1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128
129
130
131
132
133
134
135
136
137
138
139
140
|
# Copyright (C) 2007 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 2 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_example: examples of how to use kernel density and regression functions
# requires the optim and plot packages from Octave Forge
#
# usage: kernel_example;
# sample size (default n = 500) - should get better fit (on average)
# as this increases, supposing that you allow the optimal window width
# to be found, by uncommenting the relevant lines
n = 500;
# set this to greater than 0 to try parallel computations (requires MPITB)
compute_nodes = 0;
nodes = compute_nodes + 1; # count master node
close all;
hold off;
############################################################
# kernel regression example
# uniformly spaced data points on [0,2]
x = 1:n;
x = x';
x = 2*x/n;
# generate dependent variable
trueline = x + (x.^2)/2 - 3.1*(x.^3)/3 + 1.2*(x.^4)/4;
sig = 0.5;
y = trueline + sig*randn(n,1);
tic;
fit = kernel_regression(x, y, x); # use the default bandwidth
t1 = toc;
printf("\n");
printf("########################################################################\n");
printf("time for kernel regression example using %d data points and %d compute nodes: %f\n", n, nodes, t1);
plot(x, fit, ";fit;", x, trueline,";true;");
grid("on");
title("Example 1: Kernel regression fit");
############################################################
# kernel density example: univariate - fit to Chi^2(3) data
data = sumsq(randn(n,3),2);
# evaluation point are on a grid for plotting
stepsize = 0.2;
grid_x = (-1:stepsize:11)';
bandwidth = 0.55;
# get optimal bandwidth (time consuming, uncomment if you want to try it)
# bandwidth = kernel_optimal_bandwidth(data);
# get the fitted density and do a plot
tic;
dens = kernel_density(grid_x, data, bandwidth, "__kernel_normal", false, false, compute_nodes);
t1 = toc;
printf("\n");
printf("########################################################################\n");
printf("time for univariate kernel density example using %d data points and %d compute nodes: %f\n", n, nodes, t1);
printf("A rough integration under the fitted univariate density is %f\n", sum(dens)*stepsize);
figure();
plot(grid_x, dens, ";fitted density;", grid_x, chi2pdf(grid_x,3), ";true density;");
title("Example 2: Kernel density fit: Univariate Chi^2(3) data");
############################################################
# kernel density example: bivariate
# X ~ N(0,1)
# Y ~ Chi squared(3)
# X, Y are dependent
d = randn(n,3);
data = [d(:,1) sumsq(d,2)];
# evaluation points are on a grid for plotting
stepsize = 0.2;
a = (-5:stepsize:5)'; # for the N(0,1)
b = (-1:stepsize:9)'; # for the Chi squared(3)
gridsize = rows(a);
[grid_x, grid_y] = meshgrid(a, b);
eval_points = [vec(grid_x) vec(grid_y)];
bandwidth = 0.85;
# get optimal bandwidth (time consuming, uncomment if you want to try it)
# bandwidth = kernel_optimal_bandwidth(data);
# get the fitted density and do a plot
tic;
dens = kernel_density(eval_points, data, bandwidth, "__kernel_normal", false, false, compute_nodes);
t1 = toc;
printf("\n");
printf("########################################################################\n");
printf("time for multivariate kernel density example using %d data points and %d compute nodes: %f\n", n, nodes, t1);
dens = reshape(dens, gridsize, gridsize);
printf("A rough integration under the fitted bivariate density is %f\n", sum(sum(dens))*stepsize^2);
figure();
surf(grid_x, grid_y, dens);
title("Example 3: Kernel density fit: dependent bivariate data");
xlabel("true marginal density is N(0,1)");
ylabel("true marginal density is Chi^2(3)");
# more extensive test of parallel
if compute_nodes > 0 # only try this if parallel is available
ns =[4000; 8000; 10000; 12000; 16000; 20000];
printf("\n");
printf("########################################################################\n");
printf("kernel regression example with several sample sizes serial/parallel timings\n");
figure();
clg;
title("Compute time versus nodes, kernel regression with different sample sizes");
xlabel("nodes");
ylabel("time (sec)");
hold on;
ts = zeros(rows(ns),4);
for i = 1:rows(ns)
for compute_nodes = 0:3
nodes = compute_nodes + 1;
n = ns(i,:);
x = 1:n;
x = x';
x = 2*x/n;
# generate dependent variable
trueline = x + (x.^2)/2 - 3.1*(x.^3)/3 + 1.2*(x.^4)/4;
sig = 0.5;
y = trueline + sig*randn(n,1);
bandwidth = 0.45;
tic;
fit = kernel_regression(x, y, x, bandwidth, "__kernel_normal", false, false, compute_nodes);
t1 = toc;
ts(i, nodes) = t1;
plot(nodes, t1, "*");
printf(" %d data points and %d compute nodes: %f\n", n, nodes, t1);
endfor
plot(ts(i,:)');
endfor
hold off;
endif
|
