pcr.cv: Model selection for Princinpal Components regression based on cross-validation

Description

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
)

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.

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.

Author

Nicole Kraemer, Mikio L. Braun

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.

Examples

Run this code


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)

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