dvvtz {plsdof}

First derivative of the projection operator

Description

This function computes the first derivative of the projection operator

P_V z= V V^\top z

Usage

dvvtz(v, z, dv, dz)

Arguments

v

orthonormal basis of the space on which z is projected. v is either a matrix or a vector.

z

vector that is projected onto the columns of v

dv

first derivative of the the columns of v with respect to a vector y. If v is a matrix, dv is an array of dimension ncol(v)xnrow(v)xlength(y). If v is a vector, dv is a matrix of dimension nrow(v)xlength(y).

dz

first derivative of z with respect to a vector y. This is a matrix of dimension nrow(v)xlength(y).

Details

For the computation of the first derivative, we assume that the columns of v are normalized and mutually orthogonal. (Note that the function will not return an error message if these assumptionsa are not fulfilled. If we denote the columns of v by v_1,\ldots,v_l, the first derivative of the projection operator is

\frac{\partial P}{\partial y}=\sum_{j=1} ^ l \left[ \left(v_j z^ \top + v_j^ \top z I_n \right)\frac{\partial v_j}{\partial y} + v_j v_j ^ \top \frac{\partial z}{\partial y}\right]

Here, n denotes the length of the vectors v_j.

Value

The first derivative of the projection operator with respect to y. This is a matrix of dimension nrow(v)xlength(y).

Note

This is an internal function.

Author(s)

Nicole Kraemer, Mikio L. Braun

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

vvtz