This implements the Outer Product of Canonical Gradients (OPCG) in a forth coming paper Quach and Li (2021).
Usage
made(
x,
y,
d,
bw,
lambda = 0,
B_mat = NULL,
ytype = "continuous",
method = list(opcg = "newton", made = "newton"),
parallelize = F,
r_mat = NULL,
control_list = list()
)Arguments
- d
specified the reduced dimension
- bw
the bandwidth parameter for the kernel; the default kernel is gaussian
- B_mat
- ytype
specify the response as 'continuous', 'cat', or 'ord-cat'
- method
"newton" or "cg" methods; for carrying out the optimization using the standard newton-raphson (i.e. Fisher Scoring) or using Congugate Gradients
- parallelize
Default is False; to run in parallel, you will need to have foreach and some parallel backend loaded; parallelization is strongly recommended and encouraged.
- r_mat
- control_list
a list of control parameters for the Newton-Raphson or Conjugate Gradient methods
opcg - A 'pxd' matrix that estimates a basis for the central subspace.
opcg_wls - A 'pxd' matrix that estimates a basis for the central subspace based on the initial value of the optimization problem; useful for examining bad starting values.
cand_mat - A list that contains both the candidate matrix for OPCG and for the initial value; this is used in other functions for order determination
gradients - The estimated local gradients; used in regularization of OPCG
weights - The kernel weights in the local-linear GLM.
- x_matrix
a 'pxn' matrix of predictors;
- y_matrix
a 'mxn' matrix response
Value
A list containing both the estimate and candidate matrix.
opcg - A 'pxd' matrix that estimates a basis for the central subspace.
opcg_wls - A 'pxd' matrix that estimates a basis for the central subspace based on the initial value of the optimization problem; useful for examining bad starting values.
cand_mat - A list that contains both the candidate matrix for OPCG and for the initial value; this is used in other functions for order determination
gradients - The estimated local gradients; used in regularization of OPCG
weights - The kernel weights in the local-linear GLM.
Details
This version of MADE differs from that of Adragni and is currently available for continuous, cat, or ord-cat response so far.
For scalar continuous response, the estimation is identical to OPG.
The kernel for the local linear regression is fixed at a gaussian kernel.
For large 'p', we strongly recommend using the Conjugate Gradients implement, by setting method="cg". For method="cg", the hybrid conjugate gradient of Dai and Yuan is implemented, but only the armijo rule is implemented through backtracking, like in Bertsekas' "Convex Optimization Algorithms". A weak Wolfe condition can also be enforced by adding setting c_wolfe > 0 in the control_list, but since c_wolfe is usually set to 0.1 (Wikipedia) and this drastically slows down the algorithm relative to newton for small to moderate p, we leave the default as not enforcing a Wolfe condition, since we assume that our link function gives us a close enough initial point that local convergence is satisfactory. Should the initial values be suspect, then maybe enforcing the Wolfe condition is a reasonable trade-off.