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ardour/libs/qm-dsp/maths/pca/pca.c

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/*********************************/
/* Principal Components Analysis */
/*********************************/
/*********************************************************************/
/* Principal Components Analysis or the Karhunen-Loeve expansion is a
classical method for dimensionality reduction or exploratory data
analysis. One reference among many is: F. Murtagh and A. Heck,
Multivariate Data Analysis, Kluwer Academic, Dordrecht, 1987.
Author:
F. Murtagh
Phone: + 49 89 32006298 (work)
+ 49 89 965307 (home)
Earn/Bitnet: fionn@dgaeso51, fim@dgaipp1s, murtagh@stsci
Span: esomc1::fionn
Internet: murtagh@scivax.stsci.edu
F. Murtagh, Munich, 6 June 1989 */
/*********************************************************************/
#include <stdio.h>
#include <stdlib.h>
#include <math.h>
#include "pca.h"
#define SIGN(a, b) ( (b) < 0 ? -fabs(a) : fabs(a) )
/** Variance-covariance matrix: creation *****************************/
/* Create m * m covariance matrix from given n * m data matrix. */
static void covcol(double** data, int n, int m, double** symmat)
{
double *mean;
int i, j, j1, j2;
/* Allocate storage for mean vector */
mean = (double*) malloc(m*sizeof(double));
/* Determine mean of column vectors of input data matrix */
for (j = 0; j < m; j++)
{
mean[j] = 0.0;
for (i = 0; i < n; i++)
{
mean[j] += data[i][j];
}
mean[j] /= (double)n;
}
/*
printf("\nMeans of column vectors:\n");
for (j = 0; j < m; j++) {
printf("%12.1f",mean[j]); } printf("\n");
*/
/* Center the column vectors. */
for (i = 0; i < n; i++)
{
for (j = 0; j < m; j++)
{
data[i][j] -= mean[j];
}
}
/* Calculate the m * m covariance matrix. */
for (j1 = 0; j1 < m; j1++)
{
for (j2 = j1; j2 < m; j2++)
{
symmat[j1][j2] = 0.0;
for (i = 0; i < n; i++)
{
symmat[j1][j2] += data[i][j1] * data[i][j2];
}
symmat[j2][j1] = symmat[j1][j2];
}
}
free(mean);
return;
}
/** Error handler **************************************************/
static void erhand(char const* err_msg)
{
fprintf(stderr,"Run-time error:\n");
fprintf(stderr,"%s\n", err_msg);
fprintf(stderr,"Exiting to system.\n");
exit(1);
}
/** Reduce a real, symmetric matrix to a symmetric, tridiag. matrix. */
/* Householder reduction of matrix a to tridiagonal form.
Algorithm: Martin et al., Num. Math. 11, 181-195, 1968.
Ref: Smith et al., Matrix Eigensystem Routines -- EISPACK Guide
Springer-Verlag, 1976, pp. 489-494.
W H Press et al., Numerical Recipes in C, Cambridge U P,
1988, pp. 373-374. */
static void tred2(double** a, int n, double* d, double* e)
{
int l, k, j, i;
double scale, hh, h, g, f;
for (i = n-1; i >= 1; i--)
{
l = i - 1;
h = scale = 0.0;
if (l > 0)
{
for (k = 0; k <= l; k++)
scale += fabs(a[i][k]);
if (scale == 0.0)
e[i] = a[i][l];
else
{
for (k = 0; k <= l; k++)
{
a[i][k] /= scale;
h += a[i][k] * a[i][k];
}
f = a[i][l];
g = f>0 ? -sqrt(h) : sqrt(h);
e[i] = scale * g;
h -= f * g;
a[i][l] = f - g;
f = 0.0;
for (j = 0; j <= l; j++)
{
a[j][i] = a[i][j]/h;
g = 0.0;
for (k = 0; k <= j; k++)
g += a[j][k] * a[i][k];
for (k = j+1; k <= l; k++)
g += a[k][j] * a[i][k];
e[j] = g / h;
f += e[j] * a[i][j];
}
hh = f / (h + h);
for (j = 0; j <= l; j++)
{
f = a[i][j];
e[j] = g = e[j] - hh * f;
for (k = 0; k <= j; k++)
a[j][k] -= (f * e[k] + g * a[i][k]);
}
}
}
else
e[i] = a[i][l];
d[i] = h;
}
d[0] = 0.0;
e[0] = 0.0;
for (i = 0; i < n; i++)
{
l = i - 1;
if (d[i])
{
for (j = 0; j <= l; j++)
{
g = 0.0;
for (k = 0; k <= l; k++)
g += a[i][k] * a[k][j];
for (k = 0; k <= l; k++)
a[k][j] -= g * a[k][i];
}
}
d[i] = a[i][i];
a[i][i] = 1.0;
for (j = 0; j <= l; j++)
a[j][i] = a[i][j] = 0.0;
}
}
/** Tridiagonal QL algorithm -- Implicit **********************/
static void tqli(double* d, double* e, int n, double** z)
{
int m, l, iter, i, k;
double s, r, p, g, f, dd, c, b;
for (i = 1; i < n; i++)
e[i-1] = e[i];
e[n-1] = 0.0;
for (l = 0; l < n; l++)
{
iter = 0;
do
{
for (m = l; m < n-1; m++)
{
dd = fabs(d[m]) + fabs(d[m+1]);
if (fabs(e[m]) + dd == dd) break;
}
if (m != l)
{
if (iter++ == 30) erhand("No convergence in TLQI.");
g = (d[l+1] - d[l]) / (2.0 * e[l]);
r = sqrt((g * g) + 1.0);
g = d[m] - d[l] + e[l] / (g + SIGN(r, g));
s = c = 1.0;
p = 0.0;
for (i = m-1; i >= l; i--)
{
f = s * e[i];
b = c * e[i];
if (fabs(f) >= fabs(g))
{
c = g / f;
r = sqrt((c * c) + 1.0);
e[i+1] = f * r;
c *= (s = 1.0/r);
}
else
{
s = f / g;
r = sqrt((s * s) + 1.0);
e[i+1] = g * r;
s *= (c = 1.0/r);
}
g = d[i+1] - p;
r = (d[i] - g) * s + 2.0 * c * b;
p = s * r;
d[i+1] = g + p;
g = c * r - b;
for (k = 0; k < n; k++)
{
f = z[k][i+1];
z[k][i+1] = s * z[k][i] + c * f;
z[k][i] = c * z[k][i] - s * f;
}
}
d[l] = d[l] - p;
e[l] = g;
e[m] = 0.0;
}
} while (m != l);
}
}
/* In place projection onto basis vectors */
void pca_project(double** data, int n, int m, int ncomponents)
{
int i, j, k, k2;
double **symmat, /* **symmat2, */ *evals, *interm;
//TODO: assert ncomponents < m
symmat = (double**) malloc(m*sizeof(double*));
for (i = 0; i < m; i++)
symmat[i] = (double*) malloc(m*sizeof(double));
covcol(data, n, m, symmat);
/*********************************************************************
Eigen-reduction
**********************************************************************/
/* Allocate storage for dummy and new vectors. */
evals = (double*) malloc(m*sizeof(double)); /* Storage alloc. for vector of eigenvalues */
interm = (double*) malloc(m*sizeof(double)); /* Storage alloc. for 'intermediate' vector */
//MALLOC_ARRAY(symmat2,m,m,double);
//for (i = 0; i < m; i++) {
// for (j = 0; j < m; j++) {
// symmat2[i][j] = symmat[i][j]; /* Needed below for col. projections */
// }
//}
tred2(symmat, m, evals, interm); /* Triangular decomposition */
tqli(evals, interm, m, symmat); /* Reduction of sym. trid. matrix */
/* evals now contains the eigenvalues,
columns of symmat now contain the associated eigenvectors. */
/*
printf("\nEigenvalues:\n");
for (j = m-1; j >= 0; j--) {
printf("%18.5f\n", evals[j]); }
printf("\n(Eigenvalues should be strictly positive; limited\n");
printf("precision machine arithmetic may affect this.\n");
printf("Eigenvalues are often expressed as cumulative\n");
printf("percentages, representing the 'percentage variance\n");
printf("explained' by the associated axis or principal component.)\n");
printf("\nEigenvectors:\n");
printf("(First three; their definition in terms of original vbes.)\n");
for (j = 0; j < m; j++) {
for (i = 1; i <= 3; i++) {
printf("%12.4f", symmat[j][m-i]); }
printf("\n"); }
*/
/* Form projections of row-points on prin. components. */
/* Store in 'data', overwriting original data. */
for (i = 0; i < n; i++) {
for (j = 0; j < m; j++) {
interm[j] = data[i][j]; } /* data[i][j] will be overwritten */
for (k = 0; k < ncomponents; k++) {
data[i][k] = 0.0;
for (k2 = 0; k2 < m; k2++) {
data[i][k] += interm[k2] * symmat[k2][m-k-1]; }
}
}
/*
printf("\nProjections of row-points on first 3 prin. comps.:\n");
for (i = 0; i < n; i++) {
for (j = 0; j < 3; j++) {
printf("%12.4f", data[i][j]); }
printf("\n"); }
*/
/* Form projections of col.-points on first three prin. components. */
/* Store in 'symmat2', overwriting what was stored in this. */
//for (j = 0; j < m; j++) {
// for (k = 0; k < m; k++) {
// interm[k] = symmat2[j][k]; } /*symmat2[j][k] will be overwritten*/
// for (i = 0; i < 3; i++) {
// symmat2[j][i] = 0.0;
// for (k2 = 0; k2 < m; k2++) {
// symmat2[j][i] += interm[k2] * symmat[k2][m-i-1]; }
// if (evals[m-i-1] > 0.0005) /* Guard against zero eigenvalue */
// symmat2[j][i] /= sqrt(evals[m-i-1]); /* Rescale */
// else
// symmat2[j][i] = 0.0; /* Standard kludge */
// }
// }
/*
printf("\nProjections of column-points on first 3 prin. comps.:\n");
for (j = 0; j < m; j++) {
for (k = 0; k < 3; k++) {
printf("%12.4f", symmat2[j][k]); }
printf("\n"); }
*/
for (i = 0; i < m; i++)
free(symmat[i]);
free(symmat);
//FREE_ARRAY(symmat2,m);
free(evals);
free(interm);
}
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