## Copyright (C) 2011-2014 L. Markowsky <lmarkov@users.sourceforge.net>
##
## This file is part of the fuzzy-logic-toolkit.
##
## The fuzzy-logic-toolkit 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.
##
## The fuzzy-logic-toolkit 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 the fuzzy-logic-toolkit; see the file COPYING.  If not,
## see <http://www.gnu.org/licenses/>.

## -*- texinfo -*-
## @deftypefn {Function File} {@var{cluster_centers} =} gustafson_kessel (@var{input_data}, @var{num_clusters})
## @deftypefnx {Function File} {@var{cluster_centers} =} gustafson_kessel (@var{input_data}, @var{num_clusters}, @var{cluster_volume})
## @deftypefnx {Function File} {@var{cluster_centers} =} gustafson_kessel (@var{input_data}, @var{num_clusters}, @var{cluster_volume}, @var{options})
## @deftypefnx {Function File} {@var{cluster_centers} =} gustafson_kessel (@var{input_data}, @var{num_clusters}, @var{cluster_volume}, [@var{m}, @var{max_iterations}, @var{epsilon}, @var{display_intermediate_results}])
## @deftypefnx {Function File} {[@var{cluster_centers}, @var{soft_partition}, @var{obj_fcn_history}] =} gustafson_kessel (@var{input_data}, @var{num_clusters})
## @deftypefnx {Function File} {[@var{cluster_centers}, @var{soft_partition}, @var{obj_fcn_history}] =} gustafson_kessel (@var{input_data}, @var{num_clusters}, @var{cluster_volume})
## @deftypefnx {Function File} {[@var{cluster_centers}, @var{soft_partition}, @var{obj_fcn_history}] =} gustafson_kessel (@var{input_data}, @var{num_clusters}, @var{cluster_volume}, @var{options})
## @deftypefnx {Function File} {[@var{cluster_centers}, @var{soft_partition}, @var{obj_fcn_history}] =} gustafson_kessel (@var{input_data}, @var{num_clusters}, @var{cluster_volume}, [@var{m}, @var{max_iterations}, @var{epsilon}, @var{display_intermediate_results}])
##
## Using the Gustafson-Kessel algorithm, calculate and return the soft partition
## of a set of unlabeled data points.
##
## Also, if @var{display_intermediate_results} is true, display intermediate 
## results after each iteration. Note that because the initial cluster
## prototypes are randomly selected locations in the ranges determined by the
## input data, the results of this function are nondeterministic.
##
## The required arguments to gustafson_kessel are:
## @itemize @w
## @item
## @var{input_data} - a matrix of input data points; each row corresponds to one point
## @item
## @var{num_clusters} - the number of clusters to form
## @end itemize
##
## The third (optional) argument to gustafson_kessel is a vector of cluster volumes.
## If omitted, a vector of 1's will be used as the default.
##
## The fourth (optional) argument to gustafson_kessel is a vector consisting of:
## @itemize @w
## @item
## @var{m} - the parameter (exponent) in the objective function; default = 2.0
## @item
## @var{max_iterations} - the maximum number of iterations before stopping; default = 100
## @item
## @var{epsilon} - the stopping criteria; default = 1e-5
## @item
## @var{display_intermediate_results} - if 1, display results after each iteration, and if 0, do not; default = 1
## @end itemize
##
## The default values are used if any of the four elements of the vector are missing or
## evaluate to NaN.
##
## The return values are:
## @itemize @w
## @item
## @var{cluster_centers} - a matrix of the cluster centers; each row corresponds to one point
## @item
## @var{soft_partition} - a constrained soft partition matrix
## @item
## @var{obj_fcn_history} - the values of the objective function after each iteration
## @end itemize
##
## Three important matrices used in the calculation are X (the input points
## to be clustered), V (the cluster centers), and Mu (the membership of each
## data point in each cluster). Each row of X and V denotes a single point,
## and Mu(i, j) denotes the membership degree of input point X(j, :) in the
## cluster having center V(i, :).
##
## X is identical to the required argument @var{input_data}; V is identical
## to the output @var{cluster_centers}; and Mu is identical to the output
## @var{soft_partition}.
##
## If n denotes the number of input points and k denotes the number of
## clusters to be formed, then X, V, and Mu have the dimensions:
##
## @example
## @group
##                               1    2   ...  #features
##                          1 [                           ]
##    X  =  input_data  =   2 [                           ]
##                        ... [                           ]
##                          n [                           ]
## @end group
## @end example
##
## @example
## @group
##                                    1    2   ...  #features
##                               1 [                           ]
##    V  =  cluster_centers  =   2 [                           ]
##                             ... [                           ]
##                               k [                           ]
## @end group
## @end example
##
## @example
## @group
##                                    1    2   ...   n
##                               1 [                    ]
##    Mu  =  soft_partition  =   2 [                    ]
##                             ... [                    ]
##                               k [                    ]
## @end group
## @end example
##
## @seealso{fcm, partition_coeff, partition_entropy, xie_beni_index}
##
## @end deftypefn

## Author:        L. Markowsky
## Keywords:      fuzzy-logic-toolkit fuzzy partition clustering
## Directory:     fuzzy-logic-toolkit/inst/
## Filename:      gustafson_kessel.m
## Last-Modified: 5 Sep 2012

function [cluster_centers, soft_partition, obj_fcn_history] = ...
           gustafson_kessel (input_data, num_clusters, ...
           cluster_volume = [], options = [2.0, 100, 1e-5, 1])

  ## If gustafson_kessel was called with an incorrect number of
  ## arguments, or the arguments do not have the correct type, print
  ## an error message and halt.

  if ((nargin < 2) || (nargin > 4))
    puts ("Type 'help gustafson_kessel' for more information.\n");
    error ("gustafson_kessel requires 2, 3, or 4 arguments\n");
  elseif (!is_real_matrix (input_data))
    puts ("Type 'help gustafson_kessel' for more information.\n");
    error ("gustafson_kessel's 1st argument must be matrix of reals\n");
  elseif (!(is_int (num_clusters) && (num_clusters > 1)))
    puts ("Type 'help gustafson_kessel' for more information.\n");
    error ("gustafson_kessel's 2nd argument must be an integer > 1\n");
  elseif (!(isequal (cluster_volume, []) || ...
           (isreal (cluster_volume) && isvector (cluster_volume))))
    puts ("Type 'help gustafson_kessel' for more information.\n");
    error ("gustafson_kessel's 3rd arg must be a vector of reals\n");
  elseif (!(isreal (options) && isvector (options)))
    puts ("Type 'help gustafson_kessel' for more information.\n");
    error ("gustafson_kessel's 4th arg must be a vector of reals\n");
  endif

  ## If the cluster volume matrix was not entered, create a default
  ## value (a vector of 1's).

  if (isequal (cluster_volume, []))
    cluster_volume = ones (1, num_clusters);
  endif

  ## Assign options to the more readable variable names: m,
  ## max_iterations, epsilon, and display_intermediate_results.
  ## If options are missing or NaN (not a number), use the default
  ## values.

  default_options = [2.0, 100, 1e-5, 1];

  for i = 1 : 4
    if ((length (options) < i) || ...
        isna (options(i)) || isnan (options(i)))
      options(i) = default_options(i);
    endif
  endfor

  m = options(1);
  max_iterations = options(2);
  epsilon = options(3);
  display_intermediate_results = options(4);

  ## Call a private function to compute the output.

  [cluster_centers, soft_partition, obj_fcn_history] = ...
    gustafson_kessel_private (input_data, num_clusters, ...
                              cluster_volume, m, max_iterations, ...
                              epsilon, display_intermediate_results);
endfunction

##----------------------------------------------------------------------
## Function: gustafson_kessel_private
## Purpose:  Classify unlabeled data points using the Gustafson-Kessel
##           algorithm.
## Note:     This function (gustafson_kessel_private) is an
##           implementation of Algorithm 4.2 in Fuzzy and Neural
##           Control, by Robert Babuska, November 2009, p. 69.
##----------------------------------------------------------------------

function [V, Mu, obj_fcn_history] = ...
  gustafson_kessel_private (X, k, cluster_volume, m, max_iterations, ...
                            epsilon, display_intermediate_results)

  ## Initialize the prototypes and the calculation.
  V = init_cluster_prototypes (X, k);
  obj_fcn_history = zeros (max_iterations);
  convergence_criterion = epsilon + 1;
  iteration = 0;

  ## Calculate a few numbers here to reduce redundant computation.
  k = rows (V);
  n = rows (X);
  sqr_dist = square_distance_matrix (X, V);

  ## Loop until the objective function is within tolerance or the
  ## maximum number of iterations has been reached.
  while (convergence_criterion > epsilon && ...
         ++iteration <= max_iterations)
    V_previous = V;
    Mu = update_cluster_membership (V, X, m, k, n, sqr_dist);
    Mu_m = Mu .^ m;
    V = update_cluster_prototypes (Mu_m, X, k);
    sqr_dist = gk_square_distance_matrix (X, V, Mu_m, cluster_volume);
    obj_fcn_history(iteration) = ...
      compute_cluster_obj_fcn (Mu_m, sqr_dist);
    if (display_intermediate_results)
      printf ("Iteration count = %d,  Objective fcn = %8.6f\n", ...
               iteration, obj_fcn_history(iteration));
    endif
    convergence_criterion = ...
      compute_cluster_convergence (V, V_previous);
  endwhile

  ## Remove extraneous entries from the tail of the objective ...
  ## function history.
  if (convergence_criterion <= epsilon)
    obj_fcn_history = obj_fcn_history(1 : iteration);
  endif

endfunction

##----------------------------------------------------------------------
## Function: gk_square_distance_matrix
##----------------------------------------------------------------------

function sqr_dist = gk_square_distance_matrix (X, V, Mu_m, ...
                                               cluster_volume)

  k = rows (V);
  n = rows (X);
  num_features = columns (X);
  sqr_dist = zeros (k, n);

  for i = 1 : k
    Vi = V(i, :);
    covariance_matrix = compute_covariance_matrix (X, V, Mu_m, i);

    for j = 1 : n
      Vi_to_Xj = X(j, :) - Vi;
      A = cluster_volume(i) * ...
          det (covariance_matrix) ^ (1.0 / num_features) * ...
          inv (covariance_matrix);
      sqr_dist(i, j) = sum (Vi_to_Xj .* (A * Vi_to_Xj')');

    endfor

  endfor

endfunction

##----------------------------------------------------------------------
## Function: compute_covariance_matrix
##----------------------------------------------------------------------

function covariance_matrix = compute_covariance_matrix (X, V, Mu_m, i)

  num_features = columns (V);
  n = rows (X);
  num = zeros (num_features);
  denom = 0.0;
  Vi = V(i, :);

  for j = 1 : n
    Vi_to_Xj = X(j, :) - Vi;
    num += Mu_m(i, j) * Vi_to_Xj' * Vi_to_Xj;
    denom += Mu_m(i, j);
  endfor

  covariance_matrix = num / denom;

endfunction

##----------------------------------------------------------------------
## Gustafson-Kessel Demo #1
##----------------------------------------------------------------------

%!demo
%! ## This demo:
%! ##    - classifies a small set of unlabeled data points using
%! ##      the Gustafson-Kessel algorithm into two fuzzy clusters
%! ##    - plots the input points together with the cluster centers
%! ##    - evaluates the quality of the resulting clusters using
%! ##      three validity measures: the partition coefficient, the
%! ##      partition entropy, and the Xie-Beni validity index
%! ##
%! ## Note: The input_data is taken from Chapter 13, Example 17 in
%! ##       Fuzzy Logic: Intelligence, Control and Information, by
%! ##       J. Yen and R. Langari, Prentice Hall, 1999, page 381
%! ##       (International Edition). 
%! 
%! ## Use gustafson_kessel to classify the input_data.
%! input_data = [2 12; 4 9; 7 13; 11 5; 12 7; 14 4];
%! number_of_clusters = 2;
%! [cluster_centers, soft_partition, obj_fcn_history] = ...
%!   gustafson_kessel (input_data, number_of_clusters)
%! 
%! ## Plot the data points as small blue x's.
%! figure ('NumberTitle', 'off', 'Name', 'Gustafson-Kessel Demo 1');
%! for i = 1 : rows (input_data)
%!   plot (input_data(i, 1), input_data(i, 2), 'LineWidth', 2, ...
%!         'marker', 'x', 'color', 'b');
%!   hold on;
%! endfor
%! 
%! ## Plot the cluster centers as larger red *'s.
%! for i = 1 : number_of_clusters
%!   plot (cluster_centers(i, 1), cluster_centers(i, 2), ...
%!         'LineWidth', 4, 'marker', '*', 'color', 'r');
%!   hold on;
%! endfor
%! 
%! ## Make the figure look a little better:
%! ##    - scale and label the axes
%! ##    - show gridlines
%! xlim ([0 15]);
%! ylim ([0 15]);
%! xlabel ('Feature 1');
%! ylabel ('Feature 2');
%! grid
%! hold
%! 
%! ## Calculate and print the three validity measures.
%! printf ("Partition Coefficient: %f\n", ...
%!         partition_coeff (soft_partition));
%! printf ("Partition Entropy (with a = 2): %f\n", ...
%!         partition_entropy (soft_partition, 2));
%! printf ("Xie-Beni Index: %f\n\n", ...
%!         xie_beni_index (input_data, cluster_centers, ...
%!         soft_partition));
 
##----------------------------------------------------------------------
## Gustafson-Kessel Demo #2
##----------------------------------------------------------------------

%!demo
%! ## This demo:
%! ##    - classifies three-dimensional unlabeled data points using
%! ##      the Gustafson-Kessel algorithm into three fuzzy clusters
%! ##    - plots the input points together with the cluster centers
%! ##    - evaluates the quality of the resulting clusters using
%! ##      three validity measures: the partition coefficient, the
%! ##      partition entropy, and the Xie-Beni validity index
%! ##
%! ## Note: The input_data was selected to form three areas of
%! ##       different shapes.
%! 
%! ## Use gustafson_kessel to classify the input_data.
%! input_data = [1 11 5; 1 12 6; 1 13 5; 2 11 7; 2 12 6; 2 13 7;
%!               3 11 6; 3 12 5; 3 13 7; 1 1 10; 1 3 9; 2 2 11;
%!               3 1 9; 3 3 10; 3 5 11; 4 4 9; 4 6 8; 5 5 8; 5 7 9;
%!               6 6 10; 9 10 12; 9 12 13; 9 13 14; 10 9 13; 10 13 12;
%!               11 10 14; 11 12 13; 12 6 12; 12 7 15; 12 9 15;
%!               14 6 14; 14 8 13];
%! number_of_clusters = 3;
%! [cluster_centers, soft_partition, obj_fcn_history] = ...
%!   gustafson_kessel (input_data, number_of_clusters, [1 1 1], ...
%!                     [NaN NaN NaN 0])
%! 
%! ## Plot the data points in two dimensions (using features 1 & 2)
%! ## as small blue x's.
%! figure ('NumberTitle', 'off', 'Name', 'Gustafson-Kessel Demo 2');
%! for i = 1 : rows (input_data)
%!   plot (input_data(i, 1), input_data(i, 2), 'LineWidth', 2, ...
%!         'marker', 'x', 'color', 'b');
%!   hold on;
%! endfor
%! 
%! ## Plot the cluster centers in two dimensions
%! ## (using features 1 & 2) as larger red *'s.
%! for i = 1 : number_of_clusters
%!   plot (cluster_centers(i, 1), cluster_centers(i, 2), ...
%!         'LineWidth', 4, 'marker', '*', 'color', 'r');
%!   hold on;
%! endfor
%! 
%! ## Make the figure look a little better:
%! ##    - scale and label the axes
%! ##    - show gridlines
%! xlim ([0 15]);
%! ylim ([0 15]);
%! xlabel ('Feature 1');
%! ylabel ('Feature 2');
%! grid
%!  
%! ## Plot the data points in two dimensions
%! ## (using features 1 & 3) as small blue x's.
%! figure ('NumberTitle', 'off', 'Name', 'Gustafson-Kessel Demo 2');
%! for i = 1 : rows (input_data)
%!   plot (input_data(i, 1), input_data(i, 3), 'LineWidth', 2, ...
%!         'marker', 'x', 'color', 'b');
%!   hold on;
%! endfor
%! 
%! ## Plot the cluster centers in two dimensions
%! ## (using features 1 & 3) as larger red *'s.
%! for i = 1 : number_of_clusters
%!   plot (cluster_centers(i, 1), cluster_centers(i, 3), ...
%!         'LineWidth', 4, 'marker', '*', 'color', 'r');
%!   hold on;
%! endfor
%! 
%! ## Make the figure look a little better:
%! ##    - scale and label the axes
%! ##    - show gridlines
%! xlim ([0 15]);
%! ylim ([0 15]);
%! xlabel ('Feature 1');
%! ylabel ('Feature 3');
%! grid
%! hold
%! 
%! ## Calculate and print the three validity measures.
%! printf ("Partition Coefficient: %f\n", ...
%!         partition_coeff (soft_partition));
%! printf ("Partition Entropy (with a = 2): %f\n", ...
%!         partition_entropy (soft_partition, 2));
%! printf ("Xie-Beni Index: %f\n\n", ...
%!         xie_beni_index (input_data, cluster_centers, ...
%!         soft_partition));
