Computation of the Degrees of Freedom

This function computes the Degrees of Freedom using the Krylov representation of PLS.

Usage

pls.dof(pls.object, n, y, K, m, DoF.max)

Arguments

pls.object: object returned by linear.pls.fit or by kernel.pls.fit
n: number of observations
y: vector of response observations.
K: kernel matrix X X^t.
m: number of components
DoF.max: upper bound on the Degrees of Freedom.

Value

coefficients: matrix of regression coefficients
intercept: vector of regression intercepts
DoF: Degrees of Freedom
sigmahat: vector of estimated model error
Yhat: matrix of fitted values
yhat: vector of squared length of fitted values
RSS: vector of residual sum of error
TT: matrix of normalized PLS components

Details

This computation of the Degrees of Freedom is based on the equivalence of PLS regression and the projection of the response vector y onto the Krylov space spanned by $$Ky,K^2 y,...,K^m y.$$ Details can be found in Kraemer and Sugiyama (2011).

References

Kraemer, N., Sugiyama M. (2011). "The Degrees of Freedom of Partial Least Squares Regression". Journal of the American Statistical Association 106 (494) https://www.tandfonline.com/doi/abs/10.1198/jasa.2011.tm10107

Kraemer, N., Sugiyama M., Braun, M.L. (2009) "Lanczos Approximations for the Speedup of Kernel Partial Least Squares Regression." Proceedings of the Twelfth International Conference on Artificial Intelligence and Statistics (AISTATS), p. 272-279

Author

Nicole Kraemer, Mikio L. Braun

Examples


# this is an internal function