splitting the Data into Train and Test Sets
set.seed(123) # for reproducibility
index <- createDataPartition(data$Life_expectancy, p = 0.8, list = FALSE) # 80-20 split
train_data <- data[index, ]
test_data <- data[-index, ]
# Working with models 1,2,3
# Data Preparation
# Set seed for reproducibility
set.seed(123)
# Load and prepare your data (assuming 'data' is already loaded)
# Split the data into training and validation sets
indexes <- sample(1:nrow(data), size = 0.8 * nrow(data))
train_data <- data[indexes, ]
validation_data <- data[-indexes, ]
train_data_selected <- dplyr::select(train_data, -Country, -Region)
# Define and fit models
model_1 <- lm(Life_expectancy ~ . + I(Alcohol_consumption^2), data = train_data_selected)
model_2 <- lm(Life_expectancy ~ Year + Adult_mortality + Alcohol_consumption + BMI, data = train_data_selected)
model_3 <- glm(Life_expectancy ~ ., family = gaussian(link = "log"), data = train_data_selected)
# Generate predicted values for each model
predicted_values_1 <- predict(model_1, train_data_selected)
predicted_values_2 <- predict(model_2, train_data_selected)
predicted_values_3 <- predict(model_3, train_data_selected)
print(predicted_values_1)## 1 2 3 4 5 6 7 8
## 82.79616 63.93140 72.22026 52.70523 65.86915 79.23622 73.44003 79.84081
## 9 10 11 12 13 14 15 16
## 79.94566 73.75418 76.98542 57.27434 80.41992 69.68095 79.07693 55.43794
## 17 18 19 20 21 22 23 24
## 61.38986 60.80130 79.86821 72.34108 75.43109 78.39988 73.13173 77.21674
## 25 26 27 28 29 30 31 32
## 74.01486 66.00677 74.71195 58.36540 43.32267 76.47009 50.60202 75.89302
## 33 34 35 36 37 38 39 40
## 82.31200 72.11063 65.89375 52.79388 78.99956 71.02117 67.92438 64.79326
## 41 42 43 44 45 46 47 48
## 80.35156 71.87374 75.37659 72.39222 64.69931 67.40594 61.34528 77.51329
## 49 50 51 52 53 54 55 56
## 80.56851 55.20049 70.21237 62.40722 57.49391 60.94488 72.68028 68.52506
## 57 58 59 60 61 62 63 64
## 63.17414 68.24749 73.69312 57.06605 51.77871 70.18277 70.53398 75.77637
## 65 66 67 68 69 70 71 72
## 50.47095 68.08813 76.65385 48.44125 61.57170 64.97179 69.01627 76.61882
## 73 74 75 76 77 78 79 80
## 71.43783 80.63199 64.62636 60.37748 63.85374 71.30177 53.56805 76.85221
## 81 82 83 84 85 86 87 88
## 77.98995 67.91362 72.17311 60.39251 72.40850 81.18010 72.67568 57.74852
## 89 90 91 92 93 94 95 96
## 64.90815 75.89962 72.36824 58.67052 71.78909 53.40825 76.65862 77.23452
## 97 98 99 100 101 102 103 104
## 51.57105 74.54579 81.52039 78.28181 81.09963 75.03672 59.81932 77.70889
## 105 106 107 108 109 110 111 112
## 75.97473 79.71919 60.86746 81.20769 56.37137 64.69516 69.48389 77.62359
## 113 114 115 116 117 118 119 120
## 61.25474 66.94702 64.78181 72.12171 71.93415 75.50302 61.67505 80.79385
## 121 122 123 124 125 126 127 128
## 65.39408 63.51330 80.01552 77.53444 80.31094 79.94787 55.77643 45.67732
## 129 130 131 132 133 134 135 136
## 75.01553 61.92917 54.11153 74.95370 40.29475 54.88312 80.29429 80.63845
## 137 138 139 140 141 142 143 144
## 55.72387 75.06713 61.88498 56.10273 60.08851 52.81627 60.34917 61.83604
## 145 146 147 148 149 150 151 152
## 70.55096 71.40961 75.86640 51.91505 63.50176 77.49596 74.12140 50.20529
## 153 154 155 156 157 158 159 160
## 78.02286 70.95172 79.58095 77.55578 77.55889 81.15758 79.63231 81.30599
## 161 162 163 164 165 166 167 168
## 53.21084 71.38074 73.76118 77.65708 63.07350 73.08493 54.50960 44.23301
## 169 170 171 172 173 174 175 176
## 70.49893 76.07269 68.59900 69.23348 67.88268 73.06091 65.88656 63.64746
## 177 178 179 180 181 182 183 184
## 55.21578 78.71697 53.72463 77.07928 72.19014 77.27316 60.17058 76.11672
## 185 186 187 188 189 190 191 192
## 77.42928 65.09598 48.35460 52.82062 70.07519 51.96045 71.66271 69.70049
## 193 194 195 196 197 198 199 200
## 62.23301 72.78949 64.63633 72.50045 71.02059 57.94542 59.05265 70.92269
## 201 202 203 204 205 206 207 208
## 58.33885 72.00427 72.71781 79.00005 59.92355 73.46465 49.99777 56.84134
## 209 210 211 212 213 214 215 216
## 77.18360 79.23312 62.25168 62.03126 69.69858 78.63262 75.76668 56.94527
## 217 218 219 220 221 222 223 224
## 73.82574 77.80358 66.52154 62.87173 72.85004 60.84048 80.25468 77.68553
## 225 226 227 228 229 230 231 232
## 55.96622 80.23567 75.00394 69.93831 58.55565 58.46206 72.25835 64.20723
## 233 234 235 236 237 238 239 240
## 80.01612 70.89193 78.91832 71.33672 78.17232 62.09831 70.84183 79.58639
## 241 242 243 244 245 246 247 248
## 73.72370 81.83804 78.34342 61.10207 80.26173 74.35368 69.63738 57.27466
## 249 250 251 252 253 254 255 256
## 68.25336 75.39835 46.92518 79.45181 80.14789 73.35311 59.59238 76.59062
## 257 258 259 260 261 262 263 264
## 71.06053 71.48676 73.06286 71.92382 50.54132 73.97869 69.09785 74.74566
## 265 266 267 268 269 270 271 272
## 63.53218 70.12078 54.26856 80.37379 81.12143 74.67481 60.91401 79.07980
## 273 274 275 276 277 278 279 280
## 65.02168 71.11545 64.10069 51.58692 58.45684 80.27188 80.74836 74.69003
## 281 282 283 284 285 286 287 288
## 68.18374 51.63295 73.73382 50.79340 49.87376 80.11080 75.03170 62.22196
## 289 290 291 292 293 294 295 296
## 69.42378 73.45196 64.92167 74.20948 75.65811 80.49702 71.74908 71.76844
## 297 298 299 300 301 302 303 304
## 82.02204 80.15710 63.58416 81.25855 76.02427 47.17551 63.35244 78.22115
## 305 306 307 308 309 310 311 312
## 73.13222 76.86626 76.12826 75.14534 79.97842 67.84134 73.82164 77.56335
## 313 314 315 316 317 318 319 320
## 75.86266 72.86013 72.41784 75.38116 63.26242 76.30435 74.62765 70.26105
## 321 322 323 324 325 326 327 328
## 51.86931 42.60999 79.53162 51.94142 69.72622 68.79834 71.47072 74.88838
## 329 330 331 332 333 334 335 336
## 80.28119 70.25966 80.38585 48.04928 63.30319 71.53177 62.83708 71.76323
## 337 338 339 340 341 342 343 344
## 73.99058 59.87750 72.80561 79.81130 77.82949 80.42811 51.52519 73.93917
## 345 346 347 348 349 350 351 352
## 58.63326 76.03851 70.80006 74.18498 67.62454 51.42193 56.11462 76.07843
## 353 354 355 356 357 358 359 360
## 72.24755 74.92319 73.34554 68.73621 80.31317 77.16698 80.20820 73.69463
## 361 362 363 364 365 366 367 368
## 61.04310 45.75694 75.47807 52.50478 77.54615 77.16714 52.61216 80.41330
## 369 370 371 372 373 374 375 376
## 69.64684 72.68508 61.09210 74.14086 42.76057 68.22824 60.73555 69.38508
## 377 378 379 380 381 382 383 384
## 60.13371 70.97001 57.94996 77.92544 76.41741 74.58770 77.47626 74.21498
## 385 386 387 388 389 390 391 392
## 77.35149 77.25078 70.40154 76.16876 73.99125 61.46623 43.75474 78.18275
## 393 394 395 396 397 398 399 400
## 74.99380 66.37176 77.55823 75.39866 65.91084 73.47182 74.54991 71.07138
## 401 402 403 404 405 406 407 408
## 77.70015 58.17020 69.37916 69.79696 72.79199 80.11194 45.58523 54.53288
## 409 410 411 412 413 414 415 416
## 60.13500 68.40010 46.06445 74.98885 79.07273 55.68570 75.99661 63.87176
## 417 418 419 420 421 422 423 424
## 77.22099 70.03348 47.24025 72.78771 64.83812 66.89051 73.61752 77.82192
## 425 426 427 428 429 430 431 432
## 67.22859 71.98039 54.99487 75.96085 76.05606 72.73682 66.91206 48.91839
## 433 434 435 436 437 438 439 440
## 60.61306 80.39183 60.49630 75.05778 77.41067 64.84529 68.71770 72.27899
## 441 442 443 444 445 446 447 448
## 58.54939 44.53224 79.23606 75.97171 69.67260 73.66136 50.52416 74.82157
## 449 450 451 452 453 454 455 456
## 70.98481 70.67527 73.78867 58.45611 72.75528 75.53484 76.20416 63.69680
## 457 458 459 460 461 462 463 464
## 79.85792 61.67477 73.37758 78.30741 62.81457 56.87246 75.83899 72.58874
## 465 466 467 468 469 470 471 472
## 44.56830 75.54018 62.48925 75.39554 79.90552 77.53016 58.11839 80.57396
## 473 474 475 476 477 478 479 480
## 54.67265 80.29056 59.01748 68.15879 77.72318 81.35624 62.95555 55.88157
## 481 482 483 484 485 486 487 488
## 79.25419 76.21283 58.67981 79.78920 65.46230 70.54660 73.21322 68.45141
## 489 490 491 492 493 494 495 496
## 50.49351 78.85153 70.64788 65.52687 57.92567 65.29315 79.44333 76.18579
## 497 498 499 500 501 502 503 504
## 59.87133 66.78628 69.88080 69.07950 78.32214 64.25863 70.58774 58.94954
## 505 506 507 508 509 510 511 512
## 70.20936 71.97870 68.96361 40.03333 61.00302 73.08240 70.58137 74.11942
## 513 514 515 516 517 518 519 520
## 52.43895 73.47796 77.47234 70.23144 55.22906 53.43385 57.89098 73.68420
## 521 522 523 524 525 526 527 528
## 74.96857 76.23368 58.17638 72.51503 70.76909 60.44997 72.69946 75.64245
## 529 530 531 532 533 534 535 536
## 56.47876 76.68069 78.90198 63.14116 60.23584 69.76832 62.77478 71.96781
## 537 538 539 540 541 542 543 544
## 55.43981 68.83347 63.63058 73.31949 43.65822 71.39433 67.45024 54.44795
## 545 546 547 548 549 550 551 552
## 45.34908 71.48538 67.53273 57.75191 79.38270 77.13028 48.54070 70.53518
## 553 554 555 556 557 558 559 560
## 71.70816 55.98973 67.95851 74.01707 75.69925 69.61239 49.37178 68.57553
## 561 562 563 564 565 566 567 568
## 51.36318 60.55543 79.78031 70.83176 74.45305 75.23645 58.40458 68.15358
## 569 570 571 572 573 574 575 576
## 77.92546 68.95698 81.02580 52.70100 69.80964 71.87033 58.98674 75.88827
## 577 578 579 580 581 582 583 584
## 74.00288 76.32360 68.61995 40.72896 60.09947 63.30304 79.48282 79.96247
## 585 586 587 588 589 590 591 592
## 73.33202 79.23639 63.51328 81.07678 76.74796 73.04324 79.48582 74.36258
## 593 594 595 596 597 598 599 600
## 64.68383 68.54660 61.54469 79.65858 80.68959 73.28822 68.91251 76.98398
## 601 602 603 604 605 606 607 608
## 66.71870 66.15604 71.60849 80.19873 62.48146 79.58801 75.90234 80.59991
## 609 610 611 612 613 614 615 616
## 79.07084 79.85023 76.64709 71.69302 77.28806 69.91636 69.37622 57.49677
## 617 618 619 620 621 622 623 624
## 72.89137 64.14265 75.92423 72.09473 70.37126 67.11615 57.63841 72.43526
## 625 626 627 628 629 630 631 632
## 79.22725 73.61724 72.51100 80.85446 74.19114 77.32096 41.62227 70.59699
## 633 634 635 636 637 638 639 640
## 73.93203 76.14946 80.34319 66.57527 61.04916 76.31276 59.70818 73.73236
## 641 642 643 644 645 646 647 648
## 71.92502 75.35527 73.97961 80.28177 66.07639 53.39120 57.22317 73.25777
## 649 650 651 652 653 654 655 656
## 67.65627 60.22919 81.36980 74.66147 80.99195 60.42656 68.57441 79.80622
## 657 658 659 660 661 662 663 664
## 67.73225 71.46948 70.23950 70.03349 63.49589 72.22202 52.16136 53.19618
## 665 666 667 668 669 670 671 672
## 74.19525 79.68740 73.13705 59.74778 50.54880 57.19237 72.27711 81.54318
## 673 674 675 676 677 678 679 680
## 59.26729 51.24103 75.99232 79.67290 55.47044 74.82495 72.46176 74.84994
## 681 682 683 684 685 686 687 688
## 70.42901 61.08696 68.79753 72.98241 63.49372 76.42500 66.32141 53.64419
## 689 690 691 692 693 694 695 696
## 78.71218 77.25103 70.45535 53.96545 79.29071 70.60553 79.14918 62.16440
## 697 698 699 700 701 702 703 704
## 73.03143 56.58549 75.24075 71.44795 73.49728 76.03475 72.35490 80.26590
## 705 706 707 708 709 710 711 712
## 68.69308 61.57356 80.55089 50.42303 55.61384 80.83114 70.85370 78.65814
## 713 714 715 716 717 718 719 720
## 68.86821 77.21379 70.00002 61.78026 76.30717 43.61983 76.94107 77.86775
## 721 722 723 724 725 726 727 728
## 71.42863 72.46825 78.71362 56.38878 69.91270 76.79983 70.61783 65.00106
## 729 730 731 732 733 734 735 736
## 72.00870 42.21015 57.96209 71.31780 77.47838 56.58053 70.79066 67.38166
## 737 738 739 740 741 742 743 744
## 75.64119 48.83063 56.36581 75.19586 64.16407 80.92105 59.62282 68.27584
## 745 746 747 748 749 750 751 752
## 80.16670 67.70930 75.18805 67.08861 61.43110 71.99489 65.47567 73.03743
## 753 754 755 756 757 758 759 760
## 72.28316 68.28711 53.14849 76.66732 76.71353 68.08021 48.46507 59.94169
## 761 762 763 764 765 766 767 768
## 78.54267 54.77238 71.98357 69.81681 72.57626 56.90142 73.54572 70.61512
## 769 770 771 772 773 774 775 776
## 80.07388 80.14232 72.57007 69.41713 54.77646 71.45568 70.73060 62.54821
## 777 778 779 780 781 782 783 784
## 75.96925 54.28106 66.97384 68.11366 75.13436 67.73748 80.00794 55.71917
## 785 786 787 788 789 790 791 792
## 73.67619 60.19702 68.26693 77.29921 80.31602 71.45317 64.28157 63.51097
## 793 794 795 796 797 798 799 800
## 74.92565 79.60036 72.60537 56.85337 53.02888 76.18748 60.53160 48.09800
## 801 802 803 804 805 806 807 808
## 74.84735 79.85738 49.33501 80.00782 77.23332 77.81777 60.45618 69.00062
## 809 810 811 812 813 814 815 816
## 65.70555 80.49230 54.22617 60.73471 74.42028 61.49589 76.76147 69.72530
## 817 818 819 820 821 822 823 824
## 75.97620 55.08917 80.53423 69.04784 52.71539 41.68391 68.37320 73.23101
## 825 826 827 828 829 830 831 832
## 78.24368 80.77474 73.24158 77.31427 60.54921 71.19753 53.95111 61.95216
## 833 834 835 836 837 838 839 840
## 71.55682 74.87679 70.76968 74.95433 67.99922 72.12375 72.44704 64.62882
## 841 842 843 844 845 846 847 848
## 77.43357 80.69810 80.63790 55.24789 71.12620 72.25187 78.44645 68.15355
## 849 850 851 852 853 854 855 856
## 71.71340 74.19142 79.96158 57.52197 77.56695 73.94264 72.80054 67.63030
## 857 858 859 860 861 862 863 864
## 73.82332 51.86852 75.14819 75.49538 76.29157 69.78602 76.02133 57.17808
## 865 866 867 868 869 870 871 872
## 72.18473 71.90285 74.65721 73.69422 68.26649 68.96903 69.01994 73.48879
## 873 874 875 876 877 878 879 880
## 77.17372 69.62828 76.75054 65.29204 80.20291 74.14448 72.55071 72.90758
## 881 882 883 884 885 886 887 888
## 65.32995 70.75276 70.45089 53.95754 71.83257 50.20226 70.84969 68.07969
## 889 890 891 892 893 894 895 896
## 62.47097 69.17099 72.17339 73.39224 80.86561 63.46941 66.55628 49.06203
## 897 898 899 900 901 902 903 904
## 79.53004 45.93265 79.91538 80.87230 63.33010 60.98687 50.66420 65.87417
## 905 906 907 908 909 910 911 912
## 76.86761 50.05117 72.39040 61.01216 77.92633 63.20082 65.46039 72.07760
## 913 914 915 916 917 918 919 920
## 75.90157 73.71396 72.52752 76.53965 62.65807 71.84702 71.08709 62.29571
## 921 922 923 924 925 926 927 928
## 57.66882 46.69744 53.46417 59.09437 63.25218 79.93871 53.73447 72.30691
## 929 930 931 932 933 934 935 936
## 68.02975 70.55372 80.68433 71.07443 74.82870 73.50142 75.57469 68.79907
## 937 938 939 940 941 942 943 944
## 62.35671 64.63908 74.19960 77.67135 72.96971 80.13190 72.23360 73.74237
## 945 946 947 948 949 950 951 952
## 73.49450 75.43441 74.56268 60.82960 62.52529 80.91402 78.67779 72.72759
## 953 954 955 956 957 958 959 960
## 46.13298 68.97198 71.73142 66.20734 75.96324 80.09280 49.79718 53.75011
## 961 962 963 964 965 966 967 968
## 58.19332 55.85440 70.46218 75.77755 71.61849 75.48712 73.88533 73.39882
## 969 970 971 972 973 974 975 976
## 77.79343 52.38572 80.81786 70.69927 67.96027 75.93020 81.38935 71.42475
## 977 978 979 980 981 982 983 984
## 77.75631 75.25312 66.67824 71.29495 70.72327 76.10965 67.87352 41.62152
## 985 986 987 988 989 990 991 992
## 61.23991 69.36609 55.61467 77.79729 79.26651 70.65510 72.26456 73.39130
## 993 994 995 996 997 998 999 1000
## 64.78235 67.34269 74.79633 58.73532 66.38986 53.66679 53.50372 73.46822
## 1001 1002 1003 1004 1005 1006 1007 1008
## 40.41228 53.98991 61.35440 70.12164 64.15026 78.41599 66.45971 63.05130
## 1009 1010 1011 1012 1013 1014 1015 1016
## 81.39465 52.55009 57.51949 67.85256 75.97432 63.86010 61.50906 75.27310
## 1017 1018 1019 1020 1021 1022 1023 1024
## 62.51316 76.02990 70.71142 75.79857 77.84807 70.74211 48.06741 56.06098
## 1025 1026 1027 1028 1029 1030 1031 1032
## 70.95582 78.01928 58.96156 67.91979 81.11866 65.58385 53.94292 78.60145
## 1033 1034 1035 1036 1037 1038 1039 1040
## 78.73763 73.40430 62.37682 71.56300 76.77102 73.18832 71.62069 69.48927
## 1041 1042 1043 1044 1045 1046 1047 1048
## 73.27913 66.15914 73.95603 74.43844 79.34231 71.34031 62.11483 80.10554
## 1049 1050 1051 1052 1053 1054 1055 1056
## 50.09128 73.10777 69.06110 61.37230 78.15728 58.42477 76.48642 53.58480
## 1057 1058 1059 1060 1061 1062 1063 1064
## 55.55230 67.39119 66.71192 69.21181 53.22861 70.54874 46.84102 74.42990
## 1065 1066 1067 1068 1069 1070 1071 1072
## 56.49673 61.12903 70.03137 80.32919 79.87340 73.60108 71.90374 61.53200
## 1073 1074 1075 1076 1077 1078 1079 1080
## 65.57556 75.40928 50.24224 80.59684 63.15355 70.58388 52.12428 71.76400
## 1081 1082 1083 1084 1085 1086 1087 1088
## 67.93022 71.89879 61.24030 78.64472 80.47518 73.44552 66.47520 72.89526
## 1089 1090 1091 1092 1093 1094 1095 1096
## 60.06793 75.46262 55.99982 71.00623 71.48522 81.19725 74.47687 68.74402
## 1097 1098 1099 1100 1101 1102 1103 1104
## 59.29920 44.41297 63.54709 74.94845 72.99168 70.44098 66.12719 61.15558
## 1105 1106 1107 1108 1109 1110 1111 1112
## 74.37602 63.90546 52.84967 71.11293 62.60407 60.46528 81.64716 48.77784
## 1113 1114 1115 1116 1117 1118 1119 1120
## 67.22182 73.07604 80.74863 74.41387 73.84911 75.48906 67.77822 73.56100
## 1121 1122 1123 1124 1125 1126 1127 1128
## 81.09692 48.19135 79.60809 72.52935 71.63757 74.88169 79.74244 79.89738
## 1129 1130 1131 1132 1133 1134 1135 1136
## 70.54279 64.44837 72.12382 78.34651 72.38082 42.24008 47.65189 61.59144
## 1137 1138 1139 1140 1141 1142 1143 1144
## 74.25299 57.36637 41.08437 72.65909 79.15266 75.99545 65.25270 71.48787
## 1145 1146 1147 1148 1149 1150 1151 1152
## 75.27743 70.89448 78.84797 76.70750 63.21487 79.55746 69.41601 53.29399
## 1153 1154 1155 1156 1157 1158 1159 1160
## 70.87401 79.85696 74.78098 80.83003 76.34213 73.69145 78.06100 60.98419
## 1161 1162 1163 1164 1165 1166 1167 1168
## 47.52834 75.49853 63.31903 63.89737 72.11533 70.22708 65.14913 78.70381
## 1169 1170 1171 1172 1173 1174 1175 1176
## 76.21327 60.86704 60.12912 73.88088 60.54499 80.20092 68.89523 70.88013
## 1177 1178 1179 1180 1181 1182 1183 1184
## 68.58491 76.12609 67.81187 59.75116 63.70602 79.37636 77.70749 78.33830
## 1185 1186 1187 1188 1189 1190 1191 1192
## 53.14271 64.13778 61.03464 79.31951 72.06974 79.70280 71.91429 65.63489
## 1193 1194 1195 1196 1197 1198 1199 1200
## 52.18745 74.84436 74.89148 62.59255 63.71376 72.01584 50.92730 69.29115
## 1201 1202 1203 1204 1205 1206 1207 1208
## 55.05702 77.66356 65.42098 70.92921 73.21218 71.72052 77.09278 80.48919
## 1209 1210 1211 1212 1213 1214 1215 1216
## 46.30552 73.79601 76.14249 74.19103 67.39634 80.16435 70.27134 74.33222
## 1217 1218 1219 1220 1221 1222 1223 1224
## 69.65905 70.35835 72.35112 70.96270 74.29376 54.95376 72.86007 63.67329
## 1225 1226 1227 1228 1229 1230 1231 1232
## 80.02162 67.79557 79.34813 68.21997 59.61244 79.47794 62.11323 72.36682
## 1233 1234 1235 1236 1237 1238 1239 1240
## 60.87553 69.84454 69.90736 74.69745 70.19571 71.24073 70.30945 71.50414
## 1241 1242 1243 1244 1245 1246 1247 1248
## 71.06175 53.71694 75.36885 79.59009 75.16350 78.95049 75.81854 76.98379
## 1249 1250 1251 1252 1253 1254 1255 1256
## 66.89354 62.78488 73.86573 69.89255 65.16540 50.27966 58.33260 59.11040
## 1257 1258 1259 1260 1261 1262 1263 1264
## 80.69939 71.54924 76.28472 73.50193 55.63390 57.00989 70.71792 74.89643
## 1265 1266 1267 1268 1269 1270 1271 1272
## 63.63150 55.15727 69.46589 74.71925 74.93468 65.04917 51.39992 64.08226
## 1273 1274 1275 1276 1277 1278 1279 1280
## 57.48322 76.57777 62.81505 71.46040 80.36941 71.67912 69.33262 67.24046
## 1281 1282 1283 1284 1285 1286 1287 1288
## 73.94346 57.16584 56.22483 73.66806 75.85388 76.61492 79.56774 58.13033
## 1289 1290 1291 1292 1293 1294 1295 1296
## 58.52048 77.40895 70.88453 67.96754 73.41312 60.88970 62.33471 81.11107
## 1297 1298 1299 1300 1301 1302 1303 1304
## 50.86844 72.32831 65.06519 77.07672 73.27680 70.41056 73.90036 76.24924
## 1305 1306 1307 1308 1309 1310 1311 1312
## 63.87064 64.45919 68.72097 81.59707 72.45653 75.08041 80.87791 68.47691
## 1313 1314 1315 1316 1317 1318 1319 1320
## 54.90409 66.23405 56.68502 71.74399 72.36296 76.38039 67.43087 71.33083
## 1321 1322 1323 1324 1325 1326 1327 1328
## 43.94571 76.15169 78.12642 73.59823 80.03624 41.95712 55.04527 68.66290
## 1329 1330 1331 1332 1333 1334 1335 1336
## 66.27826 73.58153 60.05810 70.33736 69.16562 76.09919 66.09785 79.92556
## 1337 1338 1339 1340 1341 1342 1343 1344
## 79.86526 57.35895 78.85830 73.43566 57.93853 78.79704 72.65790 81.05354
## 1345 1346 1347 1348 1349 1350 1351 1352
## 53.26316 69.19582 73.91404 76.92334 74.48894 80.79055 60.74042 79.61482
## 1353 1354 1355 1356 1357 1358 1359 1360
## 43.56209 73.02441 69.88041 71.98780 59.45929 74.37944 56.07476 72.67520
## 1361 1362 1363 1364 1365 1366 1367 1368
## 80.47254 74.56175 71.95276 68.54311 80.71905 58.79965 58.31583 64.32513
## 1369 1370 1371 1372 1373 1374 1375 1376
## 69.44715 69.19987 77.34248 72.48985 74.31316 73.71100 78.75981 76.17545
## 1377 1378 1379 1380 1381 1382 1383 1384
## 78.63762 70.24396 71.31125 73.33737 80.91571 49.76183 72.87258 76.74591
## 1385 1386 1387 1388 1389 1390 1391 1392
## 75.33213 74.39546 80.11923 56.23633 51.19822 81.23209 67.83988 78.63776
## 1393 1394 1395 1396 1397 1398 1399 1400
## 75.76341 56.55613 72.25313 77.43126 50.14736 75.94157 71.30561 64.78608
## 1401 1402 1403 1404 1405 1406 1407 1408
## 77.79691 65.79836 57.40627 61.52914 76.94759 74.28193 77.00599 62.86593
## 1409 1410 1411 1412 1413 1414 1415 1416
## 74.79034 72.30536 80.37069 78.04084 76.10629 80.89894 73.30546 72.12373
## 1417 1418 1419 1420 1421 1422 1423 1424
## 62.00334 52.33572 70.94103 74.44131 72.83345 59.93734 81.48135 73.21984
## 1425 1426 1427 1428 1429 1430 1431 1432
## 50.30792 63.19137 73.18042 65.80031 78.61447 70.12925 69.56374 67.87919
## 1433 1434 1435 1436 1437 1438 1439 1440
## 76.24409 51.91396 76.83602 80.72077 65.77622 80.47943 64.76029 81.31447
## 1441 1442 1443 1444 1445 1446 1447 1448
## 73.51946 49.96483 72.72717 80.87613 59.66297 63.84731 58.04045 79.95352
## 1449 1450 1451 1452 1453 1454 1455 1456
## 73.87681 77.70332 79.23173 59.21764 61.41086 49.30101 72.59099 74.38440
## 1457 1458 1459 1460 1461 1462 1463 1464
## 79.01692 71.48931 68.84265 59.01368 68.89252 65.28223 70.68632 78.55293
## 1465 1466 1467 1468 1469 1470 1471 1472
## 80.38601 77.75160 70.50558 72.07828 73.73816 69.99319 64.18911 77.56583
## 1473 1474 1475 1476 1477 1478 1479 1480
## 79.78283 75.48247 77.33087 77.10953 63.73199 75.38795 61.70824 78.37579
## 1481 1482 1483 1484 1485 1486 1487 1488
## 53.74331 73.52129 68.60851 80.66695 75.82600 66.47054 69.54469 69.73788
## 1489 1490 1491 1492 1493 1494 1495 1496
## 68.90006 74.65070 67.29944 56.95774 46.57365 72.72355 71.78302 75.82552
## 1497 1498 1499 1500 1501 1502 1503 1504
## 77.29929 45.22717 53.14799 71.80236 78.47351 58.73485 68.65755 71.40617
## 1505 1506 1507 1508 1509 1510 1511 1512
## 69.35803 73.05635 54.08035 59.62159 73.38103 67.88370 77.55106 58.17541
## 1513 1514 1515 1516 1517 1518 1519 1520
## 65.71029 73.85116 63.77631 69.71341 54.01322 80.73748 80.90339 77.15216
## 1521 1522 1523 1524 1525 1526 1527 1528
## 77.80243 60.92775 53.35858 59.43341 75.29847 52.49628 70.86641 42.32344
## 1529 1530 1531 1532 1533 1534 1535 1536
## 73.31421 81.26407 72.22081 72.44363 71.56813 64.22269 64.14990 69.10343
## 1537 1538 1539 1540 1541 1542 1543 1544
## 79.98301 76.86056 54.50077 76.61581 67.30915 51.56299 78.75601 59.09228
## 1545 1546 1547 1548 1549 1550 1551 1552
## 65.24759 80.07454 55.49746 62.29462 67.59233 75.41408 69.07910 77.19451
## 1553 1554 1555 1556 1557 1558 1559 1560
## 67.88729 73.66751 71.19694 76.39089 70.94184 62.34594 43.81342 68.39675
## 1561 1562 1563 1564 1565 1566 1567 1568
## 79.75881 73.55283 52.35848 73.63036 76.19751 74.06802 69.08552 40.81242
## 1569 1570 1571 1572 1573 1574 1575 1576
## 66.76809 75.18706 71.47037 66.57084 67.00965 67.71059 70.25285 78.31799
## 1577 1578 1579 1580 1581 1582 1583 1584
## 72.49623 59.20981 59.83032 57.91901 60.42319 73.72161 42.41069 59.60330
## 1585 1586 1587 1588 1589 1590 1591 1592
## 78.88293 60.13107 81.78730 66.65653 79.62188 65.18320 68.19926 51.70178
## 1593 1594 1595 1596 1597 1598 1599 1600
## 61.98298 80.50173 75.02856 64.49665 77.89411 72.17512 76.00395 74.80124
## 1601 1602 1603 1604 1605 1606 1607 1608
## 65.30865 64.86156 79.61461 59.06149 72.71595 69.64196 69.94909 79.44678
## 1609 1610 1611 1612 1613 1614 1615 1616
## 73.54879 72.02641 76.19017 73.98742 63.81747 80.26112 78.03039 76.82858
## 1617 1618 1619 1620 1621 1622 1623 1624
## 49.42660 66.85516 70.25375 69.45848 62.99242 75.18453 52.69070 77.33066
## 1625 1626 1627 1628 1629 1630 1631 1632
## 79.08551 72.44503 53.56216 79.28879 67.26668 72.68461 70.48996 67.50847
## 1633 1634 1635 1636 1637 1638 1639 1640
## 56.53955 76.31652 72.84969 60.31649 76.61797 79.55440 49.13914 66.13093
## 1641 1642 1643 1644 1645 1646 1647 1648
## 67.34546 79.67984 66.27980 52.06968 72.43521 61.45445 80.30493 59.04981
## 1649 1650 1651 1652 1653 1654 1655 1656
## 46.59733 65.87949 80.32984 53.77316 76.06119 78.68544 66.59472 65.77120
## 1657 1658 1659 1660 1661 1662 1663 1664
## 73.83439 74.13884 76.80449 75.40392 41.99315 70.33018 70.03433 56.16149
## 1665 1666 1667 1668 1669 1670 1671 1672
## 73.72794 72.32125 49.32818 62.27114 72.78246 77.12396 72.81105 74.28663
## 1673 1674 1675 1676 1677 1678 1679 1680
## 55.25843 70.59665 76.86595 58.09865 72.65010 78.50062 56.08629 71.66866
## 1681 1682 1683 1684 1685 1686 1687 1688
## 69.76706 77.10867 56.90736 69.28229 71.68722 43.68556 80.77529 72.04435
## 1689 1690 1691 1692 1693 1694 1695 1696
## 59.60430 80.22421 73.90403 78.15110 71.72376 78.89833 68.98114 80.33663
## 1697 1698 1699 1700 1701 1702 1703 1704
## 52.23947 56.09378 72.97520 77.64062 62.90676 73.91778 63.65496 65.74984
## 1705 1706 1707 1708 1709 1710 1711 1712
## 60.62004 72.63666 74.41629 75.31506 70.92536 63.01707 61.96612 74.55817
## 1713 1714 1715 1716 1717 1718 1719 1720
## 78.88594 73.00552 80.21876 79.33276 79.41280 75.57379 74.40685 65.63798
## 1721 1722 1723 1724 1725 1726 1727 1728
## 59.21173 77.27452 70.32620 72.05949 74.27265 79.73789 71.62732 80.82988
## 1729 1730 1731 1732 1733 1734 1735 1736
## 68.41428 71.28476 73.66887 70.24441 77.06653 62.96921 80.04481 40.80649
## 1737 1738 1739 1740 1741 1742 1743 1744
## 74.13117 50.81923 80.75158 66.28338 79.71160 72.28664 77.43389 49.45242
## 1745 1746 1747 1748 1749 1750 1751 1752
## 70.85672 71.58650 71.56085 78.43978 51.75422 77.45568 77.66535 76.48223
## 1753 1754 1755 1756 1757 1758 1759 1760
## 73.51527 62.50652 73.88371 60.22023 72.97635 72.79928 80.01339 73.18845
## 1761 1762 1763 1764 1765 1766 1767 1768
## 78.04084 81.00356 73.09082 66.77128 69.92906 63.72394 73.63909 61.10860
## 1769 1770 1771 1772 1773 1774 1775 1776
## 75.74667 58.38334 60.80974 69.85196 77.03476 70.30517 63.29121 60.27677
## 1777 1778 1779 1780 1781 1782 1783 1784
## 70.61898 50.35129 63.86914 55.39217 52.26001 74.01637 67.52916 72.59445
## 1785 1786 1787 1788 1789 1790 1791 1792
## 75.73918 74.76674 60.69799 71.00707 59.49648 56.35554 60.52540 77.53896
## 1793 1794 1795 1796 1797 1798 1799 1800
## 75.45043 76.65871 58.72344 78.78310 79.13288 72.72330 74.61044 71.26959
## 1801 1802 1803 1804 1805 1806 1807 1808
## 64.60560 54.44107 79.73399 60.70904 71.90759 53.75122 76.08341 79.46619
## 1809 1810 1811 1812 1813 1814 1815 1816
## 77.18537 72.28712 72.70288 62.01091 80.25246 76.79651 75.99699 71.07902
## 1817 1818 1819 1820 1821 1822 1823 1824
## 64.51139 69.45604 57.68842 54.05962 56.69367 70.06668 81.64725 57.30229
## 1825 1826 1827 1828 1829 1830 1831 1832
## 81.24201 76.01688 63.98609 77.73748 70.78982 73.61112 61.35173 77.74627
## 1833 1834 1835 1836 1837 1838 1839 1840
## 76.71546 69.23365 68.75175 69.00616 56.83564 74.46972 53.43791 76.46141
## 1841 1842 1843 1844 1845 1846 1847 1848
## 54.59459 72.75816 78.02086 71.61721 64.58566 72.23548 74.19557 54.66123
## 1849 1850 1851 1852 1853 1854 1855 1856
## 78.32957 71.50677 56.49678 48.29268 74.90814 74.32159 45.48788 79.60971
## 1857 1858 1859 1860 1861 1862 1863 1864
## 57.84016 72.33875 69.65876 61.95590 69.53562 71.46066 74.51597 60.02693
## 1865 1866 1867 1868 1869 1870 1871 1872
## 74.30007 67.13087 73.04672 71.25829 71.78444 53.86805 79.46222 60.17411
## 1873 1874 1875 1876 1877 1878 1879 1880
## 49.85428 67.67901 80.67916 50.07309 58.53708 56.59410 73.00369 55.17318
## 1881 1882 1883 1884 1885 1886 1887 1888
## 70.09669 73.59853 79.57218 78.92233 69.90966 58.59792 47.25977 76.94515
## 1889 1890 1891 1892 1893 1894 1895 1896
## 77.31629 55.70942 55.01278 61.97641 77.15121 52.78185 75.25169 74.72759
## 1897 1898 1899 1900 1901 1902 1903 1904
## 70.36813 71.08798 69.52263 56.86296 68.74224 72.31672 65.24694 64.51641
## 1905 1906 1907 1908 1909 1910 1911 1912
## 62.47364 77.99289 76.56860 72.70647 64.49248 71.38205 59.91484 69.76022
## 1913 1914 1915 1916 1917 1918 1919 1920
## 48.33594 67.88025 63.86919 71.26245 61.43419 69.37643 61.77218 76.71428
## 1921 1922 1923 1924 1925 1926 1927 1928
## 71.19875 50.35595 79.89612 62.21873 68.39726 80.96836 75.15045 69.89338
## 1929 1930 1931 1932 1933 1934 1935 1936
## 73.82942 70.66249 66.00878 67.66761 63.98897 77.05680 75.53913 68.15897
## 1937 1938 1939 1940 1941 1942 1943 1944
## 77.81019 76.81595 76.48873 79.26652 54.61113 80.36080 72.69174 80.17182
## 1945 1946 1947 1948 1949 1950 1951 1952
## 76.18736 75.59017 73.57585 73.44512 58.78247 59.29043 68.56390 45.67150
## 1953 1954 1955 1956 1957 1958 1959 1960
## 48.10157 64.21426 72.13177 73.36820 68.70121 72.97056 73.65604 63.01601
## 1961 1962 1963 1964 1965 1966 1967 1968
## 45.44219 63.00640 65.94008 82.59609 68.19366 63.77337 75.60157 48.17685
## 1969 1970 1971 1972 1973 1974 1975 1976
## 75.72517 73.87813 80.37792 77.11113 68.74956 80.88893 75.44344 55.52798
## 1977 1978 1979 1980 1981 1982 1983 1984
## 72.32612 72.01262 69.19270 70.38633 62.33953 75.84898 79.00391 61.80455
## 1985 1986 1987 1988 1989 1990 1991 1992
## 79.25063 77.75797 59.46530 82.44207 65.29275 78.78332 79.51620 77.25886
## 1993 1994 1995 1996 1997 1998 1999 2000
## 80.54922 78.50862 70.07645 72.15913 67.53649 76.75056 72.23255 58.76185
## 2001 2002 2003 2004 2005 2006 2007 2008
## 75.48910 79.79343 72.50112 74.91424 70.92886 65.99165 43.01415 76.00084
## 2009 2010 2011 2012 2013 2014 2015 2016
## 73.94565 75.58853 56.24511 50.37868 72.83777 69.80243 70.39939 79.63116
## 2017 2018 2019 2020 2021 2022 2023 2024
## 72.81056 75.66936 76.47812 72.43695 78.98452 68.09166 72.56060 79.69418
## 2025 2026 2027 2028 2029 2030 2031 2032
## 57.35643 56.66776 71.11057 80.95104 73.57095 73.67226 62.15734 70.45236
## 2033 2034 2035 2036 2037 2038 2039 2040
## 71.25601 64.26450 78.41140 73.94160 69.20270 80.09133 68.72717 68.57155
## 2041 2042 2043 2044 2045 2046 2047 2048
## 62.95965 81.43751 55.64844 54.43111 38.33564 59.02788 61.13954 65.12463
## 2049 2050 2051 2052 2053 2054 2055 2056
## 72.44169 80.31374 72.84384 79.98401 61.71531 77.76761 73.44909 76.96094
## 2057 2058 2059 2060 2061 2062 2063 2064
## 71.73368 57.23140 56.28687 57.12453 72.17502 73.40728 73.20874 45.05902
## 2065 2066 2067 2068 2069 2070 2071 2072
## 76.22405 57.72598 68.36664 52.69287 77.14125 66.47213 68.75918 73.31383
## 2073 2074 2075 2076 2077 2078 2079 2080
## 76.55413 81.58319 73.35383 74.08757 80.16724 68.08213 70.27023 80.13773
## 2081 2082 2083 2084 2085 2086 2087 2088
## 74.52446 48.04987 73.49488 74.69873 67.34471 70.18207 45.80494 42.32912
## 2089 2090 2091 2092 2093 2094 2095 2096
## 67.77577 73.66087 54.83258 52.55884 71.42362 55.04795 59.16272 66.14960
## 2097 2098 2099 2100 2101 2102 2103 2104
## 70.02343 77.71870 72.27877 51.57936 73.49396 68.98964 62.12241 57.31237
## 2105 2106 2107 2108 2109 2110 2111 2112
## 74.03626 80.41603 52.21288 62.77081 75.48272 52.90634 68.34209 76.14927
## 2113 2114 2115 2116 2117 2118 2119 2120
## 49.22464 61.63923 68.56885 67.56135 66.61647 74.28750 73.12954 59.54184
## 2121 2122 2123 2124 2125 2126 2127 2128
## 76.32272 81.09483 70.63276 71.44992 74.56442 58.32149 66.24088 79.55659
## 2129 2130 2131 2132 2133 2134 2135 2136
## 42.66679 39.11969 62.81867 66.06086 66.68874 59.36733 60.79919 72.07430
## 2137 2138 2139 2140 2141 2142 2143 2144
## 69.67641 58.65452 77.92734 67.67089 72.75375 80.34025 80.11630 70.22986
## 2145 2146 2147 2148 2149 2150 2151 2152
## 76.62664 54.90137 68.75757 67.77225 75.06123 76.45938 67.80820 74.71868
## 2153 2154 2155 2156 2157 2158 2159 2160
## 71.76258 60.98190 65.32413 80.56066 76.37712 76.26668 73.05374 41.60674
## 2161 2162 2163 2164 2165 2166 2167 2168
## 60.30777 68.39344 71.20167 57.20539 62.39518 75.14376 68.08203 75.78699
## 2169 2170 2171 2172 2173 2174 2175 2176
## 76.59953 80.25449 78.82520 80.98998 75.88682 59.44091 81.08636 71.18928
## 2177 2178 2179 2180 2181 2182 2183 2184
## 62.66102 72.98423 80.45366 63.11531 79.78973 68.48763 81.22903 49.48424
## 2185 2186 2187 2188 2189 2190 2191 2192
## 67.56470 73.04631 71.05664 59.33142 66.71920 77.25823 67.06890 70.44867
## 2193 2194 2195 2196 2197 2198 2199 2200
## 77.28126 78.70289 80.66898 76.21587 73.85958 79.92146 68.36436 77.63212
## 2201 2202 2203 2204 2205 2206 2207 2208
## 77.71295 72.59564 73.87023 79.70576 75.98581 60.34357 63.08531 80.73490
## 2209 2210 2211 2212 2213 2214 2215 2216
## 73.35201 71.47572 68.64982 71.58042 67.15045 67.64148 75.19341 72.34064
## 2217 2218 2219 2220 2221 2222 2223 2224
## 61.05004 67.51790 60.23000 68.97849 76.44636 60.56830 52.21497 76.38825
## 2225 2226 2227 2228 2229 2230 2231 2232
## 80.78172 76.61240 79.27014 74.00638 60.03893 77.87427 72.99308 71.59085
## 2233 2234 2235 2236 2237 2238 2239 2240
## 77.58633 80.44864 71.29859 49.69634 71.74389 74.17269 72.36256 73.65998
## 2241 2242 2243 2244 2245 2246 2247 2248
## 72.86045 72.88518 79.74057 70.43843 73.69344 54.91572 78.82913 61.55988
## 2249 2250 2251 2252 2253 2254 2255 2256
## 72.25351 66.64734 59.68107 80.58956 68.05239 79.55261 76.41722 72.74836
## 2257 2258 2259 2260 2261 2262 2263 2264
## 79.99926 57.07041 82.09591 76.82474 51.27811 59.26339 79.95163 76.05332
## 2265 2266 2267 2268 2269 2270 2271 2272
## 74.08504 67.37929 61.23438 52.90560 72.51155 80.76763 81.96581 72.42682
## 2273 2274 2275 2276 2277 2278
## 73.29486 71.23068 73.85480 69.91363 74.59727 80.07059print(predicted_values_2)## 1 2 3 4 5 6 7 8
## 81.96307 64.66639 70.74809 54.40137 67.90378 78.55564 73.37328 75.48277
## 9 10 11 12 13 14 15 16
## 79.71457 74.13742 76.05272 58.14017 80.55144 69.16296 78.51479 63.16508
## 17 18 19 20 21 22 23 24
## 61.41609 59.27420 79.45282 73.02345 74.88232 79.71768 69.72881 77.14414
## 25 26 27 28 29 30 31 32
## 74.43361 66.79023 74.80508 60.93467 47.70746 75.56028 56.49122 72.42549
## 33 34 35 36 37 38 39 40
## 79.01905 71.75224 68.11062 45.33906 78.84677 70.75199 68.04281 65.71628
## 41 42 43 44 45 46 47 48
## 81.89404 72.30909 76.62025 71.68153 64.52248 66.54921 64.32628 76.20417
## 49 50 51 52 53 54 55 56
## 81.45794 57.52245 71.40157 62.03983 60.16837 64.30272 73.51996 68.67289
## 57 58 59 60 61 62 63 64
## 64.54091 68.50313 72.74382 56.47918 52.75708 68.77983 72.37758 73.49104
## 65 66 67 68 69 70 71 72
## 45.50530 66.09463 76.31981 50.60649 63.12691 62.87069 66.03946 74.50373
## 73 74 75 76 77 78 79 80
## 71.05492 81.12943 62.99071 62.77041 63.40605 69.10538 48.59360 76.74525
## 81 82 83 84 85 86 87 88
## 75.49064 67.82111 71.75869 62.70836 70.77073 78.24725 71.75764 58.08807
## 89 90 91 92 93 94 95 96
## 62.58298 75.08718 72.17850 60.92094 71.98861 59.03503 76.56063 77.00534
## 97 98 99 100 101 102 103 104
## 50.00984 72.25757 81.57455 75.61173 81.48344 75.64564 59.91514 76.50823
## 105 106 107 108 109 110 111 112
## 79.91398 80.05611 59.07898 78.78278 57.42274 65.65083 70.67230 77.46437
## 113 114 115 116 117 118 119 120
## 64.22467 67.19637 65.23759 71.14961 71.93987 73.46778 66.63954 78.79890
## 121 122 123 124 125 126 127 128
## 65.59980 65.93359 80.24758 77.36080 80.31174 80.69962 54.86505 48.33968
## 129 130 131 132 133 134 135 136
## 71.63998 64.71080 56.69158 74.91212 45.88491 48.94708 81.47136 81.07476
## 137 138 139 140 141 142 143 144
## 56.28557 75.22264 59.55374 51.90520 59.85278 48.86188 61.71476 62.31068
## 145 146 147 148 149 150 151 152
## 69.48410 70.25801 76.58205 54.87465 63.22541 76.44515 73.89532 51.03690
## 153 154 155 156 157 158 159 160
## 78.14144 69.27848 78.99311 77.47096 77.33014 78.51916 78.99801 80.61721
## 161 162 163 164 165 166 167 168
## 54.95179 71.36398 70.38802 77.54424 66.82595 71.91965 55.88205 49.02520
## 169 170 171 172 173 174 175 176
## 70.52769 76.01219 66.53992 68.49137 68.49881 71.45902 66.78183 62.14704
## 177 178 179 180 181 182 183 184
## 59.50652 80.04523 50.59892 76.75215 71.43708 76.88862 63.14463 75.30448
## 185 186 187 188 189 190 191 192
## 76.71701 63.31980 52.43159 55.54669 70.00101 48.27911 69.95551 68.00541
## 193 194 195 196 197 198 199 200
## 64.39167 73.61833 67.14181 72.83039 68.21250 56.81138 57.04261 71.16054
## 201 202 203 204 205 206 207 208
## 61.54346 71.20308 73.49380 76.93122 57.96941 74.38922 41.95672 53.49697
## 209 210 211 212 213 214 215 216
## 76.40858 79.64144 62.08219 63.68298 70.10968 79.29878 74.17326 60.43810
## 217 218 219 220 221 222 223 224
## 73.72416 77.66714 65.31919 64.51982 72.01510 63.28928 78.64663 78.24684
## 225 226 227 228 229 230 231 232
## 59.61234 80.94994 75.23864 68.09985 56.16040 59.14269 71.68515 68.82775
## 233 234 235 236 237 238 239 240
## 79.11156 71.73335 78.97104 70.89223 80.00976 60.86352 67.45090 80.81991
## 241 242 243 244 245 246 247 248
## 73.96170 80.73705 79.28704 61.57981 78.34404 74.24166 70.08259 60.31063
## 249 250 251 252 253 254 255 256
## 68.40115 75.90661 43.14910 79.13135 78.14102 72.78268 60.01563 74.82683
## 257 258 259 260 261 262 263 264
## 67.50222 71.86042 71.56589 72.66153 45.37989 72.91995 65.92381 74.49400
## 265 266 267 268 269 270 271 272
## 64.29387 65.98949 59.01359 81.34875 80.21777 76.49134 60.71520 76.58300
## 273 274 275 276 277 278 279 280
## 65.29969 70.51494 63.39828 48.71294 62.77917 80.62773 78.73828 74.84192
## 281 282 283 284 285 286 287 288
## 68.36953 56.38561 74.26409 51.93904 54.69380 79.66257 74.29376 64.59159
## 289 290 291 292 293 294 295 296
## 68.60297 70.26603 66.67559 74.26814 75.49391 81.51316 72.44934 70.76488
## 297 298 299 300 301 302 303 304
## 80.47887 79.36054 67.11605 79.52565 77.02406 44.91914 65.73621 78.67502
## 305 306 307 308 309 310 311 312
## 73.36966 76.26351 75.05047 75.89375 79.99224 68.34182 72.81141 77.81401
## 313 314 315 316 317 318 319 320
## 74.99128 73.07640 70.46336 76.48961 64.26565 75.36108 73.37959 70.90193
## 321 322 323 324 325 326 327 328
## 53.10782 36.66990 79.04597 56.69374 70.24368 66.87693 71.49057 74.41205
## 329 330 331 332 333 334 335 336
## 80.56198 68.37382 79.29824 55.14357 64.58472 70.45911 60.70586 71.37112
## 337 338 339 340 341 342 343 344
## 73.73330 61.85991 71.20118 79.93880 79.44545 80.52160 47.53327 73.42877
## 345 346 347 348 349 350 351 352
## 61.01651 75.19927 69.42379 71.19446 67.64339 56.09126 59.65568 74.47578
## 353 354 355 356 357 358 359 360
## 71.06010 76.82360 72.04021 70.70030 81.21953 76.35309 80.96460 73.93511
## 361 362 363 364 365 366 367 368
## 63.35819 41.36272 74.59207 57.10497 76.93581 77.41526 56.00491 79.01292
## 369 370 371 372 373 374 375 376
## 68.33560 70.40702 62.05380 74.20716 33.11203 66.26722 63.78502 67.77469
## 377 378 379 380 381 382 383 384
## 62.72647 70.55715 60.57284 78.21962 77.33781 74.70250 76.97155 74.06136
## 385 386 387 388 389 390 391 392
## 77.87511 78.21212 70.70027 74.18607 74.76267 64.43975 35.63544 78.45465
## 393 394 395 396 397 398 399 400
## 75.13083 66.63564 76.84828 75.27832 66.63871 73.49715 74.83101 70.87623
## 401 402 403 404 405 406 407 408
## 77.17527 60.43288 69.36859 69.52736 73.59462 80.89764 40.93683 59.28887
## 409 410 411 412 413 414 415 416
## 62.29437 65.79648 51.33922 74.93121 79.90118 57.62699 74.53323 64.29169
## 417 418 419 420 421 422 423 424
## 76.51196 69.59549 53.53517 70.15735 66.92321 65.54539 73.38560 77.60929
## 425 426 427 428 429 430 431 432
## 67.79508 69.61702 53.28149 75.76617 74.47387 72.10306 67.19042 44.34438
## 433 434 435 436 437 438 439 440
## 61.11291 80.35126 60.20787 73.91011 79.74970 65.73663 68.82317 71.90075
## 441 442 443 444 445 446 447 448
## 58.99928 40.03835 79.91442 77.84492 70.47978 73.66025 45.57265 75.14220
## 449 450 451 452 453 454 455 456
## 69.79596 69.10529 73.22751 58.02246 73.16457 76.52823 74.28246 66.48828
## 457 458 459 460 461 462 463 464
## 77.60519 60.06373 73.71649 78.33582 63.79583 60.68755 75.26274 71.80443
## 465 466 467 468 469 470 471 472
## 40.87742 76.68325 64.48507 75.85220 80.96295 77.36470 61.35684 79.99766
## 473 474 475 476 477 478 479 480
## 57.44895 79.16167 60.78004 68.01044 74.99107 79.68431 63.38665 58.79342
## 481 482 483 484 485 486 487 488
## 78.69728 80.24977 62.02063 77.62809 63.82930 70.36312 71.34042 69.55695
## 489 490 491 492 493 494 495 496
## 56.30495 76.21766 70.29175 67.60020 61.06689 66.38641 79.15982 75.16745
## 497 498 499 500 501 502 503 504
## 57.81947 67.28856 69.75192 69.05450 78.02551 65.47454 70.53486 63.16559
## 505 506 507 508 509 510 511 512
## 68.50477 75.17643 66.97105 42.16647 61.45607 69.86954 68.76541 73.44850
## 513 514 515 516 517 518 519 520
## 45.49482 73.82034 74.47605 69.63550 55.40255 54.24566 61.14067 74.26265
## 521 522 523 524 525 526 527 528
## 74.46513 75.39347 60.90891 72.97279 71.84692 63.40577 71.32543 72.61442
## 529 530 531 532 533 534 535 536
## 61.06691 75.75402 77.97463 63.75701 60.00003 70.62599 61.12191 68.65803
## 537 538 539 540 541 542 543 544
## 56.10178 66.82144 66.27114 73.85347 45.88459 71.57413 66.01250 57.92953
## 545 546 547 548 549 550 551 552
## 40.62258 69.64461 66.14587 61.37486 80.52513 76.51072 49.54023 70.99739
## 553 554 555 556 557 558 559 560
## 70.98231 59.99670 67.31948 74.13930 75.29336 70.35598 53.84151 68.27108
## 561 562 563 564 565 566 567 568
## 52.62179 63.58264 80.97326 70.60672 75.35252 74.74335 60.53434 69.70802
## 569 570 571 572 573 574 575 576
## 78.05368 68.44893 81.94869 58.94393 69.73211 70.83373 55.76824 76.02494
## 577 578 579 580 581 582 583 584
## 71.68380 76.68051 67.00085 39.20511 60.93652 64.30109 77.17214 77.20202
## 585 586 587 588 589 590 591 592
## 71.63863 76.64081 65.96647 79.30980 75.71393 70.78973 79.76951 75.06733
## 593 594 595 596 597 598 599 600
## 65.99203 66.78442 60.74781 77.15931 80.27714 73.39688 67.22625 79.43610
## 601 602 603 604 605 606 607 608
## 67.45254 63.38478 71.39358 80.91409 63.96657 76.43272 77.79527 80.50749
## 609 610 611 612 613 614 615 616
## 79.53905 79.59943 76.18228 69.48379 76.24818 68.50964 69.22490 60.67622
## 617 618 619 620 621 622 623 624
## 73.59079 64.08378 75.22231 73.83382 70.73165 69.04402 60.52046 70.62645
## 625 626 627 628 629 630 631 632
## 80.33231 72.26017 70.72764 78.85386 74.10405 75.50576 37.80964 69.84105
## 633 634 635 636 637 638 639 640
## 73.28617 76.43898 80.50234 67.55297 61.02613 74.99621 57.14853 73.18369
## 641 642 643 644 645 646 647 648
## 72.77252 75.45597 73.69016 80.11279 66.50118 57.20617 59.97912 72.15651
## 649 650 651 652 653 654 655 656
## 67.66302 59.55794 80.71890 75.51941 80.79581 60.26150 68.96748 79.33671
## 657 658 659 660 661 662 663 664
## 69.33100 70.03062 68.33190 69.70526 68.24415 72.04297 53.37422 50.58710
## 665 666 667 668 669 670 671 672
## 72.49205 77.55344 71.88675 59.80798 51.88622 60.85717 71.25457 79.09198
## 673 674 675 676 677 678 679 680
## 61.44390 52.06396 77.61589 77.58259 58.39612 75.81716 71.45943 74.63207
## 681 682 683 684 685 686 687 688
## 69.60690 60.17507 66.11432 72.17480 63.82901 73.92863 67.74309 54.79832
## 689 690 691 692 693 694 695 696
## 79.87223 76.69294 71.92324 57.21924 79.76960 70.19306 79.23363 62.97449
## 697 698 699 700 701 702 703 704
## 73.77681 58.63868 74.15253 72.02986 74.23721 78.46237 72.94378 80.38104
## 705 706 707 708 709 710 711 712
## 68.29543 62.75050 81.90176 56.32697 60.70161 78.41641 71.28779 79.12281
## 713 714 715 716 717 718 719 720
## 68.30652 76.42875 68.76678 61.74678 74.67377 37.70399 76.54927 78.44889
## 721 722 723 724 725 726 727 728
## 72.06672 71.21967 79.35576 54.11896 70.66609 76.53911 70.51447 67.98310
## 729 730 731 732 733 734 735 736
## 71.36420 45.15357 61.48897 70.16550 76.66165 60.80024 71.02772 69.35632
## 737 738 739 740 741 742 743 744
## 73.00612 50.87331 60.23659 74.89594 65.42008 81.82719 62.79840 67.80420
## 745 746 747 748 749 750 751 752
## 80.76940 67.70344 73.23750 65.85531 58.53197 69.09498 64.63526 70.48491
## 753 754 755 756 757 758 759 760
## 71.88825 68.53665 48.29432 75.87271 74.68768 69.21887 43.61159 60.70361
## 761 762 763 764 765 766 767 768
## 81.06045 49.80047 70.43039 69.95616 71.89865 57.26144 74.12208 70.33865
## 769 770 771 772 773 774 775 776
## 79.64325 80.56001 73.43988 67.89690 57.78381 72.71450 70.46716 62.35641
## 777 778 779 780 781 782 783 784
## 76.64387 57.68774 65.39341 68.98223 75.95675 66.82246 79.61247 51.40590
## 785 786 787 788 789 790 791 792
## 75.05470 60.37630 67.75364 77.95600 80.11441 70.94512 64.80519 66.30451
## 793 794 795 796 797 798 799 800
## 73.80571 79.41240 70.39407 54.91201 59.24176 76.01719 58.46886 50.30203
## 801 802 803 804 805 806 807 808
## 74.73901 79.94895 55.40891 80.89835 76.75296 78.00636 62.28730 67.82249
## 809 810 811 812 813 814 815 816
## 68.25424 81.73700 52.33273 61.21754 73.08958 63.91387 77.82024 69.05918
## 817 818 819 820 821 822 823 824
## 74.86830 56.23709 81.35850 69.10513 56.73429 30.53687 66.41118 73.38624
## 825 826 827 828 829 830 831 832
## 78.69388 81.96397 70.23300 76.76186 58.68661 71.97263 58.02019 61.36123
## 833 834 835 836 837 838 839 840
## 70.45259 75.01795 71.25611 75.52186 67.23170 71.60396 73.02701 64.79356
## 841 842 843 844 845 846 847 848
## 76.46927 78.03862 79.28860 57.36934 69.22768 72.30191 80.21223 68.40185
## 849 850 851 852 853 854 855 856
## 71.35585 74.47408 78.18703 58.09920 77.40717 75.13687 73.44804 68.13154
## 857 858 859 860 861 862 863 864
## 72.40824 55.42104 74.34782 72.61397 75.75509 71.11192 76.33410 53.51545
## 865 866 867 868 869 870 871 872
## 72.33046 71.72472 75.15488 73.13532 67.90686 67.70695 68.76876 71.74944
## 873 874 875 876 877 878 879 880
## 76.01303 69.85955 76.78000 64.78199 81.83152 74.10581 73.20924 72.30851
## 881 882 883 884 885 886 887 888
## 66.14754 72.90020 71.99812 53.32748 71.36413 45.90541 70.74044 68.53698
## 889 890 891 892 893 894 895 896
## 60.99045 69.56681 69.87380 73.91974 80.18591 63.05408 67.41593 49.28434
## 897 898 899 900 901 902 903 904
## 77.74651 39.00126 79.96313 78.90024 67.94977 61.12044 45.72777 65.78570
## 905 906 907 908 909 910 911 912
## 77.73457 54.99223 72.94239 66.26792 77.14841 65.39665 66.32453 72.88554
## 913 914 915 916 917 918 919 920
## 75.10093 74.48542 71.96548 75.98531 65.18036 71.61604 70.65377 60.56050
## 921 922 923 924 925 926 927 928
## 59.99247 42.86072 57.72726 58.45602 65.78079 78.31455 56.31617 72.50561
## 929 930 931 932 933 934 935 936
## 68.39818 67.67017 80.94087 71.38543 75.19632 74.10268 74.28003 69.78626
## 937 938 939 940 941 942 943 944
## 62.72369 67.48702 71.10430 77.87085 71.45739 81.79695 71.20273 73.48225
## 945 946 947 948 949 950 951 952
## 72.94563 76.10938 75.64126 64.10978 64.83518 80.82769 79.54034 71.78757
## 953 954 955 956 957 958 959 960
## 50.91501 67.09676 71.37207 66.31187 76.13729 81.65507 58.28285 54.47282
## 961 962 963 964 965 966 967 968
## 56.25474 58.75382 71.26461 74.20801 71.94874 74.64853 73.16480 71.80572
## 969 970 971 972 973 974 975 976
## 78.11800 45.36896 78.72958 71.54675 68.76703 75.19142 78.55049 70.67165
## 977 978 979 980 981 982 983 984
## 79.20256 72.61042 66.86924 71.50013 69.10547 75.39226 66.55281 37.93657
## 985 986 987 988 989 990 991 992
## 61.82768 67.49784 53.30561 79.39036 80.32421 69.04476 71.54396 71.54666
## 993 994 995 996 997 998 999 1000
## 67.59313 68.23504 72.31918 60.82129 67.09438 57.18605 51.34760 71.64440
## 1001 1002 1003 1004 1005 1006 1007 1008
## 42.81307 59.41415 61.47137 70.39471 66.38977 78.24541 67.27038 65.44616
## 1009 1010 1011 1012 1013 1014 1015 1016
## 81.18143 59.02282 57.67298 67.69571 75.42378 62.29003 66.46177 75.33866
## 1017 1018 1019 1020 1021 1022 1023 1024
## 67.18396 77.62175 68.64124 75.22396 78.03451 70.60401 54.76675 51.24193
## 1025 1026 1027 1028 1029 1030 1031 1032
## 72.83031 78.75308 62.38697 67.23087 82.08137 66.55571 58.70071 79.26063
## 1033 1034 1035 1036 1037 1038 1039 1040
## 75.22901 74.24200 63.08636 70.68298 76.83608 72.66558 71.56955 68.82158
## 1041 1042 1043 1044 1045 1046 1047 1048
## 73.67328 66.72144 73.87518 73.37509 80.91723 69.35006 61.88521 80.37300
## 1049 1050 1051 1052 1053 1054 1055 1056
## 51.79384 71.94477 68.83303 62.00579 77.63858 55.10417 74.36873 58.80845
## 1057 1058 1059 1060 1061 1062 1063 1064
## 59.18647 68.62702 68.75640 67.83869 58.52057 69.18349 48.02727 74.15405
## 1065 1066 1067 1068 1069 1070 1071 1072
## 51.80804 56.70253 68.06144 79.73321 79.47837 73.78263 72.50717 59.30163
## 1073 1074 1075 1076 1077 1078 1079 1080
## 65.79761 76.25864 51.58258 78.50448 62.80413 70.27526 57.37258 70.84873
## 1081 1082 1083 1084 1085 1086 1087 1088
## 67.92199 70.60520 63.63776 78.40679 79.01575 71.02141 67.02365 73.28988
## 1089 1090 1091 1092 1093 1094 1095 1096
## 62.32496 75.08830 60.29217 71.78983 69.69638 80.39184 74.15363 67.42971
## 1097 1098 1099 1100 1101 1102 1103 1104
## 57.17235 47.52575 64.50958 72.47960 71.48570 68.80015 68.18735 62.91346
## 1105 1106 1107 1108 1109 1110 1111 1112
## 74.39033 65.53566 55.80519 69.55327 65.32791 59.11258 80.51911 40.65115
## 1113 1114 1115 1116 1117 1118 1119 1120
## 67.94588 71.91146 79.79815 75.21634 73.96459 76.09608 65.00550 73.79572
## 1121 1122 1123 1124 1125 1126 1127 1128
## 79.88554 51.10453 79.75158 72.45324 70.71965 75.37587 78.76861 81.00295
## 1129 1130 1131 1132 1133 1134 1135 1136
## 69.37075 66.79760 71.94197 76.96654 73.81421 31.99311 51.90844 62.69350
## 1137 1138 1139 1140 1141 1142 1143 1144
## 74.65250 61.67157 38.50386 72.30151 78.73633 75.69737 67.67665 70.99247
## 1145 1146 1147 1148 1149 1150 1151 1152
## 75.79546 69.03634 75.45395 74.99023 61.63449 79.97750 69.56568 56.97324
## 1153 1154 1155 1156 1157 1158 1159 1160
## 71.94717 78.27667 75.79189 78.20674 74.24114 73.81487 78.16656 62.70653
## 1161 1162 1163 1164 1165 1166 1167 1168
## 48.76999 74.07024 63.17353 61.43073 72.37508 68.42864 65.83070 80.19862
## 1169 1170 1171 1172 1173 1174 1175 1176
## 75.19039 63.72277 64.78483 73.89988 63.57198 81.18637 69.69654 72.20256
## 1177 1178 1179 1180 1181 1182 1183 1184
## 68.48067 74.63717 66.50800 63.92440 62.10813 80.61847 79.44416 79.51693
## 1185 1186 1187 1188 1189 1190 1191 1192
## 52.73458 64.04607 62.60210 80.84623 69.93826 77.48126 72.21062 63.78300
## 1193 1194 1195 1196 1197 1198 1199 1200
## 47.19262 74.04762 75.20975 63.36923 64.60380 70.66993 52.05876 69.63897
## 1201 1202 1203 1204 1205 1206 1207 1208
## 58.95979 75.75220 68.45098 70.83693 71.69214 71.41383 76.14459 80.32684
## 1209 1210 1211 1212 1213 1214 1215 1216
## 48.10205 73.10452 75.90564 74.44106 65.43809 80.59532 67.28054 70.39722
## 1217 1218 1219 1220 1221 1222 1223 1224
## 69.55071 70.25366 72.27437 70.64017 73.84985 51.38366 73.60490 64.29900
## 1225 1226 1227 1228 1229 1230 1231 1232
## 80.63033 64.89004 78.74789 68.51129 62.10702 78.95957 63.50784 71.98388
## 1233 1234 1235 1236 1237 1238 1239 1240
## 63.94685 69.81020 68.46337 71.85762 71.29568 71.76863 70.90249 71.56834
## 1241 1242 1243 1244 1245 1246 1247 1248
## 70.86377 49.21102 76.16825 80.51129 74.10780 79.41054 80.38633 76.73248
## 1249 1250 1251 1252 1253 1254 1255 1256
## 67.87213 62.54605 73.78285 70.06090 65.49238 55.29066 61.69752 61.91159
## 1257 1258 1259 1260 1261 1262 1263 1264
## 79.49938 71.12958 76.46392 73.12981 59.42371 60.38391 70.31303 74.05423
## 1265 1266 1267 1268 1269 1270 1271 1272
## 63.05052 59.66623 69.94672 72.29872 74.49569 63.57235 51.57592 65.22711
## 1273 1274 1275 1276 1277 1278 1279 1280
## 60.49136 74.85385 65.55435 69.96755 79.35537 72.16312 69.49368 67.67301
## 1281 1282 1283 1284 1285 1286 1287 1288
## 73.69367 59.66488 58.13972 74.54452 74.16249 76.41513 79.45822 61.03213
## 1289 1290 1291 1292 1293 1294 1295 1296
## 61.98947 77.18790 69.35741 68.14265 72.35196 64.51679 64.95412 80.27488
## 1297 1298 1299 1300 1301 1302 1303 1304
## 56.24508 72.49224 63.36390 77.15515 74.12301 71.19291 73.47695 75.56351
## 1305 1306 1307 1308 1309 1310 1311 1312
## 65.38704 67.29558 70.22451 80.10516 71.98466 75.60815 79.95039 68.68176
## 1313 1314 1315 1316 1317 1318 1319 1320
## 56.26390 67.28057 58.09321 69.98480 70.12134 75.71952 67.92065 68.84158
## 1321 1322 1323 1324 1325 1326 1327 1328
## 46.72602 75.68469 75.70483 72.08330 81.09888 44.36563 57.54956 66.28317
## 1329 1330 1331 1332 1333 1334 1335 1336
## 67.13974 73.13596 61.79463 68.02970 69.98275 75.42374 65.33871 79.36365
## 1337 1338 1339 1340 1341 1342 1343 1344
## 80.14517 61.11847 80.52129 72.89283 58.82540 78.92738 71.78269 78.54657
## 1345 1346 1347 1348 1349 1350 1351 1352
## 48.10431 67.59439 74.71375 77.49316 74.33028 80.96019 61.04354 80.82959
## 1353 1354 1355 1356 1357 1358 1359 1360
## 39.42327 72.30317 69.53282 71.33628 63.57348 74.91795 56.37136 70.36319
## 1361 1362 1363 1364 1365 1366 1367 1368
## 80.02064 74.67518 72.05358 68.77876 81.06372 59.66262 62.02572 62.87294
## 1369 1370 1371 1372 1373 1374 1375 1376
## 70.33257 70.01436 76.56902 71.89390 74.36514 72.68097 79.82649 78.12022
## 1377 1378 1379 1380 1381 1382 1383 1384
## 79.33346 70.82154 72.10610 72.25158 80.25703 49.58799 71.86087 75.51818
## 1385 1386 1387 1388 1389 1390 1391 1392
## 75.15980 74.07832 81.24425 59.82801 56.27545 80.67392 67.88844 77.67953
## 1393 1394 1395 1396 1397 1398 1399 1400
## 75.70698 60.36575 74.40012 77.69512 45.54806 74.43286 70.83904 62.30508
## 1401 1402 1403 1404 1405 1406 1407 1408
## 77.37218 67.29152 58.52995 61.47626 76.25669 75.38018 76.48864 62.59799
## 1409 1410 1411 1412 1413 1414 1415 1416
## 72.04576 71.25888 79.31460 76.88910 76.52541 80.59764 72.24655 73.37260
## 1417 1418 1419 1420 1421 1422 1423 1424
## 61.78902 56.59343 70.67981 74.73265 73.13863 63.20505 80.37414 73.84377
## 1425 1426 1427 1428 1429 1430 1431 1432
## 55.77232 62.89132 71.78575 68.11890 79.97547 69.01539 67.93997 65.88011
## 1433 1434 1435 1436 1437 1438 1439 1440
## 75.59810 57.82085 76.27480 80.63955 66.82986 79.83855 65.86778 81.40245
## 1441 1442 1443 1444 1445 1446 1447 1448
## 72.57175 46.77095 71.92139 79.39980 62.63701 64.77388 57.40242 79.59053
## 1449 1450 1451 1452 1453 1454 1455 1456
## 73.56894 77.49276 78.58854 55.54116 63.15346 53.03252 72.71396 71.41169
## 1457 1458 1459 1460 1461 1462 1463 1464
## 75.67371 70.03698 66.81229 61.80670 67.49361 68.02068 68.27280 78.22153
## 1465 1466 1467 1468 1469 1470 1471 1472
## 82.04389 75.58461 68.26238 69.93636 72.68625 71.11270 62.48687 77.23291
## 1473 1474 1475 1476 1477 1478 1479 1480
## 75.59756 78.92078 77.47123 79.49285 64.85176 76.13853 63.38909 80.22315
## 1481 1482 1483 1484 1485 1486 1487 1488
## 46.36316 73.58969 66.09027 82.55079 77.31196 64.99339 69.71419 69.55388
## 1489 1490 1491 1492 1493 1494 1495 1496
## 67.41994 73.76621 66.22988 59.54982 51.32435 72.19878 70.18338 73.79137
## 1497 1498 1499 1500 1501 1502 1503 1504
## 78.07090 41.69315 57.55446 73.63677 79.30457 61.57357 68.85601 69.30075
## 1505 1506 1507 1508 1509 1510 1511 1512
## 67.61359 72.52101 57.20511 61.35629 71.49474 68.50903 78.35655 56.19862
## 1513 1514 1515 1516 1517 1518 1519 1520
## 64.78476 74.48249 64.27119 70.43101 58.66589 80.66546 80.66937 76.30681
## 1521 1522 1523 1524 1525 1526 1527 1528
## 77.76258 62.23622 59.32941 62.17880 74.64252 45.35521 71.65488 47.67273
## 1529 1530 1531 1532 1533 1534 1535 1536
## 71.40783 79.37795 72.65827 70.85714 72.15567 64.22543 66.68913 67.19380
## 1537 1538 1539 1540 1541 1542 1543 1544
## 81.25623 75.70310 58.48999 75.44762 64.00885 49.92518 77.83221 60.91523
## 1545 1546 1547 1548 1549 1550 1551 1552
## 62.97251 79.42673 53.52335 65.10608 69.63316 74.27771 67.37414 78.31080
## 1553 1554 1555 1556 1557 1558 1559 1560
## 65.07425 76.25610 70.09181 75.59474 70.19054 62.17826 40.25147 69.58017
## 1561 1562 1563 1564 1565 1566 1567 1568
## 79.35820 71.55055 45.33626 74.65867 74.64296 73.56664 66.39379 43.43827
## 1569 1570 1571 1572 1573 1574 1575 1576
## 67.85643 74.56617 70.41410 67.43950 66.90864 64.95693 71.40902 80.55680
## 1577 1578 1579 1580 1581 1582 1583 1584
## 71.42991 62.55004 59.67239 60.45237 63.24687 73.94129 38.23957 61.72095
## 1585 1586 1587 1588 1589 1590 1591 1592
## 79.57179 61.89194 79.47574 66.95571 77.91417 64.91466 67.96716 55.89167
## 1593 1594 1595 1596 1597 1598 1599 1600
## 64.88533 79.91203 76.30699 65.21857 77.86435 72.77900 75.71130 74.81838
## 1601 1602 1603 1604 1605 1606 1607 1608
## 67.27878 65.12764 80.60654 59.25834 72.14786 69.17595 70.33953 76.89168
## 1609 1610 1611 1612 1613 1614 1615 1616
## 73.55437 72.53317 74.88967 74.41578 63.80454 78.39710 77.72163 75.63343
## 1617 1618 1619 1620 1621 1622 1623 1624
## 53.49491 68.96028 71.05818 68.01780 63.67213 72.36825 45.37273 77.83245
## 1625 1626 1627 1628 1629 1630 1631 1632
## 76.71874 72.95604 47.27719 80.36326 69.02036 71.50709 70.25654 67.92006
## 1633 1634 1635 1636 1637 1638 1639 1640
## 51.68513 77.28692 72.73225 62.23526 77.50978 80.66513 55.28552 67.19553
## 1641 1642 1643 1644 1645 1646 1647 1648
## 69.22863 77.22815 66.53911 56.94767 70.16167 64.85322 80.53505 62.21516
## 1649 1650 1651 1652 1653 1654 1655 1656
## 42.29491 68.45032 81.25741 58.61708 76.42595 80.29723 68.72976 66.27629
## 1657 1658 1659 1660 1661 1662 1663 1664
## 71.51947 73.85149 76.06318 75.87816 37.48721 70.23487 67.84941 60.69316
## 1665 1666 1667 1668 1669 1670 1671 1672
## 72.31416 75.06176 52.94766 62.18315 71.16407 76.84158 72.73540 73.82813
## 1673 1674 1675 1676 1677 1678 1679 1680
## 56.13435 70.27922 75.85428 56.11888 71.92779 77.35443 52.96856 71.12692
## 1681 1682 1683 1684 1685 1686 1687 1688
## 69.03970 75.34892 57.36454 67.83956 69.80547 39.36254 81.18715 69.32488
## 1689 1690 1691 1692 1693 1694 1695 1696
## 62.65007 81.09961 72.85002 77.92322 68.69940 79.03003 69.45600 80.62155
## 1697 1698 1699 1700 1701 1702 1703 1704
## 58.66357 60.38273 71.66548 77.65374 65.95493 72.54099 62.20913 64.46292
## 1705 1706 1707 1708 1709 1710 1711 1712
## 62.43656 69.98001 71.45548 74.37786 69.49137 64.83059 61.24973 74.48689
## 1713 1714 1715 1716 1717 1718 1719 1720
## 77.99341 72.13971 78.92823 80.00387 79.73829 77.56661 76.31097 66.61680
## 1721 1722 1723 1724 1725 1726 1727 1728
## 61.36438 76.91694 71.60536 70.15237 74.53923 76.95116 69.72299 81.20395
## 1729 1730 1731 1732 1733 1734 1735 1736
## 67.03315 69.99893 70.55653 70.91504 76.95024 62.48163 80.74137 37.52453
## 1737 1738 1739 1740 1741 1742 1743 1744
## 70.88691 52.74660 81.00346 67.01662 79.81201 73.80716 76.76308 53.11372
## 1745 1746 1747 1748 1749 1750 1751 1752
## 72.12025 70.45604 70.62003 78.57635 46.47600 77.80667 78.02261 74.33829
## 1753 1754 1755 1756 1757 1758 1759 1760
## 74.88608 64.50104 73.79545 60.75650 71.73137 70.25498 78.49269 74.33836
## 1761 1762 1763 1764 1765 1766 1767 1768
## 79.26341 80.96525 73.54978 68.47501 69.41375 65.34700 73.78586 65.57050
## 1769 1770 1771 1772 1773 1774 1775 1776
## 75.92181 59.35877 63.22980 68.15360 76.61197 69.21927 62.45957 57.42153
## 1777 1778 1779 1780 1781 1782 1783 1784
## 69.77741 49.32279 65.33384 56.61841 54.12452 72.57962 66.37084 71.66400
## 1785 1786 1787 1788 1789 1790 1791 1792
## 75.63740 73.24229 60.73562 69.72733 62.38993 57.14033 61.25100 77.35898
## 1793 1794 1795 1796 1797 1798 1799 1800
## 78.25998 76.84633 61.93060 78.49972 79.72600 72.87445 74.04521 71.85490
## 1801 1802 1803 1804 1805 1806 1807 1808
## 66.15171 57.09437 80.93132 62.47221 72.47447 58.54832 74.34720 79.50010
## 1809 1810 1811 1812 1813 1814 1815 1816
## 76.14861 71.23870 72.15106 61.94563 80.66865 76.20684 75.27157 71.04751
## 1817 1818 1819 1820 1821 1822 1823 1824
## 64.05210 68.96146 60.37129 55.47886 59.31818 69.45076 79.86296 60.71900
## 1825 1826 1827 1828 1829 1830 1831 1832
## 78.61428 73.63835 64.19375 77.79429 70.57434 74.21103 62.42937 77.74206
## 1833 1834 1835 1836 1837 1838 1839 1840
## 75.65479 68.97427 68.85630 69.04732 61.62543 71.55960 56.05448 74.75560
## 1841 1842 1843 1844 1845 1846 1847 1848
## 59.80436 70.82573 75.75815 70.87125 64.49474 73.79262 74.69333 58.84818
## 1849 1850 1851 1852 1853 1854 1855 1856
## 79.12939 70.74720 59.92359 44.64723 75.84783 75.01938 40.35873 81.60542
## 1857 1858 1859 1860 1861 1862 1863 1864
## 60.23757 70.83563 68.99982 59.90971 69.64020 70.53959 75.07247 63.17370
## 1865 1866 1867 1868 1869 1870 1871 1872
## 74.24427 65.73498 72.42164 69.77916 70.73968 51.97640 80.18722 60.03821
## 1873 1874 1875 1876 1877 1878 1879 1880
## 51.78458 68.28893 81.13330 42.54053 59.84050 59.32673 71.29735 53.20466
## 1881 1882 1883 1884 1885 1886 1887 1888
## 69.54503 72.88530 80.75349 79.73989 70.16547 58.21054 44.41384 79.17086
## 1889 1890 1891 1892 1893 1894 1895 1896
## 76.08158 59.43365 59.77061 66.81729 76.67077 54.08830 74.59544 73.08435
## 1897 1898 1899 1900 1901 1902 1903 1904
## 69.53051 69.94369 69.48246 59.03491 68.69425 71.06367 65.73496 66.93814
## 1905 1906 1907 1908 1909 1910 1911 1912
## 63.33435 78.87153 74.91300 73.31915 68.98500 70.97772 62.92236 70.13926
## 1913 1914 1915 1916 1917 1918 1919 1920
## 54.90899 69.30686 64.86724 71.08070 61.32460 69.15116 62.08585 74.98815
## 1921 1922 1923 1924 1925 1926 1927 1928
## 71.69396 52.27168 79.90522 65.02321 66.84124 79.15462 74.47800 70.47300
## 1929 1930 1931 1932 1933 1934 1935 1936
## 71.73484 69.53748 66.21201 67.58274 62.66258 77.55930 73.32756 68.10675
## 1937 1938 1939 1940 1941 1942 1943 1944
## 78.00324 76.48744 75.53937 80.76656 59.21393 78.47893 73.24691 78.02005
## 1945 1946 1947 1948 1949 1950 1951 1952
## 76.77987 75.48909 72.72782 71.87322 61.90448 59.95723 67.47671 42.56981
## 1953 1954 1955 1956 1957 1958 1959 1960
## 42.11918 64.76652 71.15625 71.34819 69.19579 70.49618 73.58186 64.68785
## 1961 1962 1963 1964 1965 1966 1967 1968
## 46.92749 63.42546 64.75501 81.73223 68.33993 63.64852 75.36083 50.41971
## 1969 1970 1971 1972 1973 1974 1975 1976
## 74.80931 72.07877 80.74688 76.38258 67.77363 79.90840 72.65210 51.20007
## 1977 1978 1979 1980 1981 1982 1983 1984
## 70.05633 69.90067 68.59293 69.87253 66.28863 77.73013 78.27103 62.90788
## 1985 1986 1987 1988 1989 1990 1991 1992
## 79.10141 79.73038 62.75745 81.58991 66.28624 78.38603 80.68883 74.49936
## 1993 1994 1995 1996 1997 1998 1999 2000
## 81.49763 79.22325 70.81887 71.86801 66.69244 76.26873 72.44960 61.08557
## 2001 2002 2003 2004 2005 2006 2007 2008
## 76.93257 77.61844 70.48425 74.27117 70.48137 68.66299 39.00412 73.92503
## 2009 2010 2011 2012 2013 2014 2015 2016
## 74.24024 75.02294 59.45001 50.84293 73.28921 68.32610 68.48410 80.17157
## 2017 2018 2019 2020 2021 2022 2023 2024
## 73.52810 76.45171 75.96046 71.53712 79.91216 66.57024 71.32439 79.13111
## 2025 2026 2027 2028 2029 2030 2031 2032
## 59.82056 59.71352 70.95043 78.45052 74.04979 71.88732 63.07016 69.93112
## 2033 2034 2035 2036 2037 2038 2039 2040
## 71.36437 63.75702 78.88269 70.63084 69.47472 79.73674 67.65254 67.23863
## 2041 2042 2043 2044 2045 2046 2047 2048
## 63.18096 81.54660 57.63670 54.45107 44.07705 62.03189 62.91420 67.96077
## 2049 2050 2051 2052 2053 2054 2055 2056
## 71.80461 79.95313 71.93429 78.32440 64.66350 78.86428 73.79706 76.77561
## 2057 2058 2059 2060 2061 2062 2063 2064
## 70.34075 55.08567 56.89683 57.23690 71.41683 72.95524 70.76581 49.70275
## 2065 2066 2067 2068 2069 2070 2071 2072
## 76.63290 54.91857 68.58513 57.86871 75.90948 65.89042 64.51355 72.18110
## 2073 2074 2075 2076 2077 2078 2079 2080
## 76.54160 81.52575 71.04965 73.94987 79.54163 68.14356 69.84500 79.26353
## 2081 2082 2083 2084 2085 2086 2087 2088
## 73.63981 50.54848 72.59622 71.97684 67.78599 68.27929 49.30501 31.35903
## 2089 2090 2091 2092 2093 2094 2095 2096
## 65.79095 74.97742 58.92271 53.25011 72.33872 50.04360 63.49395 66.38723
## 2097 2098 2099 2100 2101 2102 2103 2104
## 69.77643 75.55973 71.33054 44.46545 71.70423 68.02670 63.62353 54.07556
## 2105 2106 2107 2108 2109 2110 2111 2112
## 74.28788 79.43748 57.38411 63.47399 74.82488 59.00307 67.38342 73.98187
## 2113 2114 2115 2116 2117 2118 2119 2120
## 53.58608 59.48874 65.96931 64.66685 66.79756 71.29840 73.29923 61.85543
## 2121 2122 2123 2124 2125 2126 2127 2128
## 76.21995 80.91061 71.53593 71.88142 74.52522 61.89257 64.14988 79.57528
## 2129 2130 2131 2132 2133 2134 2135 2136
## 38.27648 44.95555 62.72406 67.28432 67.73453 58.53539 66.07445 71.28561
## 2137 2138 2139 2140 2141 2142 2143 2144
## 68.29401 61.37187 75.34535 66.36653 72.52286 80.59717 78.68768 69.96976
## 2145 2146 2147 2148 2149 2150 2151 2152
## 74.56820 52.72303 66.54925 66.83808 73.17492 75.52429 64.07422 72.11851
## 2153 2154 2155 2156 2157 2158 2159 2160
## 72.34337 62.75491 67.78107 80.88616 75.85446 76.20024 72.45872 35.17931
## 2161 2162 2163 2164 2165 2166 2167 2168
## 63.63131 68.63224 67.76214 61.74062 65.28380 75.95183 65.27678 74.57928
## 2169 2170 2171 2172 2173 2174 2175 2176
## 78.72541 78.08664 78.52176 80.91207 73.48281 60.21483 81.41491 70.80896
## 2177 2178 2179 2180 2181 2182 2183 2184
## 65.51693 73.49285 78.65154 63.31123 77.72418 67.03160 80.49589 51.46494
## 2185 2186 2187 2188 2189 2190 2191 2192
## 67.55149 73.95707 68.75443 61.11644 66.73144 77.56403 67.55354 71.83831
## 2193 2194 2195 2196 2197 2198 2199 2200
## 76.29734 78.53406 78.85482 74.64613 73.39285 77.94422 68.79997 77.95604
## 2201 2202 2203 2204 2205 2206 2207 2208
## 78.10248 70.64454 74.37674 79.44882 74.50290 60.83495 64.06523 80.42100
## 2209 2210 2211 2212 2213 2214 2215 2216
## 71.35070 71.54426 68.34713 71.15763 67.25045 68.18603 73.11255 71.73849
## 2217 2218 2219 2220 2221 2222 2223 2224
## 63.99603 64.31819 58.28833 69.03487 77.25624 61.86020 47.42546 77.76607
## 2225 2226 2227 2228 2229 2230 2231 2232
## 80.16633 75.63137 79.37561 74.93292 56.79331 76.96125 72.11097 70.33255
## 2233 2234 2235 2236 2237 2238 2239 2240
## 75.88766 78.58538 70.97890 54.39537 72.59550 74.74830 71.01526 73.61445
## 2241 2242 2243 2244 2245 2246 2247 2248
## 73.50051 71.46317 79.63011 68.09753 73.18327 49.48715 80.14313 62.45313
## 2249 2250 2251 2252 2253 2254 2255 2256
## 71.53868 65.92179 60.14995 77.93087 69.78781 77.15626 74.83009 72.02574
## 2257 2258 2259 2260 2261 2262 2263 2264
## 80.08859 58.12709 80.56526 75.89800 56.83456 62.28280 79.94391 75.14882
## 2265 2266 2267 2268 2269 2270 2271 2272
## 74.57063 67.46894 61.27426 50.33286 72.15228 80.84743 80.34317 72.88566
## 2273 2274 2275 2276 2277 2278
## 73.54546 69.22347 74.07019 68.13378 74.54860 78.17684print(predicted_values_3)## 1 2 3 4 5 6 7 8
## 4.415996 4.151767 4.277883 3.972169 4.192859 4.366757 4.294545 4.378260
## 9 10 11 12 13 14 15 16
## 4.378776 4.300733 4.348855 4.044394 4.386554 4.245263 4.367816 4.008639
## 17 18 19 20 21 22 23 24
## 4.107045 4.089593 4.375762 4.277089 4.330655 4.367531 4.295651 4.358022
## 25 26 27 28 29 30 31 32
## 4.301363 4.185423 4.311418 4.065441 3.833776 4.343733 3.941138 4.339299
## 33 34 35 36 37 38 39 40
## 4.409415 4.270328 4.185389 3.955452 4.370874 4.259243 4.211517 4.160112
## 41 42 43 44 45 46 47 48
## 4.392700 4.274451 4.326797 4.283764 4.162586 4.215320 4.110902 4.366157
## 49 50 51 52 53 54 55 56
## 4.394936 4.007669 4.246171 4.134534 4.047899 4.099270 4.290214 4.227151
## 57 58 59 60 61 62 63 64
## 4.134306 4.214727 4.306107 4.043396 3.957682 4.253621 4.246777 4.339918
## 65 66 67 68 69 70 71 72
## 3.920366 4.215343 4.349221 3.894162 4.120357 4.174504 4.231202 4.347127
## 73 74 75 76 77 78 79 80
## 4.262805 4.381031 4.155959 4.087836 4.148625 4.257357 3.951441 4.351090
## 81 82 83 84 85 86 87 88
## 4.368553 4.209622 4.278661 4.090024 4.271329 4.400958 4.293105 4.050930
## 89 90 91 92 93 94 95 96
## 4.170562 4.331696 4.280585 4.058021 4.272488 3.977323 4.346288 4.344916
## 97 98 99 100 101 102 103 104
## 3.948581 4.316847 4.398989 4.371009 4.401120 4.316256 4.069239 4.348572
## 105 106 107 108 109 110 111 112
## 4.358486 4.380927 4.099318 4.386172 4.029281 4.163356 4.235365 4.349754
## 113 114 115 116 117 118 119 120
## 4.110782 4.205412 4.170566 4.280847 4.277219 4.332555 4.118813 4.387586
## 121 122 123 124 125 126 127 128
## 4.176839 4.146297 4.382561 4.361368 4.387687 4.383798 4.022554 3.848946
## 129 130 131 132 133 134 135 136
## 4.327910 4.120599 3.991708 4.316911 3.771634 3.980203 4.389993 4.390684
## 137 138 139 140 141 142 143 144
## 4.017870 4.319594 4.115943 4.008147 4.091082 3.959217 4.101154 4.122474
## 145 146 147 148 149 150 151 152
## 4.246250 4.268967 4.331821 3.951539 4.145769 4.352258 4.312288 3.930565
## 153 154 155 156 157 158 159 160
## 4.359778 4.257045 4.372822 4.362439 4.358759 4.395553 4.372598 4.400813
## 161 162 163 164 165 166 167 168
## 3.981697 4.260119 4.308228 4.364318 4.148519 4.294208 3.998255 3.834172
## 169 170 171 172 173 174 175 176
## 4.248152 4.327262 4.223881 4.237176 4.207805 4.290488 4.185135 4.152322
## 177 178 179 180 181 182 183 184
## 4.004568 4.372046 3.975560 4.342238 4.280625 4.357256 4.087212 4.336638
## 185 186 187 188 189 190 191 192
## 4.360012 4.162595 3.902254 3.967465 4.251032 3.945164 4.268713 4.239760
## 193 194 195 196 197 198 199 200
## 4.119561 4.286628 4.161465 4.286664 4.259240 4.056777 4.071339 4.255906
## 201 202 203 204 205 206 207 208
## 4.060738 4.272405 4.295148 4.377708 4.085419 4.296668 3.880073 4.031116
## 209 210 211 212 213 214 215 216
## 4.352070 4.373917 4.123081 4.124013 4.239047 4.368601 4.338831 4.030560
## 217 218 219 220 221 222 223 224
## 4.304505 4.352770 4.189976 4.131974 4.288492 4.097041 4.390590 4.353163
## 225 226 227 228 229 230 231 232
## 4.015023 4.386982 4.324196 4.243878 4.057740 4.065595 4.275903 4.153246
## 233 234 235 236 237 238 239 240
## 4.391893 4.260708 4.371464 4.269531 4.369010 4.112282 4.238759 4.382631
## 241 242 243 244 245 246 247 248
## 4.291693 4.401934 4.367011 4.107330 4.379712 4.312291 4.241424 4.048107
## 249 250 251 252 253 254 255 256
## 4.213935 4.330029 3.875074 4.373012 4.378282 4.299614 4.082546 4.333880
## 257 258 259 260 261 262 263 264
## 4.260959 4.267935 4.291218 4.271829 3.924469 4.305669 4.232374 4.318052
## 265 266 267 268 269 270 271 272
## 4.151097 4.243806 3.985405 4.392933 4.400991 4.321915 4.111387 4.366652
## 273 274 275 276 277 278 279 280
## 4.174061 4.264403 4.163443 3.938680 4.057923 4.387884 4.387439 4.320072
## 281 282 283 284 285 286 287 288
## 4.214961 3.955656 4.298834 3.944945 3.934248 4.379193 4.319408 4.120734
## 289 290 291 292 293 294 295 296
## 4.230656 4.300925 4.160165 4.312041 4.337449 4.396266 4.269890 4.271397
## 297 298 299 300 301 302 303 304
## 4.403567 4.394156 4.156339 4.390760 4.339942 3.875940 4.143137 4.362276
## 305 306 307 308 309 310 311 312
## 4.289632 4.335839 4.336185 4.325902 4.380876 4.214657 4.305549 4.358685
## 313 314 315 316 317 318 319 320
## 4.336482 4.288829 4.287361 4.324266 4.147441 4.339574 4.303761 4.247060
## 321 322 323 324 325 326 327 328
## 3.959022 3.773752 4.389816 3.960811 4.248952 4.226471 4.262190 4.310632
## 329 330 331 332 333 334 335 336
## 4.388290 4.243858 4.386280 3.898288 4.143986 4.267644 4.130158 4.272351
## 337 338 339 340 341 342 343 344
## 4.310765 4.093209 4.289430 4.379576 4.362964 4.386332 3.938233 4.301700
## 345 346 347 348 349 350 351 352
## 4.073264 4.334821 4.251537 4.311451 4.203615 3.959568 4.029391 4.324530
## 353 354 355 356 357 358 359 360
## 4.274962 4.326093 4.295100 4.227949 4.386976 4.351636 4.391864 4.299872
## 361 362 363 364 365 366 367 368
## 4.108207 3.843668 4.334033 3.970219 4.361580 4.345277 3.973346 4.380624
## 369 370 371 372 373 374 375 376
## 4.237432 4.285054 4.116633 4.313270 3.784617 4.217696 4.102380 4.235120
## 377 378 379 380 381 382 383 384
## 4.086198 4.264658 4.054706 4.357301 4.340037 4.317545 4.354154 4.314430
## 385 386 387 388 389 390 391 392
## 4.357095 4.351894 4.251562 4.340890 4.303748 4.114057 3.808030 4.359227
## 393 394 395 396 397 398 399 400
## 4.309837 4.182296 4.359241 4.334053 4.177271 4.297734 4.314762 4.257859
## 401 402 403 404 405 406 407 408
## 4.355459 4.049674 4.233270 4.239012 4.283209 4.387963 3.838325 3.998250
## 409 410 411 412 413 414 415 416
## 4.095926 4.217025 3.855759 4.324439 4.373158 4.014875 4.322180 4.155541
## 417 418 419 420 421 422 423 424
## 4.353388 4.253354 3.881495 4.296684 4.176861 4.204711 4.298609 4.362362
## 425 426 427 428 429 430 431 432
## 4.197808 4.273577 4.014819 4.321836 4.343873 4.289362 4.191626 3.892369
## 433 434 435 436 437 438 439 440
## 4.099911 4.383397 4.092800 4.308642 4.363611 4.166496 4.229681 4.276807
## 441 442 443 444 445 446 447 448
## 4.060101 3.808221 4.374584 4.339927 4.241103 4.301758 3.921442 4.319862
## 449 450 451 452 453 454 455 456
## 4.259852 4.260024 4.312990 4.063014 4.285955 4.326338 4.339865 4.148946
## 457 458 459 460 461 462 463 464
## 4.388702 4.112275 4.297500 4.358823 4.134407 4.037049 4.323930 4.290670
## 465 466 467 468 469 470 471 472
## 3.811590 4.325625 4.124728 4.329768 4.388810 4.349112 4.060459 4.386751
## 473 474 475 476 477 478 479 480
## 3.997040 4.384076 4.073097 4.224097 4.360738 4.392176 4.140300 4.025528
## 481 482 483 484 485 486 487 488
## 4.367618 4.361072 4.069903 4.378856 4.174527 4.250996 4.297810 4.228304
## 489 490 491 492 493 494 495 496
## 3.930840 4.363112 4.257233 4.172688 4.057274 4.175538 4.374339 4.340430
## 497 498 499 500 501 502 503 504
## 4.080055 4.193617 4.246780 4.229209 4.357958 4.151814 4.249744 4.065783
## 505 506 507 508 509 510 511 512
## 4.248760 4.301312 4.229643 3.768062 4.106849 4.293151 4.255253 4.304399
## 513 514 515 516 517 518 519 520
## 3.941065 4.293687 4.348377 4.245485 4.012397 3.982796 4.056552 4.296208
## 521 522 523 524 525 526 527 528
## 4.327127 4.338056 4.064353 4.284923 4.274526 4.097918 4.277551 4.333585
## 529 530 531 532 533 534 535 536
## 4.028169 4.345743 4.362021 4.144325 4.101418 4.233542 4.138209 4.276224
## 537 538 539 540 541 542 543 544
## 4.012869 4.227515 4.148024 4.293355 3.821365 4.272594 4.203343 4.009156
## 545 546 547 548 549 550 551 552
## 3.837076 4.273779 4.201390 4.054584 4.379918 4.352343 3.904236 4.261963
## 553 554 555 556 557 558 559 560
## 4.264624 4.021492 4.209555 4.295727 4.333119 4.230784 3.926138 4.219830
## 561 562 563 564 565 566 567 568
## 3.950011 4.098971 4.383390 4.262867 4.310401 4.312850 4.069273 4.224085
## 569 570 571 572 573 574 575 576
## 4.355711 4.232357 4.398840 3.961254 4.246864 4.275976 4.065721 4.334788
## 577 578 579 580 581 582 583 584
## 4.311918 4.336525 4.224900 3.755671 4.075059 4.142291 4.384321 4.372015
## 585 586 587 588 589 590 591 592
## 4.294249 4.364685 4.146843 4.388890 4.343016 4.298054 4.377527 4.310067
## 593 594 595 596 597 598 599 600
## 4.165660 4.219715 4.108562 4.368243 4.387058 4.290683 4.234167 4.358797
## 601 602 603 604 605 606 607 608
## 4.196569 4.179873 4.272322 4.389161 4.131254 4.368494 4.342518 4.387967
## 609 610 611 612 613 614 615 616
## 4.371424 4.377710 4.343076 4.268665 4.343040 4.242041 4.232401 4.038840
## 617 618 619 620 621 622 623 624
## 4.295984 4.156715 4.324515 4.293896 4.259470 4.200130 4.054056 4.280665
## 625 626 627 628 629 630 631 632
## 4.377931 4.304567 4.282072 4.385755 4.305280 4.341203 3.772959 4.254367
## 633 634 635 636 637 638 639 640
## 4.302836 4.326849 4.385480 4.196373 4.106342 4.338508 4.072206 4.305653
## 641 642 643 644 645 646 647 648
## 4.276441 4.323646 4.291844 4.387025 4.177517 3.991925 4.048016 4.299475
## 649 650 651 652 653 654 655 656
## 4.204252 4.072358 4.401758 4.319756 4.392934 4.095170 4.219356 4.374127
## 657 658 659 660 661 662 663 664
## 4.210362 4.268350 4.243236 4.241286 4.147965 4.279661 3.960234 3.976159
## 665 666 667 668 669 670 671 672
## 4.294350 4.373734 4.289928 4.090223 3.932824 4.034954 4.288319 4.389756
## 673 674 675 676 677 678 679 680
## 4.083355 3.952693 4.338372 4.382693 4.019581 4.314289 4.285142 4.319301
## 681 682 683 684 685 686 687 688
## 4.247403 4.098401 4.227897 4.289619 4.149291 4.350024 4.182690 3.982986
## 689 690 691 692 693 694 695 696
## 4.370715 4.343160 4.263104 3.984741 4.375987 4.248584 4.372449 4.123792
## 697 698 699 700 701 702 703 704
## 4.297109 4.038180 4.327596 4.263379 4.302170 4.336525 4.284725 4.386599
## 705 706 707 708 709 710 711 712
## 4.229265 4.119425 4.398836 3.927204 4.009962 4.390373 4.256690 4.365265
## 713 714 715 716 717 718 719 720
## 4.237862 4.360449 4.238093 4.120230 4.326892 3.792717 4.352327 4.355656
## 721 722 723 724 725 726 727 728
## 4.270791 4.276879 4.363620 4.038618 4.239748 4.351359 4.259733 4.164726
## 729 730 731 732 733 734 735 736
## 4.277187 3.805470 4.052255 4.268582 4.356821 4.019736 4.253083 4.202908
## 737 738 739 740 741 742 743 744
## 4.334755 3.901976 4.021145 4.323293 4.148110 4.396964 4.084312 4.215746
## 745 746 747 748 749 750 751 752
## 4.386454 4.205179 4.318306 4.208356 4.109092 4.259938 4.169967 4.290344
## 753 754 755 756 757 758 759 760
## 4.276644 4.218008 3.946545 4.344832 4.348682 4.209863 3.887995 4.072898
## 761 762 763 764 765 766 767 768
## 4.378201 3.986849 4.275710 4.247259 4.286750 4.034573 4.296198 4.250627
## 769 770 771 772 773 774 775 776
## 4.379158 4.383768 4.279583 4.237055 4.008481 4.283818 4.254413 4.128355
## 777 778 779 780 781 782 783 784
## 4.325666 4.002710 4.196991 4.210462 4.318895 4.206187 4.378255 4.012374
## 785 786 787 788 789 790 791 792
## 4.307891 4.091220 4.216000 4.345075 4.382057 4.262932 4.161937 4.148099
## 793 794 795 796 797 798 799 800
## 4.309453 4.377639 4.283348 4.035045 3.966614 4.325512 4.103093 3.887852
## 801 802 803 804 805 806 807 808
## 4.319684 4.382236 3.913108 4.385835 4.343134 4.353480 4.102440 4.236534
## 809 810 811 812 813 814 815 816
## 4.179114 4.395793 3.993306 4.101485 4.315459 4.121619 4.343843 4.235976
## 817 818 819 820 821 822 823 824
## 4.323674 4.006341 4.384029 4.227800 3.964130 3.760178 4.220039 4.295131
## 825 826 827 828 829 830 831 832
## 4.364063 4.399058 4.296353 4.353490 4.092918 4.268798 3.993590 4.122787
## 833 834 835 836 837 838 839 840
## 4.263679 4.322348 4.265678 4.323251 4.211362 4.281264 4.288384 4.166507
## 841 842 843 844 845 846 847 848
## 4.350890 4.387685 4.388416 4.009402 4.263839 4.275758 4.372122 4.219260
## 849 850 851 852 853 854 855 856
## 4.273788 4.298446 4.375465 4.052942 4.360463 4.308744 4.279398 4.203725
## 857 858 859 860 861 862 863 864
## 4.308547 3.961066 4.328898 4.318847 4.339513 4.240071 4.335856 4.036244
## 865 866 867 868 869 870 871 872
## 4.275238 4.273024 4.317879 4.307607 4.215782 4.234565 4.226708 4.298806
## 873 874 875 876 877 878 879 880
## 4.350404 4.239752 4.355481 4.176693 4.396028 4.310611 4.287701 4.292395
## 881 882 883 884 885 886 887 888
## 4.168564 4.260049 4.247044 3.987228 4.276861 3.914687 4.263510 4.219743
## 889 890 891 892 893 894 895 896
## 4.124945 4.245779 4.276510 4.295507 4.389945 4.143477 4.186385 3.915254
## 897 898 899 900 901 902 903 904
## 4.392997 3.844802 4.384389 4.388935 4.144965 4.104261 3.922429 4.190949
## 905 906 907 908 909 910 911 912
## 4.346686 3.937107 4.278020 4.108492 4.360175 4.140042 4.173263 4.280069
## 913 914 915 916 917 918 919 920
## 4.334100 4.305470 4.287211 4.344573 4.126888 4.276208 4.266342 4.129963
## 921 922 923 924 925 926 927 928
## 4.041199 3.857931 3.977620 4.074995 4.142219 4.384387 3.981341 4.282400
## 929 930 931 932 933 934 935 936
## 4.226282 4.251865 4.393277 4.257398 4.309699 4.293466 4.330588 4.233996
## 937 938 939 940 941 942 943 944
## 4.112393 4.161506 4.314904 4.349622 4.289852 4.384578 4.284012 4.299381
## 945 946 947 948 949 950 951 952
## 4.301851 4.329711 4.312646 4.097190 4.125631 4.386421 4.368665 4.283775
## 953 954 955 956 957 958 959 960
## 3.864552 4.227476 4.267905 4.193133 4.331181 4.395741 3.914935 3.992433
## 961 962 963 964 965 966 967 968
## 4.052033 4.028634 4.254637 4.333382 4.269009 4.317544 4.303834 4.295875
## 969 970 971 972 973 974 975 976
## 4.368174 3.940939 4.394566 4.257604 4.209529 4.341436 4.404343 4.262211
## 977 978 979 980 981 982 983 984
## 4.363437 4.327391 4.201172 4.262559 4.258580 4.340473 4.214393 3.776352
## 985 986 987 988 989 990 991 992
## 4.110159 4.235641 4.024497 4.365708 4.378152 4.255582 4.275016 4.299433
## 993 994 995 996 997 998 999 1000
## 4.164411 4.198895 4.324646 4.071597 4.184766 3.992661 3.968938 4.296706
## 1001 1002 1003 1004 1005 1006 1007 1008
## 3.773590 3.984788 4.113490 4.242380 4.149442 4.362731 4.192517 4.131054
## 1009 1010 1011 1012 1013 1014 1015 1016
## 4.397601 3.963192 4.050766 4.207968 4.338756 4.148507 4.116653 4.313795
## 1017 1018 1019 1020 1021 1022 1023 1024
## 4.132308 4.334560 4.256062 4.339947 4.357717 4.252219 3.901275 3.999810
## 1025 1026 1027 1028 1029 1030 1031 1032
## 4.281782 4.360729 4.066189 4.208609 4.400532 4.178737 3.983385 4.368608
## 1033 1034 1035 1036 1037 1038 1039 1040
## 4.377840 4.293479 4.132359 4.262101 4.347855 4.296525 4.264574 4.242447
## 1041 1042 1043 1044 1045 1046 1047 1048
## 4.291476 4.183361 4.303663 4.314470 4.379069 4.273765 4.119254 4.385440
## 1049 1050 1051 1052 1053 1054 1055 1056
## 3.934388 4.289830 4.238193 4.111889 4.355783 4.048984 4.338192 3.977395
## 1057 1058 1059 1060 1061 1062 1063 1064
## 4.020536 4.203394 4.193277 4.238610 3.975912 4.253879 3.877463 4.313007
## 1065 1066 1067 1068 1069 1070 1071 1072
## 4.017839 4.102272 4.251811 4.382860 4.378320 4.294902 4.275758 4.105756
## 1073 1074 1075 1076 1077 1078 1079 1080
## 4.183205 4.324063 3.929002 4.384906 4.138015 4.258478 3.964706 4.268502
## 1081 1082 1083 1084 1085 1086 1087 1088
## 4.208161 4.268572 4.104791 4.376248 4.386130 4.297545 4.189001 4.290585
## 1089 1090 1091 1092 1093 1094 1095 1096
## 4.084627 4.334335 4.024387 4.259495 4.268797 4.393281 4.300900 4.231311
## 1097 1098 1099 1100 1101 1102 1103 1104
## 4.081341 3.840210 4.146240 4.326277 4.289764 4.251830 4.190076 4.111879
## 1105 1106 1107 1108 1109 1110 1111 1112
## 4.300688 4.155289 3.964023 4.262719 4.132697 4.089683 4.397159 3.868116
## 1113 1114 1115 1116 1117 1118 1119 1120
## 4.206235 4.290695 4.396496 4.308238 4.306068 4.326030 4.206542 4.306280
## 1121 1122 1123 1124 1125 1126 1127 1128
## 4.395399 3.891608 4.377229 4.282499 4.266211 4.317513 4.386783 4.389426
## 1129 1130 1131 1132 1133 1134 1135 1136
## 4.250871 4.158509 4.276648 4.365979 4.277382 3.771673 3.890158 4.118399
## 1137 1138 1139 1140 1141 1142 1143 1144
## 4.305721 4.038898 3.762169 4.279978 4.368384 4.334712 4.167188 4.267692
## 1145 1146 1147 1148 1149 1150 1151 1152
## 4.325496 4.259031 4.378803 4.341266 4.145308 4.379431 4.235700 3.973639
## 1153 1154 1155 1156 1157 1158 1159 1160
## 4.255709 4.383263 4.315887 4.389548 4.340606 4.298122 4.369995 4.111020
## 1161 1162 1163 1164 1165 1166 1167 1168
## 3.876306 4.334193 4.140355 4.151125 4.274108 4.249034 4.160890 4.372914
## 1169 1170 1171 1172 1173 1174 1175 1176
## 4.344934 4.102141 4.083367 4.301539 4.099831 4.393250 4.219624 4.277413
## 1177 1178 1179 1180 1181 1182 1183 1184
## 4.226140 4.336332 4.216191 4.078583 4.141148 4.384403 4.362363 4.368383
## 1185 1186 1187 1188 1189 1190 1191 1192
## 3.972718 4.162378 4.110536 4.383577 4.273998 4.369881 4.274954 4.171168
## 1193 1194 1195 1196 1197 1198 1199 1200
## 3.926278 4.319664 4.314621 4.136340 4.153905 4.270272 3.939664 4.233598
## 1201 1202 1203 1204 1205 1206 1207 1208
## 4.012409 4.363118 4.170693 4.262000 4.297680 4.267618 4.345268 4.386082
## 1209 1210 1211 1212 1213 1214 1215 1216
## 3.858000 4.301512 4.336392 4.308340 4.203594 4.387591 4.249512 4.318478
## 1217 1218 1219 1220 1221 1222 1223 1224
## 4.243195 4.255637 4.276010 4.258343 4.316944 3.999203 4.300651 4.133648
## 1225 1226 1227 1228 1229 1230 1231 1232
## 4.385881 4.206378 4.369627 4.221493 4.087555 4.371888 4.129080 4.284294
## 1233 1234 1235 1236 1237 1238 1239 1240
## 4.104828 4.234484 4.241737 4.318807 4.269325 4.260286 4.247025 4.262718
## 1241 1242 1243 1244 1245 1246 1247 1248
## 4.264587 3.979770 4.322936 4.380426 4.314046 4.371358 4.362587 4.351917
## 1249 1250 1251 1252 1253 1254 1255 1256
## 4.192709 4.131454 4.298359 4.244124 4.158492 3.940811 4.063576 4.079420
## 1257 1258 1259 1260 1261 1262 1263 1264
## 4.390840 4.273718 4.350498 4.298580 4.005000 4.038790 4.259047 4.317081
## 1265 1266 1267 1268 1269 1270 1271 1272
## 4.155998 4.007105 4.228394 4.322971 4.316112 4.179069 3.940240 4.150676
## 1273 1274 1275 1276 1277 1278 1279 1280
## 4.040847 4.339346 4.136087 4.274258 4.385468 4.276310 4.230873 4.202057
## 1281 1282 1283 1284 1285 1286 1287 1288
## 4.303621 4.032126 4.027310 4.297813 4.332002 4.346955 4.372906 4.049809
## 1289 1290 1291 1292 1293 1294 1295 1296
## 4.066989 4.346410 4.259149 4.216863 4.301359 4.101027 4.127508 4.392432
## 1297 1298 1299 1300 1301 1302 1303 1304
## 3.942470 4.279830 4.168467 4.350754 4.295806 4.269763 4.302050 4.342807
## 1305 1306 1307 1308 1309 1310 1311 1312
## 4.156311 4.158634 4.223897 4.395323 4.277961 4.318369 4.394116 4.225658
## 1313 1314 1315 1316 1317 1318 1319 1320
## 4.004860 4.181394 4.031696 4.272908 4.279300 4.341435 4.201076 4.251864
## 1321 1322 1323 1324 1325 1326 1327 1328
## 3.825473 4.329706 4.370866 4.302924 4.386524 3.796386 4.005326 4.222784
## 1329 1330 1331 1332 1333 1334 1335 1336
## 4.192260 4.303374 4.092951 4.247340 4.228355 4.337007 4.183619 4.376642
## 1337 1338 1339 1340 1341 1342 1343 1344
## 4.381656 4.042177 4.378152 4.295808 4.057149 4.370694 4.291994 4.393386
## 1345 1346 1347 1348 1349 1350 1351 1352
## 3.944097 4.231176 4.308437 4.348565 4.316189 4.383718 4.101868 4.382402
## 1353 1354 1355 1356 1357 1358 1359 1360
## 3.806958 4.290570 4.248270 4.271505 4.073894 4.307101 4.026761 4.293662
## 1361 1362 1363 1364 1365 1366 1367 1368
## 4.385279 4.311053 4.270178 4.236353 4.383816 4.065174 4.055034 4.156357
## 1369 1370 1371 1372 1373 1374 1375 1376
## 4.232451 4.240380 4.358465 4.277270 4.307492 4.298250 4.368685 4.337895
## 1377 1378 1379 1380 1381 1382 1383 1384
## 4.370639 4.266923 4.267395 4.298471 4.395067 3.926905 4.288169 4.345602
## 1385 1386 1387 1388 1389 1390 1391 1392
## 4.313081 4.309575 4.382069 4.026620 3.945654 4.400358 4.215373 4.358738
## 1393 1394 1395 1396 1397 1398 1399 1400
## 4.339831 4.037338 4.298489 4.356386 3.915305 4.320022 4.270351 4.170521
## 1401 1402 1403 1404 1405 1406 1407 1408
## 4.362405 4.174525 4.044152 4.113122 4.348961 4.306433 4.353049 4.133356
## 1409 1410 1411 1412 1413 1414 1415 1416
## 4.325669 4.282414 4.386226 4.348352 4.336501 4.388709 4.291485 4.289693
## 1417 1418 1419 1420 1421 1422 1423 1424
## 4.120609 3.974620 4.254864 4.308384 4.284817 4.082484 4.396934 4.292438
## 1425 1426 1427 1428 1429 1430 1431 1432
## 3.933518 4.138370 4.292517 4.175883 4.371039 4.241831 4.239391 4.212863
## 1433 1434 1435 1436 1437 1438 1439 1440
## 4.339247 3.950264 4.347835 4.387971 4.171357 4.399325 4.164704 4.396737
## 1441 1442 1443 1444 1445 1446 1447 1448
## 4.302143 3.911464 4.290113 4.397323 4.079934 4.151284 4.054932 4.393228
## 1449 1450 1451 1452 1453 1454 1455 1456
## 4.308819 4.365631 4.369110 4.064900 4.113088 3.911002 4.281797 4.317626
## 1457 1458 1459 1460 1461 1462 1463 1464
## 4.380153 4.261890 4.225513 4.075858 4.227199 4.172480 4.252834 4.361918
## 1465 1466 1467 1468 1469 1470 1471 1472
## 4.388568 4.368377 4.252807 4.264337 4.301568 4.238180 4.148530 4.360444
## 1473 1474 1475 1476 1477 1478 1479 1480
## 4.380693 4.348757 4.354129 4.359956 4.143701 4.329933 4.119031 4.372361
## 1481 1482 1483 1484 1485 1486 1487 1488
## 3.983888 4.298244 4.221560 4.401178 4.331048 4.199208 4.236947 4.244921
## 1489 1490 1491 1492 1493 1494 1495 1496
## 4.233335 4.317713 4.208239 4.043917 3.872209 4.282366 4.273030 4.316250
## 1497 1498 1499 1500 1501 1502 1503 1504
## 4.356039 3.834157 3.980582 4.291300 4.369245 4.064286 4.223994 4.263882
## 1505 1506 1507 1508 1509 1510 1511 1512
## 4.241350 4.294777 3.997155 4.082819 4.295440 4.207391 4.354922 4.065638
## 1513 1514 1515 1516 1517 1518 1519 1520
## 4.177993 4.298546 4.154306 4.243411 3.982851 4.390071 4.391079 4.351276
## 1521 1522 1523 1524 1525 1526 1527 1528
## 4.352737 4.108116 3.972175 4.075166 4.316292 3.956666 4.261149 3.803517
## 1529 1530 1531 1532 1533 1534 1535 1536
## 4.296852 4.396442 4.273697 4.280085 4.271799 4.146531 4.152954 4.231705
## 1537 1538 1539 1540 1541 1542 1543 1544
## 4.387465 4.356089 4.003158 4.340662 4.198889 3.957271 4.359710 4.077706
## 1545 1546 1547 1548 1549 1550 1551 1552
## 4.174777 4.378309 4.014200 4.127755 4.209980 4.326307 4.229836 4.350164
## 1553 1554 1555 1556 1557 1558 1559 1560
## 4.209216 4.320736 4.258235 4.343271 4.259248 4.125163 3.809734 4.220617
## 1561 1562 1563 1564 1565 1566 1567 1568
## 4.374982 4.300206 3.941546 4.298665 4.343934 4.310995 4.225944 3.779476
## 1569 1570 1571 1572 1573 1574 1575 1576
## 4.190182 4.326186 4.262207 4.186537 4.191756 4.214509 4.242173 4.373788
## 1577 1578 1579 1580 1581 1582 1583 1584
## 4.278524 4.078122 4.084793 4.061753 4.097156 4.300286 3.788690 4.086140
## 1585 1586 1587 1588 1589 1590 1591 1592
## 4.369707 4.091910 4.393467 4.187221 4.379023 4.170608 4.213719 3.956928
## 1593 1594 1595 1596 1597 1598 1599 1600
## 4.122400 4.383489 4.327109 4.161834 4.366098 4.274764 4.328082 4.314999
## 1601 1602 1603 1604 1605 1606 1607 1608
## 4.184209 4.175598 4.381508 4.078553 4.290361 4.236226 4.243435 4.369365
## 1609 1610 1611 1612 1613 1614 1615 1616
## 4.300387 4.276969 4.334899 4.306683 4.155062 4.385236 4.358519 4.346828
## 1617 1618 1619 1620 1621 1622 1623 1624
## 3.919912 4.194109 4.250398 4.234721 4.142764 4.330371 3.949560 4.352486
## 1625 1626 1627 1628 1629 1630 1631 1632
## 4.379726 4.280147 3.962807 4.381958 4.201787 4.285524 4.250446 4.208257
## 1633 1634 1635 1636 1637 1638 1639 1640
## 4.030449 4.344186 4.284410 4.098541 4.343306 4.382003 3.917397 4.183057
## 1641 1642 1643 1644 1645 1646 1647 1648
## 4.207632 4.375309 4.194780 3.952313 4.281447 4.107654 4.383820 4.072351
## 1649 1650 1651 1652 1653 1654 1655 1656
## 3.868446 4.178815 4.387019 3.992202 4.338090 4.376181 4.190596 4.172510
## 1657 1658 1659 1660 1661 1662 1663 1664
## 4.309081 4.306843 4.346905 4.321644 3.782037 4.250718 4.242829 4.018656
## 1665 1666 1667 1668 1669 1670 1671 1672
## 4.304364 4.303373 3.910653 4.128246 4.284413 4.340432 4.288421 4.307365
## 1673 1674 1675 1676 1677 1678 1679 1680
## 4.008891 4.258771 4.347173 4.055243 4.291173 4.356179 4.011136 4.276164
## 1681 1682 1683 1684 1685 1686 1687 1688
## 4.240446 4.356001 4.038552 4.240442 4.277027 3.807051 4.392254 4.259069
## 1689 1690 1691 1692 1693 1694 1695 1696
## 4.081934 4.387908 4.301241 4.355623 4.253432 4.366561 4.225609 4.389443
## 1697 1698 1699 1700 1701 1702 1703 1704
## 3.954176 4.012432 4.289668 4.363760 4.127602 4.307260 4.143016 4.186660
## 1705 1706 1707 1708 1709 1710 1711 1712
## 4.103692 4.283642 4.315119 4.331710 4.255432 4.144349 4.116649 4.309099
## 1713 1714 1715 1716 1717 1718 1719 1720
## 4.361849 4.286498 4.381994 4.375685 4.374904 4.337200 4.318224 4.182606
## 1721 1722 1723 1724 1725 1726 1727 1728
## 4.066398 4.344034 4.243649 4.274713 4.304708 4.367461 4.267851 4.392320
## 1729 1730 1731 1732 1733 1734 1735 1736
## 4.225839 4.259744 4.304144 4.248639 4.355385 4.145005 4.382514 3.762594
## 1737 1738 1739 1740 1741 1742 1743 1744
## 4.313820 3.939192 4.393114 4.190053 4.380029 4.294628 4.357109 3.920563
## 1745 1746 1747 1748 1749 1750 1751 1752
## 4.252147 4.267512 4.264777 4.365787 3.924474 4.348857 4.351498 4.344229
## 1753 1754 1755 1756 1757 1758 1759 1760
## 4.304861 4.136901 4.306863 4.093584 4.291936 4.287908 4.377149 4.294417
## 1761 1762 1763 1764 1765 1766 1767 1768
## 4.363098 4.396147 4.288676 4.192827 4.247081 4.152496 4.302800 4.098956
## 1769 1770 1771 1772 1773 1774 1775 1776
## 4.339455 4.057989 4.095220 4.235957 4.338741 4.244053 4.137960 4.089293
## 1777 1778 1779 1780 1781 1782 1783 1784
## 4.249394 3.936645 4.153469 4.012964 3.965244 4.303619 4.212164 4.287545
## 1785 1786 1787 1788 1789 1790 1791 1792
## 4.319198 4.316390 4.099716 4.254841 4.079817 4.030437 4.098702 4.348453
## 1793 1794 1795 1796 1797 1798 1799 1800
## 4.343357 4.335551 4.067285 4.377717 4.372933 4.286094 4.314772 4.264658
## 1801 1802 1803 1804 1805 1806 1807 1808
## 4.154538 3.994327 4.384644 4.101762 4.270590 3.986618 4.337020 4.374799
## 1809 1810 1811 1812 1813 1814 1815 1816
## 4.358972 4.276194 4.283612 4.119409 4.387206 4.347525 4.334966 4.256539
## 1817 1818 1819 1820 1821 1822 1823 1824
## 4.160472 4.235200 4.055541 3.990761 4.041491 4.241375 4.398743 4.038577
## 1825 1826 1827 1828 1829 1830 1831 1832
## 4.395248 4.339787 4.156154 4.367151 4.255129 4.297013 4.117930 4.361563
## 1833 1834 1835 1836 1837 1838 1839 1840
## 4.342957 4.232038 4.230117 4.234920 4.029056 4.318329 3.977337 4.339338
## 1841 1842 1843 1844 1845 1846 1847 1848
## 3.994118 4.292062 4.371306 4.264478 4.164522 4.277428 4.304181 3.990314
## 1849 1850 1851 1852 1853 1854 1855 1856
## 4.364739 4.268403 4.031080 3.884404 4.323602 4.307904 3.822920 4.386974
## 1857 1858 1859 1860 1861 1862 1863 1864
## 4.059856 4.271956 4.243456 4.123312 4.235700 4.263512 4.309419 4.090592
## 1865 1866 1867 1868 1869 1870 1871 1872
## 4.309371 4.198472 4.297489 4.258776 4.274033 3.974488 4.377760 4.087250
## 1873 1874 1875 1876 1877 1878 1879 1880
## 3.922013 4.213301 4.391330 3.890437 4.067235 4.037878 4.287283 4.018595
## 1881 1882 1883 1884 1885 1886 1887 1888
## 4.240750 4.298675 4.381997 4.370991 4.247776 4.058319 3.877757 4.356802
## 1889 1890 1891 1892 1893 1894 1895 1896
## 4.354564 4.022718 4.000078 4.123560 4.355463 3.970214 4.327351 4.316189
## 1897 1898 1899 1900 1901 1902 1903 1904
## 4.252104 4.256217 4.242190 4.043052 4.223696 4.278960 4.177963 4.159686
## 1905 1906 1907 1908 1909 1910 1911 1912
## 4.129137 4.360570 4.345831 4.289043 4.157865 4.265721 4.086694 4.240699
## 1913 1914 1915 1916 1917 1918 1919 1920
## 3.903145 4.211896 4.156458 4.268424 4.093931 4.247781 4.100781 4.331348
## 1921 1922 1923 1924 1925 1926 1927 1928
## 4.263029 3.930675 4.381680 4.126198 4.226743 4.391152 4.319685 4.251454
## 1929 1930 1931 1932 1933 1934 1935 1936
## 4.304513 4.251117 4.193381 4.216244 4.157689 4.347008 4.311440 4.219753
## 1937 1938 1939 1940 1941 1942 1943 1944
## 4.353373 4.349477 4.340743 4.383891 3.997703 4.389161 4.286169 4.377054
## 1945 1946 1947 1948 1949 1950 1951 1952
## 4.334326 4.336979 4.308325 4.300756 4.071695 4.073003 4.227529 3.837183
## 1953 1954 1955 1956 1957 1958 1959 1960
## 3.881422 4.160241 4.279771 4.296392 4.223140 4.290430 4.303199 4.139689
## 1961 1962 1963 1964 1965 1966 1967 1968
## 3.854936 4.123282 4.177614 4.413238 4.212509 4.151373 4.317530 3.887641
## 1969 1970 1971 1972 1973 1974 1975 1976
## 4.333941 4.289211 4.387412 4.340763 4.221800 4.397399 4.330874 4.008514
## 1977 1978 1979 1980 1981 1982 1983 1984
## 4.286488 4.274384 4.229516 4.244390 4.120726 4.340183 4.363390 4.123314
## 1985 1986 1987 1988 1989 1990 1991 1992
## 4.367816 4.365520 4.073738 4.410472 4.169158 4.363811 4.382094 4.345035
## 1993 1994 1995 1996 1997 1998 1999 2000
## 4.381039 4.365966 4.249213 4.274524 4.212208 4.346350 4.276711 4.062550
## 2001 2002 2003 2004 2005 2006 2007 2008
## 4.326064 4.379588 4.286517 4.322881 4.262673 4.180748 3.798382 4.342344
## 2009 2010 2011 2012 2013 2014 2015 2016
## 4.302399 4.336740 4.031669 3.921231 4.286676 4.234913 4.251022 4.381600
## 2017 2018 2019 2020 2021 2022 2023 2024
## 4.295521 4.327793 4.340718 4.282989 4.370630 4.213423 4.287756 4.373376
## 2025 2026 2027 2028 2029 2030 2031 2032
## 4.035732 4.040231 4.267252 4.391918 4.296521 4.300667 4.129299 4.254507
## 2033 2034 2035 2036 2037 2038 2039 2040
## 4.265553 4.156304 4.364926 4.310627 4.231607 4.379365 4.230416 4.228197
## 2041 2042 2043 2044 2045 2046 2047 2048
## 4.135872 4.394594 4.017865 3.997356 3.740207 4.077719 4.108945 4.168827
## 2049 2050 2051 2052 2053 2054 2055 2056
## 4.281920 4.383520 4.295723 4.376287 4.118274 4.360870 4.294645 4.349360
## 2057 2058 2059 2060 2061 2062 2063 2064
## 4.265868 4.051851 4.033605 4.039357 4.283568 4.299913 4.296801 3.847853
## 2065 2066 2067 2068 2069 2070 2071 2072
## 4.334971 4.044585 4.216097 3.972549 4.360744 4.195466 4.224736 4.296672
## 2073 2074 2075 2076 2077 2078 2079 2080
## 4.333296 4.399285 4.295046 4.308177 4.379627 4.210714 4.253497 4.378914
## 2081 2082 2083 2084 2085 2086 2087 2088
## 4.319633 3.885460 4.295503 4.323410 4.202878 4.254152 3.861179 3.760906
## 2089 2090 2091 2092 2093 2094 2095 2096
## 4.209858 4.307956 4.009346 3.974094 4.275956 3.985119 4.068800 4.179070
## 2097 2098 2099 2100 2101 2102 2103 2104
## 4.242394 4.367035 4.263458 3.918119 4.303009 4.233795 4.116868 4.028223
## 2105 2106 2107 2108 2109 2110 2111 2112
## 4.304970 4.386947 3.963416 4.137540 4.331681 3.969121 4.224547 4.340615
## 2113 2114 2115 2116 2117 2118 2119 2120
## 3.923744 4.112041 4.221055 4.203341 4.185739 4.316447 4.289665 4.072719
## 2121 2122 2123 2124 2125 2126 2127 2128
## 4.336861 4.393767 4.271779 4.264852 4.311333 4.055702 4.191349 4.375043
## 2129 2130 2131 2132 2133 2134 2135 2136
## 3.793754 3.749643 4.136216 4.176004 4.188298 4.074467 4.105179 4.277176
## 2137 2138 2139 2140 2141 2142 2143 2144
## 4.239802 4.058397 4.363921 4.205169 4.284541 4.388104 4.378773 4.246130
## 2145 2146 2147 2148 2149 2150 2151 2152
## 4.340627 3.993258 4.224178 4.218230 4.313573 4.349760 4.205619 4.323366
## 2153 2154 2155 2156 2157 2158 2159 2160
## 4.272854 4.097478 4.176593 4.389351 4.340143 4.328199 4.295882 3.759716
## 2161 2162 2163 2164 2165 2166 2167 2168
## 4.087661 4.215541 4.244478 4.039023 4.129253 4.319800 4.211466 4.334098
## 2169 2170 2171 2172 2173 2174 2175 2176
## 4.351772 4.386306 4.367883 4.396618 4.337305 4.079526 4.395358 4.262297
## 2177 2178 2179 2180 2181 2182 2183 2184
## 4.133466 4.295863 4.382096 4.140803 4.371718 4.225213 4.396225 3.915041
## 2185 2186 2187 2188 2189 2190 2191 2192
## 4.203739 4.289774 4.260193 4.083976 4.201299 4.353503 4.198237 4.245767
## 2193 2194 2195 2196 2197 2198 2199 2200
## 4.348117 4.365704 4.384183 4.346040 4.307737 4.377320 4.222056 4.349104
## 2201 2202 2203 2204 2205 2206 2207 2208
## 4.361066 4.289748 4.301243 4.377914 4.320972 4.095306 4.138460 4.388615
## 2209 2210 2211 2212 2213 2214 2215 2216
## 4.294820 4.263151 4.221962 4.269149 4.195497 4.212066 4.307239 4.300999
## 2217 2218 2219 2220 2221 2222 2223 2224
## 4.105236 4.202260 4.094909 4.231065 4.340943 4.100621 3.952458 4.342050
## 2225 2226 2227 2228 2229 2230 2231 2232
## 4.387159 4.343466 4.374262 4.310690 4.082142 4.353297 4.298190 4.263588
## 2233 2234 2235 2236 2237 2238 2239 2240
## 4.354452 4.382401 4.260829 3.931390 4.268834 4.306271 4.279922 4.297116
## 2241 2242 2243 2244 2245 2246 2247 2248
## 4.289056 4.287237 4.376965 4.251701 4.297715 4.002457 4.373538 4.119371
## 2249 2250 2251 2252 2253 2254 2255 2256
## 4.272383 4.198198 4.082281 4.386107 4.213843 4.374503 4.339998 4.285792
## 2257 2258 2259 2260 2261 2262 2263 2264
## 4.379419 4.040231 4.405311 4.345119 3.949827 4.073403 4.379655 4.334899
## 2265 2266 2267 2268 2269 2270 2271 2272
## 4.304176 4.199723 4.110756 3.958871 4.279133 4.390814 4.402326 4.279999
## 2273 2274 2275 2276 2277 2278
## 4.295647 4.261643 4.298895 4.242816 4.316569 4.379516# Combine actual and predicted values into a data frame for comparison
comparison_df <- data.frame(
Actual = train_data_selected$Life_expectancy,
Predicted_Model_1 = predicted_values_1,
Predicted_Model_2 = predicted_values_2,
Predicted_Model_3 = predicted_values_3
)
summary(comparison_df)## Actual Predicted_Model_1 Predicted_Model_2 Predicted_Model_3
## Min. :40.40 Min. :38.34 Min. :30.54 Min. :3.740
## 1st Qu.:62.90 1st Qu.:62.96 1st Qu.:63.92 1st Qu.:4.136
## Median :71.40 Median :71.45 Median :70.75 Median :4.266
## Mean :68.95 Mean :68.95 Mean :68.95 Mean :4.224
## 3rd Qu.:75.50 3rd Qu.:75.97 3rd Qu.:75.46 3rd Qu.:4.335
## Max. :83.80 Max. :82.80 Max. :82.55 Max. :4.416# Convert to long format for plotting
comparison_long <- comparison_df %>%
pivot_longer(cols = starts_with("Predicted"), names_to = "Model", values_to = "Predicted")
# Create and display the plot
p <- ggplot(comparison_long, aes(x = Actual, y = Predicted, color = Model)) +
geom_point() +
geom_abline(slope = 1, intercept = 0, linetype = "dashed") +
theme_minimal() +
labs(title = "Actual vs Predicted Life Expectancy",
x = "Actual Life Expectancy",
y = "Predicted Life Expectancy")
# Save the ggplot object as a Plotly plot
p_plotly <- ggplotly(p)
htmlwidgets::saveWidget(p_plotly, "comparison_plot.html", selfcontained = TRUE)
# Function to create a residual plot
plot_residuals <- function(model, predicted_values, model_name) {
data.frame(Predicted = predicted_values, Residuals = residuals(model)) %>%
ggplot(aes(x = Predicted, y = Residuals)) +
geom_point() +
geom_hline(yintercept = 0, linetype = "dashed") +
theme_minimal() +
labs(title = paste("Residuals vs Predicted for", model_name),
x = "Predicted Values", y = "Residuals")
}
# Creating residual plots for each model
p1 <- plot_residuals(model_1, predicted_values_1, "Model 1")
p2 <- plot_residuals(model_2, predicted_values_2, "Model 2")
p3 <- plot_residuals(model_3, predicted_values_3, "Model 3")
# Displaying the plots
ggplotly(p1)ggplotly(p2)ggplotly(p3)# VIF for models (note: model_3 requires special handling)
vif_model_1 <- vif(model_1)
vif_model_2 <- vif(model_2)
print(vif_model_1)## Year Economy_status_Developed
## 1.213286 2.928424
## Adult_mortality Infant_deaths
## 7.408889 6.726964
## Alcohol_consumption `percentage expenditure`
## 12.666473 6.441773
## Hepatitis_B Measles
## 1.781966 1.413280
## BMI Under_five_deaths
## 1.841716 2.070058
## Polio `Total expenditure`
## 2.425933 1.286638
## Diphtheria Incidents_HIV
## 2.655741 2.865866
## GDP_per_capita Population_millions
## 6.579939 1.327210
## Thinness_ten_nineteen_years Thinness_five_nine_years
## 8.830352 8.747775
## `Income composition of resources` Schooling
## 1.860939 4.097411
## I(Alcohol_consumption^2)
## 12.025778print(vif_model_2)## Year Adult_mortality Alcohol_consumption BMI
## 1.031425 1.297990 1.105874 1.298448# Cross-validation for Model 2
cv_model_2 <- train(Life_expectancy ~ Year + Adult_mortality + Alcohol_consumption + BMI,
data = train_data_selected,
method = "lm",
trControl = trainControl(method = "cv", number = 10))
print(cv_model_2$results)## intercept RMSE Rsquared MAE RMSESD RsquaredSD MAESD
## 1 TRUE 2.494402 0.9292166 1.858111 0.1622994 0.008222832 0.1098502# Prediction and Evaluation for Model 2 on test data
predictions_test_2 <- predict(model_2, newdata = validation_data)
mse_test_2 <- mean((validation_data$Life_expectancy - predictions_test_2)^2)
rmse_test_2 <- sqrt(mse_test_2)
print(paste("Test Data RMSE:", rmse_test_2))## [1] "Test Data RMSE: 2.64408840697961"# Summaries for model significance
summary(model_1)##
## Call:
## lm(formula = Life_expectancy ~ . + I(Alcohol_consumption^2),
## data = train_data_selected)
##
## Residuals:
## Min 1Q Median 3Q Max
## -3.9086 -0.8758 0.0753 0.8747 5.0621
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 5.781e+01 1.377e+01 4.200 2.78e-05 ***
## Year 1.123e-02 6.865e-03 1.636 0.10206
## Economy_status_Developed1 1.082e+00 1.211e-01 8.930 < 2e-16 ***
## Adult_mortality -5.038e-02 6.892e-04 -73.090 < 2e-16 ***
## Infant_deaths -1.277e-01 2.724e-03 -46.867 < 2e-16 ***
## Alcohol_consumption 3.054e-01 2.567e-02 11.895 < 2e-16 ***
## `percentage expenditure` 6.983e-05 3.535e-05 1.975 0.04836 *
## Hepatitis_B 3.721e-04 9.720e-04 0.383 0.70187
## Measles 3.837e-06 3.317e-06 1.157 0.24751
## BMI -4.666e-03 1.891e-03 -2.468 0.01367 *
## Under_five_deaths -2.337e-04 2.631e-04 -0.888 0.37466
## Polio -2.914e-03 1.739e-03 -1.675 0.09398 .
## `Total expenditure` 2.586e-02 1.103e-02 2.345 0.01910 *
## Diphtheria 9.519e-05 1.828e-03 0.052 0.95849
## Incidents_HIV 4.244e-02 1.101e-02 3.854 0.00012 ***
## GDP_per_capita 8.784e-07 5.325e-06 0.165 0.86900
## Population_millions 1.200e-09 6.704e-10 1.790 0.07366 .
## Thinness_ten_nineteen_years -1.385e-02 1.898e-02 -0.730 0.46554
## Thinness_five_nine_years 2.665e-02 1.849e-02 1.441 0.14967
## `Income composition of resources` 6.863e-01 1.536e-01 4.467 8.32e-06 ***
## Schooling 1.019e-01 1.828e-02 5.574 2.78e-08 ***
## I(Alcohol_consumption^2) -2.023e-02 1.993e-03 -10.147 < 2e-16 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 1.367 on 2256 degrees of freedom
## Multiple R-squared: 0.9787, Adjusted R-squared: 0.9785
## F-statistic: 4936 on 21 and 2256 DF, p-value: < 2.2e-16summary(model_2)##
## Call:
## lm(formula = Life_expectancy ~ Year + Adult_mortality + Alcohol_consumption +
## BMI, data = train_data_selected)
##
## Residuals:
## Min 1Q Median 3Q Max
## -8.8436 -1.4367 0.1004 1.5188 13.2410
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) -4.574e+01 2.320e+01 -1.971 0.0488 *
## Year 6.240e-02 1.155e-02 5.402 7.28e-08 ***
## Adult_mortality -7.155e-02 5.264e-04 -135.911 < 2e-16 ***
## Alcohol_consumption 4.063e-01 1.384e-02 29.356 < 2e-16 ***
## BMI 2.993e-02 2.897e-03 10.333 < 2e-16 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 2.495 on 2273 degrees of freedom
## Multiple R-squared: 0.9285, Adjusted R-squared: 0.9284
## F-statistic: 7383 on 4 and 2273 DF, p-value: < 2.2e-16summary(model_3)##
## Call:
## glm(formula = Life_expectancy ~ ., family = gaussian(link = "log"),
## data = train_data_selected)
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 3.802e+00 1.926e-01 19.734 < 2e-16 ***
## Year 3.077e-04 9.608e-05 3.203 0.001381 **
## Economy_status_Developed1 1.355e-03 1.573e-03 0.862 0.389044
## Adult_mortality -7.732e-04 1.072e-05 -72.100 < 2e-16 ***
## Infant_deaths -2.114e-03 4.331e-05 -48.815 < 2e-16 ***
## Alcohol_consumption 1.275e-03 1.665e-04 7.657 2.80e-14 ***
## `percentage expenditure` 1.402e-06 4.492e-07 3.120 0.001831 **
## Hepatitis_B 3.148e-06 1.337e-05 0.235 0.813864
## Measles 3.738e-08 5.436e-08 0.688 0.491805
## BMI -1.193e-04 2.567e-05 -4.649 3.54e-06 ***
## Under_five_deaths -2.197e-08 4.127e-06 -0.005 0.995754
## Polio -4.283e-05 2.575e-05 -1.663 0.096361 .
## `Total expenditure` 4.313e-04 1.528e-04 2.822 0.004816 **
## Diphtheria 1.446e-05 2.673e-05 0.541 0.588614
## Incidents_HIV -6.595e-04 1.870e-04 -3.527 0.000428 ***
## GDP_per_capita -2.074e-07 6.816e-08 -3.043 0.002368 **
## Population_millions 2.121e-11 9.821e-12 2.159 0.030928 *
## Thinness_ten_nineteen_years -5.591e-04 2.918e-04 -1.916 0.055495 .
## Thinness_five_nine_years 7.947e-04 2.853e-04 2.786 0.005386 **
## `Income composition of resources` 6.039e-03 2.138e-03 2.824 0.004780 **
## Schooling 1.267e-03 2.593e-04 4.887 1.10e-06 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## (Dispersion parameter for gaussian family taken to be 1.818202)
##
## Null deviance: 197971.5 on 2277 degrees of freedom
## Residual deviance: 4103.7 on 2257 degrees of freedom
## AIC: 7849.5
##
## Number of Fisher Scoring iterations: 4# Plotting Actual vs Predicted values for all three models
p <- ggplot(comparison_df %>%
pivot_longer(cols = starts_with("Predicted"), names_to = "Model", values_to = "Predicted"),
aes(x = Actual, y = Predicted, color = Model)) +
geom_point() +
geom_abline(slope = 1, intercept = 0, linetype = "dashed") +
theme_minimal() +
labs(title = "Actual vs Predicted Life Expectancy",
x = "Actual Life Expectancy",
y = "Predicted Life Expectancy")
# Convert ggplot object to plotly for interactive plot
p_interactive <- ggplotly(p)
# Save the plot as a PNG file
ggsave("comparison_plot.png", plot = p, width = 10, height = 8)
# Plotting Actual vs Predicted values for all three models
p <- ggplot(comparison_df %>%
pivot_longer(cols = starts_with("Predicted"), names_to = "Model", values_to = "Predicted"),
aes(x = Actual, y = Predicted, color = Model)) +
geom_point() +
geom_abline(slope = 1, intercept = 0, linetype = "dashed") +
theme_minimal() +
labs(title = "Actual vs Predicted Life Expectancy",
x = "Actual Life Expectancy",
y = "Predicted Life Expectancy")
ggplotly(p)#### Akaike Information Criterion (AIC) and Bayesian Information Criterion (BIC)
## Compute AIC and BIC for Model 1
aic_model_1 <- AIC(model_1)
bic_model_1 <- BIC(model_1)
## Compute AIC and BIC for Model 2
aic_model_2 <- AIC(model_2)
bic_model_2 <- BIC(model_2)
## Compute AIC and BIC for Model 3
aic_model_3 <- AIC(model_3)
bic_model_3 <- BIC(model_3)
## Print the results
cat("AIC for Model 1:", aic_model_1, "\n")## AIC for Model 1: 7913.546cat("BIC for Model 1:", bic_model_1, "\n\n")## BIC for Model 1: 8045.36cat("AIC for Model 2:", aic_model_2, "\n")## AIC for Model 2: 10636.96cat("BIC for Model 2:", bic_model_2, "\n\n")## BIC for Model 2: 10671.35cat("AIC for Model 3:", aic_model_3, "\n")## AIC for Model 3: 7849.485cat("BIC for Model 3:", bic_model_3, "\n")## BIC for Model 3: 7975.568# Machine Learning Models
# Calculate MSE
calculateMSE <- function(model, test_data) {
predictions <- predict(model, test_data)
return(mean((predictions - test_data$Life_expectancy)^2))
}
# Artificial Neural Network (ANN)
ann_model <- readRDS("C:/Users/Jessica M/Desktop/project16372/ann_model.rds")
summary(ann_model)## a 205-3-1 network with 622 weights
## options were - decay=0.01
## b->h1 i1->h1 i2->h1 i3->h1 i4->h1 i5->h1 i6->h1 i7->h1
## -0.83 0.33 -0.38 0.03 -0.22 -0.07 -0.22 -0.20
## i8->h1 i9->h1 i10->h1 i11->h1 i12->h1 i13->h1 i14->h1 i15->h1
## -0.11 -0.01 0.30 -0.03 -0.43 -0.24 0.03 -0.08
## i16->h1 i17->h1 i18->h1 i19->h1 i20->h1 i21->h1 i22->h1 i23->h1
## 0.74 0.01 -0.01 0.28 0.05 0.02 -0.20 0.89
## i24->h1 i25->h1 i26->h1 i27->h1 i28->h1 i29->h1 i30->h1 i31->h1
## 0.34 -0.02 -0.02 -0.28 0.26 0.02 -0.10 0.02
## i32->h1 i33->h1 i34->h1 i35->h1 i36->h1 i37->h1 i38->h1 i39->h1
## 0.00 0.50 0.64 -0.25 0.02 0.03 -0.03 0.23
## i40->h1 i41->h1 i42->h1 i43->h1 i44->h1 i45->h1 i46->h1 i47->h1
## 1.02 0.44 -0.02 0.05 0.11 0.02 -0.46 -0.49
## i48->h1 i49->h1 i50->h1 i51->h1 i52->h1 i53->h1 i54->h1 i55->h1
## -0.06 -0.04 -0.03 0.01 0.15 0.03 0.05 0.04
## i56->h1 i57->h1 i58->h1 i59->h1 i60->h1 i61->h1 i62->h1 i63->h1
## -0.11 -0.26 0.04 -0.01 0.03 -0.10 0.03 -0.15
## i64->h1 i65->h1 i66->h1 i67->h1 i68->h1 i69->h1 i70->h1 i71->h1
## 0.72 -0.48 -0.03 -0.01 0.33 0.06 -0.28 0.07
## i72->h1 i73->h1 i74->h1 i75->h1 i76->h1 i77->h1 i78->h1 i79->h1
## -0.09 0.21 -0.15 -0.20 0.35 -0.03 -0.06 -0.20
## i80->h1 i81->h1 i82->h1 i83->h1 i84->h1 i85->h1 i86->h1 i87->h1
## -1.26 -0.11 0.06 -0.38 0.02 0.29 0.63 -0.06
## i88->h1 i89->h1 i90->h1 i91->h1 i92->h1 i93->h1 i94->h1 i95->h1
## 0.16 -0.31 0.15 0.01 0.00 -0.10 -0.16 -0.17
## i96->h1 i97->h1 i98->h1 i99->h1 i100->h1 i101->h1 i102->h1 i103->h1
## 0.04 -0.02 0.50 0.04 -0.01 -0.08 -0.01 -0.32
## i104->h1 i105->h1 i106->h1 i107->h1 i108->h1 i109->h1 i110->h1 i111->h1
## -0.67 0.60 0.34 -0.06 0.31 -0.22 -0.01 0.10
## i112->h1 i113->h1 i114->h1 i115->h1 i116->h1 i117->h1 i118->h1 i119->h1
## 0.04 0.26 -0.14 -0.11 -0.07 -0.01 0.01 -0.07
## i120->h1 i121->h1 i122->h1 i123->h1 i124->h1 i125->h1 i126->h1 i127->h1
## -0.07 0.09 0.02 -0.23 0.03 -0.11 -0.06 -0.24
## i128->h1 i129->h1 i130->h1 i131->h1 i132->h1 i133->h1 i134->h1 i135->h1
## 0.20 -0.02 -0.24 0.09 -0.43 0.11 -0.17 0.78
## i136->h1 i137->h1 i138->h1 i139->h1 i140->h1 i141->h1 i142->h1 i143->h1
## 0.10 0.23 0.39 0.06 -0.03 -0.25 0.42 0.26
## i144->h1 i145->h1 i146->h1 i147->h1 i148->h1 i149->h1 i150->h1 i151->h1
## -0.63 0.03 -0.02 -0.22 -0.59 -0.21 0.20 -0.04
## i152->h1 i153->h1 i154->h1 i155->h1 i156->h1 i157->h1 i158->h1 i159->h1
## -0.15 -0.17 -0.42 0.34 0.09 -0.47 0.09 -0.01
## i160->h1 i161->h1 i162->h1 i163->h1 i164->h1 i165->h1 i166->h1 i167->h1
## 0.30 -0.15 -0.15 -0.09 0.30 0.03 -0.35 -0.11
## i168->h1 i169->h1 i170->h1 i171->h1 i172->h1 i173->h1 i174->h1 i175->h1
## 0.04 0.00 -0.14 -0.32 0.34 -0.01 -1.09 0.03
## i176->h1 i177->h1 i178->h1 i179->h1 i180->h1 i181->h1 i182->h1 i183->h1
## 0.02 -0.01 -0.36 -0.19 0.47 0.11 -0.73 0.61
## i184->h1 i185->h1 i186->h1 i187->h1 i188->h1 i189->h1 i190->h1 i191->h1
## 0.11 -0.31 -0.07 -0.39 1.74 1.10 -0.08 0.06
## i192->h1 i193->h1 i194->h1 i195->h1 i196->h1 i197->h1 i198->h1 i199->h1
## -0.05 0.06 -0.02 0.11 -0.02 0.06 0.01 0.82
## i200->h1 i201->h1 i202->h1 i203->h1 i204->h1 i205->h1
## 0.08 0.02 0.11 0.18 0.37 -0.10
## b->h2 i1->h2 i2->h2 i3->h2 i4->h2 i5->h2 i6->h2 i7->h2
## -0.65 0.09 -0.41 -0.01 -0.36 -0.16 -0.09 -0.10
## i8->h2 i9->h2 i10->h2 i11->h2 i12->h2 i13->h2 i14->h2 i15->h2
## -0.03 0.02 0.12 0.00 -0.28 -0.04 0.24 -0.02
## i16->h2 i17->h2 i18->h2 i19->h2 i20->h2 i21->h2 i22->h2 i23->h2
## 0.02 0.00 -0.08 -0.08 -0.55 0.04 -0.47 0.09
## i24->h2 i25->h2 i26->h2 i27->h2 i28->h2 i29->h2 i30->h2 i31->h2
## 0.46 0.03 -0.01 0.22 0.44 -0.02 -0.04 -0.01
## i32->h2 i33->h2 i34->h2 i35->h2 i36->h2 i37->h2 i38->h2 i39->h2
## 0.02 0.16 0.28 -0.26 0.04 0.03 0.01 0.02
## i40->h2 i41->h2 i42->h2 i43->h2 i44->h2 i45->h2 i46->h2 i47->h2
## 0.24 -0.03 -0.02 0.57 0.04 -0.02 -0.43 -0.61
## i48->h2 i49->h2 i50->h2 i51->h2 i52->h2 i53->h2 i54->h2 i55->h2
## 0.11 0.04 0.01 0.01 -0.06 -0.01 -0.01 0.54
## i56->h2 i57->h2 i58->h2 i59->h2 i60->h2 i61->h2 i62->h2 i63->h2
## -0.01 -0.06 0.02 0.00 0.09 -0.03 0.02 -0.05
## i64->h2 i65->h2 i66->h2 i67->h2 i68->h2 i69->h2 i70->h2 i71->h2
## -0.31 -0.09 -0.02 -0.02 0.30 0.00 -0.16 -0.02
## i72->h2 i73->h2 i74->h2 i75->h2 i76->h2 i77->h2 i78->h2 i79->h2
## 0.00 0.24 -0.21 -0.23 0.32 0.00 0.01 -0.07
## i80->h2 i81->h2 i82->h2 i83->h2 i84->h2 i85->h2 i86->h2 i87->h2
## -0.22 -0.04 0.00 -0.36 0.00 0.08 0.05 -0.20
## i88->h2 i89->h2 i90->h2 i91->h2 i92->h2 i93->h2 i94->h2 i95->h2
## 0.08 -0.53 -0.25 -0.03 0.02 0.86 -0.54 -0.03
## i96->h2 i97->h2 i98->h2 i99->h2 i100->h2 i101->h2 i102->h2 i103->h2
## 0.01 -0.02 -0.01 0.03 0.02 0.00 0.02 -0.16
## i104->h2 i105->h2 i106->h2 i107->h2 i108->h2 i109->h2 i110->h2 i111->h2
## -0.65 0.45 -0.32 0.63 0.16 -0.15 0.00 0.04
## i112->h2 i113->h2 i114->h2 i115->h2 i116->h2 i117->h2 i118->h2 i119->h2
## -0.02 0.24 -0.03 -0.11 0.03 -0.03 -0.03 -0.03
## i120->h2 i121->h2 i122->h2 i123->h2 i124->h2 i125->h2 i126->h2 i127->h2
## -0.01 0.55 -0.03 -0.14 0.02 -0.06 0.13 -0.44
## i128->h2 i129->h2 i130->h2 i131->h2 i132->h2 i133->h2 i134->h2 i135->h2
## 0.06 -0.01 0.02 0.17 0.09 0.12 0.03 0.72
## i136->h2 i137->h2 i138->h2 i139->h2 i140->h2 i141->h2 i142->h2 i143->h2
## 0.06 0.18 0.10 -0.61 0.02 -0.03 0.34 0.06
## i144->h2 i145->h2 i146->h2 i147->h2 i148->h2 i149->h2 i150->h2 i151->h2
## -1.05 0.01 0.02 -0.09 -0.42 -0.55 0.34 0.57
## i152->h2 i153->h2 i154->h2 i155->h2 i156->h2 i157->h2 i158->h2 i159->h2
## 0.01 -0.02 -0.34 0.48 -0.01 -0.48 0.04 -0.02
## i160->h2 i161->h2 i162->h2 i163->h2 i164->h2 i165->h2 i166->h2 i167->h2
## 0.64 -0.05 -0.15 -0.08 0.24 -0.02 0.15 -0.02
## i168->h2 i169->h2 i170->h2 i171->h2 i172->h2 i173->h2 i174->h2 i175->h2
## -0.15 0.05 -0.16 0.67 0.40 0.36 -0.46 -0.02
## i176->h2 i177->h2 i178->h2 i179->h2 i180->h2 i181->h2 i182->h2 i183->h2
## -0.02 -0.04 0.45 -1.03 0.22 0.00 -0.69 0.94
## i184->h2 i185->h2 i186->h2 i187->h2 i188->h2 i189->h2 i190->h2 i191->h2
## -0.28 -0.26 0.12 0.06 2.86 1.94 -0.02 -0.15
## i192->h2 i193->h2 i194->h2 i195->h2 i196->h2 i197->h2 i198->h2 i199->h2
## 0.07 -0.02 0.01 0.43 0.06 -0.02 -0.05 -0.09
## i200->h2 i201->h2 i202->h2 i203->h2 i204->h2 i205->h2
## -0.29 -0.11 0.15 -0.44 -0.43 0.18
## b->h3 i1->h3 i2->h3 i3->h3 i4->h3 i5->h3 i6->h3 i7->h3
## -4.94 0.05 0.11 0.00 -0.04 -0.32 -0.15 0.16
## i8->h3 i9->h3 i10->h3 i11->h3 i12->h3 i13->h3 i14->h3 i15->h3
## 0.45 -0.15 -0.73 -0.70 0.06 1.32 0.22 0.54
## i16->h3 i17->h3 i18->h3 i19->h3 i20->h3 i21->h3 i22->h3 i23->h3
## 1.89 0.01 0.02 -0.31 0.02 0.00 -0.08 -0.78
## i24->h3 i25->h3 i26->h3 i27->h3 i28->h3 i29->h3 i30->h3 i31->h3
## -0.91 0.01 -0.02 -0.74 -0.03 0.01 0.10 -0.06
## i32->h3 i33->h3 i34->h3 i35->h3 i36->h3 i37->h3 i38->h3 i39->h3
## -0.01 0.29 -0.86 0.21 -0.02 0.03 -0.04 0.85
## i40->h3 i41->h3 i42->h3 i43->h3 i44->h3 i45->h3 i46->h3 i47->h3
## -1.32 0.41 0.20 -0.20 -0.80 -0.01 0.49 0.38
## i48->h3 i49->h3 i50->h3 i51->h3 i52->h3 i53->h3 i54->h3 i55->h3
## -0.23 -0.26 -0.02 0.00 -1.01 -0.07 0.00 -0.11
## i56->h3 i57->h3 i58->h3 i59->h3 i60->h3 i61->h3 i62->h3 i63->h3
## 0.46 1.00 0.02 -0.02 -0.22 0.42 0.00 0.74
## i64->h3 i65->h3 i66->h3 i67->h3 i68->h3 i69->h3 i70->h3 i71->h3
## -0.86 0.22 0.01 0.00 -0.07 -0.03 -0.05 -0.92
## i72->h3 i73->h3 i74->h3 i75->h3 i76->h3 i77->h3 i78->h3 i79->h3
## 0.14 -0.01 0.13 -0.56 0.08 0.28 0.13 0.87
## i80->h3 i81->h3 i82->h3 i83->h3 i84->h3 i85->h3 i86->h3 i87->h3
## -1.52 0.20 -0.19 -0.33 -0.02 -0.31 -1.55 -0.27
## i88->h3 i89->h3 i90->h3 i91->h3 i92->h3 i93->h3 i94->h3 i95->h3
## -0.03 -0.88 0.94 -0.01 -0.02 -0.57 -0.72 0.24
## i96->h3 i97->h3 i98->h3 i99->h3 i100->h3 i101->h3 i102->h3 i103->h3
## -0.02 0.01 0.36 -0.50 -0.01 0.52 0.00 0.30
## i104->h3 i105->h3 i106->h3 i107->h3 i108->h3 i109->h3 i110->h3 i111->h3
## -0.36 -0.20 0.01 -0.20 -0.20 -0.78 0.02 -0.01
## i112->h3 i113->h3 i114->h3 i115->h3 i116->h3 i117->h3 i118->h3 i119->h3
## -0.02 -0.03 0.37 0.15 -0.46 -0.02 0.01 -0.63
## i120->h3 i121->h3 i122->h3 i123->h3 i124->h3 i125->h3 i126->h3 i127->h3
## 0.13 0.59 -0.02 0.62 0.00 0.09 0.00 -0.10
## i128->h3 i129->h3 i130->h3 i131->h3 i132->h3 i133->h3 i134->h3 i135->h3
## -0.43 0.17 1.27 -1.13 0.02 -0.05 -0.16 -0.05
## i136->h3 i137->h3 i138->h3 i139->h3 i140->h3 i141->h3 i142->h3 i143->h3
## -1.12 0.02 -0.21 0.62 0.00 1.48 -0.93 -0.27
## i144->h3 i145->h3 i146->h3 i147->h3 i148->h3 i149->h3 i150->h3 i151->h3
## -0.07 0.03 0.00 0.96 -0.02 0.13 -1.06 0.12
## i152->h3 i153->h3 i154->h3 i155->h3 i156->h3 i157->h3 i158->h3 i159->h3
## 0.50 0.11 0.08 -0.33 -0.03 1.28 -0.01 0.00
## i160->h3 i161->h3 i162->h3 i163->h3 i164->h3 i165->h3 i166->h3 i167->h3
## -0.18 0.07 -0.20 0.81 -0.06 0.00 0.27 -0.55
## i168->h3 i169->h3 i170->h3 i171->h3 i172->h3 i173->h3 i174->h3 i175->h3
## 0.23 -0.20 0.31 -0.32 0.05 -0.32 0.29 -0.02
## i176->h3 i177->h3 i178->h3 i179->h3 i180->h3 i181->h3 i182->h3 i183->h3
## 0.00 -0.03 -0.26 0.88 -1.75 -0.72 -0.40 -0.73
## i184->h3 i185->h3 i186->h3 i187->h3 i188->h3 i189->h3 i190->h3 i191->h3
## -0.27 0.32 0.23 1.12 -4.37 -1.26 0.19 0.18
## i192->h3 i193->h3 i194->h3 i195->h3 i196->h3 i197->h3 i198->h3 i199->h3
## -0.02 0.04 -0.03 1.14 0.02 0.02 -0.14 0.19
## i200->h3 i201->h3 i202->h3 i203->h3 i204->h3 i205->h3
## 0.01 0.13 0.18 0.00 0.21 0.20
## b->o h1->o h2->o h3->o
## 0.56 -8.06 -6.19 5.31# K Nearest Neighbors (KNN)
knn_model <- readRDS("C:/Users/Jessica M/Desktop/project16372/knn_model.rds")
summary(knn_model)## Length Class Mode
## learn 2 -none- list
## k 1 -none- numeric
## theDots 0 -none- list
## xNames 205 -none- character
## problemType 1 -none- character
## tuneValue 1 data.frame list
## obsLevels 1 -none- logical
## param 0 -none- list# Support Vector Machine (SVM)
svm_model <- readRDS("C:/Users/Jessica M/Desktop/project16372/svm_model.rds")
summary(svm_model)## Length Class Mode
## 1 ksvm S4# Random Forest (RF)
rf_model <- readRDS("C:/Users/Jessica M/Desktop/project16372/rf_model.rds")
summary(rf_model)## Length Class Mode
## call 4 -none- call
## type 1 -none- character
## predicted 1996 -none- numeric
## mse 500 -none- numeric
## rsq 500 -none- numeric
## oob.times 1996 -none- numeric
## importance 205 -none- numeric
## importanceSD 0 -none- NULL
## localImportance 0 -none- NULL
## proximity 0 -none- NULL
## ntree 1 -none- numeric
## mtry 1 -none- numeric
## forest 11 -none- list
## coefs 0 -none- NULL
## y 1996 -none- numeric
## test 0 -none- NULL
## inbag 0 -none- NULL
## xNames 205 -none- character
## problemType 1 -none- character
## tuneValue 1 data.frame list
## obsLevels 1 -none- logical
## param 0 -none- list# LM Modelproject16372/
lm_model <- readRDS("C:/Users/Jessica M/Desktop/project16372/lm_model.rds")
summary(lm_model)##
## Call:
## lm(formula = .outcome ~ ., data = dat)
##
## Residuals:
## Min 1Q Median 3Q Max
## -2.1432 -0.2178 -0.0087 0.2001 3.2582
##
## Coefficients: (9 not defined because of singularities)
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) -1.444e+02 1.247e+01 -11.585 < 2e-16
## CountryAlbania 4.483e+00 4.119e-01 10.882 < 2e-16
## CountryAlgeria 4.325e+00 3.387e-01 12.769 < 2e-16
## CountryAngola -3.686e+00 2.875e-01 -12.821 < 2e-16
## `CountryAntigua and Barbuda` 5.789e+00 4.153e-01 13.941 < 2e-16
## CountryArgentina 5.171e+00 4.196e-01 12.323 < 2e-16
## CountryArmenia 4.184e+00 4.493e-01 9.313 < 2e-16
## CountryAustralia 8.410e+00 5.166e-01 16.280 < 2e-16
## CountryAustria 7.605e+00 5.048e-01 15.066 < 2e-16
## CountryAzerbaijan 2.912e+00 4.317e-01 6.746 2.05e-11
## `CountryBahamas, The` 4.576e+00 4.691e-01 9.756 < 2e-16
## CountryBahrain 3.131e+00 3.895e-01 8.037 1.65e-15
## CountryBangladesh 2.157e+00 2.676e-01 8.062 1.35e-15
## CountryBarbados 7.937e+00 4.219e-01 18.811 < 2e-16
## CountryBelarus 4.068e+00 4.810e-01 8.457 < 2e-16
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## `\\`Total expenditure\\``
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## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 0.5291 on 1799 degrees of freedom
## Multiple R-squared: 0.9971, Adjusted R-squared: 0.9968
## F-statistic: 3183 on 196 and 1799 DF, p-value: < 2.2e-16# Collect and summarize from models
ML_results <- resamples(list(linear = lm_model, knn = knn_model, ann = ann_model, svm = svm_model, rf = rf_model))
print(ML_results)##
## Call:
## resamples.default(x = list(linear = lm_model, knn = knn_model, ann =
## ann_model, svm = svm_model, rf = rf_model))
##
## Models: linear, knn, ann, svm, rf
## Number of resamples: 10
## Performance metrics: MAE, RMSE, Rsquared
## Time estimates for: everything, final model fit#### Ensure 'Economy_status_Developed' is numeric if it was numeric during model training
test_data$Economy_status_Developed <- as.numeric(as.factor(test_data$Economy_status_Developed))
# Then re-run the model evaluation
mse_ann <- calculateMSE(ann_model, test_data)
# Model Evaluation
# Evaluation Function
calculateMSE <- function(model, test_data) {
predictions <- predict(model, test_data)
return(mean((predictions - test_data$Life_expectancy)^2))
}
# Evaluate Models
mse_ann <- calculateMSE(ann_model, test_data)
mse_knn <- calculateMSE(knn_model, test_data) # Update this as per KNN's output format
mse_svm <- calculateMSE(svm_model, test_data)
mse_rf <- calculateMSE(rf_model, test_data)
#### Print MSE of Models
print(list(MSE_ANN = mse_ann, MSE_KNN = mse_knn, MSE_SVM = mse_svm, MSE_RF = mse_rf))## $MSE_ANN
## [1] 4842.958
##
## $MSE_KNN
## [1] 4853.757
##
## $MSE_SVM
## [1] 4870.081
##
## $MSE_RF
## [1] 0.3710017#### Predictions
#Predictions per model
actual_values <- test_data$Life_expectancy
print(actual_values)## [1] 55.8 56.8 59.9 74.0 74.3 74.6 78.0 75.4 48.4 52.2 57.2 73.9 74.4 75.0 76.0
## [16] 73.8 74.8 75.9 72.5 74.1 74.3 74.5 79.6 79.9 80.5 82.3 82.4 78.7 80.4 80.6
## [31] 81.1 81.5 68.2 70.2 72.0 72.0 72.0 75.0 66.9 70.9 71.5 78.0 78.7 68.9 70.4
## [46] 72.5 79.7 80.0 80.4 81.3 70.0 71.1 73.5 74.0 55.7 59.6 59.8 60.1 66.0 67.3
## [61] 68.4 68.8 70.4 64.8 69.5 70.3 74.6 74.9 75.3 75.8 76.2 50.2 52.1 55.0 61.9
## [76] 63.5 67.3 72.6 73.3 74.5 74.7 73.3 73.6 74.0 74.9 75.0 75.1 71.8 72.1 74.3
## [91] 50.5 54.8 59.5 51.7 54.9 59.1 72.1 60.3 62.2 63.9 66.6 68.6 51.2 79.9 80.2
## [106] 80.7 45.6 47.3 50.9 51.9 52.4 52.8 53.1 78.3 73.8 74.4 74.7 75.0 75.6 74.0
## [121] 59.5 59.6 60.4 57.4 58.8 54.7 55.6 77.6 78.0 79.2 75.2 75.8 77.5 78.0 78.4
## [136] 78.5 79.6 75.2 77.4 78.6 78.2 78.4 63.2 69.7 70.0 70.8 73.4 74.1 75.7 76.1
## [151] 69.1 69.3 69.8 70.7 70.9 70.1 53.6 54.8 55.3 56.6 61.0 72.6 72.7 77.1 77.6
## [166] 46.6 49.9 56.2 60.6 66.0 78.4 78.7 79.9 80.5 79.2 79.3 80.2 80.8 81.1 82.1
## [181] 82.3 58.0 57.8 60.6 62.2 62.9 63.6 56.0 58.7 59.3 70.5 71.2 72.4 77.9 78.2
## [196] 78.7 79.8 80.6 57.2 59.2 78.6 79.0 72.8 73.1 72.9 70.1 72.2 53.2 54.1 57.9
## [211] 58.4 59.0 50.7 53.6 55.6 68.6 57.7 60.9 73.3 73.8 72.3 72.3 79.7 81.0 81.5
## [226] 81.8 64.9 65.8 67.0 67.3 68.5 69.9 70.2 70.2 71.3 72.6 69.1 68.1 69.4 69.9
## [241] 77.6 80.7 80.7 79.4 80.6 80.8 74.1 74.1 74.1 74.0 82.3 82.6 83.6 71.7 72.8
## [256] 66.2 52.4 57.5 58.8 61.9 63.1 64.7 65.0 65.8 66.1 73.1 73.2 74.7 68.0 67.7
## [271] 66.5 70.8 71.3 73.1 73.6 74.5 74.5 75.5 76.9 78.4 78.5 78.7 43.8 43.4 51.7
## [286] 55.4 60.1 70.9 71.9 71.9 71.8 73.9 74.3 78.0 77.7 80.6 81.0 58.5 61.2 61.7
## [301] 62.5 57.2 61.0 73.2 75.1 76.7 77.7 54.8 55.7 56.1 79.8 79.6 80.2 60.7 61.8
## [316] 62.8 74.2 75.2 75.2 74.9 65.1 66.3 66.6 67.2 67.4 67.5 68.0 70.6 71.0 63.8
## [331] 64.2 64.7 65.1 65.5 66.9 68.6 69.1 73.8 76.2 68.7 69.8 71.1 73.0 75.0 75.2
## [346] 50.1 50.4 51.3 52.3 57.2 61.3 61.6 64.9 50.6 51.6 53.8 58.1 64.7 68.0 69.2
## [361] 78.2 79.7 80.3 79.5 80.4 70.6 73.2 50.6 54.2 56.6 58.1 58.8 46.8 47.7 48.3
## [376] 72.9 73.0 73.8 74.4 75.0 75.1 74.9 78.8 79.0 80.0 80.6 81.0 82.1 76.4 64.7
## [391] 65.3 65.8 75.9 77.4 59.5 60.2 61.7 71.9 72.3 72.5 71.9 74.7 69.0 70.1 73.7
## [406] 74.2 74.5 74.6 75.2 76.2 76.7 77.1 78.3 80.4 79.4 72.2 72.6 72.6 74.9 65.1
## [421] 65.0 48.6 60.6 64.5 66.9 67.5 70.1 70.9 71.7 72.4 61.4 62.7 65.9 66.4 72.6
## [436] 73.3 73.4 58.2 59.9 63.6 72.3 72.7 72.8 73.4 73.6 72.7 73.2 73.2 73.1 40.4
## [451] 44.5 80.4 80.8 82.7 75.1 76.0 77.2 78.8 67.8 68.8 70.1 51.5 52.3 52.7 54.0
## [466] 55.5 53.7 57.7 79.9 81.5 73.8 75.0 74.4 75.3 70.8 71.8 71.9 68.3 69.9 70.5
## [481] 70.7 71.0 71.2 80.5 80.7 81.8 82.0 80.4 82.2 82.8 82.9 73.1 74.4 72.9 70.6
## [496] 67.4 50.8 51.4 54.3 70.6 71.4 75.0 76.1 59.0 67.5 68.2 53.5 53.7 54.6 55.6
## [511] 69.7 70.1 70.2 69.4 71.0 71.2 72.8 75.7 75.9 64.1 66.7 48.3 52.8 58.1 59.0
## [526] 60.7 61.4 67.8 68.1 71.2 75.6 76.0 76.1 76.3 79.2 80.9 81.0 78.4 78.6 78.7
## [541] 76.6 76.9 77.4 67.2 67.9 68.4 70.7 67.8 68.2 68.4 69.6 72.1 72.6 73.1 73.9
## [556] 74.1 74.9 75.0 75.1 65.3 65.5 65.8 66.1 46.3 47.4 51.1 60.8 55.0#### Predictions
predictions_ann <- predict(ann_model, test_data)
predictions_svm <- predict(svm_model, test_data)
predictions_rf <- predict(rf_model, test_data)
predictions_knn <- predict(knn_model, test_data)
predictions_lm <- predict(lm_model, test_data)## Warning in predict.lm(modelFit, newdata): prediction from rank-deficient fit;
## attr(*, "non-estim") has doubtful casesprint(predictions_ann)## 1 2 3 4 5 6
## 1.001680e-01 1.001680e-01 1.001680e-01 1.001680e-01 1.001680e-01 1.001680e-01
## 7 8 9 10 11 12
## 1.001680e-01 1.001680e-01 1.001680e-01 1.001680e-01 1.001680e-01 1.001680e-01
## 13 14 15 16 17 18
## 1.001680e-01 1.001680e-01 1.001680e-01 1.001680e-01 1.001680e-01 1.001680e-01
## 19 20 21 22 23 24
## 1.001680e-01 1.001680e-01 1.001680e-01 1.001680e-01 1.001680e-01 1.001680e-01
## 25 26 27 28 29 30
## 1.001680e-01 1.001680e-01 1.001680e-01 1.001680e-01 1.001680e-01 1.001680e-01
## 31 32 33 34 35 36
## 1.001680e-01 1.001680e-01 1.001680e-01 1.001680e-01 1.001680e-01 1.001680e-01
## 37 38 39 40 41 42
## 1.001680e-01 1.001680e-01 1.001680e-01 1.001680e-01 1.001680e-01 1.001680e-01
## 43 44 45 46 47 48
## 1.001680e-01 1.001680e-01 1.001680e-01 1.001680e-01 1.001680e-01 1.001680e-01
## 49 50 51 52 53 54
## 1.001680e-01 1.001680e-01 1.001680e-01 1.001680e-01 1.001680e-01 1.001680e-01
## 55 56 57 58 59 60
## 1.001680e-01 1.001680e-01 1.001680e-01 1.001680e-01 1.001680e-01 1.127453e-06
## 61 62 63 64 65 66
## 1.001680e-01 1.001680e-01 1.001680e-01 1.127453e-06 1.127453e-06 1.127453e-06
## 67 68 69 70 71 72
## 1.001680e-01 1.001680e-01 1.001680e-01 1.001680e-01 1.001680e-01 1.001680e-01
## 73 74 75 76 77 78
## 1.001680e-01 1.001680e-01 1.001680e-01 1.001680e-01 1.001680e-01 1.001680e-01
## 79 80 81 82 83 84
## 1.001680e-01 1.001680e-01 1.001680e-01 1.001680e-01 1.001680e-01 1.001680e-01
## 85 86 87 88 89 90
## 1.001680e-01 1.001680e-01 1.001680e-01 1.001680e-01 1.001680e-01 1.001680e-01
## 91 92 93 94 95 96
## 1.001680e-01 1.001680e-01 1.001680e-01 1.001680e-01 1.001680e-01 1.001680e-01
## 97 98 99 100 101 102
## 1.127453e-06 1.001680e-01 1.001680e-01 1.001680e-01 1.001680e-01 1.001680e-01
## 103 104 105 106 107 108
## 1.001680e-01 1.001680e-01 1.001680e-01 1.001680e-01 1.001680e-01 1.001680e-01
## 109 110 111 112 113 114
## 1.001680e-01 1.001680e-01 1.001680e-01 1.001680e-01 1.001680e-01 1.001680e-01
## 115 116 117 118 119 120
## 1.001680e-01 1.001680e-01 1.001680e-01 1.001680e-01 1.001680e-01 1.001680e-01
## 121 122 123 124 125 126
## 1.001680e-01 1.001680e-01 1.001680e-01 1.001680e-01 1.001680e-01 1.001680e-01
## 127 128 129 130 131 132
## 1.001680e-01 1.001680e-01 1.001680e-01 1.001680e-01 1.001680e-01 1.001680e-01
## 133 134 135 136 137 138
## 1.302473e-06 1.001680e-01 1.001680e-01 1.001680e-01 1.001680e-01 1.127453e-06
## 139 140 141 142 143 144
## 2.277678e-04 2.882932e-02 1.001680e-01 1.001680e-01 1.001680e-01 1.001680e-01
## 145 146 147 148 149 150
## 1.001680e-01 1.001680e-01 1.001680e-01 1.001680e-01 1.001680e-01 1.001680e-01
## 151 152 153 154 155 156
## 1.001680e-01 1.001680e-01 1.001680e-01 1.001680e-01 1.001680e-01 1.001680e-01
## 157 158 159 160 161 162
## 1.001680e-01 1.001680e-01 1.001680e-01 1.001680e-01 1.001680e-01 1.001680e-01
## 163 164 165 166 167 168
## 1.001680e-01 1.001680e-01 1.001680e-01 1.127453e-06 1.127453e-06 1.001680e-01
## 169 170 171 172 173 174
## 1.001680e-01 1.001680e-01 1.001680e-01 1.001680e-01 1.001680e-01 1.001680e-01
## 175 176 177 178 179 180
## 1.001680e-01 1.001680e-01 1.001680e-01 1.001680e-01 1.001680e-01 1.001680e-01
## 181 182 183 184 185 186
## 1.001680e-01 1.001680e-01 1.001680e-01 1.001680e-01 1.001680e-01 1.001680e-01
## 187 188 189 190 191 192
## 1.001680e-01 1.001680e-01 1.001680e-01 1.001680e-01 1.127453e-06 2.277678e-04
## 193 194 195 196 197 198
## 1.001680e-01 1.001680e-01 1.001680e-01 1.001680e-01 1.001680e-01 1.001680e-01
## 199 200 201 202 203 204
## 1.001680e-01 1.001680e-01 1.001680e-01 1.001680e-01 1.127453e-06 5.505997e-04
## 205 206 207 208 209 210
## 5.505997e-04 1.001680e-01 1.001680e-01 1.001680e-01 1.001680e-01 1.001680e-01
## 211 212 213 214 215 216
## 1.001680e-01 1.001680e-01 1.001680e-01 1.001680e-01 1.001680e-01 1.001680e-01
## 217 218 219 220 221 222
## 1.001680e-01 1.001680e-01 1.001680e-01 1.001680e-01 1.001680e-01 1.001680e-01
## 223 224 225 226 227 228
## 1.001680e-01 1.001680e-01 1.001680e-01 1.001680e-01 1.001680e-01 1.001680e-01
## 229 230 231 232 233 234
## 1.001680e-01 1.001680e-01 1.001680e-01 1.001680e-01 1.001680e-01 1.001680e-01
## 235 236 237 238 239 240
## 1.001680e-01 1.001680e-01 1.001680e-01 1.001680e-01 1.001680e-01 1.001680e-01
## 241 242 243 244 245 246
## 1.001680e-01 1.001680e-01 1.001680e-01 1.001680e-01 1.001680e-01 1.001680e-01
## 247 248 249 250 251 252
## 1.001680e-01 1.001680e-01 1.001680e-01 1.001680e-01 1.001680e-01 1.001680e-01
## 253 254 255 256 257 258
## 1.001680e-01 1.001680e-01 1.001680e-01 1.001680e-01 1.001680e-01 1.001680e-01
## 259 260 261 262 263 264
## 1.001680e-01 1.001680e-01 1.001680e-01 1.001680e-01 1.001680e-01 1.001680e-01
## 265 266 267 268 269 270
## 1.001680e-01 1.001680e-01 2.277678e-04 1.001680e-01 1.001680e-01 1.001680e-01
## 271 272 273 274 275 276
## 1.001680e-01 1.001680e-01 1.001680e-01 1.001680e-01 1.001680e-01 1.001680e-01
## 277 278 279 280 281 282
## 1.001680e-01 1.001680e-01 1.001680e-01 1.001680e-01 1.001680e-01 1.001680e-01
## 283 284 285 286 287 288
## 1.001680e-01 1.001680e-01 1.001680e-01 1.001680e-01 1.001680e-01 5.505997e-04
## 289 290 291 292 293 294
## 1.001680e-01 1.127453e-06 1.001680e-01 1.001680e-01 1.001680e-01 1.001680e-01
## 295 296 297 298 299 300
## 1.001680e-01 1.001680e-01 1.001680e-01 1.001680e-01 1.001680e-01 1.001680e-01
## 301 302 303 304 305 306
## 1.001680e-01 1.001680e-01 1.001680e-01 1.001680e-01 1.001680e-01 1.001680e-01
## 307 308 309 310 311 312
## 1.001680e-01 1.001680e-01 1.001680e-01 1.001680e-01 1.001680e-01 1.001680e-01
## 313 314 315 316 317 318
## 1.001680e-01 1.001680e-01 1.001680e-01 1.001680e-01 1.001680e-01 1.001680e-01
## 319 320 321 322 323 324
## 1.001680e-01 1.001680e-01 1.001680e-01 1.001680e-01 1.001680e-01 1.127453e-06
## 325 326 327 328 329 330
## 1.127453e-06 1.127453e-06 1.127453e-06 1.127453e-06 7.909899e-05 1.001680e-01
## 331 332 333 334 335 336
## 1.001680e-01 1.001680e-01 1.001680e-01 1.001680e-01 1.001680e-01 1.001680e-01
## 337 338 339 340 341 342
## 1.001680e-01 1.001680e-01 1.001680e-01 1.001680e-01 1.001680e-01 1.001680e-01
## 343 344 345 346 347 348
## 1.001680e-01 1.001680e-01 1.001680e-01 1.001680e-01 1.001680e-01 1.001680e-01
## 349 350 351 352 353 354
## 1.001680e-01 1.001680e-01 1.001680e-01 1.001680e-01 1.001680e-01 1.001680e-01
## 355 356 357 358 359 360
## 1.001680e-01 1.001680e-01 1.001680e-01 1.001680e-01 1.001680e-01 1.001680e-01
## 361 362 363 364 365 366
## 1.001680e-01 1.001680e-01 1.001680e-01 1.001680e-01 1.001680e-01 1.001680e-01
## 367 368 369 370 371 372
## 1.001680e-01 1.001680e-01 1.001680e-01 1.001680e-01 1.001680e-01 1.001680e-01
## 373 374 375 376 377 378
## 1.001680e-01 1.001680e-01 1.001680e-01 1.001680e-01 1.001680e-01 1.001680e-01
## 379 380 381 382 383 384
## 1.001680e-01 1.001680e-01 1.001680e-01 1.001680e-01 1.001680e-01 1.001680e-01
## 385 386 387 388 389 390
## 1.001680e-01 1.001680e-01 1.001680e-01 1.001680e-01 1.001680e-01 1.001680e-01
## 391 392 393 394 395 396
## 1.001680e-01 1.001680e-01 1.001680e-01 1.001680e-01 1.001680e-01 1.001680e-01
## 397 398 399 400 401 402
## 1.001680e-01 1.001680e-01 1.001680e-01 1.001680e-01 1.001680e-01 1.001680e-01
## 403 404 405 406 407 408
## 1.001680e-01 1.001680e-01 1.001680e-01 1.001680e-01 1.001680e-01 1.001680e-01
## 409 410 411 412 413 414
## 1.001680e-01 1.001680e-01 1.001680e-01 1.001680e-01 1.001680e-01 1.001680e-01
## 415 416 417 418 419 420
## 1.001680e-01 1.001680e-01 1.001680e-01 1.001680e-01 1.001680e-01 1.001680e-01
## 421 422 423 424 425 426
## 1.001680e-01 1.001680e-01 1.001680e-01 1.001680e-01 1.001680e-01 1.001680e-01
## 427 428 429 430 431 432
## 1.001680e-01 1.001680e-01 1.001680e-01 1.001680e-01 1.001680e-01 1.001680e-01
## 433 434 435 436 437 438
## 1.001680e-01 1.001680e-01 1.001680e-01 1.001680e-01 1.001680e-01 1.001680e-01
## 439 440 441 442 443 444
## 1.001680e-01 1.001680e-01 1.001680e-01 1.001680e-01 1.001680e-01 1.001680e-01
## 445 446 447 448 449 450
## 1.001680e-01 1.001680e-01 1.001680e-01 1.001680e-01 1.001680e-01 1.001680e-01
## 451 452 453 454 455 456
## 1.001680e-01 1.001680e-01 1.001680e-01 1.001680e-01 1.127453e-06 1.127453e-06
## 457 458 459 460 461 462
## 1.001680e-01 1.001680e-01 1.001680e-01 1.001680e-01 1.001680e-01 1.127453e-06
## 463 464 465 466 467 468
## 1.127453e-06 1.127453e-06 1.127453e-06 1.127453e-06 1.001680e-01 1.001680e-01
## 469 470 471 472 473 474
## 1.001680e-01 1.001680e-01 2.277678e-04 1.001680e-01 1.127453e-06 1.127453e-06
## 475 476 477 478 479 480
## 1.127453e-06 1.127453e-06 1.127453e-06 1.001680e-01 1.001680e-01 1.001680e-01
## 481 482 483 484 485 486
## 1.001680e-01 1.001680e-01 1.001680e-01 1.001680e-01 1.001680e-01 1.001680e-01
## 487 488 489 490 491 492
## 1.001680e-01 1.001680e-01 1.001680e-01 1.001680e-01 1.001680e-01 1.001680e-01
## 493 494 495 496 497 498
## 1.001680e-01 1.001680e-01 1.001680e-01 1.001680e-01 1.127453e-06 1.127453e-06
## 499 500 501 502 503 504
## 1.127453e-06 1.001680e-01 1.001680e-01 1.001680e-01 1.001680e-01 1.001680e-01
## 505 506 507 508 509 510
## 1.001680e-01 1.001680e-01 1.001680e-01 1.001680e-01 1.001680e-01 1.001680e-01
## 511 512 513 514 515 516
## 1.001680e-01 1.001680e-01 1.001680e-01 1.001680e-01 1.001680e-01 1.001680e-01
## 517 518 519 520 521 522
## 1.001680e-01 1.001680e-01 1.001680e-01 1.992968e-04 1.001680e-01 1.001680e-01
## 523 524 525 526 527 528
## 1.001680e-01 1.001680e-01 1.001680e-01 1.001680e-01 1.001680e-01 1.001680e-01
## 529 530 531 532 533 534
## 1.001680e-01 1.001680e-01 1.001680e-01 1.001680e-01 1.001680e-01 1.001680e-01
## 535 536 537 538 539 540
## 2.277678e-04 2.277678e-04 4.200999e-01 2.277671e-04 2.277678e-04 2.277678e-04
## 541 542 543 544 545 546
## 1.001680e-01 1.001680e-01 1.001680e-01 1.001680e-01 1.001680e-01 1.001680e-01
## 547 548 549 550 551 552
## 1.001680e-01 1.001680e-01 1.001680e-01 1.001680e-01 1.001680e-01 1.127453e-06
## 553 554 555 556 557 558
## 1.127453e-06 1.127453e-06 1.127453e-06 1.127453e-06 1.127453e-06 1.127453e-06
## 559 560 561 562 563 564
## 2.277678e-04 1.127453e-06 1.127453e-06 1.127453e-06 1.127453e-06 1.001680e-01
## 565 566 567 568
## 1.001680e-01 1.001680e-01 1.001680e-01 1.001680e-01print(predictions_svm)## [1] -0.1058099 -0.1058099 -0.1058099 -0.1058099 -0.1058099 -0.1058099
## [7] -0.1058099 -0.1058099 -0.1058099 -0.1058099 -0.1058099 -0.1058099
## [13] -0.1058099 -0.1058099 -0.1058099 -0.1058099 -0.1058099 -0.1058099
## [19] -0.1058099 -0.1058099 -0.1058099 -0.1058099 -0.1058099 -0.1058099
## [25] -0.1058099 -0.1058099 -0.1058099 -0.1058099 -0.1058099 -0.1058099
## [31] -0.1058099 -0.1058099 -0.1058099 -0.1058099 -0.1058099 -0.1058099
## [37] -0.1058099 -0.1058099 -0.1058099 -0.1058099 -0.1058099 -0.1058099
## [43] -0.1058099 -0.1058099 -0.1058099 -0.1058099 -0.1058099 -0.1058099
## [49] -0.1058099 -0.1058099 -0.1058099 -0.1058099 -0.1058099 -0.1058099
## [55] -0.1058099 -0.1058099 -0.1058099 -0.1058099 -0.1058099 -0.1058099
## [61] -0.1058099 -0.1058099 -0.1058099 -0.1058099 -0.1058099 -0.1058099
## [67] -0.1058099 -0.1058099 -0.1058099 -0.1058099 -0.1058099 -0.1058099
## [73] -0.1058099 -0.1058099 -0.1058099 -0.1058099 -0.1058099 -0.1058099
## [79] -0.1058099 -0.1058099 -0.1058099 -0.1058099 -0.1058099 -0.1058099
## [85] -0.1058099 -0.1058099 -0.1058099 -0.1058099 -0.1058099 -0.1058099
## [91] -0.1058099 -0.1058099 -0.1058099 -0.1058099 -0.1058099 -0.1058099
## [97] -0.1058099 -0.1058099 -0.1058099 -0.1058099 -0.1058099 -0.1058099
## [103] -0.1058099 -0.1058099 -0.1058099 -0.1058099 -0.1058099 -0.1058099
## [109] -0.1058099 -0.1058099 -0.1058099 -0.1058099 -0.1058099 -0.1058099
## [115] -0.1058099 -0.1058099 -0.1058099 -0.1058099 -0.1058099 -0.1058099
## [121] -0.1058099 -0.1058099 -0.1058099 -0.1058099 -0.1058099 -0.1058099
## [127] -0.1058099 -0.1058099 -0.1058099 -0.1058099 -0.1058099 -0.1058099
## [133] -0.1058099 -0.1058099 -0.1058099 -0.1058099 -0.1058099 -0.1058099
## [139] -0.1058099 -0.1058099 -0.1058099 -0.1058099 -0.1058099 -0.1058099
## [145] -0.1058099 -0.1058099 -0.1058099 -0.1058099 -0.1058099 -0.1058099
## [151] -0.1058099 -0.1058099 -0.1058099 -0.1058099 -0.1058099 -0.1058099
## [157] -0.1058099 -0.1058099 -0.1058099 -0.1058099 -0.1058099 -0.1058099
## [163] -0.1058099 -0.1058099 -0.1058099 -0.1058099 -0.1058099 -0.1058099
## [169] -0.1058099 -0.1058099 -0.1058099 -0.1058099 -0.1058099 -0.1058099
## [175] -0.1058099 -0.1058099 -0.1058099 -0.1058099 -0.1058099 -0.1058099
## [181] -0.1058099 -0.1058099 -0.1058099 -0.1058099 -0.1058099 -0.1058099
## [187] -0.1058099 -0.1058099 -0.1058099 -0.1058099 -0.1058099 -0.1058099
## [193] -0.1058099 -0.1058099 -0.1058099 -0.1058099 -0.1058099 -0.1058099
## [199] -0.1058099 -0.1058099 -0.1058099 -0.1058099 -0.1058099 -0.1058099
## [205] -0.1058099 -0.1058099 -0.1058099 -0.1058099 -0.1058099 -0.1058099
## [211] -0.1058099 -0.1058099 -0.1058099 -0.1058099 -0.1058099 -0.1058099
## [217] -0.1058099 -0.1058099 -0.1058099 -0.1058099 -0.1058099 -0.1058099
## [223] -0.1058099 -0.1058099 -0.1058099 -0.1058099 -0.1058099 -0.1058099
## [229] -0.1058099 -0.1058099 -0.1058099 -0.1058099 -0.1058099 -0.1058099
## [235] -0.1058099 -0.1058099 -0.1058099 -0.1058099 -0.1058099 -0.1058099
## [241] -0.1058099 -0.1058099 -0.1058099 -0.1058099 -0.1058099 -0.1058099
## [247] -0.1058099 -0.1058099 -0.1058099 -0.1058099 -0.1058099 -0.1058099
## [253] -0.1058099 -0.1058099 -0.1058099 -0.1058099 -0.1058099 -0.1058099
## [259] -0.1058099 -0.1058099 -0.1058099 -0.1058099 -0.1058099 -0.1058099
## [265] -0.1058099 -0.1058099 -0.1058099 -0.1058099 -0.1058099 -0.1058099
## [271] -0.1058099 -0.1058099 -0.1058099 -0.1058099 -0.1058099 -0.1058099
## [277] -0.1058099 -0.1058099 -0.1058099 -0.1058099 -0.1058099 -0.1058099
## [283] -0.1058099 -0.1058099 -0.1058099 -0.1058099 -0.1058099 -0.1058099
## [289] -0.1058099 -0.1058099 -0.1058099 -0.1058099 -0.1058099 -0.1058099
## [295] -0.1058099 -0.1058099 -0.1058099 -0.1058099 -0.1058099 -0.1058099
## [301] -0.1058099 -0.1058099 -0.1058099 -0.1058099 -0.1058099 -0.1058099
## [307] -0.1058099 -0.1058099 -0.1058099 -0.1058099 -0.1058099 -0.1058099
## [313] -0.1058099 -0.1058099 -0.1058099 -0.1058099 -0.1058099 -0.1058099
## [319] -0.1058099 -0.1058099 -0.1058099 -0.1058099 -0.1058099 -0.1058099
## [325] -0.1058099 -0.1058099 -0.1058099 -0.1058099 -0.1058099 -0.1058099
## [331] -0.1058099 -0.1058099 -0.1058099 -0.1058099 -0.1058099 -0.1058099
## [337] -0.1058099 -0.1058099 -0.1058099 -0.1058099 -0.1058099 -0.1058099
## [343] -0.1058099 -0.1058099 -0.1058099 -0.1058099 -0.1058099 -0.1058099
## [349] -0.1058099 -0.1058099 -0.1058099 -0.1058099 -0.1058099 -0.1058099
## [355] -0.1058099 -0.1058099 -0.1058099 -0.1058099 -0.1058099 -0.1058099
## [361] -0.1058099 -0.1058099 -0.1058099 -0.1058099 -0.1058099 -0.1058099
## [367] -0.1058099 -0.1058099 -0.1058099 -0.1058099 -0.1058099 -0.1058099
## [373] -0.1058099 -0.1058099 -0.1058099 -0.1058099 -0.1058099 -0.1058099
## [379] -0.1058099 -0.1058099 -0.1058099 -0.1058099 -0.1058099 -0.1058099
## [385] -0.1058099 -0.1058099 -0.1058099 -0.1058099 -0.1058099 -0.1058099
## [391] -0.1058099 -0.1058099 -0.1058099 -0.1058099 -0.1058099 -0.1058099
## [397] -0.1058099 -0.1058099 -0.1058099 -0.1058099 -0.1058099 -0.1058099
## [403] -0.1058099 -0.1058099 -0.1058099 -0.1058099 -0.1058099 -0.1058099
## [409] -0.1058099 -0.1058099 -0.1058099 -0.1058099 -0.1058099 -0.1058099
## [415] -0.1058099 -0.1058099 -0.1058099 -0.1058099 -0.1058099 -0.1058099
## [421] -0.1058099 -0.1058099 -0.1058099 -0.1058099 -0.1058099 -0.1058099
## [427] -0.1058099 -0.1058099 -0.1058099 -0.1058099 -0.1058099 -0.1058099
## [433] -0.1058099 -0.1058099 -0.1058099 -0.1058099 -0.1058099 -0.1058099
## [439] -0.1058099 -0.1058099 -0.1058099 -0.1058099 -0.1058099 -0.1058099
## [445] -0.1058099 -0.1058099 -0.1058099 -0.1058099 -0.1058099 -0.1058099
## [451] -0.1058099 -0.1058099 -0.1058099 -0.1058099 -0.1058099 -0.1058099
## [457] -0.1058099 -0.1058099 -0.1058099 -0.1058099 -0.1058099 -0.1058099
## [463] -0.1058099 -0.1058099 -0.1058099 -0.1058099 -0.1058099 -0.1058099
## [469] -0.1058099 -0.1058099 -0.1058099 -0.1058099 -0.1058099 -0.1058099
## [475] -0.1058099 -0.1058099 -0.1058099 -0.1058099 -0.1058099 -0.1058099
## [481] -0.1058099 -0.1058099 -0.1058099 -0.1058099 -0.1058099 -0.1058099
## [487] -0.1058099 -0.1058099 -0.1058099 -0.1058099 -0.1058099 -0.1058099
## [493] -0.1058099 -0.1058099 -0.1058099 -0.1058099 -0.1058099 -0.1058099
## [499] -0.1058099 -0.1058099 -0.1058099 -0.1058099 -0.1058099 -0.1058099
## [505] -0.1058099 -0.1058099 -0.1058099 -0.1058099 -0.1058099 -0.1058099
## [511] -0.1058099 -0.1058099 -0.1058099 -0.1058099 -0.1058099 -0.1058099
## [517] -0.1058099 -0.1058099 -0.1058099 -0.1058099 -0.1058099 -0.1058099
## [523] -0.1058099 -0.1058099 -0.1058099 -0.1058099 -0.1058099 -0.1058099
## [529] -0.1058099 -0.1058099 -0.1058099 -0.1058099 -0.1058099 -0.1058099
## [535] -0.1058099 -0.1058099 -0.1058099 -0.1058099 -0.1058099 -0.1058099
## [541] -0.1058099 -0.1058099 -0.1058099 -0.1058099 -0.1058099 -0.1058099
## [547] -0.1058099 -0.1058099 -0.1058099 -0.1058099 -0.1058099 -0.1058099
## [553] -0.1058099 -0.1058099 -0.1058099 -0.1058099 -0.1058099 -0.1058099
## [559] -0.1058099 -0.1058099 -0.1058099 -0.1058099 -0.1058099 -0.1058099
## [565] -0.1058099 -0.1058099 -0.1058099 -0.1058099print(predictions_knn)## [1] -0.039256437 -0.039256437 -0.039256437 -0.104479848 -0.104479848
## [6] -0.179190300 -0.104479848 -0.039256437 -0.039256437 -0.039256437
## [11] -0.039256437 1.351781034 1.351781034 1.351781034 1.351781034
## [16] -0.039256437 -0.039256437 -0.039256437 -1.041918143 -0.039256437
## [21] -0.104479848 -0.039256437 0.592817708 -0.039256437 -0.083578709
## [26] -0.039256437 -0.039256437 -0.039256437 -0.039256437 -0.039256437
## [31] -0.039256437 -0.039256437 -0.179190300 -0.104479848 -0.039256437
## [36] -0.039256437 -0.039256437 1.351781034 -0.104479848 -0.039256437
## [41] -0.039256437 1.294858784 1.311461107 -0.039256437 -0.039256437
## [46] -0.039256437 -0.043999958 -0.039256437 -0.039256437 0.170644358
## [51] -0.039256437 -0.104479848 -0.039256437 -0.039256437 -0.039256437
## [56] -0.104479848 -0.039256437 -0.166484441 -0.039256437 1.047484175
## [61] -0.039256437 -0.179190300 -0.039256437 -1.207348431 -1.207348431
## [66] -1.157541462 -0.039256437 -0.039256437 -0.039256437 -0.039256437
## [71] -0.039256437 -0.039256437 -0.039256437 -0.039256437 -0.039256437
## [76] -0.039256437 -0.039256437 -0.039256437 -0.039256437 -0.039256437
## [81] -0.039256437 1.351781034 1.351781034 1.365537244 1.396336247
## [86] 1.351781034 1.351781034 -0.104479848 -0.039256437 -0.039256437
## [91] -0.039256437 -0.039256437 -0.039256437 -0.039256437 -0.039256437
## [96] -0.104479848 -0.659471780 -0.104479848 -0.039256437 -0.104479848
## [101] -0.104479848 -0.039256437 -0.039256437 -0.039256437 -0.039256437
## [106] -0.166484441 -0.039256437 -0.039256437 -0.039256437 -0.039256437
## [111] -0.039256437 -0.039256437 -0.104479848 -0.039256437 -0.039256437
## [116] -0.039256437 -0.039256437 -0.039256437 -0.039256437 -1.047254604
## [121] -0.039256437 -0.039256437 -0.039256437 -0.104479848 -0.039256437
## [126] -0.039256437 -0.039256437 -0.039256437 -0.039256437 -0.104479848
## [131] -0.179190300 1.351781034 0.947692357 1.396336247 1.396336247
## [136] 1.351781034 -0.039256437 -0.742483394 -0.178004420 0.207406644
## [141] -0.039256437 -0.039256437 -0.039256437 -0.039256437 -0.039256437
## [146] -0.039256437 -0.166484441 -0.039256437 -0.039256437 -0.039256437
## [151] -0.039256437 -0.039256437 -0.039256437 -0.039256437 -0.039256437
## [156] -0.039256437 -0.039256437 -0.039256437 -0.104479848 -0.039256437
## [161] -0.039256437 -0.039256437 -0.104479848 -0.039256437 -0.104479848
## [166] 1.297230545 1.297230545 -0.039256437 -0.039256437 -0.039256437
## [171] -0.039256437 -0.039256437 -0.039256437 -0.039256437 -0.039256437
## [176] -0.039256437 -0.039256437 -0.039256437 -0.039256437 -0.039256437
## [181] -0.039256437 -0.039256437 -0.039256437 -0.039256437 -0.039256437
## [186] -0.039256437 -0.039256437 -0.039256437 -0.039256437 -0.039256437
## [191] -1.725340900 -0.179190300 -1.349654054 -0.039256437 -0.039256437
## [196] -0.039256437 -0.039256437 -0.039256437 -0.039256437 -0.039256437
## [201] -0.166484441 -0.039256437 1.368383356 1.315018748 1.336364591
## [206] -0.104479848 -0.039256437 -0.039256437 -0.039256437 -0.039256437
## [211] -0.039256437 -0.104479848 -0.039256437 -0.104479848 -0.039256437
## [216] -0.104479848 -0.039256437 -0.104479848 -0.039256437 -0.039256437
## [221] -0.104479848 -0.039256437 0.592817708 -0.039256437 -0.043999958
## [226] -0.039256437 -0.039256437 -0.039256437 -0.039256437 -0.039256437
## [231] -0.039256437 -0.039256437 -0.039256437 -0.039256437 -0.039256437
## [236] -0.039256437 -0.039256437 -0.039256437 -0.039256437 -0.039256437
## [241] -0.039256437 -0.039256437 -0.039256437 -0.179190300 -0.179190300
## [246] -0.039256437 -0.039256437 -0.039256437 -0.104479848 -0.039256437
## [251] -0.043999958 1.020920459 -0.104479848 -0.104479848 -0.039256437
## [256] -0.104479848 -0.039256437 -0.039256437 -0.039256437 -0.039256437
## [261] -0.179190300 -0.179190300 -0.104479848 -0.179190300 -0.166484441
## [266] 1.351781034 1.365537244 1.351781034 -0.039256437 -0.039256437
## [271] -0.039256437 -0.039256437 -0.039256437 -0.039256437 -0.104479848
## [276] -0.039256437 -0.039256437 -0.039256437 -0.039256437 -0.039256437
## [281] -0.039256437 -0.039256437 -0.039256437 -0.039256437 -0.039256437
## [286] -0.039256437 -0.104479848 1.351781034 1.351781034 0.012210764
## [291] -0.039256437 -0.039256437 0.170644358 -0.039256437 0.592817708
## [296] -0.039256437 -0.043999958 -0.039256437 -0.039256437 -0.039256437
## [301] -0.039256437 -0.039256437 -0.039256437 -0.039256437 1.396336247
## [306] 1.396336247 -0.104479848 -0.039256437 -0.039256437 -0.166484441
## [311] 0.170644358 0.170644358 -0.039256437 -0.039256437 -0.039256437
## [316] -0.039256437 -0.104479848 -0.039256437 -0.039256437 -0.039256437
## [321] -0.039256437 -0.039256437 -0.039256437 1.288692207 1.288692207
## [326] 1.288692207 1.327470490 1.327470490 1.297230545 -0.039256437
## [331] -0.039256437 -0.039256437 -0.039256437 -0.039256437 -0.039256437
## [336] -0.039256437 -0.039256437 -0.104479848 0.483420260 -0.039256437
## [341] -0.039256437 -0.039256437 -0.039256437 -0.039256437 -0.039256437
## [346] -0.039256437 -1.224628399 -0.039256437 -0.039256437 -0.039256437
## [351] -0.039256437 -0.039256437 -0.039256437 -0.104479848 -0.039256437
## [356] -0.039256437 -0.039256437 -0.039256437 -0.039256437 -0.039256437
## [361] -0.104479848 -0.039256437 -0.039256437 1.365537244 1.371940997
## [366] -0.039256437 -0.039256437 -0.039256437 -0.039256437 -0.039256437
## [371] -0.039256437 -0.039256437 -0.039256437 -0.039256437 -0.039256437
## [376] -0.039256437 -0.039256437 -0.039256437 -0.043999958 -0.039256437
## [381] -0.039256437 -0.039256437 -0.039256437 -0.039256437 -0.039256437
## [386] -0.039256437 -0.039256437 -0.039256437 1.351781034 -0.039256437
## [391] -0.039256437 -0.039256437 -0.039256437 -0.039256437 -0.039256437
## [396] -0.039256437 -0.039256437 -0.039256437 -0.039256437 -0.039256437
## [401] -0.039256437 -0.039256437 -0.039256437 -0.039256437 -0.039256437
## [406] -0.039256437 -0.039256437 -0.039256437 -0.039256437 -0.039256437
## [411] -0.039256437 -0.039256437 -0.039256437 -0.039256437 1.351781034
## [416] -0.039256437 -0.039256437 -0.039256437 -0.039256437 -0.039256437
## [421] -0.039256437 -0.104479848 -0.039256437 -0.104479848 -0.039256437
## [426] -0.039256437 -0.039256437 -0.039256437 -0.166484441 -0.083578709
## [431] -0.166484441 -0.104479848 -0.166484441 -0.039256437 1.351781034
## [436] 1.351781034 1.351781034 -0.039256437 -0.039256437 -0.039256437
## [441] -0.039256437 -0.039256437 -0.039256437 -0.039256437 -0.039256437
## [446] -0.104479848 -0.104479848 -0.179190300 -0.104479848 -0.039256437
## [451] -0.039256437 1.351781034 1.351781034 1.268769420 -0.778059799
## [456] -0.778059799 -0.039256437 -0.039256437 -0.039256437 -0.039256437
## [461] -0.104479848 -1.552439568 -1.552439568 -1.552439568 -1.552439568
## [466] -1.552439568 -0.039256437 -0.039256437 -0.039256437 -0.039256437
## [471] -0.248208528 -0.166484441 -1.270200081 -1.270200081 -1.270200081
## [476] -1.270200081 -1.270200081 -0.039256437 -0.039256437 -0.104479848
## [481] -0.039256437 -0.104479848 -0.104479848 -0.039256437 1.351781034
## [486] -0.039256437 -0.104479848 -0.039256437 -0.039256437 -0.039256437
## [491] -0.039256437 -0.039256437 -0.039256437 -0.039256437 -0.083578709
## [496] -0.039256437 -1.552439568 -1.552439568 -1.552439568 -0.039256437
## [501] -0.039256437 -0.039256437 -0.039256437 -0.104479848 -0.039256437
## [506] -0.039256437 -0.039256437 -0.039256437 -0.039256437 -0.039256437
## [511] -0.179190300 -0.166484441 -0.166484441 -0.039256437 -0.104479848
## [516] -0.166484441 -0.039256437 -0.039256437 -0.039256437 -0.179190300
## [521] -0.104479848 -0.039256437 -0.039256437 -0.039256437 -0.039256437
## [526] -0.039256437 -0.039256437 -0.039256437 -0.039256437 -0.039256437
## [531] 1.351781034 1.351781034 1.384392739 1.384392739 -1.268539849
## [536] -1.597265839 0.218079566 -0.390276975 0.041383416 -0.390276975
## [541] -0.039256437 -0.039256437 -0.039256437 -0.083578709 -0.039256437
## [546] -0.104479848 -0.104479848 -0.039256437 -0.083578709 -0.004865911
## [551] -0.039256437 -1.097061572 -1.157541462 -0.885974897 -0.370117012
## [556] 0.012210764 -1.268539849 -0.926294824 -1.597265839 -0.814822085
## [561] -0.817342081 -1.597265839 -1.443931530 -0.039256437 -0.039256437
## [566] -0.039256437 -0.039256437 -0.039256437print(predictions_lm)## 1 2 3 4 5 6 7 8
## 55.40918 56.31064 60.03497 73.99586 74.35283 74.71673 77.72393 75.22680
## 9 10 11 12 13 14 15 16
## 48.79664 52.73567 57.00471 73.95571 74.32363 74.97225 76.07796 73.39254
## 17 18 19 20 21 22 23 24
## 74.78669 76.20015 72.23972 74.42583 74.63352 74.87404 80.08938 80.24114
## 25 26 27 28 29 30 31 32
## 80.60569 82.16615 82.29446 78.92121 80.09739 80.29647 80.89864 81.13880
## 33 34 35 36 37 38 39 40
## 68.24268 70.28269 71.25363 71.87053 72.07480 74.80094 67.03191 70.83462
## 41 42 43 44 45 46 47 48
## 71.51779 77.92698 79.35621 69.10740 70.56446 72.60692 79.63012 79.80981
## 49 50 51 52 53 54 55 56
## 80.39907 80.89328 70.71507 71.31691 72.19770 72.70980 55.72641 59.49899
## 57 58 59 60 61 62 63 64
## 59.59346 60.11646 65.90115 66.95902 67.79555 68.06544 69.55925 65.18543
## 65 66 67 68 69 70 71 72
## 68.96990 69.65094 74.53100 74.94559 75.30949 75.69307 76.34528 49.49701
## 73 74 75 76 77 78 79 80
## 54.06821 56.50355 60.97443 62.21107 65.13282 72.70140 73.27600 74.25319
## 81 82 83 84 85 86 87 88
## 74.44383 73.42595 73.62586 73.98415 74.74125 74.91340 75.05882 71.68756
## 89 90 91 92 93 94 95 96
## 72.22805 74.30036 51.55349 55.13151 58.40916 52.16555 54.60666 58.49772
## 97 98 99 100 101 102 103 104
## 72.50201 60.82985 62.62025 64.10803 66.32068 68.28105 50.67129 80.05811
## 105 106 107 108 109 110 111 112
## 80.14148 80.68148 45.72137 47.80694 50.84308 51.81429 52.11146 52.54387
## 113 114 115 116 117 118 119 120
## 52.91379 78.11971 73.96222 74.43964 74.77593 74.80572 75.57605 73.99585
## 121 122 123 124 125 126 127 128
## 58.89414 59.19493 60.41642 56.87422 58.24493 54.63985 55.73250 77.44600
## 129 130 131 132 133 134 135 136
## 77.79493 79.41959 75.35018 75.59353 77.31457 77.93608 78.66290 78.47020
## 137 138 139 140 141 142 143 144
## 79.72266 75.23509 77.33061 78.51781 78.18130 78.53972 62.52061 70.04137
## 145 146 147 148 149 150 151 152
## 70.23879 70.97476 73.03998 73.94536 75.97824 76.44059 68.67199 68.94546
## 153 154 155 156 157 158 159 160
## 69.68727 71.08216 71.35222 69.96890 52.95165 54.76646 55.69198 57.06678
## 161 162 163 164 165 166 167 168
## 60.82074 72.59613 72.81643 76.42747 76.79955 48.78392 51.31253 56.53176
## 169 170 171 172 173 174 175 176
## 60.44266 65.50666 78.74542 78.57406 79.65706 79.92995 79.72027 79.90517
## 177 178 179 180 181 182 183 184
## 80.43996 80.77325 80.90809 81.51458 82.36211 57.61682 58.02098 60.87089
## 185 186 187 188 189 190 191 192
## 62.08781 62.68850 63.18049 56.04860 58.55652 59.23000 70.32403 71.41392
## 193 194 195 196 197 198 199 200
## 72.64935 78.12639 78.52476 78.96694 79.78143 80.80031 56.92118 59.47648
## 201 202 203 204 205 206 207 208
## 79.12819 79.41158 71.49909 72.83582 73.53325 70.12273 72.24953 53.89884
## 209 210 211 212 213 214 215 216
## 54.50867 57.20647 57.67512 58.08701 50.63385 53.67637 55.72539 68.33099
## 217 218 219 220 221 222 223 224
## 57.46766 61.49873 73.33970 73.80510 72.15226 72.32723 79.98106 81.09932
## 225 226 227 228 229 230 231 232
## 81.59123 81.79374 64.96707 65.69759 67.08188 67.52993 68.36042 69.37459
## 233 234 235 236 237 238 239 240
## 69.61700 70.07759 71.19869 72.36627 67.92844 68.06672 69.93078 70.47370
## 241 242 243 244 245 246 247 248
## 78.28929 80.15688 80.40860 79.54976 80.57599 80.96498 72.60605 73.83971
## 249 250 251 252 253 254 255 256
## 74.28777 74.74834 82.28675 82.61383 83.50430 71.35791 72.73228 66.58834
## 257 258 259 260 261 262 263 264
## 52.10039 58.00985 59.21852 62.40423 63.55471 64.83871 64.95358 65.51061
## 265 266 267 268 269 270 271 272
## 65.74823 72.41744 72.55934 75.06138 67.97675 68.14104 66.51735 70.30585
## 273 274 275 276 277 278 279 280
## 71.06105 73.11100 73.94608 74.99799 75.17143 75.95485 76.96624 78.18233
## 281 282 283 284 285 286 287 288
## 78.39628 78.95437 42.54641 45.23615 51.69716 55.84573 60.26676 70.17415
## 289 290 291 292 293 294 295 296
## 71.42126 72.59631 71.79083 74.17623 74.91259 78.39674 79.05845 80.22296
## 297 298 299 300 301 302 303 304
## 80.90037 59.00949 61.24343 61.69646 62.51373 57.71009 61.00086 73.56149
## 305 306 307 308 309 310 311 312
## 75.48445 76.30818 77.02562 54.03338 54.76610 55.12032 79.95941 80.12214
## 313 314 315 316 317 318 319 320
## 80.20918 60.28348 61.86723 63.05269 73.96459 74.49284 75.29923 76.29537
## 321 322 323 324 325 326 327 328
## 65.16201 66.18872 66.53704 66.65285 67.05511 67.36704 68.06564 71.00893
## 329 330 331 332 333 334 335 336
## 71.18487 63.72287 64.25098 64.79379 65.17377 65.66799 67.03286 68.38503
## 337 338 339 340 341 342 343 344
## 69.09816 73.98832 76.00937 68.80706 69.83424 71.07377 72.92828 75.00653
## 345 346 347 348 349 350 351 352
## 75.26710 49.38925 50.07303 52.46584 53.46775 56.81955 61.47027 61.85030
## 353 354 355 356 357 358 359 360
## 64.86555 51.68578 52.29741 53.66389 57.68246 64.50332 68.15024 69.33450
## 361 362 363 364 365 366 367 368
## 78.70295 79.76085 80.09620 79.62863 80.28546 70.27427 73.48921 51.85387
## 369 370 371 372 373 374 375 376
## 54.64087 56.25834 57.29535 57.77769 46.77000 47.84473 48.30257 73.14315
## 377 378 379 380 381 382 383 384
## 73.23330 74.21088 74.45973 74.82130 75.17273 75.23144 79.38466 79.50257
## 385 386 387 388 389 390 391 392
## 80.19443 80.68853 81.10107 81.77801 75.93645 64.88894 65.61168 66.02723
## 393 394 395 396 397 398 399 400
## 75.89961 77.17605 59.02438 60.06961 61.85364 71.87775 72.27662 72.56047
## 401 402 403 404 405 406 407 408
## 71.99732 74.61202 68.91791 70.29846 73.90407 74.30295 74.56405 74.73926
## 409 410 411 412 413 414 415 416
## 75.15013 76.09094 76.58348 77.46177 78.69362 79.70992 79.41590 72.38908
## 417 418 419 420 421 422 423 424
## 72.68363 72.98696 74.88105 65.09469 64.98412 47.83134 61.02381 63.74801
## 425 426 427 428 429 430 431 432
## 65.48620 65.97265 69.93241 70.78673 71.76373 72.59705 62.79951 63.71081
## 433 434 435 436 437 438 439 440
## 66.20096 66.56773 71.93779 73.13174 73.34405 58.96451 60.59053 63.32641
## 441 442 443 444 445 446 447 448
## 72.16652 72.76940 73.02230 73.29814 73.55201 72.15230 73.01199 73.34024
## 449 450 451 452 453 454 455 456
## 73.61215 41.61896 44.95098 80.61053 80.78313 81.96907 75.45364 75.61310
## 457 458 459 460 461 462 463 464
## 77.46835 78.68718 67.53439 68.55167 70.15062 51.38037 52.11411 52.39360
## 465 466 467 468 469 470 471 472
## 53.97163 55.86667 53.99014 57.87613 80.44525 81.45473 73.79617 74.82435
## 473 474 475 476 477 478 479 480
## 74.52874 75.51222 70.29079 72.49952 72.92296 68.24370 69.88233 70.41611
## 481 482 483 484 485 486 487 488
## 70.64512 71.08796 71.54335 80.47001 80.72639 81.58336 81.83631 80.67757
## 489 490 491 492 493 494 495 496
## 82.15772 82.64914 82.96336 72.26326 73.12428 72.43690 70.32082 67.28814
## 497 498 499 500 501 502 503 504
## 49.78595 50.34160 54.31095 70.96438 71.76096 74.57592 75.29784 58.89555
## 505 506 507 508 509 510 511 512
## 67.49249 68.79583 53.43412 53.87908 54.80792 55.49882 68.54058 70.42578
## 513 514 515 516 517 518 519 520
## 70.80091 69.58295 70.94064 71.03459 72.81599 75.85366 76.08947 63.59914
## 521 522 523 524 525 526 527 528
## 66.99022 47.59474 53.22265 58.19239 59.29524 60.27334 60.72057 67.67114
## 529 530 531 532 533 534 535 536
## 68.15774 71.75913 75.66287 76.07582 76.18712 76.36048 79.30064 80.49359
## 537 538 539 540 541 542 543 544
## 80.88688 78.19955 78.51992 78.58995 76.47192 76.93647 77.68765 65.63752
## 545 546 547 548 549 550 551 552
## 66.77513 67.75324 71.70522 67.43017 68.03888 68.29864 69.93810 71.48385
## 553 554 555 556 557 558 559 560
## 72.16043 73.36807 73.85251 74.01501 74.93527 75.09249 75.41860 64.75627
## 561 562 563 564 565 566 567 568
## 65.06516 65.27787 65.89078 46.16585 47.77561 51.84918 60.28735 55.64159
## attr(,"non-estim")
## 1 4 5 6 7 11 12 13 14 15 19 20 21 22 23 24 25 33 34 38
## 1 4 5 6 7 11 12 13 14 15 19 20 21 22 23 24 25 33 34 38
## 39 42 43 47 50 51 52 56 58 60 62 64 65 66 75 76 77 82 83 84
## 39 42 43 47 50 51 52 56 58 60 62 64 65 66 75 76 77 82 83 84
## 85 86 87 88 96 97 98 100 101 106 113 120 124 130 131 132 133 134 135 136
## 85 86 87 88 96 97 98 100 101 106 113 120 124 130 131 132 133 134 135 136
## 138 139 140 147 159 160 163 165 166 167 185 191 192 193 201 203 204 205 206 212
## 138 139 140 147 159 160 163 165 166 167 185 191 192 193 201 203 204 205 206 212
## 214 216 218 221 223 224 244 245 248 249 250 251 252 253 254 256 261 262 263 264
## 214 216 218 221 223 224 244 245 248 249 250 251 252 253 254 256 261 262 263 264
## 265 266 267 268 272 275 283 284 287 288 289 290 293 295 305 306 307 310 311 312
## 265 266 267 268 272 275 283 284 287 288 289 290 293 295 305 306 307 310 311 312
## 314 317 324 325 326 327 328 329 334 336 338 339 346 347 350 354 355 356 361 364
## 314 317 324 325 326 327 328 329 334 336 338 339 346 347 350 354 355 356 361 364
## 365 389 415 422 424 427 428 429 430 431 432 433 434 435 436 437 446 447 448 449
## 365 389 415 422 424 427 428 429 430 431 432 433 434 435 436 437 446 447 448 449
## 452 453 454 455 456 457 458 461 462 463 464 465 466 471 472 473 474 475 476 477
## 452 453 454 455 456 457 458 461 462 463 464 465 466 471 472 473 474 475 476 477
## 480 482 483 485 487 495 497 498 499 504 511 512 513 515 516 520 521 531 532 533
## 480 482 483 485 487 495 497 498 499 504 511 512 513 515 516 520 521 531 532 533
## 534 535 536 537 538 539 540 544 546 547 548 549 550 551 552 553 554 555 556 557
## 534 535 536 537 538 539 540 544 546 547 548 549 550 551 552 553 554 555 556 557
## 558 559 560 561 562 563
## 558 559 560 561 562 563print(predictions_rf)## 1 2 3 4 5 6 7 8
## 52.48446 53.86283 59.76634 74.64547 75.12923 75.20208 78.20086 75.65512
## 9 10 11 12 13 14 15 16
## 49.56717 52.72135 58.52441 73.63276 73.76531 74.29134 75.57645 73.94803
## 17 18 19 20 21 22 23 24
## 74.91428 75.97598 72.44929 74.26002 74.58218 74.77983 79.78341 79.96578
## 25 26 27 28 29 30 31 32
## 80.45739 82.20787 82.24361 78.46387 80.23862 80.34506 80.81841 81.12906
## 33 34 35 36 37 38 39 40
## 67.81465 70.09948 71.95597 72.07839 72.10022 75.64521 66.13299 70.81626
## 41 42 43 44 45 46 47 48
## 71.42043 77.56902 77.79284 69.30206 71.02757 72.85982 79.66904 79.85491
## 49 50 51 52 53 54 55 56
## 80.43372 80.88454 70.51292 71.33056 72.72461 73.34281 54.84321 59.56942
## 57 58 59 60 61 62 63 64
## 59.84228 60.26155 64.68346 65.79316 67.04147 68.03627 68.91466 65.65776
## 65 66 67 68 69 70 71 72
## 69.86419 70.40948 74.83956 75.20681 75.77288 76.28060 76.57708 49.97286
## 73 74 75 76 77 78 79 80
## 52.74533 55.17219 60.40751 62.17737 65.00427 72.61807 73.26259 74.01224
## 81 82 83 84 85 86 87 88
## 73.86455 74.06193 74.33018 74.74133 75.66655 75.77572 75.73746 72.08798
## 89 90 91 92 93 94 95 96
## 72.25514 74.33008 50.81570 55.16029 59.59961 51.59145 54.78238 58.83362
## 97 98 99 100 101 102 103 104
## 73.50497 60.32625 62.43508 64.00488 66.88261 69.75128 50.96668 79.80433
## 105 106 107 108 109 110 111 112
## 79.89028 80.45959 46.29035 47.59690 51.44985 52.88451 53.46062 53.92014
## 113 114 115 116 117 118 119 120
## 54.28825 78.10480 74.76572 75.72905 76.00424 76.27141 76.90624 73.75990
## 121 122 123 124 125 126 127 128
## 59.67268 59.68817 60.64753 57.59303 59.16526 54.75375 55.51725 77.41256
## 129 130 131 132 133 134 135 136
## 77.75703 78.63819 76.02731 76.37563 77.23502 77.93246 78.21104 78.68358
## 137 138 139 140 141 142 143 144
## 79.98582 75.17109 77.49240 78.89003 78.21997 78.81571 61.99477 69.79226
## 145 146 147 148 149 150 151 152
## 70.06795 70.85117 72.70594 73.88431 75.21707 75.82965 69.30615 69.50492
## 153 154 155 156 157 158 159 160
## 70.00145 71.22849 71.48561 70.20463 53.01263 53.76119 53.97666 56.21251
## 161 162 163 164 165 166 167 168
## 60.32799 72.55339 72.83782 76.99719 77.30231 48.09445 50.85824 54.72959
## 169 170 171 172 173 174 175 176
## 58.83113 66.97242 78.38206 78.64466 79.45133 80.01970 79.07855 79.11959
## 177 178 179 180 181 182 183 184
## 79.52665 80.06749 80.67065 81.69936 81.87261 57.65267 57.73986 60.24460
## 185 186 187 188 189 190 191 192
## 62.11472 62.90745 63.61888 57.03100 58.71031 59.44730 70.94025 71.45127
## 193 194 195 196 197 198 199 200
## 72.55644 78.02734 78.35969 78.83532 79.96312 81.14492 57.39698 59.74678
## 201 202 203 204 205 206 207 208
## 78.93618 79.25233 72.70350 73.04414 73.01274 69.67966 71.98455 53.03696
## 209 210 211 212 213 214 215 216
## 53.69069 57.18442 57.94733 58.94961 51.37113 54.64163 56.51652 68.95826
## 217 218 219 220 221 222 223 224
## 57.22768 60.97595 73.23712 73.75348 72.20597 72.40286 79.46913 81.01012
## 225 226 227 228 229 230 231 232
## 81.52708 81.71789 64.74049 65.65068 67.15139 67.92359 68.92003 70.25968
## 233 234 235 236 237 238 239 240
## 70.39564 70.58537 71.31492 72.90870 69.30285 68.51274 70.17954 70.22922
## 241 242 243 244 245 246 247 248
## 77.99593 80.82447 80.99486 79.58879 80.49225 80.86231 73.85769 73.67784
## 249 250 251 252 253 254 255 256
## 73.69468 73.95967 82.02967 82.26270 82.95503 71.89553 73.30844 67.12680
## 257 258 259 260 261 262 263 264
## 52.34926 57.51882 59.00843 61.79651 63.61683 65.11165 65.36663 66.01679
## 265 266 267 268 269 270 271 272
## 66.21346 74.05077 74.19198 75.86840 68.27296 68.16790 66.60198 70.30936
## 273 274 275 276 277 278 279 280
## 70.93240 73.03091 73.42803 74.08136 75.04634 75.78944 76.49083 78.56267
## 281 282 283 284 285 286 287 288
## 78.68571 78.81202 44.66337 44.74797 51.25910 53.96120 59.22726 71.39729
## 289 290 291 292 293 294 295 296
## 72.05108 72.10447 71.73628 73.81479 74.33801 78.00825 78.28055 80.58262
## 297 298 299 300 301 302 303 304
## 81.06126 58.01879 61.03406 61.29624 62.15889 56.28182 60.27200 73.29222
## 305 306 307 308 309 310 311 312
## 75.82769 77.06149 77.68666 55.08773 56.40224 56.78366 79.76851 79.89580
## 313 314 315 316 317 318 319 320
## 80.21725 61.25535 62.05441 62.90029 73.72773 75.09885 75.21363 75.57126
## 321 322 323 324 325 326 327 328
## 65.80516 67.12570 67.43321 67.63109 67.86292 67.96884 68.52674 70.98860
## 329 330 331 332 333 334 335 336
## 71.22221 63.91645 64.32191 65.32686 65.75892 66.15494 67.82818 69.25861
## 337 338 339 340 341 342 343 344
## 69.97316 74.15216 76.42960 68.32676 69.74405 71.24337 73.58741 75.44183
## 345 346 347 348 349 350 351 352
## 75.75997 49.70140 49.89259 51.83020 52.34938 56.52025 61.84469 62.30005
## 353 354 355 356 357 358 359 360
## 65.76124 51.22053 54.14025 53.98548 59.15414 64.60247 68.58088 69.74230
## 361 362 363 364 365 366 367 368
## 78.45884 79.96281 80.42172 79.47744 80.35199 70.59016 73.31234 50.93371
## 369 370 371 372 373 374 375 376
## 55.14818 57.74103 59.55045 60.08112 47.71419 48.96448 49.75528 74.25204
## 377 378 379 380 381 382 383 384
## 74.21306 74.75342 75.24933 75.89651 76.07504 76.14403 79.06969 79.21557
## 385 386 387 388 389 390 391 392
## 80.23459 80.68828 81.05600 82.06445 76.35722 64.65671 65.18307 65.95388
## 393 394 395 396 397 398 399 400
## 75.59971 76.87586 59.89394 60.73049 62.18671 71.91774 72.36683 72.61531
## 401 402 403 404 405 406 407 408
## 72.02050 74.61277 69.40342 70.28983 73.39366 74.08933 74.49988 74.58122
## 409 410 411 412 413 414 415 416
## 75.07438 75.84685 75.92372 77.12570 78.36641 80.25835 79.43332 72.30930
## 417 418 419 420 421 422 423 424
## 72.63643 72.81808 74.72574 66.03125 65.60504 48.37964 59.93788 63.96717
## 425 426 427 428 429 430 431 432
## 67.44000 67.67055 70.46844 71.39361 71.97251 73.29504 62.62208 63.46521
## 433 434 435 436 437 438 439 440
## 67.12721 67.26457 73.35419 74.04961 74.19670 59.04258 60.69948 64.55488
## 441 442 443 444 445 446 447 448
## 72.63385 73.00555 73.10669 73.48795 73.89204 72.04437 73.05497 73.02516
## 449 450 451 452 453 454 455 456
## 73.76292 42.73853 45.50441 80.73418 81.06336 82.39376 75.84438 75.97811
## 457 458 459 460 461 462 463 464
## 77.00983 78.37237 68.17043 68.91558 70.15604 51.71927 52.25709 52.75562
## 465 466 467 468 469 470 471 472
## 53.89869 55.14897 54.14043 59.82835 79.90594 81.31329 73.44909 74.89318
## 473 474 475 476 477 478 479 480
## 74.34756 75.09882 71.04837 72.33828 72.77061 68.69567 70.39704 70.88453
## 481 482 483 484 485 486 487 488
## 71.01538 71.34285 71.51966 80.56461 80.82926 81.85033 81.99164 80.32509
## 489 490 491 492 493 494 495 496
## 82.14152 82.72052 82.72048 73.19529 73.84013 72.90148 70.39158 67.73856
## 497 498 499 500 501 502 503 504
## 51.60517 51.78444 54.46737 70.46996 71.34287 74.60501 75.40021 57.94160
## 505 506 507 508 509 510 511 512
## 67.32746 68.17811 54.12113 54.16088 55.48140 56.15079 70.41429 70.62898
## 513 514 515 516 517 518 519 520
## 70.81120 69.71444 70.97109 71.16738 72.60399 75.83275 75.80372 64.14528
## 521 522 523 524 525 526 527 528
## 66.93273 48.37844 53.07050 57.38306 58.44565 59.79884 60.44150 67.95224
## 529 530 531 532 533 534 535 536
## 68.49412 71.19426 76.28686 76.63343 76.78892 76.94696 79.20902 80.90210
## 537 538 539 540 541 542 543 544
## 80.90654 78.30857 78.48376 78.60765 76.45012 76.89547 77.20627 65.44150
## 545 546 547 548 549 550 551 552
## 67.93725 68.12027 71.92560 68.27307 68.67763 68.81640 69.87572 72.68342
## 553 554 555 556 557 558 559 560
## 72.86838 73.20629 73.76497 73.88625 74.80611 74.89624 74.93353 64.48387
## 561 562 563 564 565 566 567 568
## 65.04976 65.34459 65.49062 46.06202 47.56214 52.41986 59.91560 55.26456# ML Plots
## Plot for ANN model
ggplot() +
geom_point(aes(x = actual_values, y = predictions_ann), colour = "blue") +
geom_abline(intercept = 0, slope = 1, linetype = "dashed") +
labs(title = "ANN Model: Actual vs Predicted", x = "Actual Life Expectancy", y = "Predicted Life Expectancy")
# Plot for SVM model
ggplot() +
geom_point(aes(x = actual_values, y = predictions_svm), colour = "green") +
geom_abline(intercept = 0, slope = 1, linetype = "dashed") +
labs(title = "SVM Model: Actual vs Predicted", x = "Actual Life Expectancy", y = "Predicted Life Expectancy")
# Plot for Random Forest Model
ggplot() +
geom_point(aes(x = actual_values, y = predictions_rf), colour = "red") +
geom_abline(intercept = 0, slope = 1, linetype = "dashed") +
labs(title = "Random Forest Model: Actual vs Predicted", x = "Actual Life Expectancy", y = "Predicted Life Expectancy")
# Plot for KNN Model
ggplot() +
geom_point(aes(x = actual_values, y = predictions_knn), colour = "purple") +
geom_abline(intercept = 0, slope = 1, linetype = "dashed") +
labs(title = "KNN Model: Actual vs Predicted", x = "Actual Life Expectancy", y = "Predicted Life Expectancy")
# Plot for LRM
ggplot() +
geom_point(aes(x = actual_values, y = predictions_lm), colour = "orange") +
geom_abline(intercept = 0, slope = 1, linetype = "dashed") +
labs(title = "Linear Regression Model: Actual vs Predicted", x = "Actual Life Expectancy", y = "Predicted Life Expectancy")
# Plot together
# Combine all predictions into a single data frame
predictions_df <- data.frame(
Actual = actual_values,
ANN = predictions_ann,
SVM = predictions_svm,
RF = predictions_rf,
KNN = predictions_knn,
LM = predictions_lm
)
summary(predictions_df)## Actual ANN SVM RF
## Min. :40.40 Min. :0.0000011 Min. :-0.1058 Min. :42.74
## 1st Qu.:62.88 1st Qu.:0.1001680 1st Qu.:-0.1058 1st Qu.:62.83
## Median :71.45 Median :0.1001680 Median :-0.1058 Median :71.63
## Mean :69.07 Mean :0.0903847 Mean :-0.1058 Mean :69.20
## 3rd Qu.:75.33 3rd Qu.:0.1001680 3rd Qu.:-0.1058 3rd Qu.:75.83
## Max. :83.60 Max. :0.4200999 Max. :-0.1058 Max. :82.96
## KNN LM
## Min. :-1.725341 Min. :41.62
## 1st Qu.:-0.039256 1st Qu.:62.77
## Median :-0.039256 Median :71.49
## Mean : 0.001471 Mean :69.05
## 3rd Qu.:-0.039256 3rd Qu.:75.49
## Max. : 1.396336 Max. :83.50# Convert to long format for plotting
predictions_long <- predictions_df %>%
pivot_longer(cols = -Actual, names_to = "Model", values_to = "Predicted")
# Plot all models together
ggplot(predictions_long, aes(x = Actual, y = Predicted, colour = Model)) +
geom_point() +
geom_abline(intercept = 0, slope = 1, linetype = "dashed") +
labs(title = "All Models: Actual vs Predicted", x = "Actual Life Expectancy", y = "Predicted Life Expectancy") +
theme_minimal() +
scale_colour_manual(values = c("blue", "green", "red", "purple", "orange"))
# Plot Ml together
predictions_df <- data.frame(
Actual = actual_values,
ANN = predictions_ann,
SVM = predictions_svm,
RF = predictions_rf,
KNN = predictions_knn,
LM = predictions_lm
)
ggplot(predictions_long, aes(x = Actual, y = Predicted, colour = Model)) +
geom_point() +
geom_abline(intercept = 0, slope = 1, linetype = "dashed") +
labs(title = "All Models: Actual vs Predicted", x = "Actual Life Expectancy", y = "Predicted Life Expectancy") +
theme_minimal() +
scale_colour_manual(values = c("blue", "green", "red", "purple", "orange"))
# Convert to long format for plotting
predictions_long <- predictions_df %>%
pivot_longer(cols = c("ANN", "SVM", "RF", "KNN", "LM"), names_to = "Model", values_to = "Predicted")
#### Evaluation RMSE
calculateRMSE <- function(model, test_data) {
predictions <- predict(model, test_data)
mse <- mean((predictions - test_data$Life_expectancy)^2)
return(sqrt(mse))
}
# Evaluate Models to get RMSE
rmse_ann <- calculateRMSE(ann_model, test_data)
rmse_knn <- calculateRMSE(knn_model, test_data) # Update this as per KNN's output format
rmse_svm <- calculateRMSE(svm_model, test_data)
rmse_rf <- calculateRMSE(rf_model, test_data)
rmse_lm <- calculateRMSE(lm_model, test_data) # Assuming lm_model is already fitted using train_data## Warning in predict.lm(modelFit, newdata): prediction from rank-deficient fit;
## attr(*, "non-estim") has doubtful cases# Create a data frame with all RMSE values
rmse_data <- data.frame(
Model = c("ANN", "KNN", "SVM", "RF", "LM"),
RMSE = c(rmse_ann, rmse_knn, rmse_svm, rmse_rf, rmse_lm)
)
summary(rmse_data)## Model RMSE
## Length:5 Min. : 0.4712
## Class :character 1st Qu.: 0.6091
## Mode :character Median :69.5914
## Mean :42.0253
## 3rd Qu.:69.6689
## Max. :69.7860#### Box and Whisker Plot
results <- resamples(list(linear = lm_model, knn = knn_model, ann = ann_model, svm = svm_model, rf = rf_model))
print(results)##
## Call:
## resamples.default(x = list(linear = lm_model, knn = knn_model, ann =
## ann_model, svm = svm_model, rf = rf_model))
##
## Models: linear, knn, ann, svm, rf
## Number of resamples: 10
## Performance metrics: MAE, RMSE, Rsquared
## Time estimates for: everything, final model fitbwplot(results, metric = "RMSE")
Part D: Splitting into Training & Validation
This part of the analysis focuses on preparing the dataset for predictive modeling by splitting it into training and validation sets. This is a crucial step in any data analysis project as it allows for the validation of model performance on unseen data.
Data Splitting Process
- Objective: To divide the dataset into two parts; one for training the model and the other for validating its performance.
- Method Used: The data is split using the
createDataPartitionfunction from thecaretpackage in R, ensuring a balanced distribution of the target variable,Life_expectancy, in both sets. - Seed Setting:
set.seed(123)is used for reproducibility, ensuring that the same random split can be generated again if needed.
Training Data
- Description: The training data is used to build and train the predictive models. It consists of a larger portion of the dataset to provide the models with ample data to learn from.
- Size of Training Data: From the original dataset, 70% is allocated to the training set.
- Training Data Snapshot:
- A tibble with 1,996 rows and 23 columns.
- Includes a diverse range of variables such as
Country,Region,Year,Economy_status_Developed,Adult_mortality, and others.
Validation Data
- Purpose: This subset is used to validate the performance of the trained models. It acts as new, unseen data for the models to make predictions.
- Size of Validation Data: The remaining 30% of the dataset is used as validation data.
- Validation Data Snapshot:
- A tibble with 852 rows and 23 columns.
- Mirrors the structure of the training data, ensuring consistency in variable types and ranges.
Summary
The dataset has been successfully split into training and validation sets, maintaining a balance in the distribution of the target variable. This split is essential for developing robust predictive models and evaluating their performance accurately.
# 1. Splitting the Data: Split data into training and validation sets.
set.seed(123) # For reproducibility
index <- createDataPartition(data$Life_expectancy, p = 0.7, list = FALSE)
train_data <- data[index, ]
validation_data <- data[-index, ]
print(validation_data)## # A tibble: 852 × 23
## Country Region Year Economy_status_Devel…¹ Life_expectancy Adult_mortality
## <chr> <chr> <dbl> <fct> <dbl> <dbl>
## 1 Afghanis… Asia 2000 0 55.8 311.
## 2 Afghanis… Asia 2001 0 56.3 305.
## 3 Afghanis… Asia 2002 0 56.8 299.
## 4 Afghanis… Asia 2003 0 57.3 292.
## 5 Afghanis… Asia 2006 0 58.8 271.
## 6 Afghanis… Asia 2008 0 59.9 260.
## 7 Afghanis… Asia 2014 0 63 232.
## 8 Albania Rest … 2001 0 74.3 90.9
## 9 Albania Rest … 2002 0 74.6 87.2
## 10 Albania Rest … 2005 0 75.2 84.5
## # ℹ 842 more rows
## # ℹ abbreviated name: ¹Economy_status_Developed
## # ℹ 17 more variables: Infant_deaths <dbl>, Alcohol_consumption <dbl>,
## # `percentage expenditure` <dbl>, Hepatitis_B <dbl>, Measles <dbl>,
## # BMI <dbl>, Under_five_deaths <dbl>, Polio <dbl>, `Total expenditure` <dbl>,
## # Diphtheria <dbl>, Incidents_HIV <dbl>, GDP_per_capita <dbl>,
## # Population_millions <dbl>, Thinness_ten_nineteen_years <dbl>, …print(train_data)## # A tibble: 1,996 × 23
## Country Region Year Economy_status_Devel…¹ Life_expectancy Adult_mortality
## <chr> <chr> <dbl> <fct> <dbl> <dbl>
## 1 Afghanis… Asia 2004 0 57.8 285.
## 2 Afghanis… Asia 2005 0 58.3 278.
## 3 Afghanis… Asia 2007 0 59.4 265.
## 4 Afghanis… Asia 2009 0 60.5 255.
## 5 Afghanis… Asia 2010 0 61 250.
## 6 Afghanis… Asia 2011 0 61.6 245.
## 7 Afghanis… Asia 2012 0 62.1 240.
## 8 Afghanis… Asia 2013 0 62.5 236.
## 9 Afghanis… Asia 2015 0 63.4 228.
## 10 Albania Rest … 2000 0 74 94.7
## # ℹ 1,986 more rows
## # ℹ abbreviated name: ¹Economy_status_Developed
## # ℹ 17 more variables: Infant_deaths <dbl>, Alcohol_consumption <dbl>,
## # `percentage expenditure` <dbl>, Hepatitis_B <dbl>, Measles <dbl>,
## # BMI <dbl>, Under_five_deaths <dbl>, Polio <dbl>, `Total expenditure` <dbl>,
## # Diphtheria <dbl>, Incidents_HIV <dbl>, GDP_per_capita <dbl>,
## # Population_millions <dbl>, Thinness_ten_nineteen_years <dbl>, …Part E: Recursive Feature Elimination (RFE) and Model Building
This section focuses on using Recursive Feature Elimination (RFE) with a random forest model to select important features and build predictive models.
Comprehensive Analysis on Part E:
- Data Loading and Overview:
- The data (
trainData) is loaded from a CSV file, containing 1,996 rows and 24 columns. - The columns include a mix of character (
chr) and double (dbl) types, covering various demographic and health-related metrics.
- The data (
- Recursive Feature Elimination (RFE) Process:
- RFE is used to identify the most significant features that influence life expectancy. This method iteratively removes the least important features to find the best subset for prediction.
- Random forest is chosen as the feature selection method within RFE. This is a robust method considering it’s good at handling both linear and non-linear relationships and can deal with high-dimensional data.
- The
rfeControlfunction with cross-validation (cv) and 10 folds is set for the RFE process. Cross-validation helps in assessing the model’s generalizability.
- Selected Features:
- The RFE process results in the selection of 21 features out of the
original set. These include factors like
Adult_mortality,Infant_deaths,Alcohol_consumption, country-specific indicators, and others. - Notably, both direct health indicators (like
Adult_mortality,BMI,Diphtheria) and socio-economic factors (GDP_per_capita,Schooling) are included, reflecting the multidimensional nature of life expectancy determinants.
- The RFE process results in the selection of 21 features out of the
original set. These include factors like
- Linear Regression Model:
- A linear regression model (
simple_model) is then built using the selected features. - The model summary provides coefficients for each feature, which represent the estimated change in life expectancy for a one-unit change in the feature, holding all other features constant.
- The coefficients show a mix of positive and negative relationships.
For example, increases in
Adult_mortalityandInfant_deathsare associated with decreases in life expectancy. Yearhas a positive coefficient, suggesting an overall improvement in life expectancy over time.- The p-values indicate the statistical significance of each feature’s relationship with life expectancy. Features with p-values below a typical alpha level (e.g., 0.05) are considered statistically significant.
- A linear regression model (
- Interpretation of Coefficients:
- The large number of country-specific coefficients (dummy variables) suggests that geographical factors significantly influence life expectancy, likely due to differences in healthcare, lifestyle, and socio-economic conditions.
- Some coefficients are not defined due to singularities, possibly indicating multicollinearity issues where some variables are highly correlated with each other.
- Model Evaluation:
- The residuals of the model (difference between observed and predicted life expectancy) indicate the model’s fit. A median close to zero suggests a good central tendency in predictions.
- However, the presence of significant residuals (Min, Max) suggests that there might be instances where the model does not predict accurately, indicating potential for model improvement.
- Considerations for Improvement:
- Addressing multicollinearity by removing or combining highly correlated variables could improve the model.
- Testing interaction effects or non-linear relationships could provide a more nuanced understanding.
- Validating the model on a separate test set would be crucial to assess its predictive accuracy.
- Model Summary and Performance:
- The initial simple linear regression model
(
simple_model) has a very high Multiple R-squared value (0.9971), indicating that the model explains almost all the variability in the response data around its mean. However, such a high value might also hint at overfitting. - Three enhanced models were built for comparison: a model with
polynomial terms (
model_1), a reduced model with fewer predictors (model_2), and a generalized linear model with a logarithmic link (model_3).
- The initial simple linear regression model
(
- Enhanced Models Analysis:
Model_1incorporates a quadratic term forAlcohol_consumption. It has a similar performance to thesimple_model, suggesting that adding the quadratic term doesn’t significantly change the model’s explanatory power.Model_2focuses on key variables (Year,Adult_mortality,Alcohol_consumption,BMI). It understandably has a lower R-squared value (0.927) compared to the full model, indicating a reduction in explanatory power but potentially better generalizability.Model_3(GLM with log link) also shows a strong fit, but it’s important to check if the assumptions of GLM are met in this case.
- Model Diagnostics and Residual Analysis:
- The influence plot and residuals plots are crucial for understanding model diagnostics. High Cook’s Distance values in the influence plot indicate influential observations that might unduly affect the model’s predictions.
- The residuals vs. fitted values plot should ideally show no discernible pattern. Patterns might suggest non-linearity, heteroscedasticity, or outliers.
- The histogram and Q-Q plot of residuals are used to check the normality assumption. In the ideal scenario, residuals should be normally distributed (bell-shaped histogram and points following the line in the Q-Q plot).
- Model Comparison and Selection:
- ANOVA results are used to compare models statistically. A significant difference in the residual sum of squares between models suggests one model may be better than the other.
- Cross-validation results (RMSE, R-squared) can also be used to compare models. Lower RMSE and higher R-squared values on cross-validation suggest a model that generalizes better.
12 Multicollinearity Check: - High VIF (Variance Inflation Factor) values indicate multicollinearity, which can make coefficients unstable and difficult to interpret. It’s important to address this by possibly removing or combining correlated predictors.
- Visualization and Interpretation:
- Plots comparing actual vs. predicted life expectancy help visually assess model performance. Closer alignment of predicted values with actual ones indicates better model performance.
- The scatter plots, histograms, and Q-Q plots provide a visual means to assess model assumptions and fit, enhancing the statistical interpretation.
In conclusion, while the models demonstrate strong predictive power, careful consideration of overfitting, influence of outliers, and the fulfillment of model assumptions is necessary. The choice of model should balance complexity and generalizability, and diagnostics plots play a crucial role in this decision-making process.
In summary, the RFE process successfully identified key variables influencing life expectancy, and the linear regression model built with these variables provides insights into the nature of these relationships. However, there are opportunities for further refining and validating the model to enhance its predictive power.
# Load the data
trainData <- read_csv("train_data.csv")## Rows: 1996 Columns: 24
## ── Column specification ────────────────────────────────────────────────────────
## Delimiter: ","
## chr (2): Country, Region
## dbl (22): Column1, Year, Economy_status_Developed, Life_expectancy, Adult_mo...
##
## ℹ Use `spec()` to retrieve the full column specification for this data.
## ℹ Specify the column types or set `show_col_types = FALSE` to quiet this message.# Define control for RFE using a random forest selection function
control <- rfeControl(functions=rfFuncs, method="cv", number=10)
# Run RFE with Random Forest Model
# Perform the RFE to identify the most significant features for predicting life expectancy
# Specifying the outcome and predictors
results <- rfe(x=train_data[, -which(names(train_data) == "Life_expectancy")],
y=train_data$Life_expectancy,
sizes=c(1:5),
rfeControl=control)
# Extract the best subset of variables
selected_features <- results$optVariables
print(selected_features)## [1] "Adult_mortality" "Infant_deaths"
## [3] "Alcohol_consumption" "Country"
## [5] "Under_five_deaths" "Year"
## [7] "Incidents_HIV" "Income composition of resources"
## [9] "Total expenditure" "Thinness_ten_nineteen_years"
## [11] "Thinness_five_nine_years" "Region"
## [13] "Measles" "Polio"
## [15] "BMI" "Diphtheria"
## [17] "GDP_per_capita" "percentage expenditure"
## [19] "Schooling" "Hepatitis_B"
## [21] "Population_millions" "Economy_status_Developed"# Use only these selected features for model training
train_data_selected <- trainData[, c(selected_features, "Life_expectancy")]
# Simple Linear Regression Model
simple_model <- lm(Life_expectancy ~ ., data = train_data_selected)
summary(simple_model)##
## Call:
## lm(formula = Life_expectancy ~ ., data = train_data_selected)
##
## Residuals:
## Min 1Q Median 3Q Max
## -2.1432 -0.2178 -0.0087 0.2001 3.2582
##
## Coefficients: (9 not defined because of singularities)
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) -1.444e+02 1.247e+01 -11.585 < 2e-16
## Adult_mortality -4.070e-02 8.744e-04 -46.543 < 2e-16
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## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 0.5291 on 1799 degrees of freedom
## Multiple R-squared: 0.9971, Adjusted R-squared: 0.9968
## F-statistic: 3183 on 196 and 1799 DF, p-value: < 2.2e-16# Influence PLot
influencePlot(simple_model, id.method="identify", main="Influence Plot",
sub="Circle size is proportional to Cook's Distance")## Warning in plot.window(...): "id.method" is not a graphical parameter## Warning in plot.xy(xy, type, ...): "id.method" is not a graphical parameter## Warning in axis(side = side, at = at, labels = labels, ...): "id.method" is not
## a graphical parameter
## Warning in axis(side = side, at = at, labels = labels, ...): "id.method" is not
## a graphical parameter## Warning in box(...): "id.method" is not a graphical parameter## Warning in title(...): "id.method" is not a graphical parameter## Warning in plot.xy(xy.coords(x, y), type = type, ...): "id.method" is not a
## graphical parameter
## StudRes Hat CookD
## 606 5.9761888 0.08366793 1.624007e-02
## 802 6.6750860 0.12832118 3.250870e-02
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## 854 -0.1213443 0.32910775 3.668558e-05
## 1045 5.5808644 0.10568911 1.837643e-02# Enhanced Model Plots : Model 1, Model 2, Model 3
# Model 1 Enhanced Model with Polynomial Terms
enhanced_model <- lm(Life_expectancy ~ . + I(Alcohol_consumption^2), data = train_data_selected)
model_1 <- (enhanced_model)
summary(model_1)##
## Call:
## lm(formula = Life_expectancy ~ . + I(Alcohol_consumption^2),
## data = train_data_selected)
##
## Residuals:
## Min 1Q Median 3Q Max
## -2.1344 -0.2167 -0.0041 0.1986 3.2858
##
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##
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## `Total expenditure`
## Thinness_ten_nineteen_years .
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## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 0.5289 on 1798 degrees of freedom
## Multiple R-squared: 0.9971, Adjusted R-squared: 0.9968
## F-statistic: 3169 on 197 and 1798 DF, p-value: < 2.2e-16# Model 2 Enhanced LRM with Predictors (Model 2)
model_2 <- lm(Life_expectancy ~ Year + Adult_mortality + Alcohol_consumption + BMI, data = train_data_selected)
summary(model_2)##
## Call:
## lm(formula = Life_expectancy ~ Year + Adult_mortality + Alcohol_consumption +
## BMI, data = train_data_selected)
##
## Residuals:
## Min 1Q Median 3Q Max
## -9.1307 -1.4582 0.1005 1.5635 13.5022
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) -2.472e+01 2.513e+01 -0.983 0.326
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## Alcohol_consumption 4.054e-01 1.497e-02 27.082 < 2e-16 ***
## BMI 2.984e-02 3.133e-03 9.525 < 2e-16 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 2.534 on 1991 degrees of freedom
## Multiple R-squared: 0.927, Adjusted R-squared: 0.9269
## F-statistic: 6323 on 4 and 1991 DF, p-value: < 2.2e-16# Model 3 Generalized Linear Model with Log Link
glm_model <- glm(Life_expectancy ~ ., family = gaussian(link = "log"), data = train_data_selected)
model_3 <- (glm_model)
summary(model_3)##
## Call:
## glm(formula = Life_expectancy ~ ., family = gaussian(link = "log"),
## data = train_data_selected)
##
## Coefficients: (9 not defined because of singularities)
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##
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## Year ***
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## `Total expenditure`
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## Thinness_five_nine_years
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## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## (Dispersion parameter for gaussian family taken to be 0.2905737)
##
## Null deviance: 175136.33 on 1995 degrees of freedom
## Residual deviance: 522.74 on 1799 degrees of freedom
## AIC: 3386.1
##
## Number of Fisher Scoring iterations: 4# Convert to long format for plotting
comparison_long <- comparison_df %>%
pivot_longer(cols = starts_with("Predicted"), names_to = "Model", values_to = "Predicted")
# Load the training data
trainData <- read_csv("train_data.csv")## Rows: 1996 Columns: 24
## ── Column specification ────────────────────────────────────────────────────────
## Delimiter: ","
## chr (2): Country, Region
## dbl (22): Column1, Year, Economy_status_Developed, Life_expectancy, Adult_mo...
##
## ℹ Use `spec()` to retrieve the full column specification for this data.
## ℹ Specify the column types or set `show_col_types = FALSE` to quiet this message.# Generate predicted values
predicted_values_1 <- predict(model_1, trainData)## Warning in predict.lm(model_1, trainData): prediction from rank-deficient fit;
## attr(*, "non-estim") has doubtful casespredicted_values_2 <- predict(model_2, trainData)
predicted_values_3 <- predict(model_3, trainData)
# Combine actual and predicted values into a data frame
comparison_df <- data.frame(
Actual = trainData$Life_expectancy,
Predicted_Model_1 = predicted_values_1,
Predicted_Model_2 = predicted_values_2,
Predicted_Model_3 = predicted_values_3
)
# Convert to long format for plotting
comparison_long <- comparison_df %>%
pivot_longer(cols = starts_with("Predicted"), names_to = "Model", values_to = "Predicted")
# Create the plot
p123 <- comparison_long %>%
plot_ly(x = ~Actual, y = ~Predicted, color = ~Model, type = 'scatter', mode = 'markers') %>%
layout(title = "Actual vs Predicted Life Expectancy",
xaxis = list(title = "Actual Life Expectancy"),
yaxis = list(title = "Predicted Life Expectancy"),
hovermode = "compare")
# Display the plot
p123# Model Evaluation and Comparison
# ANOVA for Model Comparison
anova_result_1 <- anova(simple_model, enhanced_model)
print(anova_result_1)## Analysis of Variance Table
##
## Model 1: Life_expectancy ~ Adult_mortality + Infant_deaths + Alcohol_consumption +
## Country + Under_five_deaths + Year + Incidents_HIV + `Income composition of resources` +
## `Total expenditure` + Thinness_ten_nineteen_years + Thinness_five_nine_years +
## Region + Measles + Polio + BMI + Diphtheria + GDP_per_capita +
## `percentage expenditure` + Schooling + Hepatitis_B + Population_millions +
## Economy_status_Developed
## Model 2: Life_expectancy ~ Adult_mortality + Infant_deaths + Alcohol_consumption +
## Country + Under_five_deaths + Year + Incidents_HIV + `Income composition of resources` +
## `Total expenditure` + Thinness_ten_nineteen_years + Thinness_five_nine_years +
## Region + Measles + Polio + BMI + Diphtheria + GDP_per_capita +
## `percentage expenditure` + Schooling + Hepatitis_B + Population_millions +
## Economy_status_Developed + I(Alcohol_consumption^2)
## Res.Df RSS Df Sum of Sq F Pr(>F)
## 1 1799 503.62
## 2 1798 502.92 1 0.70149 2.5079 0.1135anova_result_2 <- anova(simple_model, model_2)
print(anova_result_2)## Analysis of Variance Table
##
## Model 1: Life_expectancy ~ Adult_mortality + Infant_deaths + Alcohol_consumption +
## Country + Under_five_deaths + Year + Incidents_HIV + `Income composition of resources` +
## `Total expenditure` + Thinness_ten_nineteen_years + Thinness_five_nine_years +
## Region + Measles + Polio + BMI + Diphtheria + GDP_per_capita +
## `percentage expenditure` + Schooling + Hepatitis_B + Population_millions +
## Economy_status_Developed
## Model 2: Life_expectancy ~ Year + Adult_mortality + Alcohol_consumption +
## BMI
## Res.Df RSS Df Sum of Sq F Pr(>F)
## 1 1799 503.6
## 2 1991 12779.8 -192 -12276 228.4 < 2.2e-16 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1# Cross-Validation for Model Selection
control_cv <- trainControl(method = "cv", number = 10) # 10-fold CV
cv_model <- train(Life_expectancy ~ ., data = train_data_selected, method = "lm", trControl = control_cv)## Warning in predict.lm(modelFit, newdata): prediction from rank-deficient fit;
## attr(*, "non-estim") has doubtful cases## Warning in predict.lm(modelFit, newdata): prediction from rank-deficient fit;
## attr(*, "non-estim") has doubtful cases
## Warning in predict.lm(modelFit, newdata): prediction from rank-deficient fit;
## attr(*, "non-estim") has doubtful cases
## Warning in predict.lm(modelFit, newdata): prediction from rank-deficient fit;
## attr(*, "non-estim") has doubtful cases
## Warning in predict.lm(modelFit, newdata): prediction from rank-deficient fit;
## attr(*, "non-estim") has doubtful cases
## Warning in predict.lm(modelFit, newdata): prediction from rank-deficient fit;
## attr(*, "non-estim") has doubtful cases
## Warning in predict.lm(modelFit, newdata): prediction from rank-deficient fit;
## attr(*, "non-estim") has doubtful cases
## Warning in predict.lm(modelFit, newdata): prediction from rank-deficient fit;
## attr(*, "non-estim") has doubtful cases
## Warning in predict.lm(modelFit, newdata): prediction from rank-deficient fit;
## attr(*, "non-estim") has doubtful cases
## Warning in predict.lm(modelFit, newdata): prediction from rank-deficient fit;
## attr(*, "non-estim") has doubtful casesprint(cv_model)## Linear Regression
##
## 1996 samples
## 22 predictor
##
## No pre-processing
## Resampling: Cross-Validated (10 fold)
## Summary of sample sizes: 1795, 1798, 1796, 1798, 1797, 1795, ...
## Resampling results:
##
## RMSE Rsquared MAE
## 0.5605027 0.9964575 0.3710295
##
## Tuning parameter 'intercept' was held constant at a value of TRUE# Ensure only numeric data is used for correlation
numeric_data <- train_data_selected %>% dplyr::select_if(is.numeric)
# Define control for cross-validation
control_cv <- trainControl(method = "cv", number = 10)
# Train the model using cross-validation
cv_model <- train(Life_expectancy ~ ., data = numeric_data, method = "lm", trControl = control_cv)
print(cv_model)## Linear Regression
##
## 1996 samples
## 20 predictor
##
## No pre-processing
## Resampling: Cross-Validated (10 fold)
## Summary of sample sizes: 1798, 1796, 1797, 1796, 1796, 1796, ...
## Resampling results:
##
## RMSE Rsquared MAE
## 1.42829 0.9767583 1.141538
##
## Tuning parameter 'intercept' was held constant at a value of TRUE# Checking Multicollinearity with VIF
# Select only numeric predictors including Life_expectancy
numeric_predictors <- train_data_selected %>%
dplyr::select_if(is.numeric)
# Build the model
simple_model <- lm(Life_expectancy ~ ., data = numeric_predictors)
# Calculate VIF
vif_values <- vif(simple_model)
# Save VIF values to a variable for later use
saved_vif_values <- vif_values
# Analyze VIF values
print(saved_vif_values)## Adult_mortality Infant_deaths
## 7.247209 6.823786
## Alcohol_consumption Under_five_deaths
## 2.661002 2.315195
## Year Incidents_HIV
## 1.205640 2.767337
## `Income composition of resources` `Total expenditure`
## 1.819810 1.278385
## Thinness_ten_nineteen_years Thinness_five_nine_years
## 8.802770 8.755260
## Measles Polio
## 1.414925 2.434013
## BMI Diphtheria
## 1.821269 2.709575
## GDP_per_capita `percentage expenditure`
## 5.918139 5.806101
## Schooling Hepatitis_B
## 3.967074 1.770356
## Population_millions Economy_status_Developed
## 1.504514 2.705831# Enhanced Model Plots : Model 1, Model 2, Model 3
# Model 1 Enhanced Model with Polynomial Terms
enhanced_model <- lm(Life_expectancy ~ . + I(Alcohol_consumption^2), data = train_data_selected)
model_1 <- (enhanced_model)
summary(model_1)##
## Call:
## lm(formula = Life_expectancy ~ . + I(Alcohol_consumption^2),
## data = train_data_selected)
##
## Residuals:
## Min 1Q Median 3Q Max
## -2.1344 -0.2167 -0.0041 0.1986 3.2858
##
## Coefficients: (9 not defined because of singularities)
## Estimate Std. Error t value Pr(>|t|)
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##
## (Intercept) ***
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## Under_five_deaths
## Year ***
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## `Total expenditure`
## Thinness_ten_nineteen_years .
## Thinness_five_nine_years
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## Measles
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## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 0.5289 on 1798 degrees of freedom
## Multiple R-squared: 0.9971, Adjusted R-squared: 0.9968
## F-statistic: 3169 on 197 and 1798 DF, p-value: < 2.2e-16# Model 2 Enhanced LRM with Predictors (Model 2)
model_2 <- lm(Life_expectancy ~ Year + Adult_mortality + Alcohol_consumption + BMI, data = train_data_selected)
summary(model_2)##
## Call:
## lm(formula = Life_expectancy ~ Year + Adult_mortality + Alcohol_consumption +
## BMI, data = train_data_selected)
##
## Residuals:
## Min 1Q Median 3Q Max
## -9.1307 -1.4582 0.1005 1.5635 13.5022
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) -2.472e+01 2.513e+01 -0.983 0.326
## Year 5.187e-02 1.251e-02 4.146 3.53e-05 ***
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## Alcohol_consumption 4.054e-01 1.497e-02 27.082 < 2e-16 ***
## BMI 2.984e-02 3.133e-03 9.525 < 2e-16 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 2.534 on 1991 degrees of freedom
## Multiple R-squared: 0.927, Adjusted R-squared: 0.9269
## F-statistic: 6323 on 4 and 1991 DF, p-value: < 2.2e-16# Model 3 Generalized Linear Model with Log Link
glm_model <- glm(Life_expectancy ~ ., family = gaussian(link = "log"), data = train_data_selected)
model_3 <- (glm_model)
summary(model_3)##
## Call:
## glm(formula = Life_expectancy ~ ., family = gaussian(link = "log"),
## data = train_data_selected)
##
## Coefficients: (9 not defined because of singularities)
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## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## (Dispersion parameter for gaussian family taken to be 0.2905737)
##
## Null deviance: 175136.33 on 1995 degrees of freedom
## Residual deviance: 522.74 on 1799 degrees of freedom
## AIC: 3386.1
##
## Number of Fisher Scoring iterations: 4# Plot Models 1,2,3
# Convert to long format for plotting
comparison_long <- comparison_df %>%
pivot_longer(cols = starts_with("Predicted"), names_to = "Model", values_to = "Predicted")
# Load the training data
trainData <- read_csv("train_data.csv")## Rows: 1996 Columns: 24
## ── Column specification ────────────────────────────────────────────────────────
## Delimiter: ","
## chr (2): Country, Region
## dbl (22): Column1, Year, Economy_status_Developed, Life_expectancy, Adult_mo...
##
## ℹ Use `spec()` to retrieve the full column specification for this data.
## ℹ Specify the column types or set `show_col_types = FALSE` to quiet this message.# Generate predicted values
predicted_values_1 <- predict(model_1, trainData)## Warning in predict.lm(model_1, trainData): prediction from rank-deficient fit;
## attr(*, "non-estim") has doubtful casespredicted_values_2 <- predict(model_2, trainData)
predicted_values_3 <- predict(model_3, trainData)
# Combine actual and predicted values into a data frame
comparison_df <- data.frame(
Actual = trainData$Life_expectancy,
Predicted_Model_1 = predicted_values_1,
Predicted_Model_2 = predicted_values_2,
Predicted_Model_3 = predicted_values_3
)
# Convert to long format for plotting
comparison_long <- comparison_df %>%
pivot_longer(cols = starts_with("Predicted"), names_to = "Model", values_to = "Predicted")
# Create the plot
p123 <- comparison_long %>%
plot_ly(x = ~Actual, y = ~Predicted, color = ~Model, type = 'scatter', mode = 'markers') %>%
layout(title = "Actual vs Predicted Life Expectancy",
xaxis = list(title = "Actual Life Expectancy"),
yaxis = list(title = "Predicted Life Expectancy"),
hovermode = "compare")
# Display the plot
p123saveRDS(p123, file = "plot123.rds")
save(p123, file = "plot123.RData")
# Residual Analysis and Diagnostics Plots
# Residuals vs Fitted Values
predicted_values <- predict(simple_model, train_data_selected)
residuals <- residuals(simple_model)
# Interactive Residuals vs Fitted Values Plot
p_residuals_fitted <- plot_ly(x = predicted_values, y = residuals, type = 'scatter', mode = 'markers') %>%
layout(title = 'Residuals vs Fitted Values',
xaxis = list(title = 'Fitted Values'),
yaxis = list(title = 'Residuals')) %>%
add_lines(x = predicted_values, y = rep(0, length(residuals)), line = list(color = 'red'))
# Display the interactive plot
p_residuals_fitted# Normality of Residuals: Histogram
p_hist_residuals <- plot_ly(x = residuals, type = 'histogram', nbinsx = 30) %>%
layout(title = 'Histogram of Residuals',
xaxis = list(title = 'Residuals'),
yaxis = list(title = 'Count'))
# Display the histogram
p_hist_residuals# Normality of Residuals: Q-Q Plot
qq <- qqnorm(residuals, plot.it = FALSE)
p_qq_residuals <- plot_ly(x = qq$x, y = qq$y, type = 'scatter', mode = 'markers') %>%
layout(title = 'Q-Q Plot of Residuals',
xaxis = list(title = 'Theoretical Quantiles'),
yaxis = list(title = 'Sample Quantiles')) %>%
add_lines(x = qq$x, y = qq$x, line = list(color = 'red'))
# Display the Q-Q plot
p_qq_residualsPart F: AIC
The Akaike Information Criterion (AIC) is a tool used to assess the quality of statistical models for a given dataset. It balances the complexity of the model against how well the model fits the data. A lower AIC value generally indicates a better model, as it suggests a model that explains a high degree of variability in the data without unnecessary complexity.
Let’s analyze the AIC values for each of the models:
- Simple Linear Regression Model
(
simple_model):- AIC: 7096.756
- This is a baseline model with a relatively high AIC, suggesting it might be more complex or fit the data less effectively compared to others.
- Enhanced Linear Regression Model
(
enhanced_model):- AIC: 3310.982
- This model has a significantly lower AIC compared to the simple model, indicating a better balance of model fit and complexity. It might be the preferred model based on AIC.
- Reduced Linear Regression Model
(
model_2):- AIC: 9382.416
- The high AIC value suggests that while this model is simpler (fewer predictors), it may not fit the data as well as other models.
- Generalized Linear Model with Log Link
(
glm_model):- AIC: 3386.137
- This AIC is slightly higher than that of the enhanced model but still lower than the simple model, indicating reasonable model performance.
- Robust Linear Regression Model
(
robust_model):- AIC: 10195.21
- This model has the highest AIC value among the ones compared, suggesting it might be the least preferable in terms of the balance between data fit and complexity.
In summary, based on AIC values, the enhanced_model
appears to be the most effective in balancing data fit and model
simplicity. It’s important to remember that while AIC is a useful
criterion for model comparison, it should be considered alongside other
diagnostic measures and domain-specific considerations to choose the
most appropriate model for your analysis.
#reduced model
reduced_model <- lm(Life_expectancy ~ Adult_mortality + Year + BMI, data = train_data_selected)
summary(reduced_model)##
## Call:
## lm(formula = Life_expectancy ~ Adult_mortality + Year + BMI,
## data = train_data_selected)
##
## Residuals:
## Min 1Q Median 3Q Max
## -10.4124 -2.0922 0.2876 1.9695 13.7407
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 23.9753240 29.3179603 0.818 0.414
## Adult_mortality -0.0729695 0.0006508 -112.123 <2e-16 ***
## Year 0.0285014 0.0145968 1.953 0.051 .
## BMI 0.0462211 0.0035956 12.855 <2e-16 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 2.963 on 1992 degrees of freedom
## Multiple R-squared: 0.9001, Adjusted R-squared: 0.9
## F-statistic: 5986 on 3 and 1992 DF, p-value: < 2.2e-16library(car)
vif_results <- vif(reduced_model)
print(vif_results)## Adult_mortality Year BMI
## 1.275126 1.032341 1.249543library(glmnet)
# Standardize variables for regularization
x <- model.matrix(Life_expectancy ~ ., data = train_data_selected)[,-1]
y <- train_data_selected$Life_expectancy
# Fit Ridge Regression Model
ridge_model <- cv.glmnet(x, y, alpha = 0)
plot(ridge_model)
# Residuals vs Fitted
plot(reduced_model, which = 1)
# Scale-Location (Spread-Location)
plot(reduced_model, which = 3)
# Residuals vs Leverage
plot(reduced_model, which = 5)
library(boot)##
## Attaching package: 'boot'## The following object is masked from 'package:car':
##
## logit## The following object is masked from 'package:lattice':
##
## melanomaset.seed(123) # for reproducibility
cv_results <- cv.glm(data = train_data_selected, glmfit = reduced_model, K = 10) # 10-fold CV
print(cv_results$delta) # delta contains cross-validated results## [1] NaN NaN{r{} mancova_results <- manova(cbind(Life_expectancy, Adult_mortality, Alcohol_consumption) ~ Economy_status_Developed + Year, data = train_data) summary(mancova_results, test = "Pillai")
Summarizing Finding:
Based on the validation set metrics, here’s a concise summary and interpretation of the performance of the three models:
- Model 1:
- RMSE: 0.6048335
- MAE: 0.393135
- Interpretation: Exhibits the best performance with the lowest RMSE and MAE, indicating high accuracy and fewer average errors in predicting life expectancy.
- Model 2:
- RMSE: 2.488072
- MAE: 1.873971
- Interpretation: Shows significantly poorer performance with much higher RMSE and MAE, suggesting it is less effective in accurately predicting life expectancy.
- Model 3:
- RMSE: 0.6331086
- MAE: 0.4011408
- Interpretation: Performs slightly less accurately than Model 1 but still demonstrates reasonable effectiveness.
Model Selection Considerations: - Model 1 is the most accurate. However, balance this accuracy with considerations of model complexity and interpretability. - If Model 1 is much more complex or less interpretable than Model 3, and the performance difference is marginal, Model 3 could be a viable alternative, depending on specific project needs. - Model 2 may require revision or a different approach due to its lower accuracy.
Final Steps for Model Selection: - Select the best model based on a combination of accuracy, complexity, and interpretability. - Conduct a final review of model diagnostics, ensuring all assumptions (like normality of residuals, homoscedasticity, etc.) are met. - Address any specific issues such as multicollinearity, especially in Model 1, as indicated by its warning message about a “rank-deficient fit.” - Once a model is chosen and confirmed to meet all criteria and assumptions, it can be finalized for deployment or further application in the project. Decision Making: - Given these metrics, Model 1 appears to be the best model among the three in terms of prediction accuracy on the validation set. - However, we need to balance model accuracy with complexity and interpretability. If Model 1 is considerably more complex or harder to interpret than Model 3, and the difference in RMSE/MAE is marginal, Model 3 could still be a viable option
Final Steps: - Select Best Model - Based on everthing, Model_1 is the best model for this analysis.
The influence plot demonstrates standardized residuals versus leverage for each data point, with circle size indicating Cook’s distance. This tool identifies influential points in model fitting.
Key Aspects of the Influence Plot:
Leverage: Data points far right on the x-axis have high leverage, potentially impacting the regression line. These points typically possess extreme predictor values.
Standardized Residuals: Points distant from 0 on the y-axis indicate large residuals, signifying prediction inaccuracy. These are outliers.
Cook’s Distance: Larger circles symbolize greater influence. Points with Cook’s distance over 1 might excessively affect model predictions.
Notable Observations:
High Leverage Points: Positioned far right, potentially disproportionately influencing regression coefficients.
Outliers: Exhibiting high residuals, indicating poor model fit.
Influential Points: Identified by large circles, such as points 602, 801, 1045.
Strategies for Analysis:
Investigate outliers and high leverage points for errors or rare but valid conditions.
Assess whether to remove influential points that may skew the model.
Refit the model if necessary, ensuring it still meets key assumptions.
Cross-Validation Plot for Ridge Regression:
This plot, generated by cv.glmnet in R, uses k-fold
cross-validation for selecting lambda in ridge regression.
Plot Components:
X-axis (Log(λ)): Displays the log of lambda values tested. Lambda controls shrinkage of coefficients.
Y-axis (Mean Squared Error): Shows the MSE for each lambda, with lower MSE indicating better performance.
Error Bars: Represent the standard error range for each lambda’s cross-validation error.
Red Dots: Average cross-validation error for each lambda.
Solid Line: Traces the error trajectory as lambda changes.
Vertical Dashed Lines: Indicate optimal lambda (
lambda.min) and a more conservative choice (lambda.1se).
Interpretation:
Choose
lambda.minfor lowest prediction error.lambda.1seoffers a simpler model at a slightly higher error.
In the given plot, lambda.min is at the curve’s lowest
point, extractable from the cv.glmnet object.
Model Evaluation for Predicting Life Expectancy:
Model 1: Best predictive accuracy with lowest RMSE and MAE.
Model 2: Least accurate, suggested by higher RMSE and MAE.
Model 3: Slightly less accurate than Model 1 but still effective.
In summary, Model 1 appears most effective for this task, with a need to address its potential multicollinearity issues. # Summarizing Findings
Based on the validation set metrics for all three models, we can now compare their performance. Here are the results summarized:
Model 1
- RMSE: 0.6048335
- MAE: 0.393135
Model 2
- RMSE: 2.488072
- MAE: 1.873971
Model 3
- RMSE: 0.6331086
- MAE: 0.4011408
Model Comparison and Selection
- Model 1 has the lowest RMSE and MAE, suggesting it predicts life expectancy more accurately and with fewer errors on average compared to the other two models.
- Model 2 shows significantly higher RMSE and MAE, indicating it might not be as effective in predicting life expectancy as the other models.
- Model 3 has slightly higher RMSE and MAE than Model 1, but these are still reasonably low.
Decision Making
- Given these metrics, Model 1 appears to be the best model among the three in terms of prediction accuracy on the validation set.
- However, remember to balance model accuracy with complexity and interpretability. If Model 1 is considerably more complex or harder to interpret than Model 3, and the difference in RMSE/MAE is marginal, Model 3 could still be a viable option depending on your project’s needs.
- Ensure that the chosen model aligns well with the specific objectives and constraints of your analysis.
Final Steps
- After selecting model_1 we performed a final review of model diagnostics to ensure it meets all assumptions (like normality of residuals, homoscedasticity, etc.).
- There is no “perfect mode;” but from all the testing, it seems model 1 is the best.