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## Copyright (C) 2006 Michael Creel <michael.creel@uab.es>
##
## This program is free software; you can redistribute it and/or modify it under
## the terms of the GNU General Public License as published by the Free Software
## Foundation; either version 3 of the License, or (at your option) any later
## version.
##
## This program is distributed in the hope that it will be useful, but WITHOUT
## ANY WARRANTY; without even the implied warranty of MERCHANTABILITY or
## FITNESS FOR A PARTICULAR PURPOSE. See the GNU General Public License for more
## details.
##
## You should have received a copy of the GNU General Public License along with
## this program; if not, see <http://www.gnu.org/licenses/>.
## kernel_density: multivariate kernel density estimator
##
## usage:
## dens = kernel_density(eval_points, data, bandwidth)
##
## inputs:
## eval_points: PxK matrix of points at which to calculate the density
## data: NxK matrix of data points
## bandwidth: positive scalar, the smoothing parameter. The fit
## is more smooth as the bandwidth increases.
## kernel (optional): string. Name of the kernel function. Default is
## Gaussian kernel.
## prewhiten bool (optional): default false. If true, rotate data
## using Choleski decomposition of inverse of covariance,
## to approximate independence after the transformation, which
## makes a product kernel a reasonable choice.
## do_cv: bool (optional). default false. If true, calculate leave-1-out
## density for cross validation
## computenodes: int (optional, default 0).
## Number of compute nodes for parallel evaluation
## debug: bool (optional, default false). show results on compute nodes if doing
## a parallel run
## outputs:
## dens: Px1 vector: the fitted density value at each of the P evaluation points.
##
## References:
## Wand, M.P. and Jones, M.C. (1995), 'Kernel smoothing'.
## http://www.xplore-stat.de/ebooks/scripts/spm/html/spmhtmlframe73.html
function z = kernel_density(eval_points, data, bandwidth, kernel, prewhiten, do_cv, computenodes, debug)
if nargin < 2; error("kernel_density: at least 2 arguments are required"); endif
n = rows(data);
k = columns(data);
# set defaults for optional args
if (nargin < 3) bandwidth = (n ^ (-1/(4+k))); endif # bandwidth - see Li and Racine pg. 26
if (nargin < 4) kernel = "kernel_normal"; endif # what kernel?
if (nargin < 5) prewhiten = false; endif # automatic prewhitening?
if (nargin < 6) do_cv = false; endif # ordinary or leave-1-out
if (nargin < 7) computenodes = 0; endif # parallel?
if (nargin < 8) debug = false; endif; # debug?
nn = rows(eval_points);
n = rows(data);
if prewhiten
H = bandwidth*chol(cov(data));
else
H = bandwidth;
endif
# Inverse bandwidth matrix H_inv
H_inv = inv(H);
# weight by inverse bandwidth matrix
eval_points = eval_points*H_inv;
data = data*H_inv;
# check if doing this parallel or serial
global PARALLEL NSLAVES NEWORLD NSLAVES TAG
PARALLEL = 0;
if computenodes > 0
PARALLEL = 1;
NSLAVES = computenodes;
LAM_Init(computenodes, debug);
endif
if !PARALLEL # ordinary serial version
points_per_node = nn; # do the all on this node
z = kernel_density_nodes(eval_points, data, do_cv, kernel, points_per_node, computenodes, debug);
else # parallel version
z = zeros(nn,1);
points_per_node = floor(nn/(NSLAVES + 1)); # number of obsns per slave
# The command that the slave nodes will execute
cmd=['z_on_node = kernel_density_nodes(eval_points, data, do_cv, kernel, points_per_node, computenodes, debug); ',...
'MPI_Send(z_on_node, 0, TAG, NEWORLD);'];
# send items to slaves
NumCmds_Send({"eval_points", "data", "do_cv", "kernel", "points_per_node", "computenodes", "debug","cmd"}, {eval_points, data, do_cv, kernel, points_per_node, computenodes, debug, cmd});
# evaluate last block on master while slaves are busy
z_on_node = kernel_density_nodes(eval_points, data, do_cv, kernel, points_per_node, computenodes, debug);
startblock = NSLAVES*points_per_node + 1;
endblock = nn;
z(startblock:endblock,:) = z(startblock:endblock,:) + z_on_node;
# collect slaves' results
z_on_node = zeros(points_per_node,1); # size may differ between master and compute nodes - reset here
for i = 1:NSLAVES
MPI_Recv(z_on_node,i,TAG,NEWORLD);
startblock = i*points_per_node - points_per_node + 1;
endblock = i*points_per_node;
z(startblock:endblock,:) = z(startblock:endblock,:) + z_on_node;
endfor
# clean up after parallel
LAM_Finalize;
endif
z = z*det(H_inv);
endfunction
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