/* statistics/covar_source.c

 * 

 * Copyright (C) 1996, 1997, 1998, 1999, 2000, 2007 Jim Davies, Brian Gough

 * 

 * 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 3 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., 51 Franklin Street, Fifth Floor, Boston, MA 02110-1301, USA.

 */

static double

FUNCTION(compute,covariance) (const BASE data1[], const size_t stride1,

                              const BASE data2[], const size_t stride2,

                              const size_t n, 

                              const double mean1, const double mean2);

static double

FUNCTION(compute,covariance) (const BASE data1[], const size_t stride1,

                              const BASE data2[], const size_t stride2,

                              const size_t n, 

                              const double mean1, const double mean2)

{

  /* takes a dataset and finds the covariance */

  long double covariance = 0 ;

  size_t i;

  /* find the sum of the squares */

  for (i = 0; i < n; i++)

    {

      const long double delta1 = (data1[i * stride1] - mean1);

      const long double delta2 = (data2[i * stride2] - mean2);

      covariance += (delta1 * delta2 - covariance) / (i + 1);

    }

  return covariance ;

}

double 

FUNCTION(gsl_stats,covariance_m) (const BASE data1[], const size_t stride1, 

                                  const BASE data2[], const size_t stride2, 

                                  const size_t n, 

                                  const double mean1, const double mean2)

{

  const double covariance = FUNCTION(compute,covariance) (data1, stride1,

                                                          data2, stride2,

                                                          n, 

                                                          mean1, mean2);

  return covariance * ((double)n / (double)(n - 1));

}

double 

FUNCTION(gsl_stats,covariance) (const BASE data1[], const size_t stride1,

                                const BASE data2[], const size_t stride2,

                                const size_t n)

{

  const double mean1 = FUNCTION(gsl_stats,mean) (data1, stride1, n);

  const double mean2 = FUNCTION(gsl_stats,mean) (data2, stride2, n);

  return FUNCTION(gsl_stats,covariance_m)(data1, stride1, 

                                          data2, stride2, 

                                          n, 

                                          mean1, mean2);

}

/*

gsl_stats_correlation()

  Calculate Pearson correlation = cov(X, Y) / (sigma_X * sigma_Y)

This routine efficiently computes the correlation in one pass of the

data and makes use of the algorithm described in:

B. P. Welford, "Note on a Method for Calculating Corrected Sums of

Squares and Products", Technometrics, Vol 4, No 3, 1962.

This paper derives a numerically stable recurrence to compute a sum

of products

S = sum_{i=1..N} [ (x_i - mu_x) * (y_i - mu_y) ]

with the relation

S_n = S_{n-1} + ((n-1)/n) * (x_n - mu_x_{n-1}) * (y_n - mu_y_{n-1})

*/

double

FUNCTION(gsl_stats,correlation) (const BASE data1[], const size_t stride1,

                                 const BASE data2[], const size_t stride2,

                                 const size_t n)

{

  size_t i;

  long double sum_xsq = 0.0;

  long double sum_ysq = 0.0;

  long double sum_cross = 0.0;

  long double ratio;

  long double delta_x, delta_y;

  long double mean_x, mean_y;

  long double r;

  /*

   * Compute:

   * sum_xsq = Sum [ (x_i - mu_x)^2 ],

   * sum_ysq = Sum [ (y_i - mu_y)^2 ] and

   * sum_cross = Sum [ (x_i - mu_x) * (y_i - mu_y) ]

   * using the above relation from Welford's paper

   */

  mean_x = data1[0 * stride1];

  mean_y = data2[0 * stride2];

  for (i = 1; i < n; ++i)

    {

      ratio = i / (i + 1.0);

      delta_x = data1[i * stride1] - mean_x;

      delta_y = data2[i * stride2] - mean_y;

      sum_xsq += delta_x * delta_x * ratio;

      sum_ysq += delta_y * delta_y * ratio;

      sum_cross += delta_x * delta_y * ratio;

      mean_x += delta_x / (i + 1.0);

      mean_y += delta_y / (i + 1.0);

    }

  r = sum_cross / (sqrt(sum_xsq) * sqrt(sum_ysq));

  return r;

}

/*

gsl_stats_spearman()

  Compute Spearman rank correlation coefficient

Inputs: data1   - data1 vector

        stride1 - stride of data1

        data2   - data2 vector

        stride2 - stride of data2

        n       - number of elements in data1 and data2

        work    - additional workspace of size 2*n

Return: Spearman rank correlation coefficient

*/

double

FUNCTION(gsl_stats,spearman) (const BASE data1[], const size_t stride1,

                              const BASE data2[], const size_t stride2,

                              const size_t n, double work[])

{

  size_t i;

  gsl_vector_view ranks1 = gsl_vector_view_array(&work[0], n);

  gsl_vector_view ranks2 = gsl_vector_view_array(&work[n], n);

  double r;

  for (i = 0; i < n; ++i)

    {

      gsl_vector_set(&ranks1.vector, i, data1[i * stride1]);

      gsl_vector_set(&ranks2.vector, i, data2[i * stride2]);

    }

  /* sort data1 and update data2 at same time; compute rank of data1 */

  gsl_sort_vector2(&ranks1.vector, &ranks2.vector);

  compute_rank(&ranks1.vector);

  /* now sort data2, updating ranks1 appropriately; compute rank of data2 */

  gsl_sort_vector2(&ranks2.vector, &ranks1.vector);

  compute_rank(&ranks2.vector);

  /* compute correlation of rank vectors in double precision */

  r = gsl_stats_correlation(ranks1.vector.data, ranks1.vector.stride,

                            ranks2.vector.data, ranks2.vector.stride,

                            n);

  return r;

}