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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. */
+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 **************************************************/
+
+void erhand(char* 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. */
+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 **********************/
+
+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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