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

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

## Arguments

pls.object object returned by linear.pls.fit or by kernel.pls.fit number of observations vector of response observations. kernel matrix X X^t. number of components 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

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