This function computes the first derivative of the projection operator $$P_V z= V V^\top z$$
dvvtz(v, z, dv, dz)orthonormal basis of the space on which z is projected. v is either a matrix or a vector.
vector that is projected onto the columns of v
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).
first derivative of z with respect to a vector y. This is a
matrix of dimension nrow(v)xlength(y).
The first derivative of the projection operator with respect to y. This is a matrix of dimension nrow(v)xlength(y).
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 \(v_j\).
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