STA 631 Project Portfolio
Prediction of House Prices using Regression
Hillary Kemei
- 1.0 Introduction.
- 2.0 Motivation for Working on the Project.
- 3.0 Methodology
- 3.1 Data Set
- 4.0. Meeting Course Objectives.
- 4.1. Objective 1: Determine and apply the appropriate generalized linear model for a specific data context.
- 4.2 Objective Two: Describe probability as a foundation of statistical modeling, including inference and maximum likelihood estimation.
- 4.3 Objective Three 3: Conduct model selection for a set of candidate models.
- 4.4 Objective 4: Communicate the results of statistical models to a general audience.
- 4.5. Objective 5 Use programming software (i.e., R) to fit and assess statistical models.
- 6.0 Conclusion.
1.0 Introduction.
Throughout the semester, the Modeling and Regression course has significantly enhanced my understanding of various concepts and techniques in statistical analysis, areas where my proficiency was initially limited. My daily learning journey has allowed me to build a robust foundation in statistical methods, equipping me with the confidence to interpret models and extract meaningful insights from data. As part of the coursework, I applied these newly acquired skills to a practical project where I built a regression model, two regression trees, and a Random Forest model to predict housing prices. This hands-on experience was instrumental in helping me determine which model was best suited for predicting housing prices, further solidifying my understanding of how different modeling techniques can be optimized for specific types of data evaluation.
Having prior experience with R, I found the tasks of data import and preprocessing relatively straightforward, which allowed me to devote more attention to the complexities of model building and evaluation. This comprehensive approach not only enhanced my technical skills but also my ability to apply theoretical knowledge in practical, real-world settings.
1.1 Project Objectives.
This project aims to create a predictive model that estimates housing prices based on a variety of property characteristics, such as square footage, number of bedrooms, location, and other amenities. By integrating this data, the model is designed to provide stakeholders, including real estate buyers and sellers, with critical insights to make informed decisions. This endeavor not only serves to demonstrate the application of course concepts such as regression analysis but also showcases the practical impact of these models in real-world scenarios. Additionally, it serves as a culmination of my learning over the course of the semester, dramatically illustrating my improved proficiency and deeper understanding of statistical analysis through a tangible, impactful project.
2.0 Motivation for Working on the Project.
The motivation for this project is derived from the significant real-world impact that accurate predictions of housing prices can have on the real estate market. The ability to provide reliable price forecasts is essential for buyers, sellers, and investors alike. By employing advanced statistical techniques, such as multiple regression analysis—a skill honed during this course—I am able to delve into the factors that influence housing prices and develop a robust tool for market analysis. This project transcends academic exercise; it is designed not only to demonstrate my learning progress and fulfill the course objectives but also to enhance market clarity and efficiency, thereby benefiting a wide range of stakeholders.
3.0 Methodology
3.1 Data Preprocessing
The project began with a thorough data preprocessing phase, essential for ensuring the quality and relevance of the data for regression analysis. This involved:
Data Cleaning: Removing inaccuracies and outliers to ensure the integrity of the analysis.
Data Transformation: Standardizing numerical and categorical variables to enhance model performance.
Data Sampling: Splitting the dataset into training and testing sets to validate the model effectively.
3.2 Model Selection and Validation.
To predict housing prices, I utilized several models:
Linear Regression: This model was chosen for its simplicity and effectiveness in establishing relationships between independent and dependent variables.
Regression Tree, Bagged Regression, and Random Forest: These models were selected to handle non-linear data and provide a more detailed understanding of the effects of data features on housing prices. Employing multiple models allows for a comparison of their efficacy, enabling the selection of the best performer based on stability and accuracy.
3.1 Data Set
The dataset for this project was sourced from Kaggle and can be accessed via the following [Link].To utilize a CSV dataset from Kaggle, one must typically download the file to a local machine. This requirement arises from Kaggle’s authentication process, which mandates that users log in before they can directly access datasets.
HousePrices_df <- read.csv(file="C:/Users/Hill85/Desktop/CSV Data/Housing_Prices.csv")
HousePrices_df$price <- HousePrices_df$price / 10
# Displaying the first few records of the dataset
head(HousePrices_df,3)## price area bedrooms bathrooms stories mainroad guestroom basement
## 1 1330000 7420 4 2 3 yes no no
## 2 1225000 8960 4 4 4 yes no no
## 3 1225000 9960 3 2 2 yes no yes
## hotwaterheating airconditioning parking prefarea furnishingstatus
## 1 no yes 2 yes furnished
## 2 no yes 3 no furnished
## 3 no no 2 yes semi-furnished3.1.1 Dataset Attributes.
The data set used for this project has the following attributes:
- price: The selling price of the house.
- area: The total area of the property, typically measured in square feet or square meters.
- bedrooms: The number of bedrooms in the house.
- bathrooms: The number of bathrooms in the house, including both full and partial baths.
- stories: The number of levels or floors the house has.
- mainroad: Indicates whether the house has access to a main road (typically a binary attribute, yes/no).
- guestroom: Indicates the presence of a dedicated guestroom (yes/no).
- basement: Specifies if the house includes a basement (yes/no).
- hotwaterheating: Denotes whether the house is equipped with hot water heating (yes/no).
- airconditioning: Indicates if the house has air conditioning (yes/no).
- parking: The number of parking spaces available with the property.
- prefarea: Indicates if the house is located in a preferred area, which is often subjective but generally implies better location qualities (yes/no).
- furnishingstatus: The furnishing status of the house, typically categorized as furnished, semi-furnished, or unfurnished. These attributes collectively offer a detailed overview of the house’s characteristics, which are essential for analyzing and predicting house prices.
3.2 Data Exploration and Cleaning.
The data exploration and cleaning stage was approached with careful attention to detail, starting with the renaming of attributes to improve their clarity and facilitate easier handling. I conducted an in-depth review to detect and quantify any missing values within all attributes, achieving a thorough grasp of the data’s integrity. Simultaneously, I examined the dataset for any duplicate records to guarantee its authenticity and uniqueness. The presence of categorical variables presented a unique challenge, which I tackled through the application of ordinal encoding. This technique transformed these variables into a more analytically compatible format. This pivotal step was instrumental in priming the dataset for deeper analysis, establishing a robust groundwork for achieving precise and trustworthy results.
HousePrices_df <- HousePrices_df %>% rename(Price=price,
Area=area,
Bedrooms=bedrooms,
Bathrooms=bathrooms,
Stories=stories,
Mainroad_Access=mainroad,
Guestroom=guestroom,
Basement=basement,
HotwaterHeating=hotwaterheating,
Airconditioning=airconditioning,
Parking=parking,
PrefferedArea=prefarea,
FurnishingStatus=furnishingstatus)
head(HousePrices_df,4)## Price Area Bedrooms Bathrooms Stories Mainroad_Access Guestroom Basement
## 1 1330000 7420 4 2 3 yes no no
## 2 1225000 8960 4 4 4 yes no no
## 3 1225000 9960 3 2 2 yes no yes
## 4 1221500 7500 4 2 2 yes no yes
## HotwaterHeating Airconditioning Parking PrefferedArea FurnishingStatus
## 1 no yes 2 yes furnished
## 2 no yes 3 no furnished
## 3 no no 2 yes semi-furnished
## 4 no yes 3 yes furnished3.2.1. Checking for Missing values and Duplicates.
I then checked for missing values in the data.All the variables in the data did not have any missing records.
apply(HousePrices_df, 2, anyNA)## Price Area Bedrooms Bathrooms
## FALSE FALSE FALSE FALSE
## Stories Mainroad_Access Guestroom Basement
## FALSE FALSE FALSE FALSE
## HotwaterHeating Airconditioning Parking PrefferedArea
## FALSE FALSE FALSE FALSE
## FurnishingStatus
## FALSEHouseprice_duplicates<-unique(HousePrices_df$car_ID)
Houseprice_duplicates## NULL3.2.2. Standardizing Categorical Variables.
The categorical attributes were then standardized to ensure consistency in capitalization and formatting, thereby improving the data’s uniformity and readiness for analysis.
HousePrices_df[HousePrices_df[, "Airconditioning"] == "yes", "Airconditioning"] <- "Yes"
HousePrices_df[HousePrices_df[, "Airconditioning"] == "no", "Airconditioning"] <- "No "
HousePrices_df[HousePrices_df[, "Mainroad_Access"] == "yes", "Mainroad_Access"] <- "Yes"
HousePrices_df[HousePrices_df[, "Mainroad_Access"] == "no", "Mainroad_Access"] <- "No "
HousePrices_df[HousePrices_df[, "Guestroom"] == "yes", "Guestroom"] <- "Yes"
HousePrices_df[HousePrices_df[, "Guestroom"] == "no", "Guestroom"] <- "No "
HousePrices_df[HousePrices_df[, "Basement"] == "yes", "Basement"] <- "Yes"
HousePrices_df[HousePrices_df[, "Basement"] == "no", "Basement"] <- "No "
HousePrices_df[HousePrices_df[, "HotwaterHeating"] == "yes", "HotwaterHeating"] <- "Yes"
HousePrices_df[HousePrices_df[, "HotwaterHeating"] == "no", "HotwaterHeating"] <- "No "
HousePrices_df[HousePrices_df[, "PrefferedArea"] == "yes", "PrefferedArea"] <- "Yes"
HousePrices_df[HousePrices_df[, "PrefferedArea"] == "no", "PrefferedArea"] <- "No "
HousePrices_df[HousePrices_df[, "FurnishingStatus"] == "furnished", "FurnishingStatus"] <- "Furnished"
HousePrices_df[HousePrices_df[, "FurnishingStatus"] == "semi-furnished", "FurnishingStatus"] <- "SemiFurnished "
HousePrices_df[HousePrices_df[, "FurnishingStatus"] == "unfurnished", "FurnishingStatus"] <- "Unfurnished"
head(HousePrices_df, 4)## Price Area Bedrooms Bathrooms Stories Mainroad_Access Guestroom Basement
## 1 1330000 7420 4 2 3 Yes No No
## 2 1225000 8960 4 4 4 Yes No No
## 3 1225000 9960 3 2 2 Yes No Yes
## 4 1221500 7500 4 2 2 Yes No Yes
## HotwaterHeating Airconditioning Parking PrefferedArea FurnishingStatus
## 1 No Yes 2 Yes Furnished
## 2 No Yes 3 No Furnished
## 3 No No 2 Yes SemiFurnished
## 4 No Yes 3 Yes Furnished4.0. Meeting Course Objectives.
4.1. Objective 1: Determine and apply the appropriate generalized linear model for a specific data context.
To achieve this course objective, I conducted comprehensive exploratory data analysis and correlation assessments, supported by detailed summary statistics. This preliminary work, illustrated through specific code snippets, provided essential insights into variable distributions, relationships, and the overall data structure. By employing summary statistics, I effectively normalized and summarized the dataset, thereby setting the stage for precise modeling. Additionally, the use of Pearson correlation, as demonstrated in the code, helped quantify the relationships between key variables, highlighting their impact on housing prices. These steps were vital for the informed selection and application of generalized linear models, ensuring they were accurately adapted to the complexities of the dataset. The code snippets used for these analyses serve as solid evidence of the meticulous process undertaken to meet this objective.
Data Description and Distribution
In this subsection, the statistical distributions of the dataset’s attributes were examined to explore their interrelationships and their connections with Price, the target variable.
Bedroom Count Distribution Across Houses.
This analysis aimed to identify the distribution of bedroom counts across the houses in the dataset.Houses with three bedrooms were the most prevalent, numbering 300 and accounting for 55.04% of the dataset. Two-bedroom houses were the next most common, with 136 instances making up 24.95%. Conversely, houses with five bedrooms and one bedroom were the least common, with counts of 10 and 2, representing 1.83% and 0.36% of the dataset, respectively.
Bedrooms_COUNT <- HousePrices_df %>%
group_by(Bedrooms) %>%
summarise(n = n()) %>%
mutate(Freq = n / sum(n) * 100) %>%
as.data.frame() %>%
arrange(desc(Freq))
#head(Bedrooms_COUNT, 2)
#Plotting the distribution of the bedrooms.
barplot(Bedrooms_COUNT$Freq, names.arg = Bedrooms_COUNT$Bedrooms,
col = "skyblue", border = "black",
main = "Bar Plot of the Frequency of The Number Bedrooms Count",
xlab = "Number of Bedrooms", ylab = "Percentage",
cex.names = 0.8)
Distribution of the Stories in the House data set.
The distribution of stories in the houses, as explored through the data, reveals a concentration around houses with fewer stories. Specifically, the majority of houses in the dataset have either 2 stories, accounting for approximately 43.7% of the total, or 1 story, making up about 41.7%. Houses with 4 and 3 stories are significantly less common, constituting roughly 7.5% and 7.2% of the dataset, respectively. This distribution indicates a strong preference or prevalence of 1- and 2-story houses within the dataset, with multi-story houses (3 or 4 stories) being relatively rare.
Stories_COUNT <- HousePrices_df %>%
group_by(Stories) %>%
summarise(n = n()) %>%
mutate(Freq = n / sum(n) * 100) %>%
arrange(desc(Freq))
#head(Stories_COUNT, 2)
ggplot(data = Stories_COUNT, aes(x = reorder(Stories, -Freq), y = Freq)) +
geom_col(fill = "chocolate", width = 0.7, position = "identity") +
geom_text(aes(label = sprintf("%.1f%%", Freq)), vjust = -0.5, hjust = 1) +
labs(y = "Stories Distribution", x = "Number of Stories in the Houses",
title = "Number of Stories Distribution") +
theme_minimal() +
theme(axis.text.x = element_text(angle = 0, hjust = 1))
Distribution of the Furnishing Status of the Houses.
The distribution of furnishing status in the dataset reveals a preference for Semi-Furnished homes, accounting for approximately 41.7% of the total. Unfurnished homes follow with about 32.7%, and Furnished homes make up around 25.7%. This distribution suggests that a significant portion of homeowners or renters might prioritize homes that offer some level of furnishing without being fully furnished, providing a balance between convenience and the opportunity to personalize their living space. The lesser preference for completely furnished or unfurnished homes indicates varied needs and preferences among the population, with a slight leaning towards semi-furnished options for their moderate convenience and customization potential.
FurnishingStatus_COUNT <- HousePrices_df %>%
group_by(FurnishingStatus) %>%
summarise(n = n()) %>%
mutate(Freq = n / sum(n) * 100) %>%
arrange(desc(Freq))
custom_colors <- c("#3498db", "purple", "#2ecc71")
# Plotting as a pie chart.
pie(FurnishingStatus_COUNT$Freq,
labels = paste(FurnishingStatus_COUNT$FurnishingStatus, sprintf("%.1f%%", FurnishingStatus_COUNT$Freq)),
col = custom_colors,
main = "Pie Chart of Furnishing Status Distribution")
Price Distribution Histogram
The histogram depicting the distribution of housing prices, shows a range that spans from lower-priced homes to higher-priced ones. The bulk of the homes are clustered around the middle price range, with a peak density suggesting that most homes are priced around 5,000,000. There are fewer homes priced at the lower end around 3,000,000 and at the higher end around 10,000,000. The shape of the distribution is somewhat bell-shaped with a slight right skew, indicating that while the majority of houses are moderately priced, there are still a notable number of houses that are priced higher, stretching the tail of the distribution to the right. The distribution is not perfectly symmetrical, which is typical in real estate markets where a variety of factors can cause a wide range of prices.
ggplot(HousePrices_df, aes(x=Price)) +
geom_histogram(color="black", fill="lightblue", bins = 30) +
labs(x='Price ', y='Price(Freq)') +
labs(x = "Price", y = "Price Density", title = "Distribution of House Prices") +
theme_minimal() +
scale_x_continuous(trans='log10', labels = scales::label_number(accuracy = 1))
Price Distribution Histogram Distribution of the Air condition status of the Houses.
The box plot reveals that houses with air conditioning tend to have a higher median price and display a wider spread in pricing, including numerous high-value outliers. In contrast, houses without air conditioning show a lower median price and a narrower price distribution, with fewer outliers. This pattern suggests that air conditioning contributes to a home’s value, as evidenced by the elevated prices and increased variance in the cost of air-conditioned properties.
ggplot(HousePrices_df, aes(x=Airconditioning, y=Price, fill=Airconditioning)) +
geom_boxplot()+
labs(x="Airconditioning",y="Bedrooms",
title=" Status of Airconditioning in the Houses ")+
labs(x="Airconditioning",y="Price")+
theme(plot.title =element_text(color="black",size=12,face="bold",
lineheight = 0.8),
axis.text.x = element_text())+
scale_y_continuous(labels = scales::comma_format(scale = 1, accuracy = 1))
Distribution of Parking Spaces relative to the Prices of the Houses.
The box plot indicates that houses with no parking spaces have the lowest price range, while the inclusion of one, two, or three parking spaces is associated with higher house prices. There’s a noticeable increase in the median price from houses with no parking to those with one parking space. The median prices for homes with two and three parking spaces are similar and are higher than those with no or one parking space. This pattern suggests that having a parking space is a valuable feature that correlates with an increase in house prices within the dataset.
ggplot(data = HousePrices_df, aes(x = as.factor(Parking), y = Price, fill = as.factor(Parking))) +
geom_boxplot() +
labs(x = "Parking", y = "Price", title = "Prices of Houses Relative to the Number of Parking Spaces.") +
theme(plot.title = element_text(color = "black", size = 12, face = "bold", lineheight = 0.8)) +
theme(axis.text.x = element_text(angle = 0, hjust = 0.5)) +
scale_y_continuous(labels = scales::comma_format())
Distribution of Furnishing Status relative to the Prices of the Houses.
From the boxplot, it can be inferred that furnished houses tend to have a higher median price compared to semi-furnished and unfurnished houses. Additionally, there’s a wider range and more outliers in the prices of furnished homes, indicating greater variability in how much buyers are willing to pay for these properties.
ggplot(HousePrices_df, aes(x = FurnishingStatus, y = Price, fill = FurnishingStatus)) +
geom_boxplot() +
labs(
title = "Prices of Houses relative to Furnishing Status",
x = "Furnishing Status",
y = "Price"
) +
scale_y_continuous(labels = scales::comma_format(scale = 1, accuracy = 1))
Data Transformation.
The dataset contained five categorical attributes: Car Make, Seller Type, Fuel Type, Transmission, and Num_Owners. Since regression analysis requires numerical variables, these attributes were converted to integers using ordinal encoding. To achieve this, str_replace was employed to normalize these categories into a numerical format suitable for regression analysis.
# Converting Mainroad_Access Attribute into ordinal.
# Replacing 'Yes' with '0' and 'No' with '1' in the Mainroad_Access column.
Mainroad_Access_ordinal <- HousePrices_df$Mainroad_Access <- str_replace(HousePrices_df$Mainroad_Access, 'Yes', '0')
Mainroad_Access <- HousePrices_df$Mainroad_Access <- str_replace(HousePrices_df$Mainroad_Access, 'No', '1')
# Making Guestroom ordinal
# Replacing 'Yes' with '0' and 'No' with '1' in the Guestrooms column.
HousePrices_df$Guestroom <- str_replace(HousePrices_df$Guestroom, 'Yes', '0')
HousePrices_df$Guestroom <- str_replace(HousePrices_df$Guestroom, 'No', '1')
# Making Basement Attribute Ordinal.
# Replacing 'Yes' with '0' and 'No' with '1' in the Basement column
HousePrices_df$Basement <- str_replace(HousePrices_df$Basement, 'Yes', '0')
HousePrices_df$Basement <- str_replace(HousePrices_df$Basement, 'No', '1')
# Making HotwaterHeating Attribute Ordinal.
# Replacing 'Yes' with '0' and 'No' with '1' in the HotwaterHeating column
HousePrices_df$HotwaterHeating <- str_replace(HousePrices_df$HotwaterHeating, 'Yes', '0')
HousePrices_df$HotwaterHeating <- str_replace(HousePrices_df$HotwaterHeating, 'No', '1')
# Making Airconditioning Attribute Ordinal.
# Replacing 'Yes' with '0' and 'No' with '1' in the Airconditioning column
HousePrices_df$Airconditioning <- str_replace(HousePrices_df$Airconditioning, 'Yes', '0')
HousePrices_df$Airconditioning <- str_replace(HousePrices_df$Airconditioning, 'No', '1')
# Making FurnishingStatus Attribute Ordinal.
# Replacing 'Yes' with '0' and 'No' with '1' in the PrefferedArea column
HousePrices_df$PrefferedArea <- str_replace(HousePrices_df$PrefferedArea, 'Yes', '0')
HousePrices_df$PrefferedArea <- str_replace(HousePrices_df$PrefferedArea, 'No', '1')
# Making PrefferedArea Attribute Ordinal.
# Replacing 'Unfurnished' with '0', 'SemiFurnished' with '1', and 'Furnished' with '2'
HousePrices_df$FurnishingStatus <- str_replace(HousePrices_df$FurnishingStatus, 'Unfurnished', '0')
HousePrices_df$FurnishingStatus <- str_replace(HousePrices_df$FurnishingStatus, 'SemiFurnished', '1')
HousePrices_df$FurnishingStatus <- str_replace(HousePrices_df$FurnishingStatus, 'Furnished', '2')Obtaining Summary Statistics of the transformed data set.
After completing the data transformation process, I proceeded to analyze the predictor variables by generating their summary statistics. This was accomplished using the describe function from the psych package in R, applied to the transformed HousePrices_df dataset. This step is crucial as it allows for a deeper understanding of the distribution, central tendency, and variability of the data. By computing metrics such as mean, standard deviation, and range for each variable, I was able to obtain a comprehensive summary of the dataset’s post-transformation characteristics. These insights are invaluable for informing further analysis and decision-making processes.
psych::describe(HousePrices_df)## vars n mean sd median trimmed mad
## Price 1 545 476672.92 187043.96 434000 455929.94 155673.00
## Area 2 545 5150.54 2170.14 4600 4908.41 2060.81
## Bedrooms 3 545 2.97 0.74 3 2.93 0.00
## Bathrooms 4 545 1.29 0.50 1 1.21 0.00
## Stories 5 545 1.81 0.87 2 1.66 1.48
## Mainroad_Access* 6 545 1.14 0.35 1 1.05 0.00
## Guestroom* 7 545 1.82 0.38 2 1.90 0.00
## Basement* 8 545 1.65 0.48 2 1.69 0.00
## HotwaterHeating* 9 545 1.95 0.21 2 2.00 0.00
## Airconditioning* 10 545 1.68 0.47 2 1.73 0.00
## Parking 11 545 0.69 0.86 0 0.59 0.00
## PrefferedArea* 12 545 1.77 0.42 2 1.83 0.00
## FurnishingStatus* 13 545 1.93 0.76 2 1.91 1.48
## min max range skew kurtosis se
## Price 175000 1330000 1155000 1.21 1.91 8012.08
## Area 1650 16200 14550 1.31 2.69 92.96
## Bedrooms 1 6 5 0.49 0.70 0.03
## Bathrooms 1 4 3 1.58 2.12 0.02
## Stories 1 4 3 1.08 0.65 0.04
## Mainroad_Access* 1 2 1 2.05 2.22 0.01
## Guestroom* 1 2 1 -1.68 0.82 0.02
## Basement* 1 2 1 -0.63 -1.61 0.02
## HotwaterHeating* 1 2 1 -4.33 16.78 0.01
## Airconditioning* 1 2 1 -0.79 -1.38 0.02
## Parking 0 3 3 0.84 -0.59 0.04
## PrefferedArea* 1 2 1 -1.25 -0.44 0.02
## FurnishingStatus* 1 3 2 0.12 -1.27 0.03Exploring the Correlation of the independent variables .
The correlation analysis of the dataset highlights a significant correlation between the price of houses and their area, with the ‘Area’ variable centrally emphasized in the diagram, underscoring its paramount importance in price determination. While the area stands out as the most influential factor, the number of bathrooms, bedrooms, and stories also exhibit notable correlations with price, albeit to a lesser extent. In contrast, parking availability presents the weakest link to house prices, indicating that its impact, though present, is considerably overshadowed by other property characteristics. This insight provides a detailed perspective on the elements that contribute to the valuation of houses, guiding stakeholders in the real estate market towards more informed decisions.
Exploring the Correlation of the independent variables Using Pearson Correlation.
The Pearson correlation between the variables was also obtained. A coefficient of 0.5 to 0.7 indicates attributes that are moderately correlated, while a coefficient below 0.5 suggests a low correlation between variables (Sedgwick, 2012).
suppressWarnings(ggcorr(HousePrices_df, label = TRUE))
Data Standardization
The numeric columns were standardized in the dataset with the use of the scale function. This transformation normalizes each feature to have a mean of zero and a standard deviation of one, ensuring that no single variable will disproportionately influence the model due to scale differences. Standardizing the data is a crucial step in preparing it for effective analysis and modeling, as it enhances both the stability and performance of subsequent statistical tests and predictive models.
# Numeric columns: Price, Area, Bedrooms, Bathrooms, Stories, Parking
# Standardizing the numeric columns
HousePrices_df <- HousePrices_df %>%
mutate(across(c(Price, Area, Bedrooms, Bathrooms, Stories, Parking), scale))
# View the standardized data
head(HousePrices_df, 4)## Price Area Bedrooms Bathrooms Stories Mainroad_Access Guestroom
## 1 4.562174 1.045766 1.40213123 1.420507 1.3769519 0 1
## 2 4.000809 1.755397 1.40213123 5.400847 2.5296997 0 1
## 3 4.000809 2.216196 0.04723492 1.420507 0.2242042 0 1
## 4 3.982096 1.082630 1.40213123 1.420507 0.2242042 0 1
## Basement HotwaterHeating Airconditioning Parking PrefferedArea
## 1 1 1 0 1.516299 0
## 2 1 1 0 2.676950 1
## 3 0 1 1 1.516299 0
## 4 0 1 0 2.676950 0
## FurnishingStatus
## 1 2
## 2 2
## 3 1
## 4 24.2 Objective Two: Describe probability as a foundation of statistical modeling, including inference and maximum likelihood estimation.
To fulfill this objective, I utilized the foundational principles of probability by systematically splitting the dataset, building the regression model, and evaluating its performance, as detailed in the code snippets and subsequent explanations. I divided the dataset into training and testing sets using an 80/20 split, employing a random sampling method to ensure each subset is representative of the overall data distribution. This approach highlights the direct application of probabilistic methods to handle data variability. In building the linear regression model, I applied maximum likelihood estimation (MLE) to optimize the regression coefficients, focusing on maximizing the likelihood of observing the data given the model, which presupposes normally distributed errors. The model’s summary, which includes p-values and confidence intervals, illustrates the statistical inference derived from probabilistic concepts, affirming the contribution of each predictor under normality assumptions. In interpreting the outputs, including p-values, and evaluating the model’s performance through metrics such as R-squared, F-statistic, Mean Squared Error (MSE), and Root Mean Squared Error (RMSE), I demonstrated how these probabilistic calculations help assess the model’s fit, accuracy, and predictive power, indicating how closely the model’s predictions align with the actual data. These steps collectively showcase the practical application and theoretical integration of probability in statistical modeling, providing a comprehensive demonstration of meeting this particular course objective the subsequent R code executions and output explanation.
set.seed(105)
HousePrices_df$id <- 1:nrow(HousePrices_df)
#splitting 80% of the dataset into training set and leaving the remaining 20% as the test set.
training_set <- HousePrices_df %>% dplyr::sample_frac(0.8)
testing_set <- dplyr::anti_join(HousePrices_df, training_set, by = 'id')
#Removing id columns from both sets.
training_data <- subset(training_set, select = -c(id))
testing_data <- subset(testing_set, select = -c(id))
dim(training_data)## [1] 436 13dim(testing_data)## [1] 109 13Building Linear Regression model.
I developed a linear regression model, designating Price as the dependent variable. The predictor variables I included were: Area, Bedrooms, Bathrooms, Stories, Mainroad Access, Guestroom, Basement, Hotwater Heating, Airconditioning, Parking, Preferred Area, and Furnishing Status.
#Bulding the MLR model.
set.seed(400) #for reproducibility
Multiple_Regression_model <- lm(
formula = Price ~ Area + Bedrooms + Bathrooms +
Stories + Mainroad_Access + Guestroom +
Basement + HotwaterHeating +Airconditioning + Parking + PrefferedArea + FurnishingStatus ,data = training_data)
summary(Multiple_Regression_model)##
## Call:
## lm(formula = Price ~ Area + Bedrooms + Bathrooms + Stories +
## Mainroad_Access + Guestroom + Basement + HotwaterHeating +
## Airconditioning + Parking + PrefferedArea + FurnishingStatus,
## data = training_data)
##
## Residuals:
## Min 1Q Median 3Q Max
## -1.4104 -0.3382 -0.0383 0.2608 2.7655
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 1.08681 0.16265 6.682 7.48e-11 ***
## Area 0.28146 0.03302 8.524 2.76e-16 ***
## Bedrooms 0.04120 0.03074 1.340 0.180896
## Bathrooms 0.25317 0.03080 8.220 2.53e-15 ***
## Stories 0.23462 0.03342 7.021 8.84e-12 ***
## Mainroad_Access1 -0.20721 0.08421 -2.461 0.014268 *
## Guestroom1 -0.18221 0.07753 -2.350 0.019215 *
## Basement1 -0.21384 0.06459 -3.311 0.001011 **
## HotwaterHeating1 -0.36845 0.12764 -2.887 0.004093 **
## Airconditioning1 -0.44549 0.06443 -6.914 1.75e-11 ***
## Parking 0.15477 0.02999 5.161 3.78e-07 ***
## PrefferedArea1 -0.35169 0.06796 -5.175 3.54e-07 ***
## FurnishingStatus1 0.23795 0.06400 3.718 0.000228 ***
## FurnishingStatus2 0.20493 0.07347 2.789 0.005521 **
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 0.56 on 422 degrees of freedom
## Multiple R-squared: 0.699, Adjusted R-squared: 0.6897
## F-statistic: 75.38 on 13 and 422 DF, p-value: < 2.2e-16Model Interpretation.
Predictor Variables and Their Influence on Price.
The coefficients of predictor variables such as Area, Bedrooms, Bathrooms, etc., represent their individual impacts on the Price, with Area notably increasing Price by approximately 24.259 per unit area increase. These variables serve to illustrate the rate of change in Price as each variable adjusts.
Statistical Significance of Predictors.
The p-values (Pr(>|t|)) assess each predictor’s significance against a standard alpha level of 0.05. Significant predictors like Area, Bathrooms, and Airconditioning have p-values below this threshold, indicating a statistically meaningful impact on Price.
The Model’s Explanation Power: R-squared of 0.699.
The R-squared value indicates that approximately 69.9% of the variance in Price is explained by the model’s predictors. This measure of model fit suggests a strong correlation between the predictors and the dependent variable.
Adjusting for Predictor Count: Adjusted R-squared of 0.6897.
The Adjusted R-squared takes into account the number of predictors used, which help to provide a more accurate depiction of the model’s explanatory power. The model indicates that With a slight adjustment for the predictor count, it still explains a significant portion of the variance in Price.
Model Significance and Predictive Accuracy: F-statistic.
The F-statistic and its associated p-value test the model’s overall significance. With a p-value well below the 0.05 threshold, the model is statistically significant, indicating that the predictors, as a set, have a substantial impact on Price. The residual standard error provides further insight into the model’s predictive accuracy, highlighting the standard deviation of the residuals and thereby the typical prediction error.
MSE.
The Mean Squared Error (MSE) for the training and testing datasets quantifies the average squared difference between the actual and the model-predicted house prices. With MSE values of 0.3035196 for the training data and 0.3844732 for the testing data, these figures highlight the variance in prediction accuracy across the datasets. Generally, a lower MSE value denotes higher precision in predictions. The model demonstrates better accuracy on the training data compared to the testing data, a common outcome due to models being optimized on training data. However, the difference in MSE values also points to the model’s capacity to generalize to unseen data, albeit with room for improvement. These MSE figures underscore the need for model refinement to enhance prediction accuracy and reduce the gap in performance between training and testing phases, aiming for a model that is both accurate and robust in predicting house prices across diverse datasets.
# Function for MSE calculation.
mse <- function(actual, predicted) {
mean((predicted - actual)^2)
}
# Generating the predictions for the training dataset
predicted_train <- predict(Multiple_Regression_model, newdata = training_data)
# Calculating MSE for the training dataset
mse_training <- mse(actual = training_data$Price, predicted = predicted_train)
# Generating predictions for the testing dataset
predicted_test <- predict(Multiple_Regression_model, newdata = testing_data)
# Calculating MSE for the testing dataset
mse_testing <- mse(actual = testing_data$Price, predicted = predicted_test)
cat("Training MSE:", mse_training, "\n")## Training MSE: 0.3035196cat("Testing MSE:", mse_testing, "\n")## Testing MSE: 0.3844732RMSE
The Root Mean Squared Error (RMSE) on both training and testing data measures the model’s accuracy. The RMSE values of 0.5509261 for training and 0.620059 for testing reflect the average differences between the predicted and actual house prices. Normally, lower RMSE values indicate better model accuracy. These results show that the model is relatively accurate, with a slightly better performance on the training dataset. This difference is expected as models typically perform better on data that they were trained on. The relatively close RMSE values between the training and testing datasets suggest the model’s good generalization ability, indicating its usefulness for predicting house prices.
# Function to calculate RMSE
rmse <- function(actual, predicted) {
sqrt(mean((predicted - actual)^2))
}
# Generate predictions for the training dataset
predicted_train <- predict(Multiple_Regression_model, newdata = training_data)
# Calculate RMSE for the training dataset
rmse_training <- rmse(actual = training_data$Price, predicted = predicted_train)
# Generate predictions for the testing dataset
predicted_test <- predict(Multiple_Regression_model, newdata = testing_data)
# Calculate RMSE for the testing dataset
rmse_testing <- rmse(actual = testing_data$Price, predicted = predicted_test)
# Print the RMSE values
cat("Training RMSE:", rmse_training, "\n")## Training RMSE: 0.5509261cat("Testing RMSE:", rmse_testing, "\n")## Testing RMSE: 0.620059R-Squared Values.
The R-squared values obtained from the analysis of both training and testing data using the Multiple_Regression_model serve as indicators of the model’s explanatory power. With an R-squared of 0.6989956 for the training data, the model explains approximately 69.9% of the variance in house prices, showcasing a strong ability to fit the data it was trained on. Transitioning to the testing dataset, the R-squared value sees a decline to 0.5925821, indicating that the model explains around 59.3% of the variance in unseen data. This reduction is anticipated, as models generally exhibit superior performance on training data. Despite the decrease, the model demonstrates commendable generalizability, retaining over half of its explanatory power when applied to new data. These findings underscore the model’s robustness and its potential as a reliable tool for predicting house prices in varying contexts.
# Calculating R-squared for the training data
# Actual prices for the training data
actual_price_train <- training_data$Price
# Using the already predicted prices for the training dataset
# predicted_train is obtained from predict(Multiple_Regression_model, newdata = training_data)
# Sum of squares of residuals for training data
r_s_train <- sum((predicted_train - actual_price_train) ^ 2)
# Total sum of squares for training data
t_s_train <- sum((actual_price_train - mean(actual_price_train)) ^ 2)
# R-squared for the training data
r_squared_train <- 1 - r_s_train / t_s_train
# Calculate R-squared for the testing data
# Use the already predicted prices for the testing dataset
# Extract actual prices for the testing data
actual_price_test <- testing_data$Price
# Sum of squares of residuals for testing data
r_s_test <- sum((predicted_test - actual_price_test) ^ 2)
# Total sum of squares for testing data
t_s_test <- sum((actual_price_test - mean(actual_price_test)) ^ 2)
# R-squared for the testing data
r_squared_test <- 1 - r_s_test / t_s_test
# Print the R-squared values
cat("Training R-squared:", r_squared_train, "\n")## Training R-squared: 0.6989956cat("Testing R-squared:", r_squared_test, "\n")## Testing R-squared: 0.5925821Plotting predicted Price vs. actual price Using the Testing data
# Adding the predicted prices to the testing data frame for plotting
testing_data$Predicted_Price <- predicted_test
# Plotting actual price vs. predicted price
ggplot(testing_data, aes(x = Predicted_Price, y = Price)) +
geom_point(aes(color = "blue"), alpha = 0.7) +
geom_abline(intercept = 0, slope = 1, linetype = "solid", color = "red") +
labs(x = "Predicted Price", y = "Actual Price", title = "Actual vs. Predicted House Prices") +
theme_minimal() +
theme(legend.position = "none") +
geom_smooth(method = lm, se = FALSE, color = "darkblue", linetype = "solid")## `geom_smooth()` using formula = 'y ~ x'
Plotting the Residuals vs Fitted plot and Normal Q-Q plot
#Plotting the Q-Q plot()
plot(Multiple_Regression_model, which = c(2))
# Plotting the Residuals vs Fitted plot.
residuals_vs_fitted <- data.frame(
Fitted = Multiple_Regression_model$fitted.values,
Residuals = residuals(Multiple_Regression_model)
)
ggplot(residuals_vs_fitted, aes(x = Fitted, y = Residuals)) +
geom_point() +
geom_smooth(se = FALSE, color = "blue") +
labs(title = "Residuals vs Fitted",
x = "Fitted Values",
y = "Residuals") +
theme_minimal()## `geom_smooth()` using method = 'loess' and formula = 'y ~ x'
4.3 Objective Three 3: Conduct model selection for a set of candidate models.
I achieved this objective by evaluating multiple regression models,
including regression trees and random forests. Initially, I adjusted
variables within the linear regression model to observe changes in the
R-squared values, gaining insights into the model’s explanatory power. I
then constructed a regression tree using the rpart package
and assessed its accuracy with the RMSE metric. This was compared to the
performance of bagged regression trees and random forest models, which
incorporated cross-validation to enhance robustness. Additionally, I
used the varImp function to identify key predictors. Guided
by these performance metrics, I was able to select the most effective
model for predicting housing prices, demonstrating a structured and
empirical approach to model selection.
Validating the Multiple Linear Regression Model.
To better understand the generalizability of my regression model, I decided to use a Regression Tree to predict the price based on thirteen key predictors: Area, Bedrooms, Bathrooms, Stories, Mainroad Access, Guestroom, Basement, Hotwater Heating, Air Conditioning, Parking, Preferred Area, and Furnishing Status. I utilized the rpart package in R for this purpose, specifically setting the method to ANOVA. This decision ensures consistency in model fitting, countering the default behavior of rpart, which automatically selects different methods that can vary with each execution.
Basic Regression Tree as a validation model.
set.seed(106)
Regression_Tree_model <- rpart(
formula = Price ~ Area + Bedrooms + Bathrooms +
Stories + Mainroad_Access + Guestroom +
Basement + HotwaterHeating +Airconditioning + Parking + PrefferedArea + FurnishingStatus,
data = training_set,
method = "anova")
Regression_Tree_model## n= 436
##
## node), split, n, deviance, yval
## * denotes terminal node
##
## 1) root 436 439.643200 -0.02619032
## 2) Area< 0.3591742 304 148.754700 -0.39978500
## 4) Airconditioning=1 233 73.569010 -0.56843240
## 8) FurnishingStatus=0 93 22.259500 -0.84035400 *
## 9) FurnishingStatus=1 ,2 140 39.864970 -0.38779880
## 18) Area< -0.6937527 70 15.526890 -0.57766060 *
## 19) Area>=-0.6937527 70 19.291440 -0.19793700 *
## 5) Airconditioning=0 71 46.810990 0.15366370
## 10) Bathrooms< 0.4254217 49 15.910450 -0.13530850 *
## 11) Bathrooms>=0.4254217 22 17.695380 0.79728360
## 22) Area< -0.2859451 9 3.080105 0.24714550 *
## 23) Area>=-0.2859451 13 10.005650 1.17814800 *
## 3) Area>=0.3591742 132 150.740400 0.83420950
## 6) Stories< -0.3521697 55 22.221680 0.16821410
## 12) Basement=1 22 5.845884 -0.29737890 *
## 13) Basement=0 33 8.427308 0.47860950 *
## 7) Stories>=-0.3521697 77 86.698310 1.30992000
## 14) Parking< -0.2246764 24 13.254870 0.64337590
## 28) Bathrooms< 0.4254217 11 4.037243 0.16991520 *
## 29) Bathrooms>=0.4254217 13 4.665353 1.04399600 *
## 15) Parking>=-0.2246764 53 57.952260 1.61175200
## 30) Parking< 0.9359742 25 8.674816 1.29091100 *
## 31) Parking>=0.9359742 28 44.406220 1.89821700
## 62) Bathrooms< 0.4254217 12 15.512080 1.27397600 *
## 63) Bathrooms>=0.4254217 16 20.710940 2.36639800 *#summary(RTree_model)Obtaining the Basic Regression Tree’s training error from the training data
#predictions for the training set using the regression tree model
predictions_train <- predict(Regression_Tree_model, training_data)
# Computing the RMSE using the predictions and the actual prices from the training set
training_rmse <- RMSE(predictions_train, training_data$Price)
# Output the training RMSE
cat("Training RMSE for the Regression Tree Model: ", training_rmse, "\n")## Training RMSE for the Regression Tree Model: 0.5942146Obtaining the Basic Regression Tree’s testing error from the testing set.
# Predicting on the testing dataset
pred_test <- predict(Regression_Tree_model, testing_data)
# RMSE for the testing dataset
rmse_test <- RMSE(pred_test, testing_data$Price)
# Printing the testing RMSE
cat("Testing RMSE for Regression Tree: ", rmse_test, "\n")## Testing RMSE for Regression Tree: 0.8671303# Finding R-Squared
pred <- predict(Regression_Tree_model, training_data)
# Calculating the R-squared
rss <- sum((training_data$Price - pred)^2)
tss <- sum((training_data$Price - mean(training_data$Price))^2)
r_squared_tree <- 1 - (rss / tss)
# Print R-squared
cat("R-squared for Regression Tree: ", r_squared_tree, "\n")## R-squared for Regression Tree: 0.6498351The regression tree model began with 436 observations in the root node, initially splitting based on Area, with branches for values below and above 0.3591742. Further splits occurred for Area values less than 0.3591742 based on Airconditioning, resulting in nodes of 233 (with air conditioning) and 71 (without). Further significant splits involved attributes like FurnishingStatus, Bathrooms, and Basement, each refining the prediction of Price. The model’s efficacy was demonstrated by a Multiple R-squared of 0.699, indicating it explained approximately 70% of the variability in Price, with training and testing RMSE of 0.5509 and 0.6201, respectively, suggesting good predictive accuracy. The tree effectively uses structural and amenity-related features to segment the housing market and predict prices.
Plotting the Regression Tree model
rpart.plot(Regression_Tree_model)
### Enhancing the Efficiency of the Regression Tree Using Bootstrap
Aggregating (Bagging).
Regression trees often exhibit high variance, which can be significantly reduced through the technique of bagging (Bootstrap aggregating), a method that decreases the likelihood of overfitting by averaging the outputs of multiple trees to smooth out predictions (Breiman, 1996). The bagging process includes three steps: generating m bootstrap samples from the training dataset to preserve the original data distribution, training a fully expanded (unpruned) regression tree on each sample, and averaging the predictions from all trees for a more robust outcome. Notably, each bootstrap sample typically contains about 67% of the training data, with the remaining 33% serving as Out of Bag (OOB) data, which acts as a built-in test set to assess the model’s generalization capabilities (Hothorn & Lausen, 2002; Breiman, 1996). To enhance the performance of the regression tree, I implemented a version using this bagging technique.
# Building the bagged Regression Tree.
set.seed(150)
validation <- trainControl(method = "cv", number=70) # 70 cross validations
crossVal_baggingModel <- train(
Price ~ Area + Bedrooms + Bathrooms +
Stories + Mainroad_Access + Guestroom +
Basement + HotwaterHeating +Airconditioning + Parking + PrefferedArea + FurnishingStatus,data = training_data,
method = "treebag",
trControl = validation,
importance = TRUE) #Rank the predictor variable importance in the model
crossVal_baggingModel## Bagged CART
##
## 436 samples
## 12 predictor
##
## No pre-processing
## Resampling: Cross-Validated (70 fold)
## Summary of sample sizes: 429, 430, 430, 430, 429, 430, ...
## Resampling results:
##
## RMSE Rsquared MAE
## 0.6021752 0.7070754 0.4671555The 70-fold cross-validated bagging model yielded a RMSE of 0.6021752, R-squared of 70.7%, and MAE of 0.4671555. These metrics indicate the model has a good predictive accuracy and reliability: RMSE measures the typical prediction error, R-squared shows that over 70% of the data’s variability is explained by the model, and MAE reflects the average magnitude of prediction errors.
Ranking the Importance of the predictor attributes in the model.
The most important predictors in the bagging model are ‘Area’ and ‘Stories’, indicating that they have the most significant influence on the target variable. Following them, ‘Bathrooms’ and ‘Airconditioning1’ are also key contributors to the model’s predictive power, albeit less influential than the leading two features.
#Determining the most important variables.
var_importance <- varImp(crossVal_baggingModel)
ggplot(var_importance, aes(x = reorder(Features, Overall), y = Overall)) +
geom_bar(stat = "identity", fill = "skyblue", color = "white") +
labs(title = "Variable Importance Plot",
x = "Features",
y = "Overall Importance") +
theme_minimal() +
theme(axis.text.x = element_text(angle = 90, hjust = 1))
RandomForest Model.
After exploring regression and bagged regression trees, I experimented with a Random Forest model to assess its performance compared to the multiple regression and regression tree models. To prepare, I converted categorical variables in the training and testing datasets into factors, meeting the algorithm’s requirements for handling categorical data. I then fitted the Random Forest model to the training data, specifying an ensemble of 60 trees. With a set seed to ensure reproducibility, the model was built and subsequently used to generate predictions on the testing data. This step aimed to determine if the Random Forest model, with its ensemble learning capabilities, could offer improved prediction accuracy over the standalone regression and bagged tree models.
# Turning categorical variables into factors
training_data$Mainroad_Access <- as.factor(training_data$Mainroad_Access)
training_data$Guestroom <- as.factor(training_data$Guestroom)
training_data$Basement <- as.factor(training_data$Basement)
training_data$HotwaterHeating <- as.factor(training_data$HotwaterHeating)
training_data$Airconditioning <- as.factor(training_data$Airconditioning)
training_data$PrefferedArea <- as.factor(training_data$PrefferedArea)
training_data$FurnishingStatus <- as.factor(training_data$FurnishingStatus)
# Fitting the Random Forest model
set.seed(400) # for reproducibility
rf_model <- randomForest(Price ~ ., data = training_data, ntree = 50)# Turning categorical variables in testing_data as factors
testing_data$Mainroad_Access <- as.factor(testing_data$Mainroad_Access)
testing_data$Guestroom <- as.factor(testing_data$Guestroom)
testing_data$Basement <- as.factor(testing_data$Basement)
testing_data$HotwaterHeating <- as.factor(testing_data$HotwaterHeating)
testing_data$Airconditioning <- as.factor(testing_data$Airconditioning)
testing_data$PrefferedArea <- as.factor(testing_data$PrefferedArea)
testing_data$FurnishingStatus <- as.factor(testing_data$FurnishingStatus)
# Generating predictions for testing data
predictions_rf <- predict(rf_model, newdata = testing_data)# Actual values from testing data
y_test <- testing_data$Price
# Calculating MSE
mse_rf <- mean((y_test - predictions_rf)^2)
# MSE result
cat("MSE for Random Forest: ", mse_rf, "\n")## MSE for Random Forest: 0.3940711Random Forest Model Performance
The Random Forest model achieved an RMSE of 0.6243498, indicating that the average prediction error is approximately 0.624 units on the Price scale. The R-squared value is 0.586924, suggesting that about 58.7% of the variability in Price is explained by the model. In summary, the model demonstrates a moderate level of accuracy in predicting housing prices, with a substantial proportion of the variance in the outcome variable accounted for by the predictors used in the model.
4.4 Objective 4: Communicate the results of statistical models to a general audience.
To meet this objective, I focused on clear visualizations and straightforward explanations. I produced graphs to clearly depict the relationships between variables and their impact on housing prices. For each model—whether linear regression, regression trees, or random forests—I presented relevant plots to make data patterns understandable for non-experts. Additionally, I explained R-squared in simple terms as a measure of how effectively the variables account for the variation in housing prices, describing it as the model’s ‘fit’ to the data. I also defined RMSE as a metric that represents the average difference between the predicted and actual prices, serving as an indicator of the model’s precision. Through these visual aids and clear explanations, I ensured the results were accessible to those without a statistical background, achieving the objective of communicating effectively with a broader audience.
Comparing the Performance of the Four Models.
In the pursuit of an optimal model for predicting housing prices, I conducted a comparative analysis of four different models: Random Forest, Bagged Tree, Regression Tree, and Multiple Regression. I crafted a bar plot to juxtapose their performance based on two key metrics: Root Mean Square Error (RMSE) and R-squared (R²). This visual comparison is crucial as RMSE reflects the prediction error magnitude, while R² measures the proportion of variability explained by the model. Through this analysis, I aimed to discern which model most accurately predicts housing prices while maintaining generalizability across the data.
models <- c("Random Forest", "Bagged Regr' Tree", "Regression Tree", "Multiple Regression")
rmse <- c(0.6277509, 0.6021752, 0.5942146, 0.620059)
rsquared <- c(0.5824114, 0.7070754, 0.6498351, 0.6989956)
model_data <- data.frame(Model = rep(models, each = 2),
Metric = rep(c("RMSE", "R-squared"), times = 4),
Value = c(rmse, rsquared))
# Plotting
ggplot(model_data, aes(x = Model, y = Value, fill = Metric)) +
geom_bar(stat = "identity", position = position_dodge(width = 0.7), colour = "black") +
scale_fill_manual(values = c("skyblue", "forestgreen")) +
labs(title = "Comparison of the Four Model Performance (RMSE & R-Squared)",
x = "Model",
y = "Metric Values") +
theme_minimal() +
theme(axis.text.x = element_text(angle = 20, hjust = 1)) 
# Display the plot
ggsave("model_comparison_plot.png", width = 10, height = 6)4.5. Objective 5 Use programming software (i.e., R) to fit and assess statistical models.
To fulfill Objective 5 of using programming software to fit and assess statistical models, my project extensively utilized R, a programming language dedicated to statistical analysis. The project involved importing data, preprocessing it, creating various visualizations, conducting regression analysis, constructing decision trees, and implementing random forest algorithms, all through R’s comprehensive suite of functions and packages, including lm() for linear regression, rpart() for regression trees, and randomForest() for random forests. Assessment of the models was meticulous, employing R to calculate key metrics such as R-squared to evaluate model fit and RMSE to gauge predictive accuracy. R’s graphical capabilities were also instrumental in illustrating variable importance and visualizing the structure of model trees, which enhanced the interpretability and presentation of the results. The inclusion of R code snippets and corresponding outputs in the project documentation clearly demonstrates my proficiency in applying R for statistical modeling, thus meeting the educational objective.
6.0 Conclusion.
In conclusion, the analysis of model performance reveals that the Bagged Tree model stands out as the superior choice for predicting housing prices, as evidenced by its lowest RMSE and highest R-squared values among the contenders. This indicates a robust balance of accuracy and variability explanation. On the other end of the spectrum, the Multiple Regression model, despite its simplicity and commonly wide application, emerged as the least effective, with relatively higher RMSE and lower R-squared values in this specific context. The other models, Random Forest and Regression Tree, occupy the middle ground, performing adequately yet not surpassing the Bagged Tree model’s proficiency in this analysis.