Model selection for Princinpal Components regression based on cross-validation

This function computes the optimal model parameter using cross-validation. Mdel selection is based on mean squared error and correlation to the response, respectively.

Usage

pcr.cv(
  X,
  y,
  k = 10,
  m = min(ncol(X), nrow(X) - 1),
  groups = NULL,
  scale = TRUE,
  eps = 1e-06,
  plot.it = FALSE,
  compute.jackknife = TRUE,
  method.cor = "pearson",
  supervised = FALSE
)

Arguments

X: matrix of predictor observations.
y: vector of response observations. The length of y is the same as the number of rows of X.
k: number of cross-validation splits. Default is 10.
m: maximal number of principal components. Default is m=min(ncol(X),nrow(X)-1).
groups: an optional vector with the same length as y. It encodes a partitioning of the data into distinct subgroups. If groups is provided, k=10 is ignored and instead, cross-validation is performed based on the partioning. Default is NULL.
scale: Should the predictor variables be scaled to unit variance? Default is TRUE.
eps: precision. Eigenvalues of the correlation matrix of X that are smaller than eps are set to 0. The default value is eps=10^{-6}.
plot.it: Logical. If TRUE, the function plots the cross-validation-error as a function of the number of components. Default is FALSE.
compute.jackknife: Logical. If TRUE, the regression coefficients on each of the cross-validation splits is stored. Default is TRUE.
method.cor: How should the correlation to the response be computed? Default is ”pearson”.
supervised: Should the principal components be sorted by decreasing squared correlation to the response? Default is FALSE.

Value

cv.error.matrix: matrix of cross-validated errors based on mean squared error. A row corresponds to one cross-validation split.
cv.error: vector of cross-validated errors based on mean squared error
m.opt: optimal number of components based on mean squared error
intercept: intercept of the optimal model, based on mean squared error
coefficients: vector of regression coefficients of the optimal model, based on mean squared error
cor.error.matrix: matrix of cross-validated errors based on correlation. A row corresponds to one cross-validation split.
cor.error: vector of cross-validated errors based on correlation
m.opt.cor: optimal number of components based on correlation
intercept.cor: intercept of the optimal model, based on correlation
coefficients.cor: vector of regression coefficients of the optimal model, based on correlation
coefficients.jackknife: Array of the regression coefficients on each of the cross-validation splits, if compute.jackknife=TRUE. In this case, the dimension is ncol(X) x (m+1) x k.

Details

The function computes the principal components on the scaled predictors. Based on the regression coefficients coefficients.jackknife computed on the cross-validation splits, we can estimate their mean and their variance using the jackknife. We remark that under a fixed design and the assumption of normally distributed y-values, we can also derive the true distribution of the regression coefficients.

Author

Nicole Kraemer, Mikio L. Braun

Examples


n<-500 # number of observations
p<-5 # number of variables
X<-matrix(rnorm(n*p),ncol=p)
y<-rnorm(n)

# compute PCR 
pcr.object<-pcr.cv(X,y,scale=FALSE,m=3)
pcr.object1<-pcr.cv(X,y,groups=sample(c(1,2,3),n,replace=TRUE),m=3)