# Principal Component Regression (PCR) 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. ```r PCR(data, threshold) ``` ## 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). ## Returns 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. ## 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 ```r ## Not run: # 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)) ## End(Not run) ``` ## See Also `lm`: For fitting linear models. `eigen`: For computing eigenvalues and eigenvectors.