dA {plsdof} | R Documentation |
This function computes the derivative of the function
v\mapsto
\frac{w}{\|w\|_A}
with respect to y.
dA(w, A, dw)
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. |
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}
the Jacobian matrix of the normalization function. This is a matrix of size nxn.
Nicole Kraemer
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
w<-rnorm(15)
dw<-diag(15)
A<-diag(1:15)
d.object<-dA(w,A,dw)