/* file: dfa.c J. Mietus, C-K Peng, and G. Moody 8 February 2001
Last revised: 25 January 2005 v4.9
-------------------------------------------------------------------------------
dfa: Detrended Fluctuation Analysis (translated from C-K Peng's Fortran code)
Copyright (C) 2001-2005 Joe Mietus, C-K Peng, and George B. Moody
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, write to the Free Software Foundation, Inc., 59 Temple
Place - Suite 330, Boston, MA 02111-1307, USA.
You may contact the authors by e-mail (peng@physionet.org) or postal mail
(Beth Israel Deaconess Medical Center, Room KS-B26, 330 Brookline Ave., Boston,
MA 02215 USA). For updates to this software, please visit PhysioNet
(http://www.physionet.org/).
_______________________________________________________________________________
This method was first proposed in:
Peng C-K, Buldyrev SV, Havlin S, Simons M, Stanley HE, Goldberger AL. Mosaic
organization of DNA nucleotides. Phys Rev E 1994;49:1685-1689. [Available
on-line at http://prola.aps.org/abstract/PRE/v49/i2/p1685_1]
A detailed description of the algorithm and its application to physiologic
signals can be found in:
Peng C-K, Havlin S, Stanley HE, Goldberger AL. Quantification of scaling
exponents and crossover phenomena in nonstationary heartbeat time series.
Chaos 1995;5:82-87. [Abstract online at http://www.ncbi.nlm.nih.gov/entrez/-
query.fcgi?cmd=Retrieve&db=PubMed&list_uids=11538314&dopt=Abstract]
If you use this program in support of published research, please include a
citation of at least one of the two references above, as well as the standard
citation for PhysioNet:
Goldberger AL, Amaral LAN, Glass L, Hausdorff JM, Ivanov PCh, Mark RG,
Mietus JE, Moody GB, Peng CK, Stanley HE. PhysioBank, PhysioToolkit, and
Physionet: Components of a New Research Resource for Complex Physiologic
Signals. Circulation 101(23):e215-e220 [Circulation Electronic Pages;
http://circ.ahajournals.org/cgi/content/full/101/23/e215]; 2000 (June 13).
*/
#include <stdio.h>
#include <stdlib.h>
#include <math.h>
#define SWAP(a,b) {temp = (a); (a) = (b); (b) = temp;}
/* Function prototypes. */
long input(void);
int rscale(long minbox, long maxbox, double boxratio);
void dfa(double *seq, long npts, int nfit, long *rs, int nr, int sw);
void setup(void);
void cleanup(void);
void help(void);
double polyfit(double **x, double *y, long ndat, int nfit);
void error(char error_text[]);
double *vector(long nl, long nh);
int *ivector(long nl, long nh);
long *lvector(long nl, long nh);
double **matrix(long nrl, long nrh, long ncl, long nch);
void free_vector(double *v, long nl, long nh);
void free_ivector(int *v, long nl, long nh);
void free_lvector(long *v, long nl, long nh);
void free_matrix(double **m, long nrl, long nrh, long ncl, long nch);
/* Global variables. */
char *pname; /* this program's name (for use in error messages) */
double *seq; /* input data buffer; allocated and filled by input() */
long *rs; /* box size array; allocated and filled by rscale() */
double *mse; /* fluctuation array; allocated by setup(), filled by dfa() */
int iflag = 1; /* integrate the input data if non-zero */
int nfit = 2; /* order of the regression fit, plus 1 */
int nr; /* number of box sizes */
main(int argc, char **argv)
{
int i, sw = 0;
long minbox = 0L, maxbox = 0L, npts, temp;
/* Read and interpret the command line. */
pname = argv[0];
for (i = 1; i < argc && *argv[i] == '-'; i++) {
switch(argv[i][1]) {
case 'd': /* set nfit (the order of the regression fit) */
if ((nfit = atoi(argv[++i])+1) < 2)
error("order must be greater than 0");
break;
case 'i': /* input data are already integrated */
iflag = 0; break;
case 'l': /* set minbox (the minimum box size) */
minbox = atol(argv[++i]); break;
case 'u': /* set maxbox (the maximum box size) */
maxbox = atol(argv[++i]); break;
case 's': /* enable sliding window mode */
sw = 1; break;
case 'h': /* print usage information and quit */
default:
help();
exit(1);
}
}
/* Allocate and fill the input data array seq[]. */
npts = input();
/* Set minimum and maximum box sizes. */
if (minbox < 2*nfit) minbox = 2*nfit;
if (maxbox == 0 || maxbox > npts/4) maxbox = npts/4;
if (minbox > maxbox) {
SWAP(minbox, maxbox);
if (minbox < 2*nfit) minbox = 2*nfit;
}
/* Allocate and fill the box size array rs[]. rscale's third argument
specifies that the ratio between successive box sizes is 2^(1/8). */
nr = rscale(minbox, maxbox, pow(2.0, 1.0/8.0));
/* Allocate memory for dfa() and the functions it calls. */
setup();
/* Measure the fluctuations of the detrended input data at each box size
using the DFA algorithm; fill mse[] with these results. */
dfa(seq, npts, nfit, rs, nr, sw);
/* Output the results. */
for (i = 1; i <= nr; i++)
printf("%g %g\n", log10((double)rs[i]), log10(mse[i])/2.0);
/* Release allocated memory. */
cleanup();
exit(0);
}
/* Read input data, allocating and filling seq[], integrating if iflag != 0.
Following the convention used for other arrays in this program, seq[0] is
unused, and the first point is stored in seq[1]. The return value is the
number of points read.
This function allows the input buffer to grow as large as necessary, up to
the available memory (assuming that a long int is large enough to address
any memory location). Note that the integration is done using double
precision arithmetic to avoid complete loss of precision when the integrated
data reach large amplitudes. */
long input()
{
long maxdat = 0L, npts = 0L;
double y, yp = 0.0;
while (scanf("%lf", &y) == 1) {
if (++npts >= maxdat) {
double *s;
maxdat += 50000; /* allow the input buffer to grow (the
increment is arbitrary) */
if ((s = realloc(seq, maxdat * sizeof(double))) == NULL) {
fprintf(stderr,
"%s: insufficient memory, truncating input at row %d\n",
pname, npts);
break;
}
seq = s;
}
seq[npts] = iflag ? (yp += y) : y;
}
if (npts < 1) error("no data read");
return (npts);
}
int rslen; /* length of rs[] */
/* rscale() allocates and fills rs[], the array of box sizes used by dfa()
below. The box sizes range from (exactly) minbox to (approximately) maxbox,
and are arranged in a geometric series such that the ratio between
consecutive box sizes is (approximately) boxratio. The return value is
the number of box sizes in rs[].
*/
int rscale(long minbox, long maxbox, double boxratio)
{
int ir, n;
long rw;
/* Determine how many scales are needed. */
rslen = log10(maxbox / (double)minbox) / log10(boxratio) + 1.5;
/* Thanks to Peter Domitrovich for pointing out that a previous version
of the above calculation undercounted the number of scales in some
situations. */
rs = lvector(1, rslen);
for (ir = 1, n = 2, rs[1] = minbox; n <= rslen && rs[n-1] < maxbox; ir++)
if ((rw = minbox * pow(boxratio, ir) + 0.5) > rs[n-1])
rs[n++] = rw;
if (rs[--n] > maxbox) --n;
return (n);
}
double **x; /* matrix of abscissas and their powers, for polyfit(). */
/* Detrended fluctuation analysis
seq: input data array
npts: number of input points
nfit: order of detrending (2: linear, 3: quadratic, etc.)
rs: array of box sizes (uniformly distributed on log scale)
nr: number of entries in rs[] and mse[]
sw: mode (0: non-overlapping windows, 1: sliding window)
This function returns the mean squared fluctuations in mse[].
*/
void dfa(double *seq, long npts, int nfit, long *rs, int nr, int sw)
{
long i, boxsize, inc, j;
double stat;
for (i = 1; i <= nr; i++) {
boxsize = rs[i];
if (sw) { inc = 1; stat = (int)(npts - boxsize + 1) * boxsize; }
else { inc = boxsize; stat = (int)(npts / boxsize) * boxsize; }
for (mse[i] = 0.0, j = 0; j <= npts - boxsize; j += inc)
mse[i] += polyfit(x, seq + j, boxsize, nfit);
mse[i] /= stat;
}
}
/* workspace for polyfit() */
double *beta, **covar, **covar0;
int *indxc, *indxr, *ipiv;
/* This function allocates workspace for dfa() and polyfit(), and sets
x[i][j] = i**(j-1), in preparation for polyfit(). */
void setup()
{
long i;
int j, k;
beta = vector(1, nfit);
covar = matrix(1, nfit, 1, nfit);
covar0 = matrix(1, nfit, 1, nfit);
indxc = ivector(1, nfit);
indxr = ivector(1, nfit);
ipiv = ivector(1, nfit);
mse = vector(1, nr);
x = matrix(1, rs[nr], 1, nfit);
for (i = 1; i <= rs[nr]; i++) {
x[i][1] = 1.0;
x[i][2] = i;
for (j = 3; j <= nfit; j++)
x[i][j] = x[i][j-1] * i;
}
}
/* This function frees all memory previously allocated by this program. */
void cleanup()
{
free_matrix(x, 1, rs[nr], 1, nfit);
free_vector(mse, 1, nr);
free_ivector(ipiv, 1, nfit);
free_ivector(indxr, 1, nfit);
free_ivector(indxc, 1, nfit);
free_matrix(covar0, 1, nfit, 1, nfit);
free_matrix(covar, 1, nfit, 1, nfit);
free_vector(beta, 1, nfit);
free_lvector(rs, 1, rslen); /* allocated by rscale() */
free(seq); /* allocated by input() */
}
static char *help_strings[] = {
"usage: %s [OPTIONS ...]\n",
"where OPTIONS may include:",
" -d K detrend using a polynomial of degree K",
" (default: K=1 -- linear detrending)",
" -h print this usage summary",
" -i input series is already integrated",
" -l MINBOX smallest box width (default: 2K+2)",
" -s sliding window DFA",
" -u MAXBOX largest box width (default: NPTS/4)",
"The standard input should contain one column of data in text format.",
"The standard output is two columns: log(n) and log(F) [base 10 logarithms],",
"where n is the box size and F is the root mean square fluctuation.",
NULL
};
void help(void)
{
int i;
(void)fprintf(stderr, help_strings[0], pname);
for (i = 1; help_strings[i] != NULL; i++)
(void)fprintf(stderr, "%s\n", help_strings[i]);
}
/* polyfit() is based on lfit() and gaussj() from Numerical Recipes in C
(Press, Teukolsky, Vetterling, and Flannery; Cambridge U. Press, 1992). It
fits a polynomial of degree (nfit-1) to a set of boxsize points given by
x[1...boxsize][2] and y[1...boxsize]. The return value is the sum of the
squared errors (chisq) between the (x,y) pairs and the fitted polynomial.
*/
double polyfit(double **x, double *y, long boxsize, int nfit)
{
int icol, irow, j, k;
double big, chisq, pivinv, temp;
long i;
static long pboxsize = 0L;
/* This block sets up the covariance matrix. Provided that boxsize
never decreases (which is true in this case), covar0 can be calculated
incrementally from the previous value. */
if (pboxsize != boxsize) { /* this will be false most of the time */
if (pboxsize > boxsize) /* this should never happen */
pboxsize = 0L;
if (pboxsize == 0L) /* this should be true the first time only */
for (j = 1; j <= nfit; j++)
for (k = 1; k <= nfit; k++)
covar0[j][k] = 0.0;
for (i = pboxsize+1; i <= boxsize; i++)
for (j = 1; j <= nfit; j++)
for (k = 1, temp = x[i][j]; k <= j; k++)
covar0[j][k] += temp * x[i][k];
for (j = 2; j <= nfit; j++)
for (k = 1; k < j; k++)
covar0[k][j] = covar0[j][k];
pboxsize = boxsize;
}
for (j = 1; j <= nfit; j++) {
beta[j] = ipiv[j] = 0;
for (k = 1; k <= nfit; k++)
covar[j][k] = covar0[j][k];
}
for (i = 1; i <= boxsize; i++) {
beta[1] += (temp = y[i]);
beta[2] += temp * i;
}
if (nfit > 2)
for (i = 1; i <= boxsize; i++)
for (j = 3, temp = y[i]; j <= nfit; j++)
beta[j] += temp * x[i][j];
for (i = 1; i <= nfit; i++) {
big = 0.0;
for (j = 1; j <= nfit; j++)
if (ipiv[j] != 1)
for (k = 1; k <= nfit; k++) {
if (ipiv[k] == 0) {
if ((temp = covar[j][k]) >= big ||
(temp = -temp) >= big) {
big = temp;
irow = j;
icol = k;
}
}
else if (ipiv[k] > 1)
error("singular matrix");
}
++(ipiv[icol]);
if (irow != icol) {
for (j = 1; j <= nfit; j++) SWAP(covar[irow][j], covar[icol][j]);
SWAP(beta[irow], beta[icol]);
}
indxr[i] = irow;
indxc[i] = icol;
if (covar[icol][icol] == 0.0) error("singular matrix");
pivinv = 1.0 / covar[icol][icol];
covar[icol][icol] = 1.0;
for (j = 1; j <= nfit; j++) covar[icol][j] *= pivinv;
beta[icol] *= pivinv;
for (j = 1; j <= nfit; j++)
if (j != icol) {
temp = covar[j][icol];
covar[j][icol] = 0.0;
for (k = 1; k <= nfit; k++) covar[j][k] -= covar[icol][k]*temp;
beta[j] -= beta[icol] * temp;
}
}
chisq = 0.0;
if (nfit <= 2)
for (i = 1; i <= boxsize; i++) {
temp = beta[1] + beta[2] * i - y[i];
chisq += temp * temp;
}
else
for (i = 1; i <= boxsize; i++) {
temp = beta[1] + beta[2] * i - y[i];
for (j = 3; j <= nfit; j++) temp += beta[j] * x[i][j];
chisq += temp * temp;
}
return (chisq);
}
/* The functions below are based on those of the same names in Numerical
Recipes (see above). */
void error(char error_text[])
{
fprintf(stderr, "%s: %s\n", pname, error_text);
exit(1);
}
double *vector(long nl, long nh)
/* allocate a double vector with subscript range v[nl..nh] */
{
double *v = (double *)malloc((size_t)((nh-nl+2) * sizeof(double)));
if (v == NULL) error("allocation failure in vector()");
return (v-nl+1);
}
int *ivector(long nl, long nh)
/* allocate an int vector with subscript range v[nl..nh] */
{
int *v = (int *)malloc((size_t)((nh-nl+2) * sizeof(int)));
if (v == NULL) error("allocation failure in ivector()");
return (v-nl+1);
}
long *lvector(long nl, long nh)
/* allocate a long int vector with subscript range v[nl..nh] */
{
long *v = (long *)malloc((size_t)((nh-nl+2) * sizeof(long)));
if (v == NULL) error("allocation failure in lvector()");
return (v-nl+1);
}
double **matrix(long nrl, long nrh, long ncl, long nch)
/* allocate a double matrix with subscript range m[nrl..nrh][ncl..nch] */
{
long i, nrow = nrh-nrl+1, ncol = nch-ncl+1;
double **m;
/* allocate pointers to rows */
m = (double **) malloc((size_t)((nrow+1) * sizeof(double*)));
if (!m) error("allocation failure 1 in matrix()");
m += 1;
m -= nrl;
/* allocate rows and set pointers to them */
m[nrl] = (double *) malloc((size_t)((nrow*ncol+1) * sizeof(double)));
if (!m[nrl]) error("allocation failure 2 in matrix()");
m[nrl] += 1;
m[nrl] -= ncl;
for (i = nrl+1; i <= nrh; i++) m[i] = m[i-1]+ncol;
/* return pointer to array of pointers to rows */
return (m);
}
void free_vector(double *v, long nl, long nh)
/* free a double vector allocated with vector() */
{
free(v+nl-1);
}
void free_ivector(int *v, long nl, long nh)
/* free an int vector allocated with ivector() */
{
free(v+nl-1);
}
void free_lvector(long *v, long nl, long nh)
/* free a long int vector allocated with lvector() */
{
free(v+nl-1);
}
void free_matrix(double **m, long nrl, long nrh, long ncl, long nch)
/* free a double matrix allocated by matrix() */
{
free(m[nrl]+ncl-1);
free(m+nrl-1);
}