plsdof (version 0.2-1)

dA: 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.

Value

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

Details

The first derivative of the normalization operator is $$\frac{\partial}{\partial y}\left(w\mapsto \frac{w}{\|w\|_A}\right)=\frac{1}{\|w\|}\left(I_n - \frac{w w^ \top A}{w^\top w}\right) \frac{\partial w}{\partial y}$$

References

Kraemer, N., Sugiyama M. (2010). "The Degrees of Freedom of Partial Least Squares Regression". preprint, http://arxiv.org/abs/1002.4112 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)