library(ISLR2)
library(glmnet)
## Loading required package: Matrix
## Loaded glmnet 4.1-8
library(leaps)
library(pls)
##
## Attaching package: 'pls'
## The following object is masked from 'package:stats':
##
## loadings
data("Boston")
View(Boston)
Boston <- na.omit(Boston)
#Least Squares Regression
set.seed(1)
train <- sample(1:nrow(Boston), nrow(Boston)*0.90)
test <- -train
y.test <- Boston$crim[test]
lm.fit <- lm(crim~., data=Boston, subset=train)
summary(lm.fit)
##
## Call:
## lm(formula = crim ~ ., data = Boston, subset = train)
##
## Residuals:
## Min 1Q Median 3Q Max
## -8.772 -2.329 -0.341 1.044 73.453
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 14.2943760 7.7759724 1.838 0.066693 .
## zn 0.0445855 0.0207506 2.149 0.032206 *
## indus -0.0693981 0.0902700 -0.769 0.442433
## chas -0.9142474 1.2970512 -0.705 0.481265
## nox -9.8495028 5.7626291 -1.709 0.088115 .
## rm 0.7508779 0.6588809 1.140 0.255059
## age -0.0008849 0.0196652 -0.045 0.964127
## dis -1.0570654 0.3149226 -3.357 0.000857 ***
## rad 0.6306703 0.0942514 6.691 6.71e-11 ***
## tax -0.0042698 0.0055458 -0.770 0.441760
## ptratio -0.3304103 0.2053000 -1.609 0.108242
## lstat 0.1332271 0.0822788 1.619 0.106114
## medv -0.2375039 0.0648955 -3.660 0.000283 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 6.756 on 442 degrees of freedom
## Multiple R-squared: 0.438, Adjusted R-squared: 0.4228
## F-statistic: 28.71 on 12 and 442 DF, p-value: < 2.2e-16
lm.pred <- predict(lm.fit, Boston[test,])
lm.error <- mean((lm.pred - y.test)^2)
lm.error
## [1] 8.179267
#Lasso Regression
x <- model.matrix(crim~., Boston)[, -1]
y <- Boston$crim
lasso_fit <- glmnet(x[train, ], y[train], alpha=1)
plot(lasso_fit)
lambda_lasso <- lasso_fit$lambda.min
lasso_pred <- predict(lasso_fit, s=lambda_lasso, newx=x[test,])
lasso_error <- mean((lasso_pred-y[test])^2)
lasso_error
## [1] 8.846331
#Ridge Regression
ridge_fit <- glmnet(x[train, ], y[train], alpha=0)
summary(ridge_fit)
## Length Class Mode
## a0 100 -none- numeric
## beta 1200 dgCMatrix S4
## df 100 -none- numeric
## dim 2 -none- numeric
## lambda 100 -none- numeric
## dev.ratio 100 -none- numeric
## nulldev 1 -none- numeric
## npasses 1 -none- numeric
## jerr 1 -none- numeric
## offset 1 -none- logical
## call 4 -none- call
## nobs 1 -none- numeric
plot(ridge_fit)
lambda_ridge <- ridge_fit$lambda.min
lambda_ridge
## NULL
ridge_pred <- predict(ridge_fit, s=lambda_ridge, newx=x[test,])
ridge_error <- mean((ridge_pred-y[test])^2)
ridge_error
## [1] 16.34827
#partial least squares
pls.fit<- plsr(crim~., data=Boston, subset=train, scale=TRUE, validation="CV")
summary(pls.fit)
## Data: X dimension: 455 12
## Y dimension: 455 1
## Fit method: kernelpls
## Number of components considered: 12
##
## VALIDATION: RMSEP
## Cross-validated using 10 random segments.
## (Intercept) 1 comps 2 comps 3 comps 4 comps 5 comps 6 comps
## CV 8.902 7.350 6.934 6.838 6.819 6.802 6.782
## adjCV 8.902 7.349 6.931 6.835 6.813 6.796 6.776
## 7 comps 8 comps 9 comps 10 comps 11 comps 12 comps
## CV 6.778 6.770 6.769 6.770 6.770 6.770
## adjCV 6.771 6.764 6.763 6.764 6.764 6.764
##
## TRAINING: % variance explained
## 1 comps 2 comps 3 comps 4 comps 5 comps 6 comps 7 comps 8 comps
## X 49.5 58.77 64.68 74.38 80.71 83.44 86.28 91.11
## crim 31.9 39.99 42.15 42.88 43.27 43.66 43.76 43.79
## 9 comps 10 comps 11 comps 12 comps
## X 94.52 96.23 98.2 100.0
## crim 43.80 43.80 43.8 43.8