Property-Price-Assessment-Using-Predictive-Modelling.Rmd

---
title: "R Notebook"
output: html_notebook
---
```{r}
library(Amelia)
library(corrplot)
library(caret)
library(plotly)
library(dplyr)
library(ggplot2)
library(GGally)
library(readr)
library(psych)
library(DescTools)
library(purrr)
library(tidyr)
```

Part B – Data Exploration and Cleaning

1. c.
```{r}
# Load the dataset
Housing_Valuation <- read.csv("HousingValuation.csv")

### CONVERTING NOMINAL VARIABLES TO NUMERICALS BY ONE-HOT ENCODING

# Variables for one-hot encoding
nominal_vars <- Housing_Valuation[, c("LandContour", "Utilities", "LotConfig", "DwellClass", "CentralAir", "GarageType", "PavedDrive")]

# One hot encoding using dummy variables
dummies <- dummyVars(~ ., data = nominal_vars, fullRank = TRUE)
categorical_dummies <- predict(dummies, newdata = nominal_vars)

# Combining  one-hot encoded variables to the rest of the data
Housing_Valuation <- cbind(Housing_Valuation[, !(names(Housing_Valuation) %in% c("LandContour", "Utilities", "LotConfig", "DwellClass", "CentralAir", "GarageType", "PavedDrive"))], categorical_dummies)


### CONVERTING ORDINAL VARIABLES TO NUMERICALS BY INTEGER ENCODING

# Define integer encoding for Slope
Housing_Valuation$Slope <- as.integer(factor(Housing_Valuation$Slope, levels = c("Gtl", "Mod", "Sev"), labels = c(1, 2, 3)))
# Define integer encoding for ExteriorCondition
Housing_Valuation$ExteriorCondition <- as.integer(factor(Housing_Valuation$ExteriorCondition, levels = c("Ex", "Gd", "TA", "Fa", "Po"), labels = c(1, 2, 3, 4, 5)))
# Define integer encoding for BasementCondition
Housing_Valuation$BasementCondition <- as.integer(factor(Housing_Valuation$BasementCondition, levels = c("Ex", "Gd", "TA", "Fa", "Po", "NB"), labels = c(1, 2, 3, 4, 5, 6)))
# Define integer encoding for KitchenQuality
Housing_Valuation$KitchenQuality <- as.integer(factor(Housing_Valuation$KitchenQuality, levels = c("Ex", "Gd", "TA", "Fa", "Po"), labels = c(1, 2, 3, 4, 5)))

print(head(Housing_Valuation))
```


2. a.
```{r}
# Load the dataset
HousingValuation <- read.csv("HousingValuation.csv")

# Calculate summary statistics for continuous variables
continuous_vars <- c("LotArea", "TotalBSF", "LowQualFinSF", "LivingArea", "PoolArea", "OpenPorchSF", "SalePrice")

summary_stats <- data.frame(
  Variable = continuous_vars,
  Mean = sapply(HousingValuation[continuous_vars], mean, na.rm = TRUE),
  Median = sapply(HousingValuation[continuous_vars], median, na.rm = TRUE),
  Max = sapply(HousingValuation[continuous_vars], max, na.rm = TRUE),
  SD = sapply(HousingValuation[continuous_vars], sd, na.rm = TRUE)
)

print(summary_stats)

# Calculate counts for categorical variables
categorical_vars <- c("LotShape", "LandContour", "Utilities", "LotConfig", 
                       "Slope", "DwellClass", "OverallQuality", "OverallCondition", 
                       "ExteriorCondition", "BasementCondition", "CentralAir", 
                       "KitchenQuality", "GarageType", "PavedDrive", "MoSold", "YrSold")

# Calculate categorical variable count occurrences of each level
for (var in categorical_vars) {
  print(var)
  print(table(HousingValuation[[var]]))
}

```

2.b.
```{r}
# Boxplots for visualizing outliers
par(mfrow = c(1, 7))  # Adjust layout to fit all plots
for (var in continuous_vars) {
  boxplot(HousingValuation[[var]], main = var, ylab = var)
}

# Calculate IQR and identify outliers
outlier_detection <- function(x) {
  Q1 <- quantile(x, 0.25, na.rm = TRUE)
  Q3 <- quantile(x, 0.75, na.rm = TRUE)
  IQR <- Q3 - Q1
  lower_bound <- Q1 - 1.5 * IQR
  upper_bound <- Q3 + 1.5 * IQR
  outliers <- x[x < lower_bound | x > upper_bound]
  return(outliers)
}

outliers_list <- lapply(HousingValuation[continuous_vars], outlier_detection)
outliers_summary <- data.frame(
  Variable = continuous_vars,
  Num_Outliers = sapply(outliers_list, length)
)

print(outliers_summary)
```


3

```{r}
# Load the dataset
HousingValuation <- read.csv("HousingValuation.csv")

# Define continuous variables
continuous_vars <- c("LotArea", "TotalBSF", "LowQualFinSF", "LivingArea", "PoolArea", "OpenPorchSF", "SalePrice")

# Create a function to plot histograms and calculate summary statistics
plot_histograms_and_summary <- function(data, vars) {
  
  # Create an empty data frame to store summary statistics
  summary_stats <- data.frame(
    Variable = vars,
    Mean = numeric(length(vars)),
    Median = numeric(length(vars)),
    Max = numeric(length(vars)),
    SD = numeric(length(vars))
  )
  
  # Loop through each variable
  for (i in 1:length(vars)) {
    var <- vars[i]
    
    # Plot histogram for each variable in a different window and color
    hist(data[[var]], main = paste("Histogram of", var), xlab = var, 
         col = rainbow(length(vars))[i], border = "black", breaks = 100)
    
    # Calculate summary statistics
    summary_stats[i, "Mean"] <- mean(data[[var]], na.rm = TRUE)
    summary_stats[i, "Median"] <- median(data[[var]], na.rm = TRUE)
    summary_stats[i, "Max"] <- max(data[[var]], na.rm = TRUE)
    summary_stats[i, "SD"] <- sd(data[[var]], na.rm = TRUE)
  }
  
  return(summary_stats)
}

# Generate histograms and summary statistics
summary_stats <- plot_histograms_and_summary(HousingValuation, continuous_vars)
print(summary_stats)


```




4. A.
```{r}
# Check for missing values
missing_values <- sapply(HousingValuation, function(x) sum(is.na(x)))
missing_vars <- missing_values[missing_values > 0]
print(missing_vars)
```

4. B.

Handling missing values of YearBuilt

```{r}
HousingValuation <- read.csv("HousingValuation.csv")
summary_original_year <- c(summary(HousingValuation$YearBuilt))
print(summary_original_year)

### Handling missing values of YearBuilt

### Fill missing values in YearBuilt with the mean
HousingValuation_filled <- HousingValuation
HousingValuation_filled$YearBuilt[is.na(HousingValuation_filled$YearBuilt)] <- mean(HousingValuation_filled$YearBuilt, na.rm = TRUE)

# Summary statistics for YearBuilt
summary_filled_year <- c(summary(HousingValuation_filled$YearBuilt), SD = sd(HousingValuation_filled$YearBuilt, na.rm = TRUE))
print(summary_filled_year)

# Transformation plot
hist(HousingValuation_filled$YearBuilt, main = "YearBuilt - Filled with Mean", xlab = "YearBuilt", col = "lightblue", breaks = 30)


### Delete records with missing values in YearBuilt
HousingValuation_deleted <- na.omit(HousingValuation)

# Summary statistics for YearBuilt
summary_deleted_year <- c(summary(HousingValuation_deleted$YearBuilt), SD = sd(HousingValuation_deleted$YearBuilt, na.rm = TRUE))
print(summary_deleted_year)

# Transformation plot
hist(HousingValuation_deleted$YearBuilt, main = "YearBuilt - Records Deleted", xlab = "YearBuilt", col = "lightgreen", breaks = 30)


### Replace missing values in YearBuilt with 0
HousingValuation_zero <- HousingValuation
HousingValuation_zero$YearBuilt[is.na(HousingValuation_zero$YearBuilt)] <- 0

# Summary statistics for YearBuilt
summary_zero_year <- c(summary(HousingValuation_zero$YearBuilt), SD = sd(HousingValuation_zero$YearBuilt, na.rm = TRUE))
print(summary_zero_year)

# Transformation plot
hist(HousingValuation_zero$YearBuilt, main = "YearBuilt - Replaced with 0", xlab = "YearBuilt", col = "lightcoral", breaks = 30)



### Original density plot
plot(density(HousingValuation$YearBuilt, na.rm = TRUE), 
     main = "Density Plot for YearBuilt (Original vs. Imputed Methods)",
     xlab = "YearBuilt", ylab = "Density", col = "black", lwd = 2)

# Fill missing values with the mean
HousingValuation_filled <- HousingValuation
HousingValuation_filled$YearBuilt[is.na(HousingValuation_filled$YearBuilt)] <- mean(HousingValuation_filled$YearBuilt, na.rm = TRUE)
lines(density(HousingValuation_filled$YearBuilt), col = "blue", lwd = 2)

# Delete records with missing values
HousingValuation_deleted <- na.omit(HousingValuation)
lines(density(HousingValuation_deleted$YearBuilt), col = "green", lwd = 2)

# Replace missing values with 0
HousingValuation_zero <- HousingValuation
HousingValuation_zero$YearBuilt[is.na(HousingValuation_zero$YearBuilt)] <- 0
lines(density(HousingValuation_zero$YearBuilt), col = "red", lwd = 2)

# Add a legend
legend("topright", 
       legend = c("Original", "Filled with Mean", "Records Deleted", "Replaced with 0"), 
       col = c("black", "blue", "green", "red"), 
       lwd = 2)
```

Handling missing values of LivingArea

```{r}
HousingValuation <- read.csv("HousingValuation.csv")
summary_original_living <- c(summary(HousingValuation$LivingArea))
print(summary_original_living)

### Handling missing values of LivingArea

### Fill missing values in LivingArea with the mean
HousingValuation_filled <- HousingValuation
HousingValuation_filled$LivingArea[is.na(HousingValuation_filled$LivingArea)] <- mean(HousingValuation_filled$LivingArea, na.rm = TRUE)

# Summary statistics for LivingArea
summary_filled_living <- c(summary(HousingValuation_filled$LivingArea), SD = sd(HousingValuation_filled$LivingArea, na.rm = TRUE))
print(summary_filled_living)

# Transformation plot
hist(HousingValuation_filled$LivingArea, main = "LivingArea - Filled with Mean", xlab = "LivingArea", col = "lightblue", breaks = 30)


### Delete records with missing values in LivingArea
HousingValuation_deleted <- na.omit(HousingValuation)

# Summary statistics for LivingArea
summary_deleted_living <- c(summary(HousingValuation_deleted$LivingArea), SD = sd(HousingValuation_deleted$LivingArea, na.rm = TRUE))
print(summary_deleted_living)

# Transformation plot
hist(HousingValuation_deleted$LivingArea, main = "LivingArea - Records Deleted", xlab = "LivingArea", col = "lightgreen", breaks = 30)


### Replace missing values in LivingArea with 0
HousingValuation_zero <- HousingValuation
HousingValuation_zero$LivingArea[is.na(HousingValuation_zero$LivingArea)] <- 0

# Summary statistics for LivingArea
summary_zero_living <- c(summary(HousingValuation_zero$LivingArea), SD = sd(HousingValuation_zero$LivingArea, na.rm = TRUE))
print(summary_zero_living)

# Transformation plot
hist(HousingValuation_zero$LivingArea, main = "LivingArea - Replaced with 0", xlab = "LivingArea", col = "lightcoral", breaks = 30)


### Original density plot
plot(density(HousingValuation$LivingArea, na.rm = TRUE), 
     main = "Density Plot for LivingArea (Original vs. Imputed Methods)",
     xlab = "LivingArea", ylab = "Density", col = "black", lwd = 2)

# Fill missing values with the mean
HousingValuation_filled <- HousingValuation
HousingValuation_filled$LivingArea[is.na(HousingValuation_filled$LivingArea)] <- mean(HousingValuation_filled$LivingArea, na.rm = TRUE)
lines(density(HousingValuation_filled$LivingArea), col = "blue", lwd = 2)

# Delete records with missing values
HousingValuation_deleted <- na.omit(HousingValuation)
lines(density(HousingValuation_deleted$LivingArea), col = "green", lwd = 2)

# Replace missing values with 0
HousingValuation_zero <- HousingValuation
HousingValuation_zero$LivingArea[is.na(HousingValuation_zero$LivingArea)] <- 0
lines(density(HousingValuation_zero$LivingArea), col = "red", lwd = 2)

# Add a legend
legend("topright", 
       legend = c("Original", "Filled with Mean", "Records Deleted", "Replaced with 0"), 
       col = c("black", "blue", "green", "red"), 
       lwd = 2)
```

Handling Missing Values for GarageType (Categorical Variable)

```{r}
HousingValuation <- read.csv("HousingValuation.csv")
summary_original_garage <- c(summary(HousingValuation$GarageType))
print(summary_original_garage)

### Handling missing values of GarageType

### Function to calculate the mode
calculate_mode <- function(x) {
  uniqx <- unique(x[!is.na(x)])
  uniqx[which.max(tabulate(match(x, uniqx)))]
}

### Fill missing values in GarageType with the mode
mode_garage <- calculate_mode(HousingValuation$GarageType)
HousingValuation_filled <- HousingValuation
HousingValuation_filled$GarageType[is.na(HousingValuation_filled$GarageType)] <- mode_garage

# Summary statistics for GarageType
summary_filled_garage <- c(summary(HousingValuation_filled$GarageType))
print(summary_filled_garage)

# Transformation plot (bar plot for categorical variable)
barplot(table(HousingValuation_filled$GarageType), 
        main = "GarageType - Filled with Mode", 
        xlab = "GarageType", col = "lightblue")


### Delete records with missing values in GarageType
HousingValuation_deleted <- na.omit(HousingValuation)

# Summary statistics for GarageType
summary_deleted_garage <- c(summary(HousingValuation_deleted$GarageType))
print(summary_deleted_garage)

# Transformation plot (bar plot for categorical variable)
barplot(table(HousingValuation_deleted$GarageType), 
        main = "GarageType - Records Deleted", 
        xlab = "GarageType", col = "lightgreen")


### Replace missing values in GarageType with "NA"
HousingValuation_zero <- HousingValuation
HousingValuation_zero$GarageType[is.na(HousingValuation_zero$GarageType)] <- "NA"

# Summary statistics for GarageType
summary_zero_garage <- c(summary(HousingValuation_zero$GarageType))
print(summary_zero_garage)

# Transformation plot (bar plot for categorical variable)
barplot(table(HousingValuation_zero$GarageType), 
        main = "GarageType - Replaced with 'NA'", 
        xlab = "GarageType", col = "lightcoral")


### Original bar plot
barplot(table(HousingValuation$GarageType), 
        main = "Bar Plot for GarageType (Original vs. Imputed Methods)",
        xlab = "GarageType", ylab = "Frequency", col = "black")

# Fill missing values with the mode
barplot(table(HousingValuation_filled$GarageType), 
        col = "blue", add = TRUE)

# Delete records with missing values
barplot(table(HousingValuation_deleted$GarageType), 
        col = "green", add = TRUE)

# Replace missing values with "NA"
barplot(table(HousingValuation_zero$GarageType), 
        col = "red", add = TRUE)

# Add a legend
legend("topright", 
       legend = c("Original", "Filled with Mode", "Records Deleted", "Replaced with 'None'"), 
       col = c("black", "blue", "green", "red"), 
       fill = c("black", "blue", "green", "red"))

```



5

```{r}
# Load necessary libraries
library(ggplot2)
library(e1071) # For skewness function
library(corrplot) # For correlation plots
library(caret) # For findCorrelation function

# Read the dataset
HousingValuation <- read.csv("HousingValuation.csv")

### TREATING MISSING VALUES

# Replace NAs with the mean for 'YearBuilt'
HousingValuation$YearBuilt[is.na(HousingValuation$YearBuilt)] <- mean(HousingValuation$YearBuilt, na.rm = TRUE)
# Convert YearBuilt to whole numbers
HousingValuation$YearBuilt <- round(HousingValuation$YearBuilt)

# Replace NAs with the mean for 'LivingArea'
HousingValuation$LivingArea[is.na(HousingValuation$LivingArea)] <- mean(HousingValuation$LivingArea, na.rm = TRUE)

# Function to calculate mode
calculate_mode <- function(x) {
  ux <- unique(x)
  ux[which.max(tabulate(match(x, ux)))]
}
# Calculate the mode for 'GarageType', excluding 'NA' as a category
mode_GarageType <- calculate_mode(HousingValuation$GarageType[HousingValuation$GarageType != "NA"])
# Replace actual NAs with the mode, without altering the 'NA' category which denoted 'No Garage' in dataset
HousingValuation$GarageType[is.na(HousingValuation$GarageType)] <- mode_GarageType


missing_values <- colSums(is.na(HousingValuation))
missing_values

### CONVERTING NOMINAL VARIABLES TO NUMERICALS BY ONE-HOT ENCODING

# Variables for one-hot encoding
nominal_vars <- HousingValuation[, c("LandContour", "Utilities", "LotConfig", "DwellClass", "CentralAir", "GarageType", "PavedDrive")]

# One hot encoding using dummy variables
dummies <- dummyVars(~ ., data = nominal_vars, fullRank = TRUE)
categorical_dummies <- predict(dummies, newdata = nominal_vars)

# Combining  one-hot encoded variables to the rest of the data
HousingValuation <- cbind(HousingValuation[, !(names(HousingValuation) %in% c("LandContour", "Utilities", "LotConfig", "DwellClass", "CentralAir", "GarageType", "PavedDrive"))], categorical_dummies)


### CONVERTING ORDINAL VARIABLES TO NUMERICALS BY INTEGER ENCODING

# Define integer encoding for Slope
HousingValuation$Slope <- as.integer(factor(HousingValuation$Slope, levels = c("Gtl", "Mod", "Sev"), labels = c(1, 2, 3)))
# Define integer encoding for ExteriorCondition
HousingValuation$ExteriorCondition <- as.integer(factor(HousingValuation$ExteriorCondition, levels = c("Ex", "Gd", "TA", "Fa", "Po"), labels = c(1, 2, 3, 4, 5)))
# Define integer encoding for BasementCondition
HousingValuation$BasementCondition <- as.integer(factor(HousingValuation$BasementCondition, levels = c("Ex", "Gd", "TA", "Fa", "Po", "NB"), labels = c(1, 2, 3, 4, 5, 6)))
# Define integer encoding for KitchenQuality
HousingValuation$KitchenQuality <- as.integer(factor(HousingValuation$KitchenQuality, levels = c("Ex", "Gd", "TA", "Fa", "Po"), labels = c(1, 2, 3, 4, 5)))

print(head(HousingValuation))


### REMOVAL OF TARGET & ID COLUMNS

# Define the target variable
target_variable <- HousingValuation$SalePrice
# Create a new dataframe excluding the target variable and the 'Id' column (because its unique)
HousingValuation_predictors <- HousingValuation[, !names(HousingValuation) %in% c("SalePrice", "Id")]

print(HousingValuation_predictors)


### BUILDING CORRELATION MATRIX

# Analyze correlation of attributes using ggcorr
cor_matrix <- cor(HousingValuation_predictors)
ggcorrplot(cor_matrix, lab = TRUE)

# Getting correlation matrix using caret package
M <- data.matrix(HousingValuation_predictors)
corrM <- cor(M)

# Correlation Matrix using caret package
corr <- cor(HousingValuation_predictors)
high_corr_vars <- findCorrelation(corr, cutoff = 0.5, names = TRUE)

print(high_corr_vars)


# Removal of variables to minimize inter-correlation
HousingValuation_predictors_reduced <- subset(HousingValuation_predictors, select = -c(FullBath, TotalRmsAbvGrd, KitchenAbvGr, LandContourLvl, Slope))

# Check dimensions
print(dim(HousingValuation_predictors_reduced))

# Evaluate Correlation of Dimensions of Reduced Dataset
ggcorr(HousingValuation_predictors_reduced, label = TRUE)
M <- data.matrix(HousingValuation_predictors_reduced)
corrM <- cor(M)

# Merge Target Variable to New Dataset
HousingValuation_reduced <- cbind(HousingValuation_predictors_reduced, SalePrice = target_variable)

# Print Final Dataset
print(head(HousingValuation_reduced))

### EXPLORING DISTRIBUTIONS AGAINST TARGET VARIABLE
# Compute the correlation matrix including the target variable
cor_matrix_with_target <- cor(HousingValuation_reduced)
# Plot the correlation matrix with the target variable
ggcorrplot(cor_matrix_with_target, method = "circle", type = "lower", 
           lab = TRUE, title = "Correlation Matrix with SalePrice")
# Extract correlation with SalePrice
cor_saleprice<-cor_matrix_with_target["SalePrice",]
# Print the correlation values with SalePrice
print(cor_saleprice)
```


PART C - BUILDING PREDICTIVE MODELS

1. Regression Modelling 
As the model for predictive modelling, Linear Regression has been chosen since the target variable (SalesPrice) to be predicted in this dataset is Continuous.

```{r}
#Setting up the sample configuration
#Here we define 2/3 as the training data sample size.
smp_size <- floor(2/3 * nrow(HousingValuation_reduced)) 
set.seed(2)

#Sample the dataset
HousingValuation_reduced <- HousingValuation_reduced[sample(nrow(HousingValuation_reduced)), ]

#Creating the ‘train’ dataset
HousingValuation.train <- HousingValuation_reduced[1:smp_size, ]
HousingValuation.test <- HousingValuation_reduced[(smp_size+1):nrow(HousingValuation_reduced), ]

#Building the predictive model

#Specifying target and input variables
formula = SalePrice ~.

#Fitting the linear regression algorithm
model <- lm(formula = formula, data = HousingValuation.train)

# Display the coefficients of the linear regression model
summary(model)$coefficients

#Regrression Equation
as.formula(
  paste0("y ~ ", round(coefficients(model)[1],2), " + ", 
         paste(sprintf("%.2f * %s",coefficients(model)[-1], 
                       names(coefficients(model)[-1])),
               collapse=" + ")
         )
  )

#Making Predictions for test and train datasets
HousingValuation.train$predicted.SalePrice <- predict(model, HousingValuation.train)
HousingValuation.test$predicted.SalePrice <- predict(model, HousingValuation.test)
 
print("Actual Values") 
head(HousingValuation.test$SalePrice[1:5])
print("Predicted Values")
head(HousingValuation.test$predicted.SalePrice[1:5])

#Plot Predicted values vs Actual values of the target variable
pl1 <-HousingValuation.test %>% 
  ggplot(aes(SalePrice,predicted.SalePrice)) + 
  geom_point(alpha=0.5) + stat_smooth(aes(colour='red')) + 
  xlab('Actual value of SalesPrice') + 
  ylab('Predicted value of SalesPrice')+ 
  theme_bw()
ggplotly(pl1)

#Calculate RMSE
error <- HousingValuation.test$SalePrice-HousingValuation.test$predicted.SalePrice 
rmse <- sqrt(mean(error^2))
print(paste("Root Mean Square Error: ", rmse))

#Calculate the R-squared value
r_squared <- summary(model)$r.squared 
print(paste("R Squared: ", r_squared))
```

MODEL 2 
```{r}
# Define the reduced dataset with selected variables
HousingValuation_reduced_vars <- HousingValuation_reduced[, c("LotArea", "OverallQuality", "YearBuilt", "LivingArea", "HalfBath", "BedroomAbvGr", "KitchenQuality", "Fireplaces", "GarageCars", "PoolArea", "OpenPorchSF", "MoSold", "YrSold", "CentralAirY", "GarageTypeAttchd", "GarageTypeBuiltIn", "GarageTypeDetchd", "PavedDriveY", "SalePrice")]

# Fit the linear regression model
model2 <- lm(SalePrice ~ ., data = HousingValuation_reduced_vars)

# Summarize the initial model
summary(model2)

# Display the coefficients of the linear regression model
summary(model2)$coefficients

#Regrression Equation
as.formula(
  paste0("y ~ ", round(coefficients(model2)[1],2), " + ", 
         paste(sprintf("%.2f * %s",coefficients(model2)[-1], 
                       names(coefficients(model2)[-1])),
               collapse=" + ")
         )
  )

#Making Predictions for test and train datasets
HousingValuation.train$predicted.SalePrice <- predict(model2, HousingValuation.train)
HousingValuation.test$predicted.SalePrice <- predict(model2, HousingValuation.test)
 
print("Actual Values") 
head(HousingValuation.test$SalePrice[1:5])
print("Predicted Values")
head(HousingValuation.test$predicted.SalePrice[1:5])

#Plot Predicted values vs Actual values of the target variable
pl1 <-HousingValuation.test %>% 
  ggplot(aes(SalePrice,predicted.SalePrice)) + 
  geom_point(alpha=0.5) + stat_smooth(aes(colour='red')) + 
  xlab('Actual value of SalesPrice') + 
  ylab('Predicted value of SalesPrice')+ 
  theme_bw()
ggplotly(pl1)

#Calculate RMSE
error <- HousingValuation.test$SalePrice-HousingValuation.test$predicted.SalePrice 
rmse <- sqrt(mean(error^2))
print(paste("Root Mean Square Error: ", rmse))

#Calculate the R-squared value
r_squared <- summary(model2)$r.squared 
print(paste("R Squared: ", r_squared))
```


MODEL 3 
```{r}
# Define the refined dataset excluding insignificant variables
HousingValuation_refined <- HousingValuation_reduced[, c("LotArea", "OverallQuality", "YearBuilt", "LivingArea", "HalfBath", "BedroomAbvGr", "KitchenQuality", "Fireplaces", "GarageCars", "GarageTypeAttchd", "GarageTypeBuiltIn", "SalePrice")]

# Fit the new linear regression model
model3 <- lm(SalePrice ~ ., data = HousingValuation_refined)

# Summarize the refined model
summary(model3)

# Display the coefficients of the linear regression model
summary(model3)$coefficients

#Regrression Equation
as.formula(
  paste0("y ~ ", round(coefficients(model3)[1],2), " + ", 
         paste(sprintf("%.2f * %s",coefficients(model3)[-1], 
                       names(coefficients(model3)[-1])),
               collapse=" + ")
         )
  )

#Making Predictions for test and train datasets
HousingValuation.train$predicted.SalePrice <- predict(model3, HousingValuation.train)
HousingValuation.test$predicted.SalePrice <- predict(model3, HousingValuation.test)
 
print("Actual Values") 
head(HousingValuation.test$SalePrice[1:5])
print("Predicted Values")
head(HousingValuation.test$predicted.SalePrice[1:5])

#Plot Predicted values vs Actual values of the target variable
pl1 <-HousingValuation.test %>% 
  ggplot(aes(SalePrice,predicted.SalePrice)) + 
  geom_point(alpha=0.5) + stat_smooth(aes(colour='red')) + 
  xlab('Actual value of SalesPrice') + 
  ylab('Predicted value of SalesPrice')+ 
  theme_bw()
ggplotly(pl1)

#Calculate RMSE
error <- HousingValuation.test$SalePrice-HousingValuation.test$predicted.SalePrice 
rmse <- sqrt(mean(error^2))
print(paste("Root Mean Square Error: ", rmse))

#Calculate the R-squared value
r_squared <- summary(model3)$r.squared 
print(paste("R Squared: ", r_squared))
```


2. Decision Tree Modelling

```{r}
#install.packages('rpart', dependencies = TRUE)
#install.packages('rpart.plot', dependencies = TRUE)

library(rpart)
library(rpart.plot)

#Partition the data
smp_size <- floor(2/3 * nrow(HousingValuation_reduced)) 
set.seed(2)

#Sample the dataset
HousingValuation_reduced.dt <- HousingValuation_reduced[sample(nrow(HousingValuation_reduced)), ]

HousingValuation.train.dt <- HousingValuation_reduced.dt[1:smp_size, ]
HousingValuation.test.dt <- HousingValuation_reduced.dt[(smp_size+1):nrow(HousingValuation_reduced.dt), ]

#Build the decision tree model

#Specifying target and input variables
formula = SalePrice ~.

#Define the decision tree parameters
dtree <- rpart(formula, data= HousingValuation.train.dt, method="anova")

#Check feature importance
dtree$variable.importance

#Visualize the decision tree
rpart.plot(dtree, type = 4, fallen.leaves = FALSE)

print(dtree)

#Predictions
predicted.SalePrice <- predict(dtree, HousingValuation.test.dt) 
print("Actual Values") 
head(HousingValuation.test.dt$SalePrice[1:5])
print("Predicted Values") 
head(predicted.SalePrice[1:5])

#Model assessment
error <- HousingValuation.test.dt$SalePrice - predicted.SalePrice
rmse <- sqrt(mean(error^2))
print(paste("Root Mean Square Error: ", rmse))

#Finding the best CP value
printcp(dtree)
dtree$cptable[which.min(dtree$cptable[,"xerror"]),"CP"]

#Prune the tree with best cp value (complexity parameter)
 pruned_dtree <- prune(dtree, cp = 0.01) 
 rpart.plot(pruned_dtree, type = 4, fallen.leaves = FALSE)
 
#Model Assessment 
predicted_pruned.SalePrice <- predict(pruned_dtree, HousingValuation.test) 
error_new <- HousingValuation.test$SalePrice - predicted_pruned.SalePrice
rmse_new <- sqrt(mean(error_new^2))
print(paste("Root Mean Square Error: ", rmse_new))
```

DECISION TREE 2
```{r}
#Prune the tree with a value greater than best cp value
 pruned_dtree <- prune(dtree, cp = 0.03) 
 rpart.plot(pruned_dtree, type = 4, fallen.leaves = FALSE)
 
#Model Assessment 
predicted_pruned.SalePrice <- predict(pruned_dtree, HousingValuation.test) 
error_new <- HousingValuation.test$SalePrice - predicted_pruned.SalePrice
rmse_new <- sqrt(mean(error_new^2))
print(paste("Root Mean Square Error: ", rmse_new))
```

DECISION TREE 3
```{r}
#Prune the tree with a value lesser than best cp value
 pruned_dtree <- prune(dtree, cp = 0.001) 
 rpart.plot(pruned_dtree, type = 4, fallen.leaves = FALSE)
 
#Model Assessment 
predicted_pruned.SalePrice <- predict(pruned_dtree, HousingValuation.test) 
error_new <- HousingValuation.test$SalePrice - predicted_pruned.SalePrice
rmse_new <- sqrt(mean(error_new^2))
print(paste("Root Mean Square Error: ", rmse_new))
```