For this assignment I wanted to cover a topic that was of interest to me. Housing prices and the variables that cause the prices to fluctuate and attempt to predict the prices of homes utilizing basic linear regression. With a few variables price, square footage, zip code, and a 3d plot with beds, baths, and price.
2024-03-21
library(tidyverse)
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library(plotly)
## ## Attaching package: 'plotly' ## ## The following object is masked from 'package:ggplot2': ## ## last_plot ## ## The following object is masked from 'package:stats': ## ## filter ## ## The following object is masked from 'package:graphics': ## ## layout
library(caret)
## Loading required package: lattice ## ## Attaching package: 'caret' ## ## The following object is masked from 'package:purrr': ## ## lift
data(Sacramento) str(Sacramento)
## 'data.frame': 932 obs. of 9 variables: ## $ city : Factor w/ 37 levels "ANTELOPE","AUBURN",..: 34 34 34 34 34 34 34 34 29 31 ... ## $ zip : Factor w/ 68 levels "z95603","z95608",..: 64 52 44 44 53 65 66 49 24 25 ... ## $ beds : int 2 3 2 2 2 3 3 3 2 3 ... ## $ baths : num 1 1 1 1 1 1 2 1 2 2 ... ## $ sqft : int 836 1167 796 852 797 1122 1104 1177 941 1146 ... ## $ type : Factor w/ 3 levels "Condo","Multi_Family",..: 3 3 3 3 3 1 3 3 1 3 ... ## $ price : int 59222 68212 68880 69307 81900 89921 90895 91002 94905 98937 ... ## $ latitude : num 38.6 38.5 38.6 38.6 38.5 ... ## $ longitude: num -121 -121 -121 -121 -121 ...
sum(is.na(Sacramento))
## [1] 0
Introduction
Building of the Linear Regression model for Price and SQFT
set.seed(123) trainIndex <- createDataPartition(Sacramento$price, p = .8, list = FALSE, times = 1) data_train <- Sacramento[trainIndex, ] data_test <- Sacramento[-trainIndex, ] model <- train(price ~ sqft, data = data_train, method = "lm" ) options(scipen = 999) predictions <- predict(model, newdata = data_test) RMSE <- sqrt(mean((predictions - data_test$price)^2)) RMSE
## [1] 77113.89
plot_data <- data.frame(Actual = data_test$price, Predicted = predictions)
Linear Regression Model graph For SQFT and Price
## `geom_smooth()` using formula = 'y ~ x'
Regression Formula for the Price VS Sqft
The linear regression equation: \(\text{Price} = \beta_{\text{sqft}} \times \text{sqft} + \beta_0\)
Price: This is the predicted price of the property.
\(\beta_{\text{sqft}}\) :This is the coefficient for the predictor variable sqft. It represents the change in the predicted price for each unit change in square footage.
sqft: This is the square footage of the property, which is the predictor variable used to predict the price.
\(\beta_0\) : This is the intercept term of the regression equation. It represents the predicted price when the square footage is zero.
Regression with Coefficients Price vs Sqft
## ## Call: ## lm(formula = .outcome ~ ., data = dat) ## ## Residuals: ## Min 1Q Median 3Q Max ## -237741 -54886 -12563 37993 598733 ## ## Coefficients: ## Estimate Std. Error t value Pr(>|t|) ## (Intercept) 11638.268 7959.589 1.462 0.144 ## sqft 140.781 4.382 32.127 <0.0000000000000002 *** ## --- ## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1 ## ## Residual standard error: 85810 on 745 degrees of freedom ## Multiple R-squared: 0.5808, Adjusted R-squared: 0.5802 ## F-statistic: 1032 on 1 and 745 DF, p-value: < 0.00000000000000022
Regressions with Coefficients Continued
Now that we have the coefficients we can plug them into our equation \(\text{Price} = 11638.268 - 140.781 \times \text{sqft} + \beta_0\) The equation is now showing that for every additional square foot the price decreases by $140.781 on average
The Standard error of the estimates show an accuracy of \(\beta = 7959.589\) and for Sqft = 4.382
Building of the Linear Regression model for Price and ZIP
set.seed(123) trainIndex <- createDataPartition(Sacramento$price, p = .8, list = FALSE, times = 1) data_train <- Sacramento[trainIndex, ] data_test <- Sacramento[-trainIndex, ] # Train linear regression model with price and zip code model_zip <- train(price ~ zip, data = data_train, method = "lm")
## Warning in predict.lm(modelFit, newdata): prediction from rank-deficient fit; ## attr(*, "non-estim") has doubtful cases ## Warning in predict.lm(modelFit, newdata): prediction from rank-deficient fit; ## attr(*, "non-estim") has doubtful cases ## Warning in predict.lm(modelFit, newdata): prediction from rank-deficient fit; ## attr(*, "non-estim") has doubtful cases ## Warning in predict.lm(modelFit, newdata): prediction from rank-deficient fit; ## attr(*, "non-estim") has doubtful cases ## Warning in predict.lm(modelFit, newdata): prediction from rank-deficient fit; ## attr(*, "non-estim") has doubtful cases ## Warning in predict.lm(modelFit, newdata): prediction from rank-deficient fit; ## attr(*, "non-estim") has doubtful cases ## Warning in predict.lm(modelFit, newdata): prediction from rank-deficient fit; ## attr(*, "non-estim") has doubtful cases ## Warning in predict.lm(modelFit, newdata): prediction from rank-deficient fit; ## attr(*, "non-estim") has doubtful cases ## Warning in predict.lm(modelFit, newdata): prediction from rank-deficient fit; ## attr(*, "non-estim") has doubtful cases ## Warning in predict.lm(modelFit, newdata): prediction from rank-deficient fit; ## attr(*, "non-estim") has doubtful cases ## Warning in predict.lm(modelFit, newdata): prediction from rank-deficient fit; ## attr(*, "non-estim") has doubtful cases ## Warning in predict.lm(modelFit, newdata): prediction from rank-deficient fit; ## attr(*, "non-estim") has doubtful cases ## Warning in predict.lm(modelFit, newdata): prediction from rank-deficient fit; ## attr(*, "non-estim") has doubtful cases ## Warning in predict.lm(modelFit, newdata): prediction from rank-deficient fit; ## attr(*, "non-estim") has doubtful cases ## Warning in predict.lm(modelFit, newdata): prediction from rank-deficient fit; ## attr(*, "non-estim") has doubtful cases ## Warning in predict.lm(modelFit, newdata): prediction from rank-deficient fit; ## attr(*, "non-estim") has doubtful cases ## Warning in predict.lm(modelFit, newdata): prediction from rank-deficient fit; ## attr(*, "non-estim") has doubtful cases ## Warning in predict.lm(modelFit, newdata): prediction from rank-deficient fit; ## attr(*, "non-estim") has doubtful cases ## Warning in predict.lm(modelFit, newdata): prediction from rank-deficient fit; ## attr(*, "non-estim") has doubtful cases ## Warning in predict.lm(modelFit, newdata): prediction from rank-deficient fit; ## attr(*, "non-estim") has doubtful cases ## Warning in predict.lm(modelFit, newdata): prediction from rank-deficient fit; ## attr(*, "non-estim") has doubtful cases ## Warning in predict.lm(modelFit, newdata): prediction from rank-deficient fit; ## attr(*, "non-estim") has doubtful cases ## Warning in predict.lm(modelFit, newdata): prediction from rank-deficient fit; ## attr(*, "non-estim") has doubtful cases ## Warning in predict.lm(modelFit, newdata): prediction from rank-deficient fit; ## attr(*, "non-estim") has doubtful cases ## Warning in predict.lm(modelFit, newdata): prediction from rank-deficient fit; ## attr(*, "non-estim") has doubtful cases
options(scipen = 999) # Make predictions predictions_zip <- predict(model_zip, newdata = data_test)
## Warning in predict.lm(modelFit, newdata): prediction from rank-deficient fit; ## attr(*, "non-estim") has doubtful cases
# Calculate RMSE RMSE_zip <- sqrt(mean((predictions_zip - data_test$price)^2)) RMSE_zip
## [1] 95259.28
# Create plot data frame plot_data_zip <- data.frame(Actual = data_test$price, Predicted = predictions_zip)
Regression Formula for the Price VS ZIP
The linear regression formula: \(\text{Price} = \beta_{\text{zip}} \times \text{zip} + \beta_0\)
\(\beta_{\text{zip}}\) :This is the coefficient for the predictor variable zip It represents the change in the predicted price for each unit change for zip codes.
zip: This is the zip code of the property, which is the predictor variable used to predict the price.
\(\beta_0\) : This is the intercept term of the regression equation. It represents the predicted price when the zip code is zero.
Results of the Linear Regression Model For ZIP and Price
## `geom_smooth()` using formula = 'y ~ x'
