smo.c

支持向量机工具箱

/* --------------------------------------------------------------------
 [Alpha,bias,nsv,kercnt,trnerr,margin]=
       smo(data,labels,ker,arg,C,eps,tol,Alpha,bias )
 Sequential Minimal Optimizer for Support Vector Machines.

 Make executable file by command 'mex smo.c kernel.c'.

  mandatory inputs:
   data [D x N ] matrix of N training D-dimensional patterns.
   labels [1 x N] pattern labels (1 for 1st class, 2 for 2nd class ).
   ker [string] kernel identifier: 'linear', 'poly', 'rbf'.
   arg [real] kernel argument.
   C [real] or [2 x real] one trade-off constant for both the classes
     or two constants for the first and the second class.

  optional inputs:
   eps [real] tolerance of KKT-conditions fulfilment (default 0.001).
   tol [real] minimal change of optimized Lagrangeians (default 0.001).
   Alpha [1xL] initial values of optimized Lagrangeians. If not given
     then SMO starts up from Alpha = zeros(1,L).
   bias [real] initial value of the threshold. If not given then SMO
     starts from bias = 0.

  mandatory outputs:
   Alpha [1 x N] found Lagrangeian multipliers.
   bias [real] found bias.

  optional outputs:
   nsv [real] number of Support Vectors (number of Alpha > ZERO_LIM).
   kercnt [int] number of kernel evaluations.
   trnerr [real] classification error on training data.
   margin [real] margin between classes and the found hyperplane.

 Statistical Pattern Recognition Toolbox, Vojtech Franc, Vaclav Hlavac
 (c) Czech Technical University Prague, http://cmp.felk.cvut.cz

 Modifications:
 23-october-2001, V.Franc
 16-October-2001, V.Franc
 27-september-2001, V.Franc, roundig of a2 in takeStep removed.
 23-September-2001, V.Franc, different trade-off C1 and C2.
 22-September-2001, V.Franc, kernel.c used.
 19-September-2001, V.Franc, computation of nsv and nerr added.
 17-September-2001, V.Franc, created.
 -------------------------------------------------------------------- */

#include "mex.h"
#include "matrix.h"
#include <math.h>
#include <stdlib.h>
#include <string.h>

#include "kernel.h"

/* case insensitive string comparision */
#ifdef __BORLANDC__
   #define STR_COMPARE(A,B,C)      strncmpi(A,B,C)  /* Borland */
#else
  #define STR_COMPARE(A,B,C)      strncmp(A,B,C) /* Linux gcc */
#endif

/* if RANDOM is defined then a random element is used within
 optimization procedure as originally suggested. */
#define RANDOM

#define ZERO_LIM   1e-9     /* patterns with alpha > ZERO_LIM are SV */

#define MAX(A,B)   (((A) > (B)) ? (A) : (B) )
#define MIN(A,B)   (((A) < (B)) ? (A) : (B) )

#define C(arg)   (target[arg] > 0 ? C1 : C2)


/* --- Global variables ---------------------------------------------- */

long N = 0;                /* number of training patterns */
double C1=1;               /* trade-off constant for the 1st class */
double C2=1;               /* trade-off constant for the 2nd class */
double tolerance=0.001;    /* tolerance in KKT fulfilment  */
double eps=0.001;          /* minimal Lagrangeian change */
double *data;              /* pointer at patterns */
double *target;            /* pointer at labels */
double *error_cache;       /* error cache */

double *alpha;             /* Lagrange multipliers */
double *b;                 /* Bias (threshold) */


/* ==============================================================
 Implementation of Sequential Minimal Optimizer (SMO)
============================================================== */

/* --------------------------------------------------------------
 Computes value of the learned function for k-th pattern.
-------------------------------------------------------------- */
double learned_func( long k )
{
   double s = 0.;
   long i;
   for( i = 0; i < N; i++ ) {
      if( alpha[i] > 0 )
         s += alpha[i]*target[i]*kernel(i,k);
   }
  s -= *b;
  return( s );
}

/* --------------------------------------------------------------
 Optimizes objective function for i1-th and i2-th pattern.
-------------------------------------------------------------- */
long takeStep( long i1, long i2 ) {
   double y1, y2, s;
   long i;
   double alpha1, alpha2; 
   double a1, a2;
   double E1, E2, L, H, k11, k22, k12, eta, Lobj, Hobj;
   double gamma;
   double c1, c2;
   double t;
   double b1, b2, bnew;
   double delta_b;
   double t1, t2;

   if( i1 == i2 ) return( 0 );

   alpha1 = alpha[i1];
   y1 = target[i1];
   if( alpha1 > 0 && alpha1 < C(i1) )
      E1 = error_cache[i1];
   else
      E1 = learned_func(i1) - y1;

   alpha2 = alpha[i2];
   y2 = target[i2];
   if( alpha2 > 0 && alpha2 < C(i2) )
      E2 = error_cache[i2];
   else
      E2 = learned_func(i2) - y2;

   s = y1 * y2;
   if(s < 0)
   {
      L = MAX(0, alpha2 - alpha1);
      H = MIN(C(i2), C(i1) + alpha2 - alpha1);
   }
   else
   {
     L = MAX(0, alpha2 + alpha1 - C(i1) );
     H = MIN(C(i2), alpha2 + alpha1);
   }

   if( L == H ) return( 0 );

   k11 = kernel(i1,i1);
   k12 = kernel(i1,i2);
   k22 = kernel(i2,i2);
   eta = 2 * k12 - k11 - k22;

   if( eta < 0 ) {
      a2 = alpha2 + y2 * (E2 - E1) / eta;
      if( a2 < L )
         a2 = L;
      else if( a2 > H )
         a2 = H;
   }
   else {
      c1 = eta/2;
      c2 = y2 * (E1-E2)- eta * alpha2;
      Lobj = c1 * L * L + c2 * L;
      Hobj = c1 * H * H + c2 * H;

      if( Lobj > Hobj+eps )
         a2 = L;
      else if( Lobj < Hobj-eps )
         a2 = H;
      else
         a2 = alpha2;
   }

   if( fabs(a2-alpha2) < eps*(a2+alpha2+eps )) return( 0 );

   a1 = alpha1 - s * (a2 - alpha2 );
   if( a1 < 0 ) {
      a2 += s * a1;
      a1 = 0;
   }
   else if( a1 > C(i1) ) {
      t = a1-C(i1);
      a2 += s * t;
      a1 = C(i1);
   }

   if( a1 > 0 && a1 < C(i1) )
      bnew = *b + E1 + y1 * (a1 - alpha1) * k11 + y2 * (a2 - alpha2) * k12;
   else {
      if( a2 > 0 && a2 < C(i2) )
         bnew = *b + E2 + y1 *(a1 - alpha1)*k12 + y2*(a2 - alpha2) * k22;
      else {
         b1 = *b + E1 + y1 * (a1 - alpha1) * k11 + y2 * (a2 - alpha2) * k12;
         b2 = *b + E2 + y1 * (a1 - alpha1) * k12 + y2 * (a2 - alpha2) * k22;
         bnew = (b1 + b2) / 2;
      }
   }

   delta_b = bnew - *b;
   *b = bnew;

   t1 = y1 * (a1-alpha1);
   t2 = y2 * (a2-alpha2);

   for( i = 0; i < N; i++ ) {
     if (0 < alpha[i] && alpha[i] < C(i)) {
        error_cache[i] +=  t1 * kernel(i1,i) + t2 * kernel(i2,i) - delta_b;
     }
   }

   error_cache[i1] = 0;
   error_cache[i2] = 0;

   alpha[i1] = a1;  /* Store a1 in the alpha array.*/
   alpha[i2] = a2;  /* Store a2 in the alpha array.*/

   return( 1 );
}


/* --------------------------------------------------------------
 Finds the second Lagrange multiplayer to be optimize.
-------------------------------------------------------------- */
long examineExample( long i1 )
{
   double y1, alpha1, E1, r1;
   double tmax;
   double E2, temp;
   long k, i2;
   long k0;

   y1 = target[i1];
   alpha1 = alpha[i1];

   if( alpha1 > 0 && alpha1 < C(i1) )
      E1 = error_cache[i1];
   else
      E1 = learned_func(i1) - y1;

   r1 = y1 * E1;
   if(( r1 < -tolerance && alpha1 < C(i1) )
      || (r1 > tolerance && alpha1 > 0)) {
    /* Try i2 by three ways; if successful, then immediately return 1; */

      for( i2 = (-1), tmax = 0, k = 0; k < N; k++ ) {
         if( alpha[k] > 0 && alpha[k] < C(k) ) {
            E2 = error_cache[k];
            temp = fabs(E1 - E2);
            if( temp > tmax ) {
               tmax = temp;
               i2 = k;
            }
         }
      }
      if( i2 >= 0 ) {
         if( takeStep(i1,i2) )
            return( 1 );
      }

#ifdef RANDOM
      for( k0 = rand(), k = k0; k < N + k0; k++ ) {
         i2 = k % N;
#else
      for( k = 0; k < N; k++) {
         i2 = k;
#endif
         if( alpha[i2] > 0 && alpha[i2] < C(i2) ) {
            if( takeStep(i1,i2) )
               return( 1 );
         }
      }

#ifdef RANDOM