dA {plsdof}

Derivative of normalization function

Description

This function computes the derivative of the function

v\mapsto \frac{w}{\|w\|_A}

with respect to y.

Usage

dA(w, A, dw)

Arguments

w

vector of length n.

A

square matrix that defines the norm

dw

derivative of w with respect to y. As y is a vector of length n, the derivative is a matrix of size nxn.

Details

The first derivative of the normalization operator is

\frac{\partial}{\partial y}\left(w\mapsto \frac{w}{\|w\|_A}\right)=\frac{1}{\|w\|}\left(I_n - \frac{w w^ \top A}{w^\top w}\right) \frac{\partial w}{\partial y}

Value

the Jacobian matrix of the normalization function. This is a matrix of size nxn.

Author(s)

Nicole Kraemer

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., Braun, M.L. (2007) "Kernelizing PLS, Degrees of Freedom, and Efficient Model Selection", Proceedings of the 24th International Conference on Machine Learning, Omni Press, 441 - 448

See Also

normalize, dnormalize

Examples

w<-rnorm(15)
dw<-diag(15)
A<-diag(1:15)
d.object<-dA(w,A,dw)