annotate maths/pca/PCA.c @ 242:6060110dc3c6

* Build PCA and HMM

/*********************************/
/* 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:     [e-mail]
   
   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);

}