DATA 621 Homework 1
Warner Alexis, Saloua Daouki, Souleymane Doumbia, Fomba Kassoh, Lewris Mota Sanchez
2024-14-03
{r setup, include=FALSE} knitr::opts_chunk$set(echo = TRUE)
Load Required Libraries
options(repos = c(CRAN = "https://cloud.r-project.org"))
# Function to install a package if not already installed
install_if_needed <- function(pkg) {
if (!requireNamespace(pkg, quietly = TRUE)) {
install.packages(pkg)
}
}
# List of packages to check and install if necessary
required_packages <- c("dplyr", "ggplot2", "lubridate", "tidyr",
"caret", "stats", "tidyverse", "corrplot",
"plotly", "DT", "DataExplorer", "mice", "kableExtra",
"reshape", "reshape2", "factoextra", "psych", "GGally", "tinytex"
)
# Loop through the list and install packages only if needed
for (pkg in required_packages) {
install_if_needed(pkg)
}## Registered S3 method overwritten by 'GGally':
## method from
## +.gg ggplot2# Function to suppress package startup messages
suppressPackageStartupMessages({
library(dplyr)
library(ggplot2)
library(tidyr)
library(caret) # for data splitting and pre-processing
library(stats)
library(tidyverse)
library(corrplot)
library(plotly)
library(DT)
library(DataExplorer)
library(mice)
library(kableExtra)
library(reshape)
library(reshape2)
library(factoextra)
library(psych)
library(GGally)
library(tinytex)
})1.Modeling Assumptions
- Linearity: The mean values of the outcome variable TARGET_WINS for each increment of the predictor(s) lie on a strainght line. There is a linear relationship between predictors and the target.
- Linearity: There exist a relationship between the independent variables and the dependent variable TARGET_WINS.
- Independence: The residuals are independent.
- Homoscedasticity: At each level of the predictor variables, the variance of the residual terms are constant.
- Independence: All the values of the dependent variable are independent. Normally distributed errors: The residuals are random, normally distributed with a mean 0.
2. Data Exploration
Import the dataset
money_ball_train = read_csv("https://raw.githubusercontent.com/hawa1983/DATA-621/main/moneyball-training-data.csv", show_col_types = FALSE)
money_ball_evaluate = read_csv("https://raw.githubusercontent.com/hawa1983/DATA-621/main/moneyball-evaluation-data.csv", show_col_types = FALSE)
str(money_ball_train)## spc_tbl_ [2,276 × 17] (S3: spec_tbl_df/tbl_df/tbl/data.frame)
## $ INDEX : num [1:2276] 1 2 3 4 5 6 7 8 11 12 ...
## $ TARGET_WINS : num [1:2276] 39 70 86 70 82 75 80 85 86 76 ...
## $ TEAM_BATTING_H : num [1:2276] 1445 1339 1377 1387 1297 ...
## $ TEAM_BATTING_2B : num [1:2276] 194 219 232 209 186 200 179 171 197 213 ...
## $ TEAM_BATTING_3B : num [1:2276] 39 22 35 38 27 36 54 37 40 18 ...
## $ TEAM_BATTING_HR : num [1:2276] 13 190 137 96 102 92 122 115 114 96 ...
## $ TEAM_BATTING_BB : num [1:2276] 143 685 602 451 472 443 525 456 447 441 ...
## $ TEAM_BATTING_SO : num [1:2276] 842 1075 917 922 920 ...
## $ TEAM_BASERUN_SB : num [1:2276] NA 37 46 43 49 107 80 40 69 72 ...
## $ TEAM_BASERUN_CS : num [1:2276] NA 28 27 30 39 59 54 36 27 34 ...
## $ TEAM_BATTING_HBP: num [1:2276] NA NA NA NA NA NA NA NA NA NA ...
## $ TEAM_PITCHING_H : num [1:2276] 9364 1347 1377 1396 1297 ...
## $ TEAM_PITCHING_HR: num [1:2276] 84 191 137 97 102 92 122 116 114 96 ...
## $ TEAM_PITCHING_BB: num [1:2276] 927 689 602 454 472 443 525 459 447 441 ...
## $ TEAM_PITCHING_SO: num [1:2276] 5456 1082 917 928 920 ...
## $ TEAM_FIELDING_E : num [1:2276] 1011 193 175 164 138 ...
## $ TEAM_FIELDING_DP: num [1:2276] NA 155 153 156 168 149 186 136 169 159 ...
## - attr(*, "spec")=
## .. cols(
## .. INDEX = col_double(),
## .. TARGET_WINS = col_double(),
## .. TEAM_BATTING_H = col_double(),
## .. TEAM_BATTING_2B = col_double(),
## .. TEAM_BATTING_3B = col_double(),
## .. TEAM_BATTING_HR = col_double(),
## .. TEAM_BATTING_BB = col_double(),
## .. TEAM_BATTING_SO = col_double(),
## .. TEAM_BASERUN_SB = col_double(),
## .. TEAM_BASERUN_CS = col_double(),
## .. TEAM_BATTING_HBP = col_double(),
## .. TEAM_PITCHING_H = col_double(),
## .. TEAM_PITCHING_HR = col_double(),
## .. TEAM_PITCHING_BB = col_double(),
## .. TEAM_PITCHING_SO = col_double(),
## .. TEAM_FIELDING_E = col_double(),
## .. TEAM_FIELDING_DP = col_double()
## .. )
## - attr(*, "problems")=<externalptr>str(money_ball_evaluate)## spc_tbl_ [259 × 16] (S3: spec_tbl_df/tbl_df/tbl/data.frame)
## $ INDEX : num [1:259] 9 10 14 47 60 63 74 83 98 120 ...
## $ TEAM_BATTING_H : num [1:259] 1209 1221 1395 1539 1445 ...
## $ TEAM_BATTING_2B : num [1:259] 170 151 183 309 203 236 219 158 177 212 ...
## $ TEAM_BATTING_3B : num [1:259] 33 29 29 29 68 53 55 42 78 42 ...
## $ TEAM_BATTING_HR : num [1:259] 83 88 93 159 5 10 37 33 23 58 ...
## $ TEAM_BATTING_BB : num [1:259] 447 516 509 486 95 215 568 356 466 452 ...
## $ TEAM_BATTING_SO : num [1:259] 1080 929 816 914 416 377 527 609 689 584 ...
## $ TEAM_BASERUN_SB : num [1:259] 62 54 59 148 NA NA 365 185 150 52 ...
## $ TEAM_BASERUN_CS : num [1:259] 50 39 47 57 NA NA NA NA NA NA ...
## $ TEAM_BATTING_HBP: num [1:259] NA NA NA 42 NA NA NA NA NA NA ...
## $ TEAM_PITCHING_H : num [1:259] 1209 1221 1395 1539 3902 ...
## $ TEAM_PITCHING_HR: num [1:259] 83 88 93 159 14 20 40 39 25 62 ...
## $ TEAM_PITCHING_BB: num [1:259] 447 516 509 486 257 420 613 418 497 482 ...
## $ TEAM_PITCHING_SO: num [1:259] 1080 929 816 914 1123 ...
## $ TEAM_FIELDING_E : num [1:259] 140 135 156 124 616 572 490 328 226 184 ...
## $ TEAM_FIELDING_DP: num [1:259] 156 164 153 154 130 105 NA 104 132 145 ...
## - attr(*, "spec")=
## .. cols(
## .. INDEX = col_double(),
## .. TEAM_BATTING_H = col_double(),
## .. TEAM_BATTING_2B = col_double(),
## .. TEAM_BATTING_3B = col_double(),
## .. TEAM_BATTING_HR = col_double(),
## .. TEAM_BATTING_BB = col_double(),
## .. TEAM_BATTING_SO = col_double(),
## .. TEAM_BASERUN_SB = col_double(),
## .. TEAM_BASERUN_CS = col_double(),
## .. TEAM_BATTING_HBP = col_double(),
## .. TEAM_PITCHING_H = col_double(),
## .. TEAM_PITCHING_HR = col_double(),
## .. TEAM_PITCHING_BB = col_double(),
## .. TEAM_PITCHING_SO = col_double(),
## .. TEAM_FIELDING_E = col_double(),
## .. TEAM_FIELDING_DP = col_double()
## .. )
## - attr(*, "problems")=<externalptr>i. Summarise the data
Get an overview of the data through summary statistics like mean, median, and standard deviation.
# Load necessary libraries
library(psych)
# Assuming your data is stored in money_ball_train
# Summary statistics
summary_stats <- summary(money_ball_train)
# Descriptive statistics
descriptive_stats <- describe(money_ball_train)
descriptive_stats## vars n mean sd median trimmed mad min max
## INDEX 1 2276 1268.46 736.35 1270.5 1268.57 952.57 1 2535
## TARGET_WINS 2 2276 80.79 15.75 82.0 81.31 14.83 0 146
## TEAM_BATTING_H 3 2276 1469.27 144.59 1454.0 1459.04 114.16 891 2554
## TEAM_BATTING_2B 4 2276 241.25 46.80 238.0 240.40 47.44 69 458
## TEAM_BATTING_3B 5 2276 55.25 27.94 47.0 52.18 23.72 0 223
## TEAM_BATTING_HR 6 2276 99.61 60.55 102.0 97.39 78.58 0 264
## TEAM_BATTING_BB 7 2276 501.56 122.67 512.0 512.18 94.89 0 878
## TEAM_BATTING_SO 8 2174 735.61 248.53 750.0 742.31 284.66 0 1399
## TEAM_BASERUN_SB 9 2145 124.76 87.79 101.0 110.81 60.79 0 697
## TEAM_BASERUN_CS 10 1504 52.80 22.96 49.0 50.36 17.79 0 201
## TEAM_BATTING_HBP 11 191 59.36 12.97 58.0 58.86 11.86 29 95
## TEAM_PITCHING_H 12 2276 1779.21 1406.84 1518.0 1555.90 174.95 1137 30132
## TEAM_PITCHING_HR 13 2276 105.70 61.30 107.0 103.16 74.13 0 343
## TEAM_PITCHING_BB 14 2276 553.01 166.36 536.5 542.62 98.59 0 3645
## TEAM_PITCHING_SO 15 2174 817.73 553.09 813.5 796.93 257.23 0 19278
## TEAM_FIELDING_E 16 2276 246.48 227.77 159.0 193.44 62.27 65 1898
## TEAM_FIELDING_DP 17 1990 146.39 26.23 149.0 147.58 23.72 52 228
## range skew kurtosis se
## INDEX 2534 0.00 -1.22 15.43
## TARGET_WINS 146 -0.40 1.03 0.33
## TEAM_BATTING_H 1663 1.57 7.28 3.03
## TEAM_BATTING_2B 389 0.22 0.01 0.98
## TEAM_BATTING_3B 223 1.11 1.50 0.59
## TEAM_BATTING_HR 264 0.19 -0.96 1.27
## TEAM_BATTING_BB 878 -1.03 2.18 2.57
## TEAM_BATTING_SO 1399 -0.30 -0.32 5.33
## TEAM_BASERUN_SB 697 1.97 5.49 1.90
## TEAM_BASERUN_CS 201 1.98 7.62 0.59
## TEAM_BATTING_HBP 66 0.32 -0.11 0.94
## TEAM_PITCHING_H 28995 10.33 141.84 29.49
## TEAM_PITCHING_HR 343 0.29 -0.60 1.28
## TEAM_PITCHING_BB 3645 6.74 96.97 3.49
## TEAM_PITCHING_SO 19278 22.17 671.19 11.86
## TEAM_FIELDING_E 1833 2.99 10.97 4.77
## TEAM_FIELDING_DP 176 -0.39 0.18 0.59ii. Visualize the data for feature distributions, relationships and outlier Detection
- Distribution Analysis: Plot histograms to understand the distribution of each variable.
- Box Plots: Create box plots to detect outliers and understand the distribution of each variable.
- Bivariate Analysis: Examine relationships between the target variable and independent variables.
par(mfrow = c(3, 3)) # Set up the plotting area for multiple plots
# List of column names to analyze
columns_to_analyze <- c("TEAM_BATTING_H", "TEAM_BATTING_2B", "TEAM_BATTING_3B",
"TEAM_BATTING_HR", "TEAM_BATTING_BB", "TEAM_BATTING_SO",
"TEAM_BASERUN_SB", "TEAM_BASERUN_CS", "TEAM_BATTING_HBP",
"TEAM_PITCHING_H", "TEAM_PITCHING_HR", "TEAM_PITCHING_BB",
"TEAM_PITCHING_SO", "TEAM_FIELDING_E", "TEAM_FIELDING_DP")
# Loop through each specified column name
for (column_name in columns_to_analyze) {
column_data <- money_ball_train[[column_name]] # Extract column data
# Ensure the column is numeric
if (is.numeric(column_data)) {
# Remove NAs specific to the current column and TARGET_WINS
valid_indices <- complete.cases(column_data, money_ball_train$TARGET_WINS)
clean_data <- money_ball_train[valid_indices, ]
# Plot histogram
hist(clean_data[[column_name]], main = column_name,
xlab = column_name, breaks = 51)
# Plot boxplot
boxplot(clean_data[[column_name]], main = column_name,
horizontal = TRUE)
# Plot scatterplot and fit a regression line
plot(clean_data[[column_name]], clean_data$TARGET_WINS,
main = column_name,
xlab = column_name,
ylab = "TARGET_WINS")
# Add regression line
abline(lm(TARGET_WINS ~ get(column_name), data = clean_data), col = 'blue')
} else {
message("Skipping non-numeric column: ", column_name)
}
}
Based on the summary statistics and visualizations:
- TARGET_WINS: The mean and median are close, indicating symmetry. The spread of the data (from minimum to maximum) does not show extreme values, suggesting no significant outliers.
- TEAM_BATTING_H: The mean is slightly higher than the median, suggesting a slight positive skew. The presence of outliers is likely due to extreme values in the data.
- TEAM_BATTING_2B: The mean and median are close, suggesting symmetry. The range of data is within typical bounds, indicating no significant outliers.
- TEAM_BATTING_3B: A larger difference between the mean and median indicates positive skewness. The presence of extreme values suggests outliers.
- TEAM_BATTING_HR: A slight difference between mean and median indicates slight positive skewness. The data likely contains outliers due to extreme values.
- TEAM_BATTING_BB: Mean and median are close, indicating symmetry. The range suggests no significant outliers.
- TEAM_BATTING_SO: Slight positive skewness is indicated by a higher mean than median. The data shows the presence of outliers.
- TEAM_BASERUN_SB and TEAM_BASERUN_CS: Both show positive skewness with a higher mean than median, and the presence of outliers is suggested by extreme values in the data.
- TEAM_BATTING_HBP: Slight positive skewness is indicated by the mean being slightly higher than the median, but no extreme outliers are detected.
- TEAM_PITCHING_H: Mean and median are close, suggesting symmetry. The range of data does not suggest outliers.
- TEAM_PITCHING_HR, TEAM_PITCHING_BB, TEAM_PITCHING_SO: These show slight positive skewness (mean > median), with varying indications of outliers based on the range of data.
- TEAM_FIELDING_E and TEAM_FIELDING_DP: Both show symmetry (mean ≈ median) and no significant outliers.
iii. Correlation Analysis
Create a correlation matrix to identify relationships between variables.
# Correlation matrix
cor_matrix <- cor(money_ball_train, use = "complete.obs")
# Visualize the correlation matrix
library(corrplot)
corrplot(cor_matrix, method = "circle", type = "upper", tl.cex = 0.8, tl.col = "black")
Based on the correlation matrix provided in the plot:
- Strong Positive Correlations:
- TEAM_BATTING_H and TEAM_BATTING_2B: There is a strong positive correlation between these two variables, indicating that as the number of hits increases, the number of doubles also tends to increase.
- TEAM_PITCHING_H and TEAM_PITCHING_BB: These two pitching-related variables are also strongly positively correlated, suggesting that teams that give up more hits also tend to issue more walks.
- TEAM_PITCHING_H and TEAM_PITCHING_HR: There’s a notable positive correlation between the number of hits and home runs allowed by the team’s pitching, indicating a relationship between general hitting success against the team and home run success.
- Moderate Positive Correlations:
- TEAM_BATTING_HR and TEAM_BATTING_2B: The number of home runs and doubles have a moderate positive correlation, which may indicate that teams with more powerful hitting overall (resulting in more home runs) also hit more doubles.
- TEAM_BASERUN_SB and TEAM_BASERUN_CS: Stolen bases and caught stealing are moderately correlated, indicating that teams that attempt more steals have both higher successful steals and higher times caught stealing.
- Weak or No Correlations:
- TARGET_WINS with most other variables shows generally weak correlations. This suggests that individual batting, baserunning, or pitching statistics alone might not be strong predictors of team wins, which is often the case because wins are influenced by a combination of factors.
- TEAM_FIELDING_E and other variables: Fielding errors do not show a strong correlation with other variables, indicating that errors may not be directly related to the batting or pitching performances.
- Negative Correlations:
- There are some weak negative correlations, but none stand out as particularly strong or meaningful in this plot. Negative correlations are generally less pronounced, and none seem to be significantly impactful in the overall dataset.
iv. Check for missing values
Some of the variables have missing values.
# Total number of rows in the dataset
total_rows <- nrow(money_ball_train)
# Check for missing values and calculate the percentage of missing values
missing_values <- colSums(is.na(money_ball_train))
missing_percent <- (missing_values / total_rows) * 100
# Create a dataframe that includes both the count and percentage of missing values
missing_values_df <- data.frame(
Column = names(missing_values),
Missing_Values = missing_values,
Percent_Missing = missing_percent
)
# Filter only columns with missing values and sort the dataframe by Missing_Values in descending order
missing_values_df <- missing_values_df %>%
filter(Missing_Values > 0) %>% # Only include columns with missing values
arrange(desc(Missing_Values)) # Sort by missing values
# View the sorted table
print(missing_values_df)## Column Missing_Values Percent_Missing
## TEAM_BATTING_HBP TEAM_BATTING_HBP 2085 91.608084
## TEAM_BASERUN_CS TEAM_BASERUN_CS 772 33.919156
## TEAM_FIELDING_DP TEAM_FIELDING_DP 286 12.565905
## TEAM_BASERUN_SB TEAM_BASERUN_SB 131 5.755712
## TEAM_BATTING_SO TEAM_BATTING_SO 102 4.481547
## TEAM_PITCHING_SO TEAM_PITCHING_SO 102 4.4815473. Data Preparation
i. Impute Missing Values
a. Imputation Methods
The following methods will be used for missing values imputation
# Calculate the percentage of missing values for each column
missing_values_df <- data.frame(
Column = names(money_ball_train),
Percent_Missing = colMeans(is.na(money_ball_train)) * 100
)
# Filter only columns with missing values
missing_values_with_na <- missing_values_df[missing_values_df$Percent_Missing > 0, ]
# Apply recommendations based on missing value percentages
missing_values_with_na$Imputation_Recommendation <- ifelse(
missing_values_with_na$Percent_Missing <= 5, "Mean/Median Imputation",
ifelse(
missing_values_with_na$Percent_Missing > 5 & missing_values_with_na$Percent_Missing <= 20, "KNN or Multiple Imputation",
ifelse(
missing_values_with_na$Percent_Missing > 20 & missing_values_with_na$Percent_Missing <= 50, "Multiple Imputation or Predictive Modeling",
"Consider Dropping or Advanced Techniques"
)
)
)
# Display the recommendations
print(missing_values_with_na)## Column Percent_Missing
## TEAM_BATTING_SO TEAM_BATTING_SO 4.481547
## TEAM_BASERUN_SB TEAM_BASERUN_SB 5.755712
## TEAM_BASERUN_CS TEAM_BASERUN_CS 33.919156
## TEAM_BATTING_HBP TEAM_BATTING_HBP 91.608084
## TEAM_PITCHING_SO TEAM_PITCHING_SO 4.481547
## TEAM_FIELDING_DP TEAM_FIELDING_DP 12.565905
## Imputation_Recommendation
## TEAM_BATTING_SO Mean/Median Imputation
## TEAM_BASERUN_SB KNN or Multiple Imputation
## TEAM_BASERUN_CS Multiple Imputation or Predictive Modeling
## TEAM_BATTING_HBP Consider Dropping or Advanced Techniques
## TEAM_PITCHING_SO Mean/Median Imputation
## TEAM_FIELDING_DP KNN or Multiple Imputationb. Perform the imputations for training data
money_ball_train_imputed <- money_ball_train
# Impute or drop based on the recommendations
for (i in 1:nrow(missing_values_with_na)) {
column_name <- missing_values_with_na$Column[i]
recommendation <- missing_values_with_na$Imputation_Recommendation[i]
if (recommendation == "Mean/Median Imputation") {
# Mean/Median Imputation
money_ball_train_imputed[[column_name]][is.na(money_ball_train_imputed[[column_name]])] <-
median(money_ball_train_imputed[[column_name]], na.rm = TRUE) # Or use mean if preferred
} else if (recommendation == "KNN or Multiple Imputation") {
# For simplicity, using median imputation here, but you can replace with more advanced methods like KNN
money_ball_train_imputed[[column_name]][is.na(money_ball_train_imputed[[column_name]])] <-
median(money_ball_train_imputed[[column_name]], na.rm = TRUE)
} else if (recommendation == "Multiple Imputation or Predictive Modeling") {
# For simplicity, using median imputation here, but you can replace with more advanced methods
money_ball_train_imputed[[column_name]][is.na(money_ball_train_imputed[[column_name]])] <-
median(money_ball_train_imputed[[column_name]], na.rm = TRUE)
} else if (recommendation == "Consider Dropping") {
# Drop the variable if recommended
money_ball_train_imputed[[column_name]] <- NULL
}
}
# Display the updated data frame
head(money_ball_train_imputed)## # A tibble: 6 × 17
## INDEX TARGET_WINS TEAM_BATTING_H TEAM_BATTING_2B TEAM_BATTING_3B
## <dbl> <dbl> <dbl> <dbl> <dbl>
## 1 1 39 1445 194 39
## 2 2 70 1339 219 22
## 3 3 86 1377 232 35
## 4 4 70 1387 209 38
## 5 5 82 1297 186 27
## 6 6 75 1279 200 36
## # ℹ 12 more variables: TEAM_BATTING_HR <dbl>, TEAM_BATTING_BB <dbl>,
## # TEAM_BATTING_SO <dbl>, TEAM_BASERUN_SB <dbl>, TEAM_BASERUN_CS <dbl>,
## # TEAM_BATTING_HBP <dbl>, TEAM_PITCHING_H <dbl>, TEAM_PITCHING_HR <dbl>,
## # TEAM_PITCHING_BB <dbl>, TEAM_PITCHING_SO <dbl>, TEAM_FIELDING_E <dbl>,
## # TEAM_FIELDING_DP <dbl>c. Perform the imputations for evaluation data
money_ball_evaluate_imputed <- money_ball_evaluate
# Impute or drop based on the recommendations
for (i in 1:nrow(missing_values_with_na)) {
column_name <- missing_values_with_na$Column[i]
recommendation <- missing_values_with_na$Imputation_Recommendation[i]
if (recommendation == "Mean/Median Imputation") {
# Mean/Median Imputation
money_ball_evaluate_imputed[[column_name]][is.na(money_ball_evaluate_imputed[[column_name]])] <-
median(money_ball_evaluate_imputed[[column_name]], na.rm = TRUE) # Or use mean if preferred
} else if (recommendation == "KNN or Multiple Imputation") {
# For simplicity, using median imputation here, but you can replace with more advanced methods like KNN
money_ball_evaluate_imputed[[column_name]][is.na(money_ball_evaluate_imputed[[column_name]])] <-
median(money_ball_evaluate_imputed[[column_name]], na.rm = TRUE)
} else if (recommendation == "Multiple Imputation or Predictive Modeling") {
# For simplicity, using median imputation here, but you can replace with more advanced methods
money_ball_evaluate_imputed[[column_name]][is.na(money_ball_evaluate_imputed[[column_name]])] <-
median(money_ball_evaluate_imputed[[column_name]], na.rm = TRUE)
} else if (recommendation == "Consider Dropping") {
# Drop the variable if recommended
money_ball_evaluate_imputed[[column_name]] <- NULL
}
}
# Display the updated data frame
head(money_ball_evaluate_imputed)## # A tibble: 6 × 16
## INDEX TEAM_BATTING_H TEAM_BATTING_2B TEAM_BATTING_3B TEAM_BATTING_HR
## <dbl> <dbl> <dbl> <dbl> <dbl>
## 1 9 1209 170 33 83
## 2 10 1221 151 29 88
## 3 14 1395 183 29 93
## 4 47 1539 309 29 159
## 5 60 1445 203 68 5
## 6 63 1431 236 53 10
## # ℹ 11 more variables: TEAM_BATTING_BB <dbl>, TEAM_BATTING_SO <dbl>,
## # TEAM_BASERUN_SB <dbl>, TEAM_BASERUN_CS <dbl>, TEAM_BATTING_HBP <dbl>,
## # TEAM_PITCHING_H <dbl>, TEAM_PITCHING_HR <dbl>, TEAM_PITCHING_BB <dbl>,
## # TEAM_PITCHING_SO <dbl>, TEAM_FIELDING_E <dbl>, TEAM_FIELDING_DP <dbl>ii. Transform the data based on the summary statistic and visualizations
Transform the data to correct skewness, non-linearity, or to stabilize variance (i.e., address heteroscedasticity).
a. Data transformation methods
Base on the summary statistics and visualizations, the following transformations will be applied to the data
# Create a data frame with variables, transformation methods, and reasons
transformation_recommendations <- data.frame(
Variable = c("TEAM_BATTING_H", "TEAM_BATTING_2B", "TEAM_BATTING_3B",
"TEAM_BATTING_HR", "TEAM_BATTING_BB", "TEAM_BATTING_SO",
"TEAM_BASERUN_SB", "TEAM_BASERUN_CS", "TEAM_BATTING_HBP",
"TEAM_PITCHING_H", "TEAM_PITCHING_HR", "TEAM_PITCHING_BB",
"TEAM_PITCHING_SO", "TEAM_FIELDING_E", "TEAM_FIELDING_DP"),
Transformation_Method = c("Logarithmic", "None", "Square Root",
"Logarithmic", "None", "Logarithmic",
"Logarithmic", "Square Root", "None",
"Logarithmic", "Logarithmic", "Logarithmic",
"Logarithmic", "Logarithmic", "None"),
Reason = c("The distribution appears right-skewed.",
"The distribution is fairly normal, and the relationship with TARGET_WINS seems linear.",
"The distribution is right-skewed with some extreme values.",
"The distribution is right-skewed.",
"The distribution and the relationship with TARGET_WINS are acceptable.",
"The distribution is right-skewed.",
"The distribution is highly right-skewed.",
"The distribution is right-skewed.",
"The distribution is relatively normal.",
"The distribution is extremely right-skewed.",
"The distribution is right-skewed.",
"The distribution is right-skewed with outliers.",
"The distribution is right-skewed.",
"The distribution is right-skewed.",
"The distribution is fairly normal, and the relationship with TARGET_WINS appears linear.")
)
# Display the data frame
print(transformation_recommendations)## Variable Transformation_Method
## 1 TEAM_BATTING_H Logarithmic
## 2 TEAM_BATTING_2B None
## 3 TEAM_BATTING_3B Square Root
## 4 TEAM_BATTING_HR Logarithmic
## 5 TEAM_BATTING_BB None
## 6 TEAM_BATTING_SO Logarithmic
## 7 TEAM_BASERUN_SB Logarithmic
## 8 TEAM_BASERUN_CS Square Root
## 9 TEAM_BATTING_HBP None
## 10 TEAM_PITCHING_H Logarithmic
## 11 TEAM_PITCHING_HR Logarithmic
## 12 TEAM_PITCHING_BB Logarithmic
## 13 TEAM_PITCHING_SO Logarithmic
## 14 TEAM_FIELDING_E Logarithmic
## 15 TEAM_FIELDING_DP None
## Reason
## 1 The distribution appears right-skewed.
## 2 The distribution is fairly normal, and the relationship with TARGET_WINS seems linear.
## 3 The distribution is right-skewed with some extreme values.
## 4 The distribution is right-skewed.
## 5 The distribution and the relationship with TARGET_WINS are acceptable.
## 6 The distribution is right-skewed.
## 7 The distribution is highly right-skewed.
## 8 The distribution is right-skewed.
## 9 The distribution is relatively normal.
## 10 The distribution is extremely right-skewed.
## 11 The distribution is right-skewed.
## 12 The distribution is right-skewed with outliers.
## 13 The distribution is right-skewed.
## 14 The distribution is right-skewed.
## 15 The distribution is fairly normal, and the relationship with TARGET_WINS appears linear.b. Perform the data transformations for testing data
# Perform the recommended transformations
money_ball_train_transformed <- money_ball_train_imputed %>%
mutate(
# Logarithmic Transformations
TEAM_BATTING_H = log(TEAM_BATTING_H + 1),
TEAM_BATTING_HR = log(TEAM_BATTING_HR + 1),
TEAM_BATTING_SO = log(TEAM_BATTING_SO + 1),
TEAM_BASERUN_SB = log(TEAM_BASERUN_SB + 1),
TEAM_PITCHING_H = log(TEAM_PITCHING_H + 1),
TEAM_PITCHING_HR = log(TEAM_PITCHING_HR + 1),
TEAM_PITCHING_BB = log(TEAM_PITCHING_BB + 1),
TEAM_PITCHING_SO = log(TEAM_PITCHING_SO + 1),
TEAM_FIELDING_E = log(TEAM_FIELDING_E + 1),
# Square Root Transformations
TEAM_BATTING_3B = sqrt(TEAM_BATTING_3B),
TEAM_BASERUN_CS = sqrt(TEAM_BASERUN_CS),
# No Transformation Needed
TEAM_BATTING_2B = TEAM_BATTING_2B, # No transformation
TEAM_BATTING_BB = TEAM_BATTING_BB, # No transformation
TEAM_BATTING_HBP = TEAM_BATTING_HBP, # No transformation
TEAM_FIELDING_DP = TEAM_FIELDING_DP # No transformation
)
# Display the updated data frame to check the first few rows
head(money_ball_train_transformed)## # A tibble: 6 × 17
## INDEX TARGET_WINS TEAM_BATTING_H TEAM_BATTING_2B TEAM_BATTING_3B
## <dbl> <dbl> <dbl> <dbl> <dbl>
## 1 1 39 7.28 194 6.24
## 2 2 70 7.20 219 4.69
## 3 3 86 7.23 232 5.92
## 4 4 70 7.24 209 6.16
## 5 5 82 7.17 186 5.20
## 6 6 75 7.15 200 6
## # ℹ 12 more variables: TEAM_BATTING_HR <dbl>, TEAM_BATTING_BB <dbl>,
## # TEAM_BATTING_SO <dbl>, TEAM_BASERUN_SB <dbl>, TEAM_BASERUN_CS <dbl>,
## # TEAM_BATTING_HBP <dbl>, TEAM_PITCHING_H <dbl>, TEAM_PITCHING_HR <dbl>,
## # TEAM_PITCHING_BB <dbl>, TEAM_PITCHING_SO <dbl>, TEAM_FIELDING_E <dbl>,
## # TEAM_FIELDING_DP <dbl>c. Perform the data transformations for evaluation data
# Perform the recommended transformations
money_ball_evaluate_transformed <- money_ball_evaluate_imputed %>%
mutate(
# Logarithmic Transformations
TEAM_BATTING_H = log(TEAM_BATTING_H + 1),
TEAM_BATTING_HR = log(TEAM_BATTING_HR + 1),
TEAM_BATTING_SO = log(TEAM_BATTING_SO + 1),
TEAM_BASERUN_SB = log(TEAM_BASERUN_SB + 1),
TEAM_PITCHING_H = log(TEAM_PITCHING_H + 1),
TEAM_PITCHING_HR = log(TEAM_PITCHING_HR + 1),
TEAM_PITCHING_BB = log(TEAM_PITCHING_BB + 1),
TEAM_PITCHING_SO = log(TEAM_PITCHING_SO + 1),
TEAM_FIELDING_E = log(TEAM_FIELDING_E + 1),
# Square Root Transformations
TEAM_BATTING_3B = sqrt(TEAM_BATTING_3B),
TEAM_BASERUN_CS = sqrt(TEAM_BASERUN_CS),
# No Transformation Needed
TEAM_BATTING_2B = TEAM_BATTING_2B, # No transformation
TEAM_BATTING_BB = TEAM_BATTING_BB, # No transformation
TEAM_BATTING_HBP = TEAM_BATTING_HBP, # No transformation
TEAM_FIELDING_DP = TEAM_FIELDING_DP # No transformation
)
# Display the updated data frame to check the first few rows
head(money_ball_evaluate_transformed)## # A tibble: 6 × 16
## INDEX TEAM_BATTING_H TEAM_BATTING_2B TEAM_BATTING_3B TEAM_BATTING_HR
## <dbl> <dbl> <dbl> <dbl> <dbl>
## 1 9 7.10 170 5.74 4.43
## 2 10 7.11 151 5.39 4.49
## 3 14 7.24 183 5.39 4.54
## 4 47 7.34 309 5.39 5.08
## 5 60 7.28 203 8.25 1.79
## 6 63 7.27 236 7.28 2.40
## # ℹ 11 more variables: TEAM_BATTING_BB <dbl>, TEAM_BATTING_SO <dbl>,
## # TEAM_BASERUN_SB <dbl>, TEAM_BASERUN_CS <dbl>, TEAM_BATTING_HBP <dbl>,
## # TEAM_PITCHING_H <dbl>, TEAM_PITCHING_HR <dbl>, TEAM_PITCHING_BB <dbl>,
## # TEAM_PITCHING_SO <dbl>, TEAM_FIELDING_E <dbl>, TEAM_FIELDING_DP <dbl>iii. Scale the variables
a. Scale the variables for training data
Scaling is to ensure that all features have the same scale.
# Create a named vector with column names and their indexes
column_indexes <- setNames(seq_along(names(money_ball_train_transformed)), names(money_ball_train_transformed))
# Drop the INDEX and TEAM_BATTING_HBP columns
money_ball_train_transformed <- money_ball_train_transformed[, -c(column_indexes["INDEX"], column_indexes["TEAM_BATTING_HBP"])]
# Min-Max scaling function
min_max_scale <- function(x) {
return((x - min(x)) / (max(x) - min(x)))
}
# Apply Min-Max scaling to all numeric columns in the dataset
money_ball_train_scaled <- money_ball_train_transformed %>%
mutate(across(everything(), min_max_scale))
# Display the first few rows of the scaled dataset
head(money_ball_train_scaled)## # A tibble: 6 × 15
## TARGET_WINS TEAM_BATTING_H TEAM_BATTING_2B TEAM_BATTING_3B TEAM_BATTING_HR
## <dbl> <dbl> <dbl> <dbl> <dbl>
## 1 0.267 0.459 0.321 0.418 0.473
## 2 0.479 0.387 0.386 0.314 0.941
## 3 0.589 0.413 0.419 0.396 0.883
## 4 0.479 0.420 0.360 0.413 0.820
## 5 0.562 0.356 0.301 0.348 0.831
## 6 0.514 0.343 0.337 0.402 0.812
## # ℹ 10 more variables: TEAM_BATTING_BB <dbl>, TEAM_BATTING_SO <dbl>,
## # TEAM_BASERUN_SB <dbl>, TEAM_BASERUN_CS <dbl>, TEAM_PITCHING_H <dbl>,
## # TEAM_PITCHING_HR <dbl>, TEAM_PITCHING_BB <dbl>, TEAM_PITCHING_SO <dbl>,
## # TEAM_FIELDING_E <dbl>, TEAM_FIELDING_DP <dbl>b. Scale the variables for evaluation data
Scaling is to ensure that all features have the same scale.
# Create a named vector with column names and their indexes
column_indexes <- setNames(seq_along(names(money_ball_evaluate_transformed)), names(money_ball_evaluate_transformed))
# Drop the INDEX and TEAM_BATTING_HBP columns
money_ball_evaluate_transformed <- money_ball_evaluate_transformed[, -c(column_indexes["INDEX"], column_indexes["TEAM_BATTING_HBP"])]
# Min-Max scaling function
min_max_scale <- function(x) {
return((x - min(x)) / (max(x) - min(x)))
}
# Apply Min-Max scaling to all numeric columns in the dataset
money_ball_evaluate_scaled <- money_ball_evaluate_transformed %>%
mutate(across(everything(), min_max_scale))
# Display the first few rows of the scaled dataset
head(money_ball_evaluate_scaled)## # A tibble: 6 × 14
## TEAM_BATTING_H TEAM_BATTING_2B TEAM_BATTING_3B TEAM_BATTING_HR TEAM_BATTING_BB
## <dbl> <dbl> <dbl> <dbl> <dbl>
## 1 0.400 0.380 0.230 0.807 0.556
## 2 0.410 0.322 0.189 0.817 0.645
## 3 0.546 0.419 0.189 0.827 0.636
## 4 0.647 0.798 0.189 0.924 0.606
## 5 0.583 0.479 0.517 0.326 0.103
## 6 0.573 0.578 0.406 0.437 0.257
## # ℹ 9 more variables: TEAM_BATTING_SO <dbl>, TEAM_BASERUN_SB <dbl>,
## # TEAM_BASERUN_CS <dbl>, TEAM_PITCHING_H <dbl>, TEAM_PITCHING_HR <dbl>,
## # TEAM_PITCHING_BB <dbl>, TEAM_PITCHING_SO <dbl>, TEAM_FIELDING_E <dbl>,
## # TEAM_FIELDING_DP <dbl>4. Check for Multicollinearity
Check for multicollinearity among independent variables.
# Load necessary library
library(car)## Loading required package: carData##
## Attaching package: 'car'## The following object is masked from 'package:psych':
##
## logit## The following object is masked from 'package:purrr':
##
## some## The following object is masked from 'package:dplyr':
##
## recode# Calculate VIF values
vif_values <- vif(lm(TARGET_WINS ~ ., data = money_ball_train_scaled))
# Convert VIF values into a data frame
vif_df <- data.frame(
Variable = names(vif_values),
VIF = vif_values
)
# Sort the data frame by VIF in descending order
vif_df <- vif_df[order(-vif_df$VIF), ]
# Display the sorted data frame
print(vif_df)## Variable VIF
## TEAM_BATTING_SO TEAM_BATTING_SO 232.326564
## TEAM_PITCHING_SO TEAM_PITCHING_SO 161.458827
## TEAM_BATTING_HR TEAM_BATTING_HR 102.754253
## TEAM_PITCHING_HR TEAM_PITCHING_HR 68.910298
## TEAM_PITCHING_H TEAM_PITCHING_H 19.701617
## TEAM_FIELDING_E TEAM_FIELDING_E 6.035903
## TEAM_BATTING_BB TEAM_BATTING_BB 5.875380
## TEAM_PITCHING_BB TEAM_PITCHING_BB 4.451019
## TEAM_BATTING_H TEAM_BATTING_H 3.541817
## TEAM_BATTING_3B TEAM_BATTING_3B 2.824945
## TEAM_BATTING_2B TEAM_BATTING_2B 2.490810
## TEAM_BASERUN_SB TEAM_BASERUN_SB 1.906257
## TEAM_FIELDING_DP TEAM_FIELDING_DP 1.455789
## TEAM_BASERUN_CS TEAM_BASERUN_CS 1.312160Based on the Variance Inflation Factor (VIF) values above, there is a presence of severe multicollinearity in the dataset, with some variables. Here’s the breakdown:
Variables with high multicollinearity: The VIF in
TEAM_BATTING_H(VIF = 119,071.0),TEAM_BATTING_BB(VIF = 196,285.5),TEAM_PITCHING_HR(VIF = 308,757.4),TEAM_PITCHING_BB(VIF = 196,404.2),TEAM_PITCHING_SO(VIF = 197,267.5) are extremely high indicating severe multicollinearity. This level of multicollinearity can cause problems in estimating the coefficients reliably and can inflate the standard errors of the coefficients.Other Variables (VIF values close to 1): For other variables like
TEAM_BATTING_2B,TEAM_BATTING_3B,TEAM_BASERUN_SB,TEAM_FIELDING_E, andTEAM_FIELDING_DP, the VIF values are low, indicating low or negligible multicollinearity.
Based on the multicollinearity, we will create multiple linear regression models with the following approaches:
5 Feature Engineering and Model Building
i. Baseline Model
- Model 1: Use all variables except those with
extremely high VIF values.
- Variables: All except
TEAM_BATTING_BB,TEAM_PITCHING_SO,TEAM_PITCHING_BB, andTEAM_PITCHING_HR. - Rationale: This model tests the impact of removing variables with the highest multicollinearity while keeping the rest to ensure comprehensive coverage of all aspects of the game.
- Variables: All except
a. Feature engineer training data for Baseline Model
# Create a named vector with column names and their indexes
column_indexes <- setNames(seq_along(names(money_ball_train_scaled)), names(money_ball_train_scaled))
# Drop the columns with high VIF values using their names
money_ball_train_baseline <- money_ball_train_scaled[, !names(money_ball_train_scaled) %in% c("TEAM_BATTING_SO", "TEAM_PITCHING_SO", "TEAM_BATTING_HR ")]
# Check for any NA values and remove rows with NA values if they exist
money_ball_train_baseline <- na.omit(money_ball_train_baseline)b. Feature engineer evaluation data for Baseline Model
# Create a named vector with column names and their indexes
column_indexes <- setNames(seq_along(names(money_ball_evaluate_scaled)), names(money_ball_evaluate_scaled))
# Drop the columns with high VIF values using their names
money_ball_evaluate_baseline <- money_ball_evaluate_scaled[, !names(money_ball_evaluate_scaled) %in% c("TEAM_BATTING_SO", "TEAM_PITCHING_SO", "TEAM_BATTING_HR ")]
# Check for any NA values and remove rows with NA values if they exist
money_ball_evaluate_baseline <- na.omit(money_ball_evaluate_baseline)c. Fit the Base Line Model
# Fit the multiple linear regression model using the remaining variables
baseline_model <- lm(TARGET_WINS ~ ., data = money_ball_train_baseline)
# Summary of the model
summary(baseline_model)##
## Call:
## lm(formula = TARGET_WINS ~ ., data = money_ball_train_baseline)
##
## Residuals:
## Min 1Q Median 3Q Max
## -0.44085 -0.05726 0.00031 0.05676 0.55197
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 0.35193 0.06251 5.630 2.02e-08 ***
## TEAM_BATTING_H 0.51586 0.03831 13.465 < 2e-16 ***
## TEAM_BATTING_2B -0.06981 0.02392 -2.919 0.00355 **
## TEAM_BATTING_3B 0.15745 0.02617 6.017 2.07e-09 ***
## TEAM_BATTING_HR -0.19824 0.10667 -1.859 0.06322 .
## TEAM_BATTING_BB 0.16926 0.02943 5.751 1.01e-08 ***
## TEAM_BASERUN_SB 0.19394 0.02580 7.516 8.09e-14 ***
## TEAM_BASERUN_CS -0.07419 0.02545 -2.915 0.00360 **
## TEAM_PITCHING_H 0.09147 0.04775 1.915 0.05556 .
## TEAM_PITCHING_HR 0.24917 0.10022 2.486 0.01298 *
## TEAM_PITCHING_BB -0.22280 0.09313 -2.392 0.01682 *
## TEAM_FIELDING_E -0.26004 0.02528 -10.286 < 2e-16 ***
## TEAM_FIELDING_DP -0.13717 0.01608 -8.532 < 2e-16 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 0.08972 on 2263 degrees of freedom
## Multiple R-squared: 0.3121, Adjusted R-squared: 0.3084
## F-statistic: 85.55 on 12 and 2263 DF, p-value: < 2.2e-16d. Interpretation of the significant coefficients
- TEAM_BATTING_H (Hits): Positive impact on wins (+0.51586). More hits generally lead to more runs and wins.
- TEAM_BATTING_2B (Doubles): Slight negative impact on wins (-0.06981). Possibly due to multicollinearity.
- TEAM_BATTING_BB (Walks): Positive impact on wins (+0.35319). Walks contribute to more scoring opportunities.
- TEAM_BASERUN_SB (Stolen Bases): Positive impact on wins (+0.19349). Successful steals lead to more runs.
- TEAM_FIELDING_E (Errors): Negative impact on wins (-0.26004). More errors generally result in fewer wins.
e. Residual analysis
# Residual analysis of the transformed model
par(mfrow = c(2, 2))
plot(baseline_model)
Analysis of Residual Plots:
- Residuals vs Fitted: The spread of residuals around the horizontal line is somewhat uneven, suggesting non-linearity and possible heteroscedasticity (non-constant variance).
- Q-Q Plot: The tails deviate from the straight line, indicating potential non-normality in the residuals, particularly in the extreme values.
- Scale-Location: The red line is slightly curved, and residuals seem more dispersed at higher fitted values, confirming heteroscedasticity.
- Residuals vs Leverage: A few points have high leverage, which may indicate influential outliers affecting the model’s stability.
ii. Model with Composite Variables
- Model 2: Create composite variables for highly
correlated variables and drop individual variables.
- Variables:
- Composite Variable 1: Average of
TEAM_BATTING_H,TEAM_BATTING_2B, andTEAM_BATTING_HR. - Composite Variable 2: Average of
TEAM_PITCHING_H,TEAM_PITCHING_HR, andTEAM_PITCHING_BB. - Retained Variables: Other variables that aren’t
highly correlated or have significant unique contributions, like
TEAM_FIELDING_DPandTEAM_FIELDING_E.
- Composite Variable 1: Average of
- Rationale: This model reduces multicollinearity by combining highly correlated variables into composite variables.
- Variables:
a. Feature engineer training data for Model with Composite Variables
# Create a named vector with column names and their indexes
column_indexes <- setNames(seq_along(names(money_ball_train_scaled)), names(money_ball_train_scaled))
# Calculate composite variables using base R
Composite_Batting <- rowMeans(money_ball_train_scaled[, c(column_indexes["TEAM_BATTING_H"],
column_indexes["TEAM_BATTING_2B"])])
Composite_Pitching <- money_ball_train_scaled[[column_indexes["TEAM_PITCHING_H"]]]
# Ensure the composite variables are numeric vectors
Composite_Batting <- as.numeric(Composite_Batting)
Composite_Pitching <- as.numeric(Composite_Pitching)
# Add the composite variables to the dataset
money_ball_train_composite <- money_ball_train_scaled %>%
mutate(Composite_Batting = Composite_Batting, Composite_Pitching = Composite_Pitching)
# Drop the original individual variables that were used to create the composites
money_ball_train_composite <- money_ball_train_composite[, !names(money_ball_train_composite) %in%
c("TEAM_BATTING_H", "TEAM_BATTING_2B",
"TEAM_BATTING_HR", "TEAM_PITCHING_H",
"TEAM_PITCHING_HR", "TEAM_PITCHING_SO",
"TEAM_PITCHING_BB")]b. Feature engineer evaluation data for Model with Composite Variables
# Create a named vector with column names and their indexes
column_indexes <- setNames(seq_along(names(money_ball_evaluate_scaled)), names(money_ball_evaluate_scaled))
# Calculate composite variables using base R
Composite_Batting <- rowMeans(money_ball_evaluate_scaled[, c(column_indexes["TEAM_BATTING_H"],
column_indexes["TEAM_BATTING_2B"])])
Composite_Pitching <- money_ball_evaluate_scaled[[column_indexes["TEAM_PITCHING_H"]]]
# Ensure the composite variables are numeric vectors
Composite_Batting <- as.numeric(Composite_Batting)
Composite_Pitching <- as.numeric(Composite_Pitching)
# Add the composite variables to the dataset
money_ball_evaluate_composite <- money_ball_evaluate_scaled %>%
mutate(Composite_Batting = Composite_Batting, Composite_Pitching = Composite_Pitching)
# Drop the original individual variables that were used to create the composites
money_ball_evaluate_composite <- money_ball_evaluate_composite[, !names(money_ball_evaluate_composite) %in%
c("TEAM_BATTING_H", "TEAM_BATTING_2B",
"TEAM_BATTING_HR", "TEAM_PITCHING_H",
"TEAM_PITCHING_HR", "TEAM_PITCHING_SO",
"TEAM_PITCHING_BB")]c. Fit the Model with Composite Variables
# Fit the multiple linear regression model using the remaining variables including the composite variables
composite_model <- lm(TARGET_WINS ~ Composite_Batting + Composite_Pitching + TEAM_FIELDING_DP + TEAM_FIELDING_E,
data = money_ball_train_composite)
# Summary of the model
summary(composite_model)##
## Call:
## lm(formula = TARGET_WINS ~ Composite_Batting + Composite_Pitching +
## TEAM_FIELDING_DP + TEAM_FIELDING_E, data = money_ball_train_composite)
##
## Residuals:
## Min 1Q Median 3Q Max
## -0.35865 -0.06276 0.00128 0.06521 0.34910
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 0.43041 0.01505 28.589 < 2e-16 ***
## Composite_Batting 0.49004 0.02548 19.233 < 2e-16 ***
## Composite_Pitching -0.08204 0.03396 -2.416 0.015783 *
## TEAM_FIELDING_DP -0.13093 0.01591 -8.230 3.12e-16 ***
## TEAM_FIELDING_E -0.06568 0.01742 -3.770 0.000168 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 0.09765 on 2271 degrees of freedom
## Multiple R-squared: 0.1823, Adjusted R-squared: 0.1808
## F-statistic: 126.5 on 4 and 2271 DF, p-value: < 2.2e-16d. Interpretation of the significant coefficients
- Composite_Batting: Positive impact on wins (+0.49004). Better batting performance leads to more wins.
- Composite_Pitching: Slight negative impact on wins (-0.08204). Higher composite pitching values may indicate allowing more hits, reducing wins.
- TEAM_FIELDING_DP (Double Plays): Negative impact on wins (-0.13093). Could suggest that reliance on double plays is correlated with weaker overall defense.
- TEAM_FIELDING_E (Errors): Negative impact on wins (-0.05668). More errors reduce the likelihood of winning.
e. Residual analysis
# Residual analysis of the transformed model
par(mfrow = c(2, 2))
plot(composite_model)
Analysis of Residual Plots:
- Residuals vs Fitted: The residuals still show some pattern, especially at higher fitted values, indicating that the model may not fully capture the relationship between the variables. There’s also slight evidence of heteroscedasticity.
- Q-Q Plot: The residuals generally follow the line but deviate at the extremes, suggesting potential issues with normality, particularly in the tails.
- Scale-Location: The red line shows a slight curve, and the spread of residuals increases as fitted values rise, indicating heteroscedasticity.
- Residuals vs Leverage: Some high-leverage points are identified, but they don’t appear overly influential on the model. However, they could still be potential outliers that affect the model’s accuracy.
iii. Model with Interaction Terms
- Model 3: Incorporate interaction terms between key
variables.
- Variables:
- Include interaction terms like
TEAM_BATTING_H * TEAM_BATTING_HRorTEAM_PITCHING_H * TEAM_PITCHING_BB. - Drop individual variables contributing to interaction terms to prevent redundancy.
- Retain core independent variables that might have unique, significant effects.
- Include interaction terms like
- Rationale: Interaction terms can capture synergistic effects between variables, offering a more nuanced understanding of how these variables collectively impact the target variable.
- Variables:
a. Feature engineer training data for Model with Interaction Terms
# Step 1: Create a named vector with column names and their indexes
column_indexes <- setNames(seq_along(names(money_ball_train_scaled)), names(money_ball_train_scaled))
# Step 2: Ensure the columns are numeric vectors
TEAM_BATTING_H <- as.numeric(money_ball_train_scaled[[column_indexes["TEAM_BATTING_H"]]])
TEAM_BATTING_2B <- as.numeric(money_ball_train_scaled[[column_indexes["TEAM_BATTING_2B"]]])
TEAM_PITCHING_H <- as.numeric(money_ball_train_scaled[[column_indexes["TEAM_PITCHING_H"]]])
TEAM_PITCHING_BB <- as.numeric(money_ball_train_scaled[[column_indexes["TEAM_PITCHING_BB"]]])
# Step 3: Calculate interaction terms using numeric vectors
Interaction_Batting <- TEAM_BATTING_H * TEAM_BATTING_2B
Interaction_Pitching <- TEAM_PITCHING_H * TEAM_PITCHING_BB
# Step 4: Add the interaction terms to the dataset
money_ball_train_interaction <- money_ball_train_scaled %>%
mutate(Interaction_Batting = Interaction_Batting, Interaction_Pitching = Interaction_Pitching)
# Step 5: Drop the original individual variables used to create interaction terms
money_ball_train_interaction <- money_ball_train_interaction[, !names(money_ball_train_interaction) %in%
c("TEAM_BATTING_H", "TEAM_BATTING_2B",
"TEAM_PITCHING_H", "TEAM_PITCHING_BB")]b. Feature engineer evaluation data for Model with Interaction Terms
# Step 1: Create a named vector with column names and their indexes
column_indexes <- setNames(seq_along(names(money_ball_evaluate_scaled)), names(money_ball_evaluate_scaled))
# Step 2: Ensure the columns are numeric vectors
TEAM_BATTING_H <- as.numeric(money_ball_evaluate_scaled[[column_indexes["TEAM_BATTING_H"]]])
TEAM_BATTING_2B <- as.numeric(money_ball_evaluate_scaled[[column_indexes["TEAM_BATTING_2B"]]])
TEAM_PITCHING_H <- as.numeric(money_ball_evaluate_scaled[[column_indexes["TEAM_PITCHING_H"]]])
TEAM_PITCHING_BB <- as.numeric(money_ball_evaluate_scaled[[column_indexes["TEAM_PITCHING_BB"]]])
# Step 3: Calculate interaction terms using numeric vectors
Interaction_Batting <- TEAM_BATTING_H * TEAM_BATTING_2B
Interaction_Pitching <- TEAM_PITCHING_H * TEAM_PITCHING_BB
# Step 4: Add the interaction terms to the dataset
money_ball_evaluate_interaction <- money_ball_evaluate_scaled %>%
mutate(Interaction_Batting = Interaction_Batting, Interaction_Pitching = Interaction_Pitching)
# Step 5: Drop the original individual variables used to create interaction terms
money_ball_evaluate_interaction <- money_ball_evaluate_interaction[, !names(money_ball_evaluate_interaction) %in%
c("TEAM_BATTING_H", "TEAM_BATTING_2B",
"TEAM_PITCHING_H", "TEAM_PITCHING_BB")]c. Fit the Model with Interaction Terms
# Fit the multiple linear regression model using the remaining variables including the interaction terms
interaction_model <- lm(TARGET_WINS ~ Interaction_Batting + Interaction_Pitching +
TEAM_BASERUN_SB + TEAM_BASERUN_CS +
TEAM_FIELDING_DP + TEAM_FIELDING_E,
data = money_ball_train_interaction)
# Summary of the model
summary(interaction_model)##
## Call:
## lm(formula = TARGET_WINS ~ Interaction_Batting + Interaction_Pitching +
## TEAM_BASERUN_SB + TEAM_BASERUN_CS + TEAM_FIELDING_DP + TEAM_FIELDING_E,
## data = money_ball_train_interaction)
##
## Residuals:
## Min 1Q Median 3Q Max
## -0.42116 -0.06460 -0.00040 0.06391 0.34775
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 0.40778 0.02130 19.144 < 2e-16 ***
## Interaction_Batting 0.45338 0.02644 17.150 < 2e-16 ***
## Interaction_Pitching 0.03646 0.04163 0.876 0.38121
## TEAM_BASERUN_SB 0.26207 0.02588 10.127 < 2e-16 ***
## TEAM_BASERUN_CS -0.08464 0.02649 -3.195 0.00142 **
## TEAM_FIELDING_DP -0.09110 0.01607 -5.669 1.62e-08 ***
## TEAM_FIELDING_E -0.15119 0.01719 -8.797 < 2e-16 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 0.09614 on 2269 degrees of freedom
## Multiple R-squared: 0.208, Adjusted R-squared: 0.206
## F-statistic: 99.34 on 6 and 2269 DF, p-value: < 2.2e-16d. Interpretation of the significant coefficients
- Interaction_Batting: Positive impact on wins (+0.45338). Indicates that the interaction between batting variables has a strong positive effect on the number of wins.
- TEAM_BASERUN_SB: Positive impact on wins (+0.12765). Successful stolen bases contribute positively to the number of wins.
- TEAM_BASERUN_CS: Negative impact on wins (-0.08464). Caught stealing negatively impacts the number of wins.
- TEAM_FIELDING_DP: Negative impact on wins (-0.10570). Similar to previous models, reliance on double plays correlates with fewer wins.
- TEAM_FIELDING_E: Negative impact on wins (-0.15119). More errors strongly reduce the likelihood of winning.
e. Residual analysis
# Residual analysis of the transformed model
par(mfrow = c(2, 2))
plot(interaction_model)
Analysis of Residual Plots:
- Residuals vs Fitted: The residuals show a pattern, particularly at higher fitted values, indicating potential issues with model fit. There’s also slight evidence of heteroscedasticity, as the spread of residuals increases with fitted values.
- Q-Q Plot: The residuals mostly follow the expected line, but there are deviations at the extremes, suggesting some issues with normality, particularly in the tails.
- Scale-Location: The red line shows a curve, indicating that the variance of the residuals is not constant across all levels of fitted values. This suggests heteroscedasticity, where the spread of residuals increases as fitted values rise.
- Residuals vs Leverage: There are a few high-leverage points, notably point 13420. While they don’t appear overly influential, they could still affect the model’s accuracy and should be examined further.
iv. Feature Selection Model
- Model 4: Perform backward stepwise regression or
Lasso/Ridge regression to select the most impactful variables.
- Variables: Start with all the variables, including composite and interaction terms if applicable, and use a method to select the best subset.
- Rationale: This model allows the data itself to determine which variables should be included, ensuring only the most significant predictors remain.
a. Feature engineer training data for Model with Interaction Terms
# Load the necessary package
library(MASS)##
## Attaching package: 'MASS'## The following object is masked from 'package:plotly':
##
## select## The following object is masked from 'package:dplyr':
##
## select# Step 1: Create a named vector with column names and their indexes
column_indexes <- setNames(seq_along(names(money_ball_train_scaled)), names(money_ball_train_scaled))
# Step 2: Calculate composite variables using base R and ensure they are numeric vectors
Composite_Batting <- rowMeans(money_ball_train_scaled[, c(column_indexes["TEAM_BATTING_H"],
column_indexes["TEAM_BATTING_2B"])])
Composite_Pitching <- as.numeric(money_ball_train_scaled[[column_indexes["TEAM_PITCHING_H"]]])
# Step 3: Add the composite and interaction terms to the dataset
money_ball_train_feature_selection <- money_ball_train_scaled %>%
mutate(Composite_Batting = Composite_Batting,
Composite_Pitching = Composite_Pitching,
Interaction_Batting = Composite_Batting * money_ball_train_scaled[[column_indexes["TEAM_BATTING_2B"]]],
Interaction_Pitching = Composite_Pitching * money_ball_train_scaled[[column_indexes["TEAM_PITCHING_BB"]]])
# Step 4: Drop the original individual variables that were used to create composite and interaction terms
money_ball_train_feature_selection <- money_ball_train_feature_selection[, !names(money_ball_train_feature_selection) %in%
c("TEAM_BATTING_H", "TEAM_BATTING_2B",
"TEAM_PITCHING_H", "TEAM_PITCHING_BB")]b. Feature engineer evaluation data for Model with Interaction Terms
# Load the necessary package
library(MASS)
# Step 1: Create a named vector with column names and their indexes
column_indexes <- setNames(seq_along(names(money_ball_evaluate_scaled)), names(money_ball_evaluate_scaled))
# Step 2: Calculate composite variables using base R and ensure they are numeric vectors
Composite_Batting <- rowMeans(money_ball_evaluate_scaled[, c(column_indexes["TEAM_BATTING_H"],
column_indexes["TEAM_BATTING_2B"])])
Composite_Pitching <- as.numeric(money_ball_evaluate_scaled[[column_indexes["TEAM_PITCHING_H"]]])
# Step 3: Add the composite and interaction terms to the dataset
money_ball_evaluate_feature_selection <- money_ball_evaluate_scaled %>%
mutate(Composite_Batting = Composite_Batting,
Composite_Pitching = Composite_Pitching,
Interaction_Batting = Composite_Batting * money_ball_evaluate_scaled[[column_indexes["TEAM_BATTING_2B"]]],
Interaction_Pitching = Composite_Pitching * money_ball_evaluate_scaled[[column_indexes["TEAM_PITCHING_BB"]]])
# Step 4: Drop the original individual variables that were used to create composite and interaction terms
money_ball_evaluate_feature_selection <- money_ball_evaluate_feature_selection[, !names(money_ball_evaluate_feature_selection) %in%
c("TEAM_BATTING_H", "TEAM_BATTING_2B",
"TEAM_PITCHING_H", "TEAM_PITCHING_BB")]c. Fit the Model with Interaction Terms
# Step 1: Fit the full model with all variables
full_model <- lm(TARGET_WINS ~ ., data = money_ball_train_feature_selection)
# Step 2: Perform backward stepwise regression to select the best model
stepwise_model <- stepAIC(full_model, direction = "backward", trace = FALSE)
# Summary of the selected model
summary(stepwise_model)##
## Call:
## lm(formula = TARGET_WINS ~ TEAM_BATTING_3B + TEAM_BATTING_BB +
## TEAM_BATTING_SO + TEAM_BASERUN_SB + TEAM_BASERUN_CS + TEAM_PITCHING_HR +
## TEAM_PITCHING_SO + TEAM_FIELDING_E + TEAM_FIELDING_DP + Composite_Batting +
## Interaction_Batting + Interaction_Pitching, data = money_ball_train_feature_selection)
##
## Residuals:
## Min 1Q Median 3Q Max
## -0.46522 -0.05826 0.00088 0.05540 0.46866
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 0.25301 0.04314 5.865 5.14e-09 ***
## TEAM_BATTING_3B 0.20141 0.02464 8.173 4.94e-16 ***
## TEAM_BATTING_BB 0.12328 0.01958 6.297 3.63e-10 ***
## TEAM_BATTING_SO -0.87475 0.24581 -3.559 0.000380 ***
## TEAM_BASERUN_SB 0.22077 0.02669 8.272 2.22e-16 ***
## TEAM_BASERUN_CS -0.09336 0.02570 -3.633 0.000286 ***
## TEAM_PITCHING_HR 0.09591 0.02276 4.213 2.62e-05 ***
## TEAM_PITCHING_SO 0.89486 0.29088 3.076 0.002121 **
## TEAM_FIELDING_E -0.24703 0.02470 -10.000 < 2e-16 ***
## TEAM_FIELDING_DP -0.14908 0.01623 -9.186 < 2e-16 ***
## Composite_Batting 0.97713 0.09301 10.505 < 2e-16 ***
## Interaction_Batting -0.62306 0.08121 -7.673 2.49e-14 ***
## Interaction_Pitching -0.20507 0.10688 -1.919 0.055161 .
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 0.09045 on 2263 degrees of freedom
## Multiple R-squared: 0.3009, Adjusted R-squared: 0.2972
## F-statistic: 81.16 on 12 and 2263 DF, p-value: < 2.2e-16c. Interpretation of the significant coefficients
- TEAM_BATTING_3B: Positive impact on wins (+0.12141). Triples contribute positively to the number of wins.
- TEAM_BATTING_BB: Positive impact on wins (+0.12328). Walks (BB) positively influence wins, indicating the importance of plate discipline.
- TEAM_BATTING_SO: Negative impact on wins (-0.24745). Strikeouts negatively affect the number of wins, as they represent lost opportunities to score.
- TEAM_BASERUN_SB: Positive impact on wins (+0.22077). Stolen bases significantly increase the chances of winning.
- TEAM_BASERUN_CS: Negative impact on wins (-0.09336). Being caught stealing has a strong negative impact on wins.
- TEAM_PITCHING_HR: Negative impact on wins (-0.20482). Allowing home runs negatively impacts the number of wins.
- TEAM_FIELDING_E: Negative impact on wins (-0.14040). More errors reduce the likelihood of winning.
- TEAM_FIELDING_DP: Negative impact on wins (-0.11498). Again, relying on double plays correlates with fewer wins.
- Composite_Batting: Positive impact on wins (+0.44821). The composite batting variable significantly increases the chances of winning.
- Interaction_Batting: Negative impact on wins (-0.22306). Interaction between batting variables reduces wins, possibly due to overcompensation or diminishing returns.
d. Residual analysis
# Residual analysis of the transformed model
par(mfrow = c(2, 2))
plot(stepwise_model)
Analysis of Residual Plots:
- Residuals vs Fitted: The residuals are scattered around the zero line with some spread, particularly at higher fitted values, which suggests potential issues with heteroscedasticity.
- Q-Q Plot: The residuals follow the theoretical quantiles fairly well, though there are slight deviations at the tails, indicating possible issues with normality, especially at the extremes.
- Scale-Location: This plot shows a relatively constant spread of residuals, but there is a slight upward trend indicating mild heteroscedasticity.
- Residuals vs Leverage: Points 13420, 20120, and 415 have high leverage, with 13420 and 415 close to the Cook’s distance threshold, suggesting they might disproportionately influence the model.
v. Fielding-Focused Model
- Model 5: Focus specifically on the fielding-related
variables (
TEAM_FIELDING_DPandTEAM_FIELDING_E) alongside a few selected pitching and batting variables.- Variables:
TEAM_FIELDING_DP,TEAM_FIELDING_E, a few selected batting variables (e.g.,TEAM_BATTING_H), and pitching variables (e.g.,TEAM_PITCHING_H). - Rationale: This model emphasizes the impact of fielding on wins while controlling for key batting and pitching effects.
- Variables:
a. Feature engineer training data for Fielding-Focused Model
# Step 1: Create a named vector with column names and their indexes
column_indexes <- setNames(seq_along(names(money_ball_train_scaled)), names(money_ball_train_scaled))
# Step 2: Select fielding-related variables along with selected batting and pitching variables
# We will use the following variables:
# - TEAM_FIELDING_DP
# - TEAM_FIELDING_E
# - TEAM_BATTING_H
# - TEAM_BATTING_2B
# - TEAM_PITCHING_H
selected_columns <- c(column_indexes["TEAM_FIELDING_DP"],
column_indexes["TEAM_FIELDING_E"],
column_indexes["TEAM_BATTING_H"],
column_indexes["TEAM_BATTING_2B"],
column_indexes["TEAM_PITCHING_H"])
# Step 3: Subset the dataset to include only these selected columns
money_ball_train_fielding <- money_ball_train_scaled[, selected_columns]
# Step 4: Ensure the TARGET_WINS column is included in the data for the regression model
money_ball_train_fielding$TARGET_WINS <- money_ball_train_scaled$TARGET_WINSb. Feature engineer evaluation data for Fielding-Focused Model
# Step 1: Create a named vector with column names and their indexes
column_indexes <- setNames(seq_along(names(money_ball_evaluate_scaled)), names(money_ball_evaluate_scaled))
# Step 2: Select fielding-related variables along with selected batting and pitching variables
# We will use the following variables:
# - TEAM_FIELDING_DP
# - TEAM_FIELDING_E
# - TEAM_BATTING_H
# - TEAM_BATTING_2B
# - TEAM_PITCHING_H
selected_columns <- c(column_indexes["TEAM_FIELDING_DP"],
column_indexes["TEAM_FIELDING_E"],
column_indexes["TEAM_BATTING_H"],
column_indexes["TEAM_BATTING_2B"],
column_indexes["TEAM_PITCHING_H"])
# Step 3: Subset the dataset to include only these selected columns
money_ball_evaluate_fielding <- money_ball_evaluate_scaled[, selected_columns]
# Step 4: Ensure the TARGET_WINS column is included in the data for the regression model
money_ball_evaluate_fielding$TARGET_WINS <- money_ball_evaluate_scaled$TARGET_WINS## Warning: Unknown or uninitialised column: `TARGET_WINS`.c. Fit the Fielding-Focused Model
# Step 5: Fit the multiple linear regression model using the selected variables
fielding_model <- lm(TARGET_WINS ~ ., data = money_ball_train_fielding)
# Step 6: Summary of the model
summary(fielding_model)##
## Call:
## lm(formula = TARGET_WINS ~ ., data = money_ball_train_fielding)
##
## Residuals:
## Min 1Q Median 3Q Max
## -0.37106 -0.06042 0.00006 0.06075 0.49694
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 0.40109 0.01447 27.721 < 2e-16 ***
## TEAM_FIELDING_DP -0.14599 0.01519 -9.613 < 2e-16 ***
## TEAM_FIELDING_E -0.16930 0.01793 -9.440 < 2e-16 ***
## TEAM_BATTING_H 0.70595 0.03256 21.684 < 2e-16 ***
## TEAM_BATTING_2B -0.07462 0.02421 -3.082 0.002081 **
## TEAM_PITCHING_H -0.12239 0.03246 -3.771 0.000167 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 0.09302 on 2270 degrees of freedom
## Multiple R-squared: 0.2583, Adjusted R-squared: 0.2567
## F-statistic: 158.1 on 5 and 2270 DF, p-value: < 2.2e-16d. Interpretation of the significant coefficients
- TEAM_FIELDING_DP: Negative impact on wins (-0.14599). A higher number of double plays correlates with fewer wins.
- TEAM_FIELDING_E: Negative impact on wins (-0.16930). More fielding errors significantly reduce the number of wins.
- TEAM_BATTING_H: Positive impact on wins (+0.17593). Hits significantly increase the chances of winning.
- TEAM_BATTING_2B: Negative impact on wins (-0.07462). Doubles surprisingly have a negative impact on wins, possibly due to confounding factors.
- TEAM_PITCHING_H: Negative impact on wins (-0.12239). Allowing more hits leads to fewer wins.
e. Residual analysis
# Residual analysis of the transformed model
par(mfrow = c(2, 2))
plot(fielding_model)
Analysis of Residual Plots:
- Residuals vs Fitted: The residuals are generally scattered around the zero line with no obvious pattern, indicating a decent fit. However, some spread is evident at higher fitted values, suggesting potential heteroscedasticity.
- Q-Q Plot: The residuals closely follow the theoretical quantiles, but slight deviations at the tails indicate some issues with normality, particularly at the extremes.
- Scale-Location: The plot shows a fairly constant spread of residuals, which suggests that heteroscedasticity is not a significant issue.
- Residuals vs Leverage: Similar to the previous model, points 13420 and 2138 show higher leverage and Cook’s distance near the threshold, implying these points might have an outsized influence on the model.
vi. Model with Polynomial Terms
- Variables:
TEAM_BATTING_H,TEAM_PITCHING_H(with polynomial terms),TEAM_FIELDING_DP, andTEAM_FIELDING_E. - Rationale:
TEAM_BATTING_H: Captures potential non-linear effects of team hits on wins, where additional hits may have diminishing or accelerating returns.TEAM_PITCHING_H: Models non-linear effects of hits allowed by pitching, reflecting how reducing hits might impact wins differently at various levels.TEAM_FIELDING_DPandTEAM_FIELDING_E: Included as linear terms to control for the overall defensive quality while assessing the non-linear batting and pitching effects.
a. Feature engineer training data for Fielding-Focused Model
# Step 1: Create a named vector with column names and their indexes
money_ball_train_polynomial <- money_ball_train_scaled
column_indexes <- setNames(seq_along(names(money_ball_train_polynomial)), names(money_ball_train_polynomial))
# Step 2: Select variables for polynomial terms
# We'll use TEAM_BATTING_H and TEAM_PITCHING_H and create polynomial terms for them
# Ensure the variables are numeric
money_ball_train_polynomial$TEAM_BATTING_H <- as.numeric(money_ball_train_polynomial[[column_indexes["TEAM_BATTING_H"]]])
money_ball_train_polynomial$TEAM_PITCHING_H <- as.numeric(money_ball_train_polynomial[[column_indexes["TEAM_PITCHING_H"]]])b. Feature engineer evaluation data for Fielding-Focused Model
# Step 1: Create a named vector with column names and their indexes
money_ball_evaluate_polynomial <- money_ball_evaluate_scaled
column_indexes <- setNames(seq_along(names(money_ball_evaluate_polynomial)), names(money_ball_evaluate_polynomial))
# Step 2: Select variables for polynomial terms
# We'll use TEAM_BATTING_H and TEAM_PITCHING_H and create polynomial terms for them
# Ensure the variables are numeric
money_ball_evaluate_polynomial$TEAM_BATTING_H <- as.numeric(money_ball_evaluate_polynomial[[column_indexes["TEAM_BATTING_H"]]])
money_ball_evaluate_polynomial$TEAM_PITCHING_H <- as.numeric(money_ball_evaluate_polynomial[[column_indexes["TEAM_PITCHING_H"]]])c. Fit the Fielding-Focused Model
# Fit a model with polynomial terms
# We'll use quadratic (degree 2) polynomial terms for TEAM_BATTING_H and TEAM_PITCHING_H
polynomial_model <- lm(TARGET_WINS ~ poly(TEAM_BATTING_H, 2) + poly(TEAM_PITCHING_H, 2) +
TEAM_FIELDING_DP + TEAM_FIELDING_E,
data = money_ball_train_polynomial)
# Step 4: Summary of the model
summary(polynomial_model)##
## Call:
## lm(formula = TARGET_WINS ~ poly(TEAM_BATTING_H, 2) + poly(TEAM_PITCHING_H,
## 2) + TEAM_FIELDING_DP + TEAM_FIELDING_E, data = money_ball_train_polynomial)
##
## Residuals:
## Min 1Q Median 3Q Max
## -0.34066 -0.06262 0.00083 0.06195 0.52719
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 0.69178 0.01129 61.255 < 2e-16 ***
## poly(TEAM_BATTING_H, 2)1 2.37904 0.13169 18.065 < 2e-16 ***
## poly(TEAM_BATTING_H, 2)2 0.44967 0.11395 3.946 8.18e-05 ***
## poly(TEAM_PITCHING_H, 2)1 -0.35234 0.16446 -2.142 0.0323 *
## poly(TEAM_PITCHING_H, 2)2 -0.74150 0.12806 -5.790 8.00e-09 ***
## TEAM_FIELDING_DP -0.14335 0.01511 -9.484 < 2e-16 ***
## TEAM_FIELDING_E -0.18983 0.01750 -10.850 < 2e-16 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 0.09248 on 2269 degrees of freedom
## Multiple R-squared: 0.2672, Adjusted R-squared: 0.2652
## F-statistic: 137.9 on 6 and 2269 DF, p-value: < 2.2e-16c. Interpretation of the significant coefficients
- **poly(TEAM_BATTING_H, 2)_1:** Positive quadratic effect on wins (+2.37904). The first-order term indicates that the relationship between hits and wins is not strictly linear, with increasing hits leading to more wins, but at a varying rate.
- **poly(TEAM_BATTING_H, 2)_2:** Positive effect (+0.44967), reinforcing the non-linear relationship.
- **poly(TEAM_PITCHING_H, 2)_1:** Negative quadratic effect on wins (-0.53243). As the number of hits allowed increases, the negative impact on wins becomes more pronounced.
- **poly(TEAM_PITCHING_H, 2)_2:** Negative effect (-0.35234), further suggesting a non-linear, detrimental effect of pitching hits on wins.
- TEAM_FIELDING_DP: Negative impact on wins (-0.14335). More double plays are correlated with fewer wins.
- TEAM_FIELDING_E: Negative impact on wins (-0.18983). Fielding errors significantly reduce the number of wins.
d. Residual analysis
# Residual analysis of the transformed model
par(mfrow = c(2, 2))
plot(polynomial_model)
Analysis of Residual Plots:
- Residuals vs Fitted: The residuals do not appear to follow a clear pattern, indicating a reasonable model fit. However, there is some spread at higher fitted values, which may suggest slight heteroscedasticity.
- Q-Q Plot: The residuals mostly follow the theoretical quantiles, but there are deviations at the tails, suggesting some issues with the normality of residuals, particularly at the extremes.
- Scale-Location: The red line is mostly flat, indicating that the variance of residuals is relatively constant across fitted values. This suggests that heteroscedasticity is not a major issue in this model.
- Residuals vs Leverage: There are a few points with higher leverage, notably point 13420, and one point (2138) with a Cook’s distance near the threshold, indicating it could be influential. These points should be further investigated to ensure they are not unduly affecting the model.
6. Model Selection
a. Model Selection based on matrix
# Calculate the R-squared, Adjusted R-squared, AIC, BIC, and Residual Standard Error (RSE) for each model
model_list <- list(baseline_model, composite_model, interaction_model, stepwise_model, fielding_model, polynomial_model)
# Create a data frame to store the comparison metrics
model_comparison <- data.frame(
Model = c("Baseline Model", "Composite Variables Model", "Interaction Terms Model", "Feature Selection Model", "Fielding-Focused Model", "Polynomial Model"),
R_squared = sapply(model_list, function(model) summary(model)$r.squared),
Adjusted_R_squared = sapply(model_list, function(model) summary(model)$adj.r.squared),
AIC = sapply(model_list, AIC),
BIC = sapply(model_list, BIC),
RSE = sapply(model_list, function(model) sqrt(sum(residuals(model)^2) / model$df.residual))
)
# Print the comparison data frame
print(model_comparison)## Model R_squared Adjusted_R_squared AIC BIC
## 1 Baseline Model 0.3120711 0.3084232 -4500.993 -4420.770
## 2 Composite Variables Model 0.1822507 0.1808104 -4123.540 -4089.159
## 3 Interaction Terms Model 0.2080464 0.2059522 -4192.493 -4146.651
## 4 Feature Selection Model 0.3008771 0.2971698 -4464.256 -4384.033
## 5 Fielding-Focused Model 0.2583120 0.2566783 -4343.739 -4303.628
## 6 Polynomial Model 0.2671662 0.2652284 -4369.074 -4323.232
## RSE
## 1 0.08972372
## 2 0.09765158
## 3 0.09614139
## 4 0.09045077
## 5 0.09301981
## 6 0.09248328# Perform ANOVA to compare models and store the result
anova_results <- anova(baseline_model, composite_model, interaction_model, stepwise_model, fielding_model, polynomial_model)
# Extract the ANOVA results into a data frame
anova_df <- data.frame(
Model = c("Baseline Model", "Composite Variables Model", "Interaction Terms Model", "Feature Selection Model", "Fielding-Focused Model", "Polynomial Model"),
Res_Df = anova_results$Res.Df,
RSS = anova_results$RSS,
Df = c(NA, diff(anova_results$Res.Df)), # Compute the difference in Res.Df for Df column
Sum_of_Sq = c(NA, anova_results$`Sum of Sq`[-1]), # Sum of Sq starting from second model
F_value = c(NA, anova_results$F[-1]), # F-values starting from second model
Pr_F = c(NA, anova_results$`Pr(>F)`[-1]) # p-values starting from second model
)
# Display the ANOVA results data frame
print(anova_df)## Model Res_Df RSS Df Sum_of_Sq F_value Pr_F
## 1 Baseline Model 2263 18.21793 NA NA NA NA
## 2 Composite Variables Model 2271 21.65587 8 -3.4379395 53.38186 1.107219e-79
## 3 Interaction Terms Model 2269 20.97274 -2 0.6831282 42.42850 8.139747e-19
## 4 Feature Selection Model 2263 18.51438 -6 2.4583685 50.89571 5.763087e-59
## 5 Fielding-Focused Model 2270 19.64160 7 -1.1272205 20.00305 3.480850e-26
## 6 Polynomial Model 2269 19.40711 -1 0.2344812 29.12684 7.486338e-08Based on the above comparison of the models.
- Best Model:
- The Baseline Model performs well, with the highest R-squared and Adjusted R-squared values (0.312 and 0.308 respectively). It also has the lowest AIC and BIC values, indicating that it provides a good balance between model complexity and fit. This model explains the most variance.
- Second Best:
- The Feature Selection Model is very close to the Baseline Model in terms of R-squared and Adjusted R-squared values, with a slightly lower complexity as shown by its similar AIC and BIC. This suggests that the Feature Selection Model may be a suitable alternative if model simplicity is a priority.
- Other Models:
The Polynomial Model offers some improvement over the Composite Variables Model and the Fielding-Focused Model in terms of R-squared and Adjusted R-squared, but it is still not as effective as the Baseline or Feature Selection models. It shows that introducing polynomial terms adds some value, but not as much as expected.
The Interaction Terms Model performs worse than the Baseline and Feature Selection models but slightly better than the Composite Variables Model. This indicates that while interaction terms are important, they may not be as impactful as other model adjustments like feature selection.
The Composite Variables Model and Fielding-Focused Model are less effective overall, with lower R-squared and Adjusted R-squared values. This was somewhat expected given their design to focus on specific aspects of the game rather than capturing the overall complexity.
7. Prediction using the models
# Assuming the evaluation datasets are already transformed and scaled:
# money_ball_evaluate_baseline
# money_ball_evaluate_composite
# money_ball_evaluate_interaction
# money_ball_evaluate_feature_selection
# money_ball_evaluate_fielding
# money_ball_evaluate_polynomial
# Also assuming the models are:
# baseline_model, composite_model, interaction_model, stepwise_model, fielding_model, polynomial_model
# Step 1: Get the min and max values of TARGET_WINS from the original (unscaled) training data
min_target <- min(money_ball_train$TARGET_WINS)
max_target <- max(money_ball_train$TARGET_WINS)
# Step 2: Reverse the Min-Max scaling function
reverse_min_max_scale <- function(pred, min_val, max_val) {
return(pred * (max_val - min_val) + min_val)
}
# Step 3: Make predictions on the evaluation datasets
predictions_combined <- data.frame(
Baseline_Predictions = predict(baseline_model, newdata = money_ball_evaluate_baseline),
Composite_Predictions = predict(composite_model, newdata = money_ball_evaluate_composite),
Interaction_Predictions = predict(interaction_model, newdata = money_ball_evaluate_interaction),
Feature_Selection_Predictions = predict(stepwise_model, newdata = money_ball_evaluate_feature_selection),
Fielding_Predictions = predict(fielding_model, newdata = money_ball_evaluate_fielding),
Polynomial_Predictions = predict(polynomial_model, newdata = money_ball_evaluate_polynomial)
)
# Step 4: Reverse the scaling of the predictions to bring them back to their original scale
predictions_combined_original_scale <- data.frame(
Baseline_Predictions = reverse_min_max_scale(predictions_combined$Baseline_Predictions, min_target, max_target),
Composite_Predictions = reverse_min_max_scale(predictions_combined$Composite_Predictions, min_target, max_target),
Interaction_Predictions = reverse_min_max_scale(predictions_combined$Interaction_Predictions, min_target, max_target),
Feature_Selection_Predictions = reverse_min_max_scale(predictions_combined$Feature_Selection_Predictions, min_target, max_target),
Fielding_Predictions = reverse_min_max_scale(predictions_combined$Fielding_Predictions, min_target, max_target),
Polynomial_Predictions = reverse_min_max_scale(predictions_combined$Polynomial_Predictions, min_target, max_target)
)
# Step 5: Display the reversed predictions
print(predictions_combined_original_scale)## Baseline_Predictions Composite_Predictions Interaction_Predictions
## 1 78.17705 76.18385 74.24624
## 2 78.51413 73.44010 72.44069
## 3 88.12298 82.35093 78.92087
## 4 98.76405 99.71393 104.37911
## 5 88.69616 80.62631 77.42297
## 6 84.30510 88.94025 84.01810
## 7 88.41887 83.85999 86.79870
## 8 92.39348 83.39301 82.22149
## 9 89.86488 79.90986 80.21275
## 10 87.76419 85.88388 80.85467
## 11 82.54949 84.33077 82.01044
## 12 94.89749 87.04892 86.06607
## 13 93.39764 85.46037 82.03634
## 14 90.83109 88.61556 86.43144
## 15 96.56320 90.48951 86.54169
## 16 87.09654 80.60579 78.42253
## 17 85.10411 86.04610 83.05763
## 18 90.64991 92.61806 94.24988
## 19 89.82556 84.80113 81.46832
## 20 101.08645 99.63857 101.40889
## 21 99.29625 96.06732 96.21803
## 22 95.95819 95.20084 95.51877
## 23 95.20387 90.83665 85.97590
## 24 77.63303 83.21838 74.77962
## 25 96.56614 94.87525 98.21592
## 26 100.72033 98.13508 104.10688
## 27 72.93475 73.98791 69.63759
## 28 89.45513 92.09789 86.80482
## 29 100.29234 102.51898 100.77321
## 30 89.16934 99.76861 95.13044
## 31 103.29758 101.89016 101.97287
## 32 98.61237 91.69847 90.27121
## 33 96.89310 92.76724 91.71510
## 34 95.19904 91.69451 89.06064
## 35 95.83279 92.22164 88.85851
## 36 92.99360 94.16530 89.25981
## 37 90.14829 88.33958 87.43335
## 38 102.45857 93.38000 93.21653
## 39 95.03866 91.31846 91.22002
## 40 105.45706 98.79395 95.63925
## 41 91.27145 101.02975 98.95233
## 42 109.56606 105.73752 107.38785
## 43 58.89194 56.38334 54.64717
## 44 113.55826 84.94436 90.81760
## 45 100.48467 81.12756 85.17998
## 46 107.56729 89.66017 93.71117
## 47 111.74202 89.10293 92.52956
## 48 90.30035 90.33871 84.15297
## 49 87.07779 83.77065 81.71446
## 50 93.48384 91.98038 85.25010
## 51 90.90507 90.75607 88.30687
## 52 98.48108 97.04324 98.32476
## 53 92.02002 89.02285 86.09103
## 54 89.37692 84.08440 80.77517
## 55 88.47775 90.27641 87.98151
## 56 97.10792 89.56134 88.31903
## 57 109.01530 89.39839 91.83270
## 58 94.16911 87.52230 87.92501
## 59 77.82097 72.66348 74.72945
## 60 99.89554 86.45910 86.12826
## 61 107.33419 94.64413 97.63116
## 62 93.18991 81.46490 80.18280
## 63 100.92898 97.00399 97.84974
## 64 92.44078 94.59260 96.09028
## 65 94.73330 90.89451 94.24462
## 66 108.87416 93.35399 102.00651
## 67 95.24478 86.49169 84.12904
## 68 102.29157 89.56052 88.53768
## 69 94.68427 80.75344 83.11651
## 70 108.92928 87.80819 89.74354
## 71 107.02291 92.02756 89.71150
## 72 86.11216 82.83831 82.00997
## 73 88.12520 81.49305 81.23513
## 74 98.30429 87.63679 86.54724
## 75 97.94957 98.36148 96.37022
## 76 98.20945 89.44309 87.62042
## 77 98.56919 87.78581 89.89645
## 78 96.07118 93.93380 94.49354
## 79 91.08924 86.84248 85.82119
## 80 96.87154 87.30901 87.72116
## 81 95.93966 102.85298 105.89415
## 82 104.69396 104.22670 106.00452
## 83 110.33516 109.44438 115.68388
## 84 88.69665 88.87434 88.41528
## 85 97.91140 88.67890 84.68835
## 86 90.45449 86.76111 87.51732
## 87 94.85271 87.96325 88.42625
## 88 98.78717 90.10911 87.31547
## 89 102.27317 92.46009 93.48286
## 90 101.75629 103.72164 108.54706
## 91 91.93779 77.11361 79.34120
## 92 103.01974 91.74177 107.64509
## 93 89.55447 80.13047 78.48615
## 94 108.41620 93.10567 91.81506
## 95 102.79201 94.24958 95.76311
## 96 103.39308 91.65413 91.57515
## 97 103.57581 90.96327 87.22066
## 98 120.83876 106.04523 104.54819
## 99 105.06330 97.62010 97.63466
## 100 104.07997 94.23601 97.84688
## 101 97.48281 94.28743 94.46305
## 102 84.25547 86.40986 80.22546
## 103 97.26661 95.12448 91.53720
## 104 95.49506 87.80564 84.50085
## 105 88.58209 90.62811 90.29441
## 106 86.73152 85.09769 82.45571
## 107 65.14375 69.63642 64.93304
## 108 92.34623 97.48263 100.95230
## 109 102.31137 99.51690 101.93765
## 110 72.08048 81.01120 77.19468
## 111 97.72403 92.10733 93.01296
## 112 98.81818 97.30893 100.72568
## 113 111.70348 97.18547 103.37735
## 114 109.83301 101.44575 105.86788
## 115 98.52207 93.73198 94.55480
## 116 95.17655 98.14052 100.16362
## 117 102.19718 94.28895 101.34176
## 118 94.57018 94.37537 95.99595
## 119 90.47786 94.19861 93.31651
## 120 86.00232 79.29996 82.04607
## 121 107.33159 85.42234 88.39190
## 122 84.37666 81.82360 81.08287
## 123 88.29264 80.34874 78.33083
## 124 85.08595 77.47224 76.68738
## 125 87.45108 78.99502 79.32675
## 126 105.64739 94.48449 92.19980
## 127 108.93739 98.68859 95.06365
## 128 96.46367 87.27903 84.76176
## 129 108.71795 103.46045 103.07789
## 130 105.77894 100.36055 102.72447
## 131 100.62730 99.81222 98.98442
## 132 93.28199 93.53184 93.84177
## 133 87.66268 91.63756 90.50732
## 134 100.24419 89.70523 91.55420
## 135 104.53858 91.74324 95.01120
## 136 87.98890 91.63499 84.52474
## 137 88.89822 84.27938 83.80725
## 138 91.10158 88.92932 85.11696
## 139 102.40840 96.54681 100.93806
## 140 92.06019 90.46396 90.06462
## 141 85.56891 83.15165 80.93536
## 142 93.47927 80.57785 79.98433
## 143 108.79245 98.67833 104.07108
## 144 88.41700 89.93250 85.92641
## 145 88.21201 86.28367 83.40037
## 146 80.54888 83.93286 73.98848
## 147 90.64526 89.33513 89.57293
## 148 94.76796 89.95467 90.78778
## 149 93.27245 86.92926 88.34727
## 150 97.58230 97.15041 96.04798
## 151 93.49682 90.24472 90.02986
## 152 99.02187 97.99320 100.69664
## 153 63.42127 53.60365 31.08794
## 154 84.49508 82.76482 77.48973
## 155 94.94241 87.37643 87.26619
## 156 83.10965 76.76255 78.59016
## 157 104.89398 103.50440 108.57936
## 158 79.77823 76.73243 71.54244
## 159 115.98167 95.58947 95.05196
## 160 92.45301 81.94832 80.87735
## 161 119.23709 104.86370 107.30157
## 162 121.03266 103.25494 107.54499
## 163 110.92798 100.37165 99.30395
## 164 117.08621 105.38130 111.25689
## 165 111.10936 100.90948 101.16870
## 166 104.00918 92.86793 89.67950
## 167 95.65582 82.03015 81.33140
## 168 92.27878 84.95438 84.89566
## 169 84.42858 82.39544 84.54098
## 170 90.92121 93.77780 95.54035
## 171 111.68652 91.34888 93.95922
## 172 108.99257 94.48826 96.82933
## 173 96.66547 95.29596 91.86693
## 174 110.38353 102.99567 103.73976
## 175 97.10043 98.60469 96.91617
## 176 89.13761 84.58259 80.77510
## 177 89.98276 85.69921 82.64846
## 178 81.91901 78.98417 76.46273
## 179 86.39962 85.79834 79.52468
## 180 92.01782 84.50780 85.79619
## 181 93.44381 86.83747 88.97276
## 182 98.93896 96.50975 96.64355
## 183 98.07821 92.99579 95.92571
## 184 97.44144 96.55731 96.13919
## 185 111.11939 93.90085 107.61383
## 186 103.61745 94.25837 102.36101
## 187 99.37364 94.98195 99.92262
## 188 70.77130 88.13272 81.97551
## 189 69.09341 80.05886 74.54998
## 190 121.20009 108.05291 122.58096
## 191 89.65111 85.32858 81.96058
## 192 103.31011 89.81268 88.94707
## 193 91.64633 90.09568 86.03500
## 194 91.07632 90.52914 88.03304
## 195 92.82522 99.62923 98.28430
## 196 76.61401 81.38434 76.15151
## 197 90.59324 90.91765 86.76310
## 198 100.47571 91.20546 87.48440
## 199 94.38483 86.53946 83.87912
## 200 102.77546 93.53959 91.75804
## 201 92.05851 84.48094 80.81918
## 202 99.37530 89.84253 87.76346
## 203 94.13245 88.27500 85.16062
## 204 102.72031 93.62103 97.68719
## 205 93.77991 92.45904 92.38067
## 206 97.92234 89.30723 87.89251
## 207 96.69000 96.21958 99.47798
## 208 95.34705 97.68652 97.11785
## 209 92.84318 88.57758 91.43110
## 210 87.03879 77.33190 77.72205
## 211 120.22309 97.69676 105.20913
## 212 112.90955 89.75202 90.26458
## 213 102.93591 87.01468 87.54581
## 214 73.69973 75.77800 71.69358
## 215 81.21571 79.88430 73.94542
## 216 100.74120 90.93782 89.33035
## 217 98.19047 85.26099 86.52119
## 218 105.87253 98.52615 101.25293
## 219 93.33503 90.73987 89.65048
## 220 95.85995 91.47023 90.99658
## 221 91.05244 93.10447 93.35235
## 222 90.65922 91.03762 94.07691
## 223 93.55006 97.87159 100.64178
## 224 90.33101 88.58631 89.83867
## 225 107.88805 92.68147 90.48246
## 226 91.44302 81.83602 81.44303
## 227 93.65891 92.46812 90.68035
## 228 93.59290 93.14625 94.87655
## 229 99.61657 95.63026 96.55251
## 230 87.18515 83.11752 77.71921
## 231 96.91146 87.23674 87.12285
## 232 108.02665 92.99137 92.53835
## 233 89.75444 81.23254 78.21316
## 234 97.76426 91.27418 89.75353
## 235 92.21308 89.01693 86.05493
## 236 91.85923 86.24661 84.06315
## 237 95.71778 92.09926 90.67448
## 238 94.68096 86.03436 86.05902
## 239 107.75020 86.25135 87.13246
## 240 90.19941 79.51920 80.04038
## 241 106.57410 96.45743 96.35673
## 242 101.48066 96.10058 94.64443
## 243 99.10892 94.96642 93.93778
## 244 99.12934 96.00532 100.21682
## 245 69.33191 74.29938 74.62336
## 246 97.68638 93.92954 96.10741
## 247 95.02981 91.02884 89.52541
## 248 97.27347 96.69182 97.23354
## 249 84.26666 84.39881 77.40492
## 250 97.38026 92.77951 99.11458
## 251 97.63192 97.24347 100.38817
## 252 85.80779 87.01311 79.39076
## 253 106.55936 88.03743 91.56509
## 254 33.15306 39.25295 56.81013
## 255 80.16673 77.72555 75.10942
## 256 84.77039 81.34989 78.17446
## 257 99.56753 97.02074 98.82328
## 258 95.01419 93.28984 97.08887
## 259 89.94374 100.77913 100.15117
## Feature_Selection_Predictions Fielding_Predictions Polynomial_Predictions
## 1 72.15273 76.38488 73.48109
## 2 71.87236 77.02300 73.38248
## 3 80.49442 89.84173 86.11691
## 4 86.92991 97.19790 98.73778
## 5 82.44192 79.30133 81.51843
## 6 81.84048 83.73845 87.23966
## 7 86.59711 82.23539 79.91001
## 8 84.83630 90.73168 87.89559
## 9 87.51721 79.75160 77.14638
## 10 82.74758 88.58588 86.71343
## 11 76.74111 83.09745 82.11334
## 12 89.09590 91.04376 90.56467
## 13 90.04664 88.83104 87.49823
## 14 84.92933 93.14973 92.33800
## 15 89.93631 99.19281 96.08222
## 16 78.54530 88.56793 85.39546
## 17 80.58782 86.17695 83.52416
## 18 83.11239 89.48646 89.80609
## 19 83.84839 87.04820 83.23763
## 20 88.21163 99.03772 103.76238
## 21 94.00780 93.05982 94.64591
## 22 86.75936 96.14845 98.53488
## 23 93.03208 89.98327 89.85957
## 24 73.07962 83.41576 81.97252
## 25 85.56155 96.50074 96.95362
## 26 88.98518 100.23429 101.46700
## 27 69.04438 73.76334 72.57377
## 28 82.90187 89.64137 89.22274
## 29 92.06685 97.07428 99.25078
## 30 78.53803 93.61554 95.24153
## 31 94.44365 98.51972 101.45303
## 32 93.50982 94.58853 95.28785
## 33 91.42505 91.80710 92.91331
## 34 89.21939 92.18186 93.19826
## 35 90.39712 92.20068 91.56684
## 36 87.64222 94.90994 93.57873
## 37 87.77861 86.29477 85.32660
## 38 94.27314 98.20042 98.43099
## 39 88.91977 97.07234 101.52067
## 40 95.86660 105.55278 105.94756
## 41 81.35924 94.69973 96.30375
## 42 99.41534 106.29278 108.39539
## 43 51.79958 44.32493 50.44626
## 44 112.40502 94.46321 94.16662
## 45 97.67076 90.16896 88.34127
## 46 106.23105 94.60883 96.72640
## 47 106.93937 101.33498 103.86746
## 48 88.52424 86.10400 84.13236
## 49 84.71489 80.63803 78.13778
## 50 89.84064 91.76251 90.14863
## 51 83.72639 89.83317 90.15702
## 52 87.72043 99.65914 102.15200
## 53 87.35554 88.36138 86.41435
## 54 84.30104 86.40171 83.68332
## 55 82.02697 88.50158 86.92475
## 56 89.82060 94.62359 92.48187
## 57 108.23926 96.24099 95.65390
## 58 91.71831 84.86920 82.47314
## 59 71.20000 70.92626 68.49547
## 60 95.39773 91.06894 87.42243
## 61 102.98886 98.47582 99.81039
## 62 90.89553 84.53038 83.03349
## 63 93.30810 99.95238 99.65843
## 64 82.68549 93.85741 94.30693
## 65 85.84239 93.41706 92.71168
## 66 105.40016 96.79783 99.92027
## 67 91.14757 88.69130 87.03213
## 68 93.74446 97.00442 97.10452
## 69 91.15696 84.71473 80.71921
## 70 106.69018 94.30300 91.21628
## 71 100.95424 98.08020 99.19429
## 72 80.36912 84.07766 83.54437
## 73 84.45932 83.68962 82.77419
## 74 92.75980 96.31337 94.95687
## 75 84.18590 100.67510 101.50915
## 76 88.20196 98.13929 96.81446
## 77 95.89244 90.29561 89.09200
## 78 89.66986 94.59908 93.28457
## 79 87.92941 84.72062 82.06231
## 80 90.05042 90.66664 87.98265
## 81 81.04743 95.12933 99.85568
## 82 89.86632 100.16472 104.93277
## 83 88.71552 105.81486 114.11219
## 84 80.41181 87.13388 88.38365
## 85 94.19668 94.05850 92.68665
## 86 83.04618 92.50443 91.74642
## 87 89.13128 93.95055 90.93332
## 88 92.08116 97.27391 95.49414
## 89 97.90801 98.57044 96.62380
## 90 89.85141 98.01660 100.37821
## 91 88.75938 83.59645 81.14474
## 92 102.72445 88.56250 81.55585
## 93 86.81735 85.17477 82.88502
## 94 99.47717 101.77495 99.27296
## 95 97.51316 96.52957 94.56948
## 96 101.13414 94.15468 91.02509
## 97 102.15779 93.63327 91.39177
## 98 109.72336 115.68215 121.09048
## 99 96.91458 97.95816 100.41294
## 100 96.79654 93.26203 95.84332
## 101 89.33895 91.57474 93.35542
## 102 81.96320 85.56621 83.91499
## 103 89.17895 97.87148 99.04124
## 104 91.74118 94.10635 92.17650
## 105 81.69915 89.22355 88.11951
## 106 85.55365 79.42284 83.23989
## 107 63.15883 60.80869 62.63969
## 108 80.82157 89.81313 91.46665
## 109 95.82837 96.78182 97.31100
## 110 71.81845 68.70329 70.34761
## 111 94.66265 90.12679 89.13045
## 112 92.01527 91.31200 91.93339
## 113 102.07257 104.12524 105.32548
## 114 98.49148 103.74196 105.53576
## 115 89.55688 94.28908 94.01458
## 116 85.27965 91.36147 92.76187
## 117 91.04725 95.27922 97.22747
## 118 88.96372 89.38060 89.22656
## 119 81.03644 90.07916 90.24347
## 120 84.92968 76.23888 74.15258
## 121 104.70162 95.24423 93.69232
## 122 81.94394 78.53748 75.59765
## 123 82.79117 81.61236 78.65102
## 124 77.82082 77.18507 75.19597
## 125 85.32967 77.45101 74.73043
## 126 96.49748 102.62685 103.73170
## 127 99.43993 106.80784 109.55518
## 128 92.81311 89.60115 87.10604
## 129 100.65214 104.01266 106.53187
## 130 91.12815 102.43929 107.70300
## 131 90.30248 99.61705 102.50510
## 132 88.22759 91.54785 92.11796
## 133 81.43405 91.06693 88.69839
## 134 94.26134 96.41551 95.84996
## 135 97.56078 99.64027 98.23425
## 136 87.27899 89.12857 91.71722
## 137 85.33620 83.26253 81.29669
## 138 83.33015 92.77904 91.15562
## 139 94.06932 98.22998 98.31055
## 140 85.44922 92.35784 90.74263
## 141 78.64247 80.42307 77.70477
## 142 86.20488 85.05081 80.84403
## 143 98.61415 99.90052 104.06344
## 144 83.38636 85.57247 85.60701
## 145 86.55758 84.35942 83.45963
## 146 77.55691 83.06338 81.86987
## 147 86.81047 85.91025 84.53424
## 148 89.20996 90.40045 89.60993
## 149 84.60791 90.90172 90.81994
## 150 91.10612 94.97666 94.92460
## 151 86.71871 90.87142 90.06111
## 152 89.88080 96.63724 97.37305
## 153 59.40415 71.20215 28.02007
## 154 75.02409 87.13132 83.72504
## 155 89.37275 92.51933 89.70731
## 156 74.74361 81.65344 77.77128
## 157 96.39615 97.13513 98.64104
## 158 77.63683 75.32646 75.72177
## 159 112.50693 104.05868 104.85131
## 160 86.41545 86.84650 82.53119
## 161 108.07138 108.51330 113.85188
## 162 111.11790 110.56227 115.43074
## 163 106.18242 103.48886 105.40087
## 164 103.17095 107.12479 113.60542
## 165 104.07935 102.47339 105.16831
## 166 99.92961 97.03444 98.07485
## 167 90.51757 90.10417 88.78115
## 168 86.84149 89.66465 88.22169
## 169 77.80953 83.63730 81.05749
## 170 82.52777 92.07442 92.18304
## 171 107.14433 98.04412 96.61330
## 172 105.95374 97.91392 96.85098
## 173 93.61945 91.72578 92.48174
## 174 101.22982 106.38763 109.73160
## 175 90.52772 94.14155 95.69832
## 176 87.59646 84.66548 83.40119
## 177 88.87285 86.64493 85.16170
## 178 77.26141 80.54714 78.74734
## 179 82.84014 87.08521 85.69803
## 180 87.00205 88.93584 86.25817
## 181 90.83740 88.02689 85.34147
## 182 94.00236 95.47923 94.48967
## 183 90.74799 96.60247 95.82689
## 184 91.24952 95.38293 95.38856
## 185 108.53307 97.86502 93.45793
## 186 96.04409 92.70711 95.81969
## 187 93.59683 91.11009 94.17214
## 188 65.14683 70.15325 74.46463
## 189 69.01579 69.03933 71.32997
## 190 101.82218 114.49909 127.88569
## 191 84.58669 86.73597 82.93454
## 192 97.84957 95.27291 92.56010
## 193 83.25139 90.42977 89.36643
## 194 82.49588 92.69532 92.99390
## 195 79.64364 96.99816 100.07728
## 196 72.59365 80.57662 78.13273
## 197 85.83005 91.08309 89.76998
## 198 100.47244 93.32640 92.05330
## 199 93.08106 88.59143 86.89521
## 200 97.59923 97.12980 97.02125
## 201 86.60677 88.35786 85.48401
## 202 92.72838 95.75507 93.85441
## 203 91.57104 87.35670 85.54438
## 204 100.18402 97.67338 101.76800
## 205 87.33982 93.77832 92.34497
## 206 95.70905 91.75671 88.80741
## 207 89.98982 92.31661 94.72722
## 208 89.25900 92.15115 92.26166
## 209 91.84161 83.63314 84.52521
## 210 85.95931 78.02837 75.76286
## 211 115.11658 104.22853 110.27073
## 212 105.71819 102.09789 102.90162
## 213 102.45944 88.64099 86.45980
## 214 70.74308 74.16965 72.00439
## 215 79.58487 77.56077 76.04045
## 216 92.67075 97.58195 98.19389
## 217 90.00076 91.24373 90.20444
## 218 96.44173 100.35305 100.79351
## 219 86.25680 92.60503 90.85098
## 220 93.53688 89.42844 88.06028
## 221 84.52230 90.10952 91.57649
## 222 82.81499 86.56102 85.85210
## 223 82.65054 91.80347 93.36908
## 224 81.45042 87.35677 87.18884
## 225 94.89549 109.31769 116.15448
## 226 88.29015 84.72886 81.65888
## 227 86.11233 95.31497 93.66905
## 228 88.24554 88.46802 88.03779
## 229 94.95976 94.07409 93.52108
## 230 83.12563 83.50593 84.48666
## 231 94.40099 90.54093 87.00451
## 232 100.38154 101.45025 103.18142
## 233 86.96677 85.20273 83.62867
## 234 89.50834 98.60530 97.01481
## 235 88.59416 89.80573 87.34298
## 236 88.30962 87.95089 84.91346
## 237 86.90777 97.86388 96.04321
## 238 90.23441 88.41820 86.10906
## 239 107.34924 93.68731 94.11004
## 240 88.73048 80.83120 80.30063
## 241 99.13935 100.07723 101.61897
## 242 93.54107 98.39605 100.60819
## 243 90.85781 98.80079 100.06441
## 244 87.99482 94.59576 95.93433
## 245 66.32968 69.20037 68.42053
## 246 86.21524 100.14554 101.30677
## 247 86.66418 96.07934 94.26038
## 248 89.16134 94.61762 94.68638
## 249 78.86195 86.22890 83.94393
## 250 87.64569 91.85666 92.54253
## 251 89.39584 92.25856 93.15241
## 252 78.24607 82.35898 84.83688
## 253 104.25803 95.11872 96.31235
## 254 35.23215 13.51457 36.16923
## 255 73.87920 81.46579 78.47868
## 256 80.73676 85.14293 81.62669
## 257 94.38506 94.37000 93.36460
## 258 90.52887 89.32203 88.43965
## 259 78.75336 88.58339 90.21146# Step 6: Optionally, view the first few rows of the reversed predictions
head(predictions_combined_original_scale)## Baseline_Predictions Composite_Predictions Interaction_Predictions
## 1 78.17705 76.18385 74.24624
## 2 78.51413 73.44010 72.44069
## 3 88.12298 82.35093 78.92087
## 4 98.76405 99.71393 104.37911
## 5 88.69616 80.62631 77.42297
## 6 84.30510 88.94025 84.01810
## Feature_Selection_Predictions Fielding_Predictions Polynomial_Predictions
## 1 72.15273 76.38488 73.48109
## 2 71.87236 77.02300 73.38248
## 3 80.49442 89.84173 86.11691
## 4 86.92991 97.19790 98.73778
## 5 82.44192 79.30133 81.51843
## 6 81.84048 83.73845 87.23966