PCR: Principal Component Regression (PCR)

Description

The PCR function performs Principal Component Regression (PCR) on a given dataset. It standardizes the predictor variables, determines the number of principal components to retain based on a specified threshold, and fits a linear regression model using the principal component scores.

Usage

PCR(data, threshold)

Value

A list containing the following elements:

Bhat: The estimated regression coefficients, including the intercept.
RMSE: The Root Mean Square Error of the regression model.
summary: The summary of the linear regression model.
yhat: The predicted values of the response variable.

Arguments

data: A data frame where the first column is the response variable and the remaining columns are predictor variables.
threshold: The proportion of variance to retain in the principal components (default is 0.95).

Details

The function performs the following steps: 1. Standardize the predictor variables. 2. Compute the covariance matrix of the standardized predictors. 3. Perform eigen decomposition on the covariance matrix to obtain principal components. 4. Determine the number of principal components to retain based on the cumulative explained variance exceeding the specified threshold. 5. Project the standardized predictors onto the retained principal components. 6. Fit a linear regression model using the principal component scores. 7. Back-transform the regression coefficients to the original scale.

Examples

Run this code

if (FALSE) {
# Example data
set.seed(1234)
n <- 2000
p <- 10
mu0 <- as.matrix(runif(p, 0))
sigma0 <- as.matrix(runif(p, 0, 10))
ro <- as.matrix(c(runif(round(p / 2), -1, -0.8), runif(p - round(p / 2), 0.8, 1)))
R0 <- ro %*% t(ro)
diag(R0) <- 1
Sigma0 <- sigma0 %*% t(sigma0) * R0
x <- mvrnorm(n, mu0, Sigma0)
colnames(x) <- paste("x", 1:p, sep = "")
e <- rnorm(n, 0, 1)
B <- sample(1:3, (p + 1), replace = TRUE)
en <- matrix(rep(1, n * 1), ncol = 1)
y <- cbind(en, x) %*% B + e
colnames(y) <- paste("y")
data <- data.frame(cbind(y, x))

# Call the PCR function
result <- PCR(data, threshold = 0.9)

# Access the estimated regression coefficients
print(Bhat <- result$Bhat)

# Access the predicted values
print(yhat <- result$yhat)

# Print the summary of the regression model
print(result$summary)

# Print the RMSE
print(paste("RMSE:", result$RMSE))
}