svm.cpp

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#include <math.h>
#include <stdio.h>
#include <stdlib.h>
#include <ctype.h>
#include <float.h>
#include <string.h>
#include <stdarg.h>
#include <limits.h>
#include <locale.h>
#include "svm.h"
int libsvm_version = LIBSVM_VERSION;
typedef float Qfloat;
typedef signed char schar;
#ifndef min
template <class T> static inline T min(T x,T y) { return (x<y)?x:y; }
#endif
#ifndef max
template <class T> static inline T max(T x,T y) { return (x>y)?x:y; }
#endif
template <class T> static inline void swap(T& x, T& y) { T t=x; x=y; y=t; }
template <class S, class T> static inline void clone(T*& dst, S* src, int n)
{
        dst = new T[n];
        memcpy((void *)dst,(void *)src,sizeof(T)*n);
}
static inline double powi(double base, int times)
{
        double tmp = base, ret = 1.0;

        for(int t=times; t>0; t/=2)
        {
                if(t%2==1) ret*=tmp;
                tmp = tmp * tmp;
        }
        return ret;
}
#define INF HUGE_VAL
#define TAU 1e-12
#define Malloc(type,n) (type *)malloc((n)*sizeof(type))

static void print_string_stdout(const char *s)
{
        fputs(s,stdout);
        fflush(stdout);
}
static void (*svm_print_string) (const char *) = &print_string_stdout;
#if 1
static void info(const char *fmt,...)
{
        char buf[BUFSIZ];
        va_list ap;
        va_start(ap,fmt);
        vsprintf(buf,fmt,ap);
        va_end(ap);
        (*svm_print_string)(buf);
}
#else
static void info(const char *fmt,...) {}
#endif

//
// Kernel Cache
//
// l is the number of total data items
// size is the cache size limit in bytes
//
class Cache
{
public:
        Cache(int l,long int size);
        ~Cache();

        // request data [0,len)
        // return some position p where [p,len) need to be filled
        // (p >= len if nothing needs to be filled)
        int get_data(const int index, Qfloat **data, int len);
        void swap_index(int i, int j);  
private:
        int l;
        long int size;
        struct head_t
        {
                head_t *prev, *next;    // a circular list
                Qfloat *data;
                int len;                // data[0,len) is cached in this entry
        };

        head_t *head;
        head_t lru_head;
        void lru_delete(head_t *h);
        void lru_insert(head_t *h);
};

Cache::Cache(int l_,long int size_):l(l_),size(size_)
{
        head = (head_t *)calloc(l,sizeof(head_t));      // initialized to 0
        size /= sizeof(Qfloat);
        size -= l * sizeof(head_t) / sizeof(Qfloat);
        size = max(size, 2 * (long int) l);     // cache must be large enough for two columns
        lru_head.next = lru_head.prev = &lru_head;
}

Cache::~Cache()
{
        for(head_t *h = lru_head.next; h != &lru_head; h=h->next)
                free(h->data);
        free(head);
}

void Cache::lru_delete(head_t *h)
{
        // delete from current location
        h->prev->next = h->next;
        h->next->prev = h->prev;
}

void Cache::lru_insert(head_t *h)
{
        // insert to last position
        h->next = &lru_head;
        h->prev = lru_head.prev;
        h->prev->next = h;
        h->next->prev = h;
}

int Cache::get_data(const int index, Qfloat **data, int len)
{
        head_t *h = &head[index];
        if(h->len) lru_delete(h);
        int more = len - h->len;

        if(more > 0)
        {
                // free old space
                while(size < more)
                {
                        head_t *old = lru_head.next;
                        lru_delete(old);
                        free(old->data);
                        size += old->len;
                        old->data = 0;
                        old->len = 0;
                }

                // allocate new space
                h->data = (Qfloat *)realloc(h->data,sizeof(Qfloat)*len);
                size -= more;
                swap(h->len,len);
        }

        lru_insert(h);
        *data = h->data;
        return len;
}

void Cache::swap_index(int i, int j)
{
        if(i==j) return;

        if(head[i].len) lru_delete(&head[i]);
        if(head[j].len) lru_delete(&head[j]);
        swap(head[i].data,head[j].data);
        swap(head[i].len,head[j].len);
        if(head[i].len) lru_insert(&head[i]);
        if(head[j].len) lru_insert(&head[j]);

        if(i>j) swap(i,j);
        for(head_t *h = lru_head.next; h!=&lru_head; h=h->next)
        {
                if(h->len > i)
                {
                        if(h->len > j)
                                swap(h->data[i],h->data[j]);
                        else
                        {
                                // give up
                                lru_delete(h);
                                free(h->data);
                                size += h->len;
                                h->data = 0;
                                h->len = 0;
                        }
                }
        }
}

//
// Kernel evaluation
//
// the static method k_function is for doing single kernel evaluation
// the constructor of Kernel prepares to calculate the l*l kernel matrix
// the member function get_Q is for getting one column from the Q Matrix
//
class QMatrix {
public:
        virtual Qfloat *get_Q(int column, int len) const = 0;
        virtual double *get_QD() const = 0;
        virtual void swap_index(int i, int j) const = 0;
        virtual ~QMatrix() {}
};

class Kernel: public QMatrix {
public:
        Kernel(int l, svm_node * const * x, const svm_parameter& param);
        virtual ~Kernel();

        static double k_function(const svm_node *x, const svm_node *y,
                                 const svm_parameter& param);
        virtual Qfloat *get_Q(int column, int len) const = 0;
        virtual double *get_QD() const = 0;
        virtual void swap_index(int i, int j) const     // no so const...
        {
                swap(x[i],x[j]);
                if(x_square) swap(x_square[i],x_square[j]);
        }
protected:

        double (Kernel::*kernel_function)(int i, int j) const;

private:
        const svm_node **x;
        double *x_square;

        // svm_parameter
        const int kernel_type;
        const int degree;
        const double gamma;
        const double coef0;

        static double dot(const svm_node *px, const svm_node *py);
        double kernel_linear(int i, int j) const
        {
                return dot(x[i],x[j]);
        }
        double kernel_poly(int i, int j) const
        {
                return powi(gamma*dot(x[i],x[j])+coef0,degree);
        }
        double kernel_rbf(int i, int j) const
        {
                return exp(-gamma*(x_square[i]+x_square[j]-2*dot(x[i],x[j])));
        }
        double kernel_sigmoid(int i, int j) const
        {
                return tanh(gamma*dot(x[i],x[j])+coef0);
        }
        double kernel_precomputed(int i, int j) const
        {
                return x[i][(int)(x[j][0].value)].value;
        }
};

Kernel::Kernel(int l, svm_node * const * x_, const svm_parameter& param)
:kernel_type(param.kernel_type), degree(param.degree),
 gamma(param.gamma), coef0(param.coef0)
{
        switch(kernel_type)
        {
                case LINEAR:
                        kernel_function = &Kernel::kernel_linear;
                        break;
                case POLY:
                        kernel_function = &Kernel::kernel_poly;
                        break;
                case RBF:
                        kernel_function = &Kernel::kernel_rbf;
                        break;
                case SIGMOID:
                        kernel_function = &Kernel::kernel_sigmoid;
                        break;
                case PRECOMPUTED:
                        kernel_function = &Kernel::kernel_precomputed;
                        break;
        }

        clone(x,x_,l);

        if(kernel_type == RBF)
        {
                x_square = new double[l];
                for(int i=0;i<l;i++)
                        x_square[i] = dot(x[i],x[i]);
        }
        else
                x_square = 0;
}

Kernel::~Kernel()
{
        delete[] x;
        delete[] x_square;
}

double Kernel::dot(const svm_node *px, const svm_node *py)
{
        double sum = 0;
        while(px->index != -1 && py->index != -1)
        {
                if(px->index == py->index)
                {
                        sum += px->value * py->value;
                        ++px;
                        ++py;
                }
                else
                {
                        if(px->index > py->index)
                                ++py;
                        else
                                ++px;
                }                       
        }
        return sum;
}

double Kernel::k_function(const svm_node *x, const svm_node *y,
                          const svm_parameter& param)
{
        switch(param.kernel_type)
        {
                case LINEAR:
                        return dot(x,y);
                case POLY:
                        return powi(param.gamma*dot(x,y)+param.coef0,param.degree);
                case RBF:
                {
                        double sum = 0;
                        while(x->index != -1 && y->index !=-1)
                        {
                                if(x->index == y->index)
                                {
                                        double d = x->value - y->value;
                                        sum += d*d;
                                        ++x;
                                        ++y;
                                }
                                else
                                {
                                        if(x->index > y->index)
                                        {       
                                                sum += y->value * y->value;
                                                ++y;
                                        }
                                        else
                                        {
                                                sum += x->value * x->value;
                                                ++x;
                                        }
                                }
                        }

                        while(x->index != -1)
                        {
                                sum += x->value * x->value;
                                ++x;
                        }

                        while(y->index != -1)
                        {
                                sum += y->value * y->value;
                                ++y;
                        }
                        
                        return exp(-param.gamma*sum);
                }
                case SIGMOID:
                        return tanh(param.gamma*dot(x,y)+param.coef0);
                case PRECOMPUTED:  //x: test (validation), y: SV
                        return x[(int)(y->value)].value;
                default:
                        return 0;  // Unreachable 
        }
}

// An SMO algorithm in Fan et al., JMLR 6(2005), p. 1889--1918
// Solves:
//
//      min 0.5(\alpha^T Q \alpha) + p^T \alpha
//
//              y^T \alpha = \delta
//              y_i = +1 or -1
//              0 <= alpha_i <= Cp for y_i = 1
//              0 <= alpha_i <= Cn for y_i = -1
//
// Given:
//
//      Q, p, y, Cp, Cn, and an initial feasible point \alpha
//      l is the size of vectors and matrices
//      eps is the stopping tolerance
//
// solution will be put in \alpha, objective value will be put in obj
//
class Solver {
public:
        Solver() {};
        virtual ~Solver() {};

        struct SolutionInfo {
                double obj;
                double rho;
                double upper_bound_p;
                double upper_bound_n;
                double r;       // for Solver_NU
        };

        void Solve(int l, const QMatrix& Q, const double *p_, const schar *y_,
                   double *alpha_, double Cp, double Cn, double eps,
                   SolutionInfo* si, int shrinking);
protected:
        int active_size;
        schar *y;
        double *G;              // gradient of objective function
        enum { LOWER_BOUND, UPPER_BOUND, FREE };
        char *alpha_status;     // LOWER_BOUND, UPPER_BOUND, FREE
        double *alpha;
        const QMatrix *Q;
        const double *QD;
        double eps;
        double Cp,Cn;
        double *p;
        int *active_set;
        double *G_bar;          // gradient, if we treat free variables as 0
        int l;
        bool unshrink;  // XXX

        double get_C(int i)
        {
                return (y[i] > 0)? Cp : Cn;
        }
        void update_alpha_status(int i)
        {
                if(alpha[i] >= get_C(i))
                        alpha_status[i] = UPPER_BOUND;
                else if(alpha[i] <= 0)
                        alpha_status[i] = LOWER_BOUND;
                else alpha_status[i] = FREE;
        }
        bool is_upper_bound(int i) { return alpha_status[i] == UPPER_BOUND; }
        bool is_lower_bound(int i) { return alpha_status[i] == LOWER_BOUND; }
        bool is_free(int i) { return alpha_status[i] == FREE; }
        void swap_index(int i, int j);
        void reconstruct_gradient();
        virtual int select_working_set(int &i, int &j);
        virtual double calculate_rho();
        virtual void do_shrinking();
private:
        bool be_shrunk(int i, double Gmax1, double Gmax2);      
};

void Solver::swap_index(int i, int j)
{
        Q->swap_index(i,j);
        swap(y[i],y[j]);
        swap(G[i],G[j]);
        swap(alpha_status[i],alpha_status[j]);
        swap(alpha[i],alpha[j]);
        swap(p[i],p[j]);
        swap(active_set[i],active_set[j]);
        swap(G_bar[i],G_bar[j]);
}

void Solver::reconstruct_gradient()
{
        // reconstruct inactive elements of G from G_bar and free variables

        if(active_size == l) return;

        int i,j;
        int nr_free = 0;

        for(j=active_size;j<l;j++)
                G[j] = G_bar[j] + p[j];

        for(j=0;j<active_size;j++)
                if(is_free(j))
                        nr_free++;

        if(2*nr_free < active_size)
                info("\nWARNING: using -h 0 may be faster\n");

        if (nr_free*l > 2*active_size*(l-active_size))
        {
                for(i=active_size;i<l;i++)
                {
                        const Qfloat *Q_i = Q->get_Q(i,active_size);
                        for(j=0;j<active_size;j++)
                                if(is_free(j))
                                        G[i] += alpha[j] * Q_i[j];
                }
        }
        else
        {
                for(i=0;i<active_size;i++)
                        if(is_free(i))
                        {
                                const Qfloat *Q_i = Q->get_Q(i,l);
                                double alpha_i = alpha[i];
                                for(j=active_size;j<l;j++)
                                        G[j] += alpha_i * Q_i[j];
                        }
        }
}

void Solver::Solve(int l, const QMatrix& Q, const double *p_, const schar *y_,
                   double *alpha_, double Cp, double Cn, double eps,
                   SolutionInfo* si, int shrinking)
{
        this->l = l;
        this->Q = &Q;
        QD=Q.get_QD();
        clone(p, p_,l);
        clone(y, y_,l);
        clone(alpha,alpha_,l);
        this->Cp = Cp;
        this->Cn = Cn;
        this->eps = eps;
        unshrink = false;

        // initialize alpha_status
        {
                alpha_status = new char[l];
                for(int i=0;i<l;i++)
                        update_alpha_status(i);
        }

        // initialize active set (for shrinking)
        {
                active_set = new int[l];
                for(int i=0;i<l;i++)
                        active_set[i] = i;
                active_size = l;
        }

        // initialize gradient
        {
                G = new double[l];
                G_bar = new double[l];
                int i;
                for(i=0;i<l;i++)
                {
                        G[i] = p[i];
                        G_bar[i] = 0;
                }
                for(i=0;i<l;i++)
                        if(!is_lower_bound(i))
                        {
                                const Qfloat *Q_i = Q.get_Q(i,l);
                                double alpha_i = alpha[i];
                                int j;
                                for(j=0;j<l;j++)
                                        G[j] += alpha_i*Q_i[j];
                                if(is_upper_bound(i))
                                        for(j=0;j<l;j++)
                                                G_bar[j] += get_C(i) * Q_i[j];
                        }
        }

        // optimization step

        int iter = 0;
        int max_iter = max(10000000, l>INT_MAX/100 ? INT_MAX : 100*l);
        int counter = min(l,1000)+1;
        
        while(iter < max_iter)
        {
                // show progress and do shrinking

                if(--counter == 0)
                {
                        counter = min(l,1000);
                        if(shrinking) do_shrinking();
                        info(".");
                }

                int i,j;
                if(select_working_set(i,j)!=0)
                {
                        // reconstruct the whole gradient
                        reconstruct_gradient();
                        // reset active set size and check
                        active_size = l;
                        info("*");
                        if(select_working_set(i,j)!=0)
                                break;
                        else
                                counter = 1;    // do shrinking next iteration
                }
                
                ++iter;

                // update alpha[i] and alpha[j], handle bounds carefully
                
                const Qfloat *Q_i = Q.get_Q(i,active_size);
                const Qfloat *Q_j = Q.get_Q(j,active_size);

                double C_i = get_C(i);
                double C_j = get_C(j);

                double old_alpha_i = alpha[i];
                double old_alpha_j = alpha[j];

                if(y[i]!=y[j])
                {
                        double quad_coef = QD[i]+QD[j]+2*Q_i[j];
                        if (quad_coef <= 0)
                                quad_coef = TAU;
                        double delta = (-G[i]-G[j])/quad_coef;
                        double diff = alpha[i] - alpha[j];
                        alpha[i] += delta;
                        alpha[j] += delta;
                        
                        if(diff > 0)
                        {
                                if(alpha[j] < 0)
                                {
                                        alpha[j] = 0;
                                        alpha[i] = diff;
                                }
                        }
                        else
                        {
                                if(alpha[i] < 0)
                                {
                                        alpha[i] = 0;
                                        alpha[j] = -diff;
                                }
                        }
                        if(diff > C_i - C_j)
                        {
                                if(alpha[i] > C_i)
                                {
                                        alpha[i] = C_i;
                                        alpha[j] = C_i - diff;
                                }
                        }
                        else
                        {
                                if(alpha[j] > C_j)
                                {
                                        alpha[j] = C_j;
                                        alpha[i] = C_j + diff;
                                }
                        }
                }
                else
                {
                        double quad_coef = QD[i]+QD[j]-2*Q_i[j];
                        if (quad_coef <= 0)
                                quad_coef = TAU;
                        double delta = (G[i]-G[j])/quad_coef;
                        double sum = alpha[i] + alpha[j];
                        alpha[i] -= delta;
                        alpha[j] += delta;

                        if(sum > C_i)
                        {
                                if(alpha[i] > C_i)
                                {
                                        alpha[i] = C_i;
                                        alpha[j] = sum - C_i;
                                }
                        }
                        else
                        {
                                if(alpha[j] < 0)
                                {
                                        alpha[j] = 0;
                                        alpha[i] = sum;
                                }
                        }
                        if(sum > C_j)
                        {
                                if(alpha[j] > C_j)
                                {
                                        alpha[j] = C_j;
                                        alpha[i] = sum - C_j;
                                }
                        }
                        else
                        {
                                if(alpha[i] < 0)
                                {
                                        alpha[i] = 0;
                                        alpha[j] = sum;
                                }
                        }
                }

                // update G

                double delta_alpha_i = alpha[i] - old_alpha_i;
                double delta_alpha_j = alpha[j] - old_alpha_j;
                
                for(int k=0;k<active_size;k++)
                {
                        G[k] += Q_i[k]*delta_alpha_i + Q_j[k]*delta_alpha_j;
                }

                // update alpha_status and G_bar

                {
                        bool ui = is_upper_bound(i);
                        bool uj = is_upper_bound(j);
                        update_alpha_status(i);
                        update_alpha_status(j);
                        int k;
                        if(ui != is_upper_bound(i))
                        {
                                Q_i = Q.get_Q(i,l);
                                if(ui)
                                        for(k=0;k<l;k++)
                                                G_bar[k] -= C_i * Q_i[k];
                                else
                                        for(k=0;k<l;k++)
                                                G_bar[k] += C_i * Q_i[k];
                        }

                        if(uj != is_upper_bound(j))
                        {
                                Q_j = Q.get_Q(j,l);
                                if(uj)
                                        for(k=0;k<l;k++)
                                                G_bar[k] -= C_j * Q_j[k];
                                else
                                        for(k=0;k<l;k++)
                                                G_bar[k] += C_j * Q_j[k];
                        }
                }
        }

        if(iter >= max_iter)
        {
                if(active_size < l)
                {
                        // reconstruct the whole gradient to calculate objective value
                        reconstruct_gradient();
                        active_size = l;
                        info("*");
                }
                info("\nWARNING: reaching max number of iterations");
        }

        // calculate rho

        si->rho = calculate_rho();

        // calculate objective value
        {
                double v = 0;
                int i;
                for(i=0;i<l;i++)
                        v += alpha[i] * (G[i] + p[i]);

                si->obj = v/2;
        }

        // put back the solution
        {
                for(int i=0;i<l;i++)
                        alpha_[active_set[i]] = alpha[i];
        }

        // juggle everything back
        /*{
                for(int i=0;i<l;i++)
                        while(active_set[i] != i)
                                swap_index(i,active_set[i]);
                                // or Q.swap_index(i,active_set[i]);
        }*/

        si->upper_bound_p = Cp;
        si->upper_bound_n = Cn;

        info("\noptimization finished, #iter = %d\n",iter);

        delete[] p;
        delete[] y;
        delete[] alpha;
        delete[] alpha_status;
        delete[] active_set;
        delete[] G;
        delete[] G_bar;
}

// return 1 if already optimal, return 0 otherwise
int Solver::select_working_set(int &out_i, int &out_j)
{
        // return i,j such that
        // i: maximizes -y_i * grad(f)_i, i in I_up(\alpha)
        // j: minimizes the decrease of obj value
        //    (if quadratic coefficeint <= 0, replace it with tau)
        //    -y_j*grad(f)_j < -y_i*grad(f)_i, j in I_low(\alpha)
        
        double Gmax = -INF;
        double Gmax2 = -INF;
        int Gmax_idx = -1;
        int Gmin_idx = -1;
        double obj_diff_min = INF;

        for(int t=0;t<active_size;t++)
                if(y[t]==+1)    
                {
                        if(!is_upper_bound(t))
                                if(-G[t] >= Gmax)
                                {
                                        Gmax = -G[t];
                                        Gmax_idx = t;
                                }
                }
                else
                {
                        if(!is_lower_bound(t))
                                if(G[t] >= Gmax)
                                {
                                        Gmax = G[t];
                                        Gmax_idx = t;
                                }
                }

        int i = Gmax_idx;
        const Qfloat *Q_i = NULL;
        if(i != -1) // NULL Q_i not accessed: Gmax=-INF if i=-1
                Q_i = Q->get_Q(i,active_size);

        for(int j=0;j<active_size;j++)
        {
                if(y[j]==+1)
                {
                        if (!is_lower_bound(j))
                        {
                                double grad_diff=Gmax+G[j];
                                if (G[j] >= Gmax2)
                                        Gmax2 = G[j];
                                if (grad_diff > 0)
                                {
                                        double obj_diff; 
                                        double quad_coef = QD[i]+QD[j]-2.0*y[i]*Q_i[j];
                                        if (quad_coef > 0)
                                                obj_diff = -(grad_diff*grad_diff)/quad_coef;
                                        else
                                                obj_diff = -(grad_diff*grad_diff)/TAU;

                                        if (obj_diff <= obj_diff_min)
                                        {
                                                Gmin_idx=j;
                                                obj_diff_min = obj_diff;
                                        }
                                }
                        }
                }
                else
                {
                        if (!is_upper_bound(j))
                        {
                                double grad_diff= Gmax-G[j];
                                if (-G[j] >= Gmax2)
                                        Gmax2 = -G[j];
                                if (grad_diff > 0)
                                {
                                        double obj_diff; 
                                        double quad_coef = QD[i]+QD[j]+2.0*y[i]*Q_i[j];
                                        if (quad_coef > 0)
                                                obj_diff = -(grad_diff*grad_diff)/quad_coef;
                                        else
                                                obj_diff = -(grad_diff*grad_diff)/TAU;

                                        if (obj_diff <= obj_diff_min)
                                        {
                                                Gmin_idx=j;
                                                obj_diff_min = obj_diff;
                                        }
                                }
                        }
                }
        }

        if(Gmax+Gmax2 < eps)
                return 1;

        out_i = Gmax_idx;
        out_j = Gmin_idx;
        return 0;
}

bool Solver::be_shrunk(int i, double Gmax1, double Gmax2)
{
        if(is_upper_bound(i))
        {
                if(y[i]==+1)
                        return(-G[i] > Gmax1);
                else
                        return(-G[i] > Gmax2);
        }
        else if(is_lower_bound(i))
        {
                if(y[i]==+1)
                        return(G[i] > Gmax2);
                else    
                        return(G[i] > Gmax1);
        }
        else
                return(false);
}

void Solver::do_shrinking()
{
        int i;
        double Gmax1 = -INF;            // max { -y_i * grad(f)_i | i in I_up(\alpha) }
        double Gmax2 = -INF;            // max { y_i * grad(f)_i | i in I_low(\alpha) }

        // find maximal violating pair first
        for(i=0;i<active_size;i++)
        {
                if(y[i]==+1)    
                {
                        if(!is_upper_bound(i))  
                        {
                                if(-G[i] >= Gmax1)
                                        Gmax1 = -G[i];
                        }
                        if(!is_lower_bound(i))  
                        {
                                if(G[i] >= Gmax2)
                                        Gmax2 = G[i];
                        }
                }
                else    
                {
                        if(!is_upper_bound(i))  
                        {
                                if(-G[i] >= Gmax2)
                                        Gmax2 = -G[i];
                        }
                        if(!is_lower_bound(i))  
                        {
                                if(G[i] >= Gmax1)
                                        Gmax1 = G[i];
                        }
                }
        }

        if(unshrink == false && Gmax1 + Gmax2 <= eps*10) 
        {
                unshrink = true;
                reconstruct_gradient();
                active_size = l;
                info("*");
        }

        for(i=0;i<active_size;i++)
                if (be_shrunk(i, Gmax1, Gmax2))
                {
                        active_size--;
                        while (active_size > i)
                        {
                                if (!be_shrunk(active_size, Gmax1, Gmax2))
                                {
                                        swap_index(i,active_size);
                                        break;
                                }
                                active_size--;
                        }
                }
}

double Solver::calculate_rho()
{
        double r;
        int nr_free = 0;
        double ub = INF, lb = -INF, sum_free = 0;
        for(int i=0;i<active_size;i++)
        {
                double yG = y[i]*G[i];

                if(is_upper_bound(i))
                {
                        if(y[i]==-1)
                                ub = min(ub,yG);
                        else
                                lb = max(lb,yG);
                }
                else if(is_lower_bound(i))
                {
                        if(y[i]==+1)
                                ub = min(ub,yG);
                        else
                                lb = max(lb,yG);
                }
                else
                {
                        ++nr_free;
                        sum_free += yG;
                }
        }

        if(nr_free>0)
                r = sum_free/nr_free;
        else
                r = (ub+lb)/2;

        return r;
}

//
// Solver for nu-svm classification and regression
//
// additional constraint: e^T \alpha = constant
//
class Solver_NU : public Solver
{
public:
        Solver_NU() {}
        void Solve(int l, const QMatrix& Q, const double *p, const schar *y,
                   double *alpha, double Cp, double Cn, double eps,
                   SolutionInfo* si, int shrinking)
        {
                this->si = si;
                Solver::Solve(l,Q,p,y,alpha,Cp,Cn,eps,si,shrinking);
        }
private:
        SolutionInfo *si;
        int select_working_set(int &i, int &j);
        double calculate_rho();
        bool be_shrunk(int i, double Gmax1, double Gmax2, double Gmax3, double Gmax4);
        void do_shrinking();
};

// return 1 if already optimal, return 0 otherwise
int Solver_NU::select_working_set(int &out_i, int &out_j)
{
        // return i,j such that y_i = y_j and
        // i: maximizes -y_i * grad(f)_i, i in I_up(\alpha)
        // j: minimizes the decrease of obj value
        //    (if quadratic coefficeint <= 0, replace it with tau)
        //    -y_j*grad(f)_j < -y_i*grad(f)_i, j in I_low(\alpha)

        double Gmaxp = -INF;
        double Gmaxp2 = -INF;
        int Gmaxp_idx = -1;

        double Gmaxn = -INF;
        double Gmaxn2 = -INF;
        int Gmaxn_idx = -1;

        int Gmin_idx = -1;
        double obj_diff_min = INF;

        for(int t=0;t<active_size;t++)
                if(y[t]==+1)
                {
                        if(!is_upper_bound(t))
                                if(-G[t] >= Gmaxp)
                                {
                                        Gmaxp = -G[t];
                                        Gmaxp_idx = t;
                                }
                }
                else
                {
                        if(!is_lower_bound(t))
                                if(G[t] >= Gmaxn)
                                {
                                        Gmaxn = G[t];
                                        Gmaxn_idx = t;
                                }
                }

        int ip = Gmaxp_idx;
        int in = Gmaxn_idx;
        const Qfloat *Q_ip = NULL;
        const Qfloat *Q_in = NULL;
        if(ip != -1) // NULL Q_ip not accessed: Gmaxp=-INF if ip=-1
                Q_ip = Q->get_Q(ip,active_size);
        if(in != -1)
                Q_in = Q->get_Q(in,active_size);

        for(int j=0;j<active_size;j++)
        {
                if(y[j]==+1)
                {
                        if (!is_lower_bound(j)) 
                        {
                                double grad_diff=Gmaxp+G[j];
                                if (G[j] >= Gmaxp2)
                                        Gmaxp2 = G[j];
                                if (grad_diff > 0)
                                {
                                        double obj_diff; 
                                        double quad_coef = QD[ip]+QD[j]-2*Q_ip[j];
                                        if (quad_coef > 0)
                                                obj_diff = -(grad_diff*grad_diff)/quad_coef;
                                        else
                                                obj_diff = -(grad_diff*grad_diff)/TAU;

                                        if (obj_diff <= obj_diff_min)
                                        {
                                                Gmin_idx=j;
                                                obj_diff_min = obj_diff;
                                        }
                                }
                        }
                }
                else
                {
                        if (!is_upper_bound(j))
                        {
                                double grad_diff=Gmaxn-G[j];
                                if (-G[j] >= Gmaxn2)
                                        Gmaxn2 = -G[j];
                                if (grad_diff > 0)
                                {
                                        double obj_diff; 
                                        double quad_coef = QD[in]+QD[j]-2*Q_in[j];
                                        if (quad_coef > 0)
                                                obj_diff = -(grad_diff*grad_diff)/quad_coef;
                                        else
                                                obj_diff = -(grad_diff*grad_diff)/TAU;

                                        if (obj_diff <= obj_diff_min)
                                        {
                                                Gmin_idx=j;
                                                obj_diff_min = obj_diff;
                                        }
                                }
                        }
                }
        }

        if(max(Gmaxp+Gmaxp2,Gmaxn+Gmaxn2) < eps)
                return 1;

        if (y[Gmin_idx] == +1)
                out_i = Gmaxp_idx;
        else
                out_i = Gmaxn_idx;
        out_j = Gmin_idx;

        return 0;
}

bool Solver_NU::be_shrunk(int i, double Gmax1, double Gmax2, double Gmax3, double Gmax4)
{
        if(is_upper_bound(i))
        {
                if(y[i]==+1)
                        return(-G[i] > Gmax1);
                else    
                        return(-G[i] > Gmax4);
        }
        else if(is_lower_bound(i))
        {
                if(y[i]==+1)
                        return(G[i] > Gmax2);
                else    
                        return(G[i] > Gmax3);
        }
        else
                return(false);
}

void Solver_NU::do_shrinking()
{
        double Gmax1 = -INF;    // max { -y_i * grad(f)_i | y_i = +1, i in I_up(\alpha) }
        double Gmax2 = -INF;    // max { y_i * grad(f)_i | y_i = +1, i in I_low(\alpha) }
        double Gmax3 = -INF;    // max { -y_i * grad(f)_i | y_i = -1, i in I_up(\alpha) }
        double Gmax4 = -INF;    // max { y_i * grad(f)_i | y_i = -1, i in I_low(\alpha) }

        // find maximal violating pair first
        int i;
        for(i=0;i<active_size;i++)
        {
                if(!is_upper_bound(i))
                {
                        if(y[i]==+1)
                        {
                                if(-G[i] > Gmax1) Gmax1 = -G[i];
                        }
                        else    if(-G[i] > Gmax4) Gmax4 = -G[i];
                }
                if(!is_lower_bound(i))
                {
                        if(y[i]==+1)
                        {       
                                if(G[i] > Gmax2) Gmax2 = G[i];
                        }
                        else    if(G[i] > Gmax3) Gmax3 = G[i];
                }
        }

        if(unshrink == false && max(Gmax1+Gmax2,Gmax3+Gmax4) <= eps*10) 
        {
                unshrink = true;
                reconstruct_gradient();
                active_size = l;
        }

        for(i=0;i<active_size;i++)
                if (be_shrunk(i, Gmax1, Gmax2, Gmax3, Gmax4))
                {
                        active_size--;
                        while (active_size > i)
                        {
                                if (!be_shrunk(active_size, Gmax1, Gmax2, Gmax3, Gmax4))
                                {
                                        swap_index(i,active_size);
                                        break;
                                }
                                active_size--;
                        }
                }
}

double Solver_NU::calculate_rho()
{
        int nr_free1 = 0,nr_free2 = 0;
        double ub1 = INF, ub2 = INF;
        double lb1 = -INF, lb2 = -INF;
        double sum_free1 = 0, sum_free2 = 0;

        for(int i=0;i<active_size;i++)
        {
                if(y[i]==+1)
                {
                        if(is_upper_bound(i))
                                lb1 = max(lb1,G[i]);
                        else if(is_lower_bound(i))
                                ub1 = min(ub1,G[i]);
                        else
                        {
                                ++nr_free1;
                                sum_free1 += G[i];
                        }
                }
                else
                {
                        if(is_upper_bound(i))
                                lb2 = max(lb2,G[i]);
                        else if(is_lower_bound(i))
                                ub2 = min(ub2,G[i]);
                        else
                        {
                                ++nr_free2;
                                sum_free2 += G[i];
                        }
                }
        }

        double r1,r2;
        if(nr_free1 > 0)
                r1 = sum_free1/nr_free1;
        else
                r1 = (ub1+lb1)/2;
        
        if(nr_free2 > 0)
                r2 = sum_free2/nr_free2;
        else
                r2 = (ub2+lb2)/2;
        
        si->r = (r1+r2)/2;
        return (r1-r2)/2;
}

//
// Q matrices for various formulations
//
class SVC_Q: public Kernel
{ 
public:
        SVC_Q(const svm_problem& prob, const svm_parameter& param, const schar *y_)
        :Kernel(prob.l, prob.x, param)
        {
                clone(y,y_,prob.l);
                cache = new Cache(prob.l,(long int)(param.cache_size*(1<<20)));
                QD = new double[prob.l];
                for(int i=0;i<prob.l;i++)
                        QD[i] = (this->*kernel_function)(i,i);
        }
        
        Qfloat *get_Q(int i, int len) const
        {
                Qfloat *data;
                int start, j;
                if((start = cache->get_data(i,&data,len)) < len)
                {
                        for(j=start;j<len;j++)
                                data[j] = (Qfloat)(y[i]*y[j]*(this->*kernel_function)(i,j));
                }
                return data;
        }

        double *get_QD() const
        {
                return QD;
        }

        void swap_index(int i, int j) const
        {
                cache->swap_index(i,j);
                Kernel::swap_index(i,j);
                swap(y[i],y[j]);
                swap(QD[i],QD[j]);
        }

        ~SVC_Q()
        {
                delete[] y;
                delete cache;
                delete[] QD;
        }
private:
        schar *y;
        Cache *cache;
        double *QD;
};

class ONE_CLASS_Q: public Kernel
{
public:
        ONE_CLASS_Q(const svm_problem& prob, const svm_parameter& param)
        :Kernel(prob.l, prob.x, param)
        {
                cache = new Cache(prob.l,(long int)(param.cache_size*(1<<20)));
                QD = new double[prob.l];
                for(int i=0;i<prob.l;i++)
                        QD[i] = (this->*kernel_function)(i,i);
        }
        
        Qfloat *get_Q(int i, int len) const
        {
                Qfloat *data;
                int start, j;
                if((start = cache->get_data(i,&data,len)) < len)
                {
                        for(j=start;j<len;j++)
                                data[j] = (Qfloat)(this->*kernel_function)(i,j);
                }
                return data;
        }

        double *get_QD() const
        {
                return QD;
        }

        void swap_index(int i, int j) const
        {
                cache->swap_index(i,j);
                Kernel::swap_index(i,j);
                swap(QD[i],QD[j]);
        }

        ~ONE_CLASS_Q()
        {
                delete cache;
                delete[] QD;
        }
private:
        Cache *cache;
        double *QD;
};

class SVR_Q: public Kernel
{ 
public:
        SVR_Q(const svm_problem& prob, const svm_parameter& param)
        :Kernel(prob.l, prob.x, param)
        {
                l = prob.l;
                cache = new Cache(l,(long int)(param.cache_size*(1<<20)));
                QD = new double[2*l];
                sign = new schar[2*l];
                index = new int[2*l];
                for(int k=0;k<l;k++)
                {
                        sign[k] = 1;
                        sign[k+l] = -1;
                        index[k] = k;
                        index[k+l] = k;
                        QD[k] = (this->*kernel_function)(k,k);
                        QD[k+l] = QD[k];
                }
                buffer[0] = new Qfloat[2*l];
                buffer[1] = new Qfloat[2*l];
                next_buffer = 0;
        }

        void swap_index(int i, int j) const
        {
                swap(sign[i],sign[j]);
                swap(index[i],index[j]);
                swap(QD[i],QD[j]);
        }
        
        Qfloat *get_Q(int i, int len) const
        {
                Qfloat *data;
                int j, real_i = index[i];
                if(cache->get_data(real_i,&data,l) < l)
                {
                        for(j=0;j<l;j++)
                                data[j] = (Qfloat)(this->*kernel_function)(real_i,j);
                }

                // reorder and copy
                Qfloat *buf = buffer[next_buffer];
                next_buffer = 1 - next_buffer;
                schar si = sign[i];
                for(j=0;j<len;j++)
                        buf[j] = (Qfloat) si * (Qfloat) sign[j] * data[index[j]];
                return buf;
        }

        double *get_QD() const
        {
                return QD;
        }

        ~SVR_Q()
        {
                delete cache;
                delete[] sign;
                delete[] index;
                delete[] buffer[0];
                delete[] buffer[1];
                delete[] QD;
        }
private:
        int l;
        Cache *cache;
        schar *sign;
        int *index;
        mutable int next_buffer;
        Qfloat *buffer[2];
        double *QD;
};

//
// construct and solve various formulations
//
static void solve_c_svc(
        const svm_problem *prob, const svm_parameter* param,
        double *alpha, Solver::SolutionInfo* si, double Cp, double Cn)
{
        int l = prob->l;
        double *minus_ones = new double[l];
        schar *y = new schar[l];

        int i;

        for(i=0;i<l;i++)
        {
                alpha[i] = 0;
                minus_ones[i] = -1;
                if(prob->y[i] > 0) y[i] = +1; else y[i] = -1;
        }

        Solver s;
        s.Solve(l, SVC_Q(*prob,*param,y), minus_ones, y,
                alpha, Cp, Cn, param->eps, si, param->shrinking);

        double sum_alpha=0;
        for(i=0;i<l;i++)
                sum_alpha += alpha[i];

        if (Cp==Cn)
                info("nu = %f\n", sum_alpha/(Cp*prob->l));

        for(i=0;i<l;i++)
                alpha[i] *= y[i];

        delete[] minus_ones;
        delete[] y;
}

static void solve_nu_svc(
        const svm_problem *prob, const svm_parameter *param,
        double *alpha, Solver::SolutionInfo* si)
{
        int i;
        int l = prob->l;
        double nu = param->nu;

        schar *y = new schar[l];

        for(i=0;i<l;i++)
                if(prob->y[i]>0)
                        y[i] = +1;
                else
                        y[i] = -1;

        double sum_pos = nu*l/2;
        double sum_neg = nu*l/2;

        for(i=0;i<l;i++)
                if(y[i] == +1)
                {
                        alpha[i] = min(1.0,sum_pos);
                        sum_pos -= alpha[i];
                }
                else
                {
                        alpha[i] = min(1.0,sum_neg);
                        sum_neg -= alpha[i];
                }

        double *zeros = new double[l];

        for(i=0;i<l;i++)
                zeros[i] = 0;

        Solver_NU s;
        s.Solve(l, SVC_Q(*prob,*param,y), zeros, y,
                alpha, 1.0, 1.0, param->eps, si,  param->shrinking);
        double r = si->r;

        info("C = %f\n",1/r);

        for(i=0;i<l;i++)
                alpha[i] *= y[i]/r;

        si->rho /= r;
        si->obj /= (r*r);
        si->upper_bound_p = 1/r;
        si->upper_bound_n = 1/r;

        delete[] y;
        delete[] zeros;
}

static void solve_one_class(
        const svm_problem *prob, const svm_parameter *param,
        double *alpha, Solver::SolutionInfo* si)
{
        int l = prob->l;
        double *zeros = new double[l];
        schar *ones = new schar[l];
        int i;

        int n = (int)(param->nu*prob->l);       // # of alpha's at upper bound

        for(i=0;i<n;i++)
                alpha[i] = 1;
        if(n<prob->l)
                alpha[n] = param->nu * prob->l - n;
        for(i=n+1;i<l;i++)
                alpha[i] = 0;

        for(i=0;i<l;i++)
        {
                zeros[i] = 0;
                ones[i] = 1;
        }

        Solver s;
        s.Solve(l, ONE_CLASS_Q(*prob,*param), zeros, ones,
                alpha, 1.0, 1.0, param->eps, si, param->shrinking);

        delete[] zeros;
        delete[] ones;
}

static void solve_epsilon_svr(
        const svm_problem *prob, const svm_parameter *param,
        double *alpha, Solver::SolutionInfo* si)
{
        int l = prob->l;
        double *alpha2 = new double[2*l];
        double *linear_term = new double[2*l];
        schar *y = new schar[2*l];
        int i;

        for(i=0;i<l;i++)
        {
                alpha2[i] = 0;
                linear_term[i] = param->p - prob->y[i];
                y[i] = 1;

                alpha2[i+l] = 0;
                linear_term[i+l] = param->p + prob->y[i];
                y[i+l] = -1;
        }

        Solver s;
        s.Solve(2*l, SVR_Q(*prob,*param), linear_term, y,
                alpha2, param->C, param->C, param->eps, si, param->shrinking);

        double sum_alpha = 0;
        for(i=0;i<l;i++)
        {
                alpha[i] = alpha2[i] - alpha2[i+l];
                sum_alpha += fabs(alpha[i]);
        }
        info("nu = %f\n",sum_alpha/(param->C*l));

        delete[] alpha2;
        delete[] linear_term;
        delete[] y;
}

static void solve_nu_svr(
        const svm_problem *prob, const svm_parameter *param,
        double *alpha, Solver::SolutionInfo* si)
{
        int l = prob->l;
        double C = param->C;
        double *alpha2 = new double[2*l];
        double *linear_term = new double[2*l];
        schar *y = new schar[2*l];
        int i;

        double sum = C * param->nu * l / 2;
        for(i=0;i<l;i++)
        {
                alpha2[i] = alpha2[i+l] = min(sum,C);
                sum -= alpha2[i];

                linear_term[i] = - prob->y[i];
                y[i] = 1;

                linear_term[i+l] = prob->y[i];
                y[i+l] = -1;
        }

        Solver_NU s;
        s.Solve(2*l, SVR_Q(*prob,*param), linear_term, y,
                alpha2, C, C, param->eps, si, param->shrinking);

        info("epsilon = %f\n",-si->r);

        for(i=0;i<l;i++)
                alpha[i] = alpha2[i] - alpha2[i+l];

        delete[] alpha2;
        delete[] linear_term;
        delete[] y;
}

//
// decision_function
//
struct decision_function
{
        double *alpha;
        double rho;     
};

static decision_function svm_train_one(
        const svm_problem *prob, const svm_parameter *param,
        double Cp, double Cn)
{
        double *alpha = Malloc(double,prob->l);
        Solver::SolutionInfo si;
        switch(param->svm_type)
        {
                case C_SVC:
                        solve_c_svc(prob,param,alpha,&si,Cp,Cn);
                        break;
                case NU_SVC:
                        solve_nu_svc(prob,param,alpha,&si);
                        break;
                case ONE_CLASS:
                        solve_one_class(prob,param,alpha,&si);
                        break;
                case EPSILON_SVR:
                        solve_epsilon_svr(prob,param,alpha,&si);
                        break;
                case NU_SVR:
                        solve_nu_svr(prob,param,alpha,&si);
                        break;
        }

        info("obj = %f, rho = %f\n",si.obj,si.rho);

        // output SVs

        int nSV = 0;
        int nBSV = 0;
        for(int i=0;i<prob->l;i++)
        {
                if(fabs(alpha[i]) > 0)
                {
                        ++nSV;
                        if(prob->y[i] > 0)
                        {
                                if(fabs(alpha[i]) >= si.upper_bound_p)
                                        ++nBSV;
                        }
                        else
                        {
                                if(fabs(alpha[i]) >= si.upper_bound_n)
                                        ++nBSV;
                        }
                }
        }

        info("nSV = %d, nBSV = %d\n",nSV,nBSV);

        decision_function f;
        f.alpha = alpha;
        f.rho = si.rho;
        return f;
}

// Platt's binary SVM Probablistic Output: an improvement from Lin et al.
static void sigmoid_train(
        int l, const double *dec_values, const double *labels, 
        double& A, double& B)
{
        double prior1=0, prior0 = 0;
        int i;

        for (i=0;i<l;i++)
                if (labels[i] > 0) prior1+=1;
                else prior0+=1;
        
        int max_iter=100;       // Maximal number of iterations
        double min_step=1e-10;  // Minimal step taken in line search
        double sigma=1e-12;     // For numerically strict PD of Hessian
        double eps=1e-5;
        double hiTarget=(prior1+1.0)/(prior1+2.0);
        double loTarget=1/(prior0+2.0);
        double *t=Malloc(double,l);
        double fApB,p,q,h11,h22,h21,g1,g2,det,dA,dB,gd,stepsize;
        double newA,newB,newf,d1,d2;
        int iter; 
        
        // Initial Point and Initial Fun Value
        A=0.0; B=log((prior0+1.0)/(prior1+1.0));
        double fval = 0.0;

        for (i=0;i<l;i++)
        {
                if (labels[i]>0) t[i]=hiTarget;
                else t[i]=loTarget;
                fApB = dec_values[i]*A+B;
                if (fApB>=0)
                        fval += t[i]*fApB + log(1+exp(-fApB));
                else
                        fval += (t[i] - 1)*fApB +log(1+exp(fApB));
        }
        for (iter=0;iter<max_iter;iter++)
        {
                // Update Gradient and Hessian (use H' = H + sigma I)
                h11=sigma; // numerically ensures strict PD
                h22=sigma;
                h21=0.0;g1=0.0;g2=0.0;
                for (i=0;i<l;i++)
                {
                        fApB = dec_values[i]*A+B;
                        if (fApB >= 0)
                        {
                                p=exp(-fApB)/(1.0+exp(-fApB));
                                q=1.0/(1.0+exp(-fApB));
                        }
                        else
                        {
                                p=1.0/(1.0+exp(fApB));
                                q=exp(fApB)/(1.0+exp(fApB));
                        }
                        d2=p*q;
                        h11+=dec_values[i]*dec_values[i]*d2;
                        h22+=d2;
                        h21+=dec_values[i]*d2;
                        d1=t[i]-p;
                        g1+=dec_values[i]*d1;
                        g2+=d1;
                }

                // Stopping Criteria
                if (fabs(g1)<eps && fabs(g2)<eps)
                        break;

                // Finding Newton direction: -inv(H') * g
                det=h11*h22-h21*h21;
                dA=-(h22*g1 - h21 * g2) / det;
                dB=-(-h21*g1+ h11 * g2) / det;
                gd=g1*dA+g2*dB;


                stepsize = 1;           // Line Search
                while (stepsize >= min_step)
                {
                        newA = A + stepsize * dA;
                        newB = B + stepsize * dB;

                        // New function value
                        newf = 0.0;
                        for (i=0;i<l;i++)
                        {
                                fApB = dec_values[i]*newA+newB;
                                if (fApB >= 0)
                                        newf += t[i]*fApB + log(1+exp(-fApB));
                                else
                                        newf += (t[i] - 1)*fApB +log(1+exp(fApB));
                        }
                        // Check sufficient decrease
                        if (newf<fval+0.0001*stepsize*gd)
                        {
                                A=newA;B=newB;fval=newf;
                                break;
                        }
                        else
                                stepsize = stepsize / 2.0;
                }

                if (stepsize < min_step)
                {
                        info("Line search fails in two-class probability estimates\n");
                        break;
                }
        }

        if (iter>=max_iter)
                info("Reaching maximal iterations in two-class probability estimates\n");
        free(t);
}

static double sigmoid_predict(double decision_value, double A, double B)
{
        double fApB = decision_value*A+B;
        // 1-p used later; avoid catastrophic cancellation
        if (fApB >= 0)
                return exp(-fApB)/(1.0+exp(-fApB));
        else
                return 1.0/(1+exp(fApB)) ;
}

// Method 2 from the multiclass_prob paper by Wu, Lin, and Weng
static void multiclass_probability(int k, double **r, double *p)
{
        int t,j;
        int iter = 0, max_iter=max(100,k);
        double **Q=Malloc(double *,k);
        double *Qp=Malloc(double,k);
        double pQp, eps=0.005/k;
        
        for (t=0;t<k;t++)
        {
                p[t]=1.0/k;  // Valid if k = 1
                Q[t]=Malloc(double,k);
                Q[t][t]=0;
                for (j=0;j<t;j++)
                {
                        Q[t][t]+=r[j][t]*r[j][t];
                        Q[t][j]=Q[j][t];
                }
                for (j=t+1;j<k;j++)
                {
                        Q[t][t]+=r[j][t]*r[j][t];
                        Q[t][j]=-r[j][t]*r[t][j];
                }
        }
        for (iter=0;iter<max_iter;iter++)
        {
                // stopping condition, recalculate QP,pQP for numerical accuracy
                pQp=0;
                for (t=0;t<k;t++)
                {
                        Qp[t]=0;
                        for (j=0;j<k;j++)
                                Qp[t]+=Q[t][j]*p[j];
                        pQp+=p[t]*Qp[t];
                }
                double max_error=0;
                for (t=0;t<k;t++)
                {
                        double error=fabs(Qp[t]-pQp);
                        if (error>max_error)
                                max_error=error;
                }
                if (max_error<eps) break;
                
                for (t=0;t<k;t++)
                {
                        double diff=(-Qp[t]+pQp)/Q[t][t];
                        p[t]+=diff;
                        pQp=(pQp+diff*(diff*Q[t][t]+2*Qp[t]))/(1+diff)/(1+diff);
                        for (j=0;j<k;j++)
                        {
                                Qp[j]=(Qp[j]+diff*Q[t][j])/(1+diff);
                                p[j]/=(1+diff);
                        }
                }
        }
        if (iter>=max_iter)
                info("Exceeds max_iter in multiclass_prob\n");
        for(t=0;t<k;t++) free(Q[t]);
        free(Q);
        free(Qp);
}

// Cross-validation decision values for probability estimates
static void svm_binary_svc_probability(
        const svm_problem *prob, const svm_parameter *param,
        double Cp, double Cn, double& probA, double& probB)
{
        int i;
        int nr_fold = 5;
        int *perm = Malloc(int,prob->l);
        double *dec_values = Malloc(double,prob->l);

        // random shuffle
        for(i=0;i<prob->l;i++) perm[i]=i;
        for(i=0;i<prob->l;i++)
        {
                int j = i+rand()%(prob->l-i);
                swap(perm[i],perm[j]);
        }
        for(i=0;i<nr_fold;i++)
        {
                int begin = i*prob->l/nr_fold;
                int end = (i+1)*prob->l/nr_fold;
                int j,k;
                struct svm_problem subprob;

                subprob.l = prob->l-(end-begin);
                subprob.x = Malloc(struct svm_node*,subprob.l);
                subprob.y = Malloc(double,subprob.l);
                        
                k=0;
                for(j=0;j<begin;j++)
                {
                        subprob.x[k] = prob->x[perm[j]];
                        subprob.y[k] = prob->y[perm[j]];
                        ++k;
                }
                for(j=end;j<prob->l;j++)
                {
                        subprob.x[k] = prob->x[perm[j]];
                        subprob.y[k] = prob->y[perm[j]];
                        ++k;
                }
                int p_count=0,n_count=0;
                for(j=0;j<k;j++)
                        if(subprob.y[j]>0)
                                p_count++;
                        else
                                n_count++;

                if(p_count==0 && n_count==0)
                        for(j=begin;j<end;j++)
                                dec_values[perm[j]] = 0;
                else if(p_count > 0 && n_count == 0)
                        for(j=begin;j<end;j++)
                                dec_values[perm[j]] = 1;
                else if(p_count == 0 && n_count > 0)
                        for(j=begin;j<end;j++)
                                dec_values[perm[j]] = -1;
                else
                {
                        svm_parameter subparam = *param;
                        subparam.probability=0;
                        subparam.C=1.0;
                        subparam.nr_weight=2;
                        subparam.weight_label = Malloc(int,2);
                        subparam.weight = Malloc(double,2);
                        subparam.weight_label[0]=+1;
                        subparam.weight_label[1]=-1;
                        subparam.weight[0]=Cp;
                        subparam.weight[1]=Cn;
                        struct svm_model *submodel = svm_train(&subprob,&subparam);
                        for(j=begin;j<end;j++)
                        {
                                svm_predict_values(submodel,prob->x[perm[j]],&(dec_values[perm[j]])); 
                                // ensure +1 -1 order; reason not using CV subroutine
                                dec_values[perm[j]] *= submodel->label[0];
                        }               
                        svm_free_and_destroy_model(&submodel);
                        svm_destroy_param(&subparam);
                }
                free(subprob.x);
                free(subprob.y);
        }               
        sigmoid_train(prob->l,dec_values,prob->y,probA,probB);
        free(dec_values);
        free(perm);
}

// Return parameter of a Laplace distribution 
static double svm_svr_probability(
        const svm_problem *prob, const svm_parameter *param)
{
        int i;
        int nr_fold = 5;
        double *ymv = Malloc(double,prob->l);
        double mae = 0;

        svm_parameter newparam = *param;
        newparam.probability = 0;
        svm_cross_validation(prob,&newparam,nr_fold,ymv);
        for(i=0;i<prob->l;i++)
        {
                ymv[i]=prob->y[i]-ymv[i];
                mae += fabs(ymv[i]);
        }               
        mae /= prob->l;
        double std=sqrt(2*mae*mae);
        int count=0;
        mae=0;
        for(i=0;i<prob->l;i++)
                if (fabs(ymv[i]) > 5*std) 
                        count=count+1;
                else 
                        mae+=fabs(ymv[i]);
        mae /= (prob->l-count);
        info("Prob. model for test data: target value = predicted value + z,\nz: Laplace distribution e^(-|z|/sigma)/(2sigma),sigma= %g\n",mae);
        free(ymv);
        return mae;
}


// label: label name, start: begin of each class, count: #data of classes, perm: indices to the original data
// perm, length l, must be allocated before calling this subroutine
static void svm_group_classes(const svm_problem *prob, int *nr_class_ret, int **label_ret, int **start_ret, int **count_ret, int *perm)
{
        int l = prob->l;
        int max_nr_class = 16;
        int nr_class = 0;
        int *label = Malloc(int,max_nr_class);
        int *count = Malloc(int,max_nr_class);
        int *data_label = Malloc(int,l);        
        int i;

        for(i=0;i<l;i++)
        {
                int this_label = (int)prob->y[i];
                int j;
                for(j=0;j<nr_class;j++)
                {
                        if(this_label == label[j])
                        {
                                ++count[j];
                                break;
                        }
                }
                data_label[i] = j;
                if(j == nr_class)
                {
                        if(nr_class == max_nr_class)
                        {
                                max_nr_class *= 2;
                                label = (int *)realloc(label,max_nr_class*sizeof(int));
                                count = (int *)realloc(count,max_nr_class*sizeof(int));
                        }
                        label[nr_class] = this_label;
                        count[nr_class] = 1;
                        ++nr_class;
                }
        }

        int *start = Malloc(int,nr_class);
        start[0] = 0;
        for(i=1;i<nr_class;i++)
                start[i] = start[i-1]+count[i-1];
        for(i=0;i<l;i++)
        {
                perm[start[data_label[i]]] = i;
                ++start[data_label[i]];
        }
        start[0] = 0;
        for(i=1;i<nr_class;i++)
                start[i] = start[i-1]+count[i-1];

        *nr_class_ret = nr_class;
        *label_ret = label;
        *start_ret = start;
        *count_ret = count;
        free(data_label);
}

//
// Interface functions
//
svm_model *svm_train(const svm_problem *prob, const svm_parameter *param)
{
        svm_model *model = Malloc(svm_model,1);
        model->param = *param;
        model->free_sv = 0;     // XXX

        if(param->svm_type == ONE_CLASS ||
           param->svm_type == EPSILON_SVR ||
           param->svm_type == NU_SVR)
        {
                // regression or one-class-svm
                model->nr_class = 2;
                model->label = NULL;
                model->nSV = NULL;
                model->probA = NULL; model->probB = NULL;
                model->sv_coef = Malloc(double *,1);

                if(param->probability && 
                   (param->svm_type == EPSILON_SVR ||
                    param->svm_type == NU_SVR))
                {
                        model->probA = Malloc(double,1);
                        model->probA[0] = svm_svr_probability(prob,param);
                }

                decision_function f = svm_train_one(prob,param,0,0);
                model->rho = Malloc(double,1);
                model->rho[0] = f.rho;

                int nSV = 0;
                int i;
                for(i=0;i<prob->l;i++)
                        if(fabs(f.alpha[i]) > 0) ++nSV;
                model->l = nSV;
                model->SV = Malloc(svm_node *,nSV);
                model->sv_coef[0] = Malloc(double,nSV);
                int j = 0;
                for(i=0;i<prob->l;i++)
                        if(fabs(f.alpha[i]) > 0)
                        {
                                model->SV[j] = prob->x[i];
                                model->sv_coef[0][j] = f.alpha[i];
                                ++j;
                        }               

                free(f.alpha);
        }
        else
        {
                // classification
                int l = prob->l;
                int nr_class;
                int *label = NULL;
                int *start = NULL;
                int *count = NULL;
                int *perm = Malloc(int,l);

                // group training data of the same class
                svm_group_classes(prob,&nr_class,&label,&start,&count,perm);
                if(nr_class == 1) 
                        info("WARNING: training data in only one class. See README for details.\n");
                
                svm_node **x = Malloc(svm_node *,l);
                int i;
                for(i=0;i<l;i++)
                        x[i] = prob->x[perm[i]];

                // calculate weighted C

                double *weighted_C = Malloc(double, nr_class);
                for(i=0;i<nr_class;i++)
                        weighted_C[i] = param->C;
                for(i=0;i<param->nr_weight;i++)
                {       
                        int j;
                        for(j=0;j<nr_class;j++)
                                if(param->weight_label[i] == label[j])
                                        break;
                        if(j == nr_class)
                                fprintf(stderr,"WARNING: class label %d specified in weight is not found\n", param->weight_label[i]);
                        else
                                weighted_C[j] *= param->weight[i];
                }

                // train k*(k-1)/2 models
                
                bool *nonzero = Malloc(bool,l);
                for(i=0;i<l;i++)
                        nonzero[i] = false;
                decision_function *f = Malloc(decision_function,nr_class*(nr_class-1)/2);

                double *probA=NULL,*probB=NULL;
                if (param->probability)
                {
                        probA=Malloc(double,nr_class*(nr_class-1)/2);
                        probB=Malloc(double,nr_class*(nr_class-1)/2);
                }

                int p = 0;
                for(i=0;i<nr_class;i++)
                        for(int j=i+1;j<nr_class;j++)
                        {
                                svm_problem sub_prob;
                                int si = start[i], sj = start[j];
                                int ci = count[i], cj = count[j];
                                sub_prob.l = ci+cj;
                                sub_prob.x = Malloc(svm_node *,sub_prob.l);
                                sub_prob.y = Malloc(double,sub_prob.l);
                                int k;
                                for(k=0;k<ci;k++)
                                {
                                        sub_prob.x[k] = x[si+k];
                                        sub_prob.y[k] = +1;
                                }
                                for(k=0;k<cj;k++)
                                {
                                        sub_prob.x[ci+k] = x[sj+k];
                                        sub_prob.y[ci+k] = -1;
                                }

                                if(param->probability)
                                        svm_binary_svc_probability(&sub_prob,param,weighted_C[i],weighted_C[j],probA[p],probB[p]);

                                f[p] = svm_train_one(&sub_prob,param,weighted_C[i],weighted_C[j]);
                                for(k=0;k<ci;k++)
                                        if(!nonzero[si+k] && fabs(f[p].alpha[k]) > 0)
                                                nonzero[si+k] = true;
                                for(k=0;k<cj;k++)
                                        if(!nonzero[sj+k] && fabs(f[p].alpha[ci+k]) > 0)
                                                nonzero[sj+k] = true;
                                free(sub_prob.x);
                                free(sub_prob.y);
                                ++p;
                        }

                // build output

                model->nr_class = nr_class;
                
                model->label = Malloc(int,nr_class);
                for(i=0;i<nr_class;i++)
                        model->label[i] = label[i];
                
                model->rho = Malloc(double,nr_class*(nr_class-1)/2);
                for(i=0;i<nr_class*(nr_class-1)/2;i++)
                        model->rho[i] = f[i].rho;

                if(param->probability)
                {
                        model->probA = Malloc(double,nr_class*(nr_class-1)/2);
                        model->probB = Malloc(double,nr_class*(nr_class-1)/2);
                        for(i=0;i<nr_class*(nr_class-1)/2;i++)
                        {
                                model->probA[i] = probA[i];
                                model->probB[i] = probB[i];
                        }
                }
                else
                {
                        model->probA=NULL;
                        model->probB=NULL;
                }

                int total_sv = 0;
                int *nz_count = Malloc(int,nr_class);
                model->nSV = Malloc(int,nr_class);
                for(i=0;i<nr_class;i++)
                {
                        int nSV = 0;
                        for(int j=0;j<count[i];j++)
                                if(nonzero[start[i]+j])
                                {       
                                        ++nSV;
                                        ++total_sv;
                                }
                        model->nSV[i] = nSV;
                        nz_count[i] = nSV;
                }
                
                info("Total nSV = %d\n",total_sv);

                model->l = total_sv;
                model->SV = Malloc(svm_node *,total_sv);
                p = 0;
                for(i=0;i<l;i++)
                        if(nonzero[i]) model->SV[p++] = x[i];

                int *nz_start = Malloc(int,nr_class);
                nz_start[0] = 0;
                for(i=1;i<nr_class;i++)
                        nz_start[i] = nz_start[i-1]+nz_count[i-1];

                model->sv_coef = Malloc(double *,nr_class-1);
                for(i=0;i<nr_class-1;i++)
                        model->sv_coef[i] = Malloc(double,total_sv);

                p = 0;
                for(i=0;i<nr_class;i++)
                        for(int j=i+1;j<nr_class;j++)
                        {
                                // classifier (i,j): coefficients with
                                // i are in sv_coef[j-1][nz_start[i]...],
                                // j are in sv_coef[i][nz_start[j]...]

                                int si = start[i];
                                int sj = start[j];
                                int ci = count[i];
                                int cj = count[j];
                                
                                int q = nz_start[i];
                                int k;
                                for(k=0;k<ci;k++)
                                        if(nonzero[si+k])
                                                model->sv_coef[j-1][q++] = f[p].alpha[k];
                                q = nz_start[j];
                                for(k=0;k<cj;k++)
                                        if(nonzero[sj+k])
                                                model->sv_coef[i][q++] = f[p].alpha[ci+k];
                                ++p;
                        }
                
                free(label);
                free(probA);
                free(probB);
                free(count);
                free(perm);
                free(start);
                free(x);
                free(weighted_C);
                free(nonzero);
                for(i=0;i<nr_class*(nr_class-1)/2;i++)
                        free(f[i].alpha);
                free(f);
                free(nz_count);
                free(nz_start);
        }
        return model;
}

// Stratified cross validation
void svm_cross_validation(const svm_problem *prob, const svm_parameter *param, int nr_fold, double *target)
{
        int i;
        int *fold_start = Malloc(int,nr_fold+1);
        int l = prob->l;
        int *perm = Malloc(int,l);
        int nr_class;

        // stratified cv may not give leave-one-out rate
        // Each class to l folds -> some folds may have zero elements
        if((param->svm_type == C_SVC ||
            param->svm_type == NU_SVC) && nr_fold < l)
        {
                int *start = NULL;
                int *label = NULL;
                int *count = NULL;
                svm_group_classes(prob,&nr_class,&label,&start,&count,perm);

                // random shuffle and then data grouped by fold using the array perm
                int *fold_count = Malloc(int,nr_fold);
                int c;
                int *index = Malloc(int,l);
                for(i=0;i<l;i++)
                        index[i]=perm[i];
                for (c=0; c<nr_class; c++) 
                        for(i=0;i<count[c];i++)
                        {
                                int j = i+rand()%(count[c]-i);
                                swap(index[start[c]+j],index[start[c]+i]);
                        }
                for(i=0;i<nr_fold;i++)
                {
                        fold_count[i] = 0;
                        for (c=0; c<nr_class;c++)
                                fold_count[i]+=(i+1)*count[c]/nr_fold-i*count[c]/nr_fold;
                }
                fold_start[0]=0;
                for (i=1;i<=nr_fold;i++)
                        fold_start[i] = fold_start[i-1]+fold_count[i-1];
                for (c=0; c<nr_class;c++)
                        for(i=0;i<nr_fold;i++)
                        {
                                int begin = start[c]+i*count[c]/nr_fold;
                                int end = start[c]+(i+1)*count[c]/nr_fold;
                                for(int j=begin;j<end;j++)
                                {
                                        perm[fold_start[i]] = index[j];
                                        fold_start[i]++;
                                }
                        }
                fold_start[0]=0;
                for (i=1;i<=nr_fold;i++)
                        fold_start[i] = fold_start[i-1]+fold_count[i-1];
                free(start);    
                free(label);
                free(count);    
                free(index);
                free(fold_count);
        }
        else
        {
                for(i=0;i<l;i++) perm[i]=i;
                for(i=0;i<l;i++)
                {
                        int j = i+rand()%(l-i);
                        swap(perm[i],perm[j]);
                }
                for(i=0;i<=nr_fold;i++)
                        fold_start[i]=i*l/nr_fold;
        }

        for(i=0;i<nr_fold;i++)
        {
                int begin = fold_start[i];
                int end = fold_start[i+1];
                int j,k;
                struct svm_problem subprob;

                subprob.l = l-(end-begin);
                subprob.x = Malloc(struct svm_node*,subprob.l);
                subprob.y = Malloc(double,subprob.l);
                        
                k=0;
                for(j=0;j<begin;j++)
                {
                        subprob.x[k] = prob->x[perm[j]];
                        subprob.y[k] = prob->y[perm[j]];
                        ++k;
                }
                for(j=end;j<l;j++)
                {
                        subprob.x[k] = prob->x[perm[j]];
                        subprob.y[k] = prob->y[perm[j]];
                        ++k;
                }
                struct svm_model *submodel = svm_train(&subprob,param);
                if(param->probability && 
                   (param->svm_type == C_SVC || param->svm_type == NU_SVC))
                {
                        double *prob_estimates=Malloc(double,svm_get_nr_class(submodel));
                        for(j=begin;j<end;j++)
                                target[perm[j]] = svm_predict_probability(submodel,prob->x[perm[j]],prob_estimates);
                        free(prob_estimates);                   
                }
                else
                        for(j=begin;j<end;j++)
                                target[perm[j]] = svm_predict(submodel,prob->x[perm[j]]);
                svm_free_and_destroy_model(&submodel);
                free(subprob.x);
                free(subprob.y);
        }               
        free(fold_start);
        free(perm);     
}


int svm_get_svm_type(const svm_model *model)
{
        return model->param.svm_type;
}

int svm_get_nr_class(const svm_model *model)
{
        return model->nr_class;
}

void svm_get_labels(const svm_model *model, int* label)
{
        if (model->label != NULL)
                for(int i=0;i<model->nr_class;i++)
                        label[i] = model->label[i];
}

double svm_get_svr_probability(const svm_model *model)
{
        if ((model->param.svm_type == EPSILON_SVR || model->param.svm_type == NU_SVR) &&
            model->probA!=NULL)
                return model->probA[0];
        else
        {
                fprintf(stderr,"Model doesn't contain information for SVR probability inference\n");
                return 0;
        }
}

double svm_predict_values(const svm_model *model, const svm_node *x, double* dec_values)
{
        int i;
        if(model->param.svm_type == ONE_CLASS ||
           model->param.svm_type == EPSILON_SVR ||
           model->param.svm_type == NU_SVR)
        {
                double *sv_coef = model->sv_coef[0];
                double sum = 0;
                for(i=0;i<model->l;i++)
                        sum += sv_coef[i] * Kernel::k_function(x,model->SV[i],model->param);
                sum -= model->rho[0];
                *dec_values = sum;

                if(model->param.svm_type == ONE_CLASS)
                        return (sum>0)?1:-1;
                else
                        return sum;
        }
        else
        {
                int nr_class = model->nr_class;
                int l = model->l;
                
                double *kvalue = Malloc(double,l);
                for(i=0;i<l;i++)
                        kvalue[i] = Kernel::k_function(x,model->SV[i],model->param);

                int *start = Malloc(int,nr_class);
                start[0] = 0;
                for(i=1;i<nr_class;i++)
                        start[i] = start[i-1]+model->nSV[i-1];

                int *vote = Malloc(int,nr_class);
                for(i=0;i<nr_class;i++)
                        vote[i] = 0;

                int p=0;
                for(i=0;i<nr_class;i++)
                        for(int j=i+1;j<nr_class;j++)
                        {
                                double sum = 0;
                                int si = start[i];
                                int sj = start[j];
                                int ci = model->nSV[i];
                                int cj = model->nSV[j];
                                
                                int k;
                                double *coef1 = model->sv_coef[j-1];
                                double *coef2 = model->sv_coef[i];
                                for(k=0;k<ci;k++)
                                        sum += coef1[si+k] * kvalue[si+k];
                                for(k=0;k<cj;k++)
                                        sum += coef2[sj+k] * kvalue[sj+k];
                                sum -= model->rho[p];
                                dec_values[p] = sum;

                                if(dec_values[p] > 0)
                                        ++vote[i];
                                else
                                        ++vote[j];
                                p++;
                        }

                int vote_max_idx = 0;
                for(i=1;i<nr_class;i++)
                        if(vote[i] > vote[vote_max_idx])
                                vote_max_idx = i;

                free(kvalue);
                free(start);
                free(vote);
                return model->label[vote_max_idx];
        }
}

double svm_predict(const svm_model *model, const svm_node *x)
{
        int nr_class = model->nr_class;
        double *dec_values;
        if(model->param.svm_type == ONE_CLASS ||
           model->param.svm_type == EPSILON_SVR ||
           model->param.svm_type == NU_SVR)
                dec_values = Malloc(double, 1);
        else 
                dec_values = Malloc(double, nr_class*(nr_class-1)/2);
        double pred_result = svm_predict_values(model, x, dec_values);
        free(dec_values);
        return pred_result;
}

double svm_predict_probability(
        const svm_model *model, const svm_node *x, double *prob_estimates)
{
        if ((model->param.svm_type == C_SVC || model->param.svm_type == NU_SVC) &&
            model->probA!=NULL && model->probB!=NULL)
        {
                int i;
                int nr_class = model->nr_class;
                double *dec_values = Malloc(double, nr_class*(nr_class-1)/2);
                svm_predict_values(model, x, dec_values);

                double min_prob=1e-7;
                double **pairwise_prob=Malloc(double *,nr_class);
                for(i=0;i<nr_class;i++)
                        pairwise_prob[i]=Malloc(double,nr_class);
                int k=0;
                for(i=0;i<nr_class;i++)
                        for(int j=i+1;j<nr_class;j++)
                        {
                                pairwise_prob[i][j]=min(max(sigmoid_predict(dec_values[k],model->probA[k],model->probB[k]),min_prob),1-min_prob);
                                pairwise_prob[j][i]=1-pairwise_prob[i][j];
                                k++;
                        }
                multiclass_probability(nr_class,pairwise_prob,prob_estimates);

                int prob_max_idx = 0;
                for(i=1;i<nr_class;i++)
                        if(prob_estimates[i] > prob_estimates[prob_max_idx])
                                prob_max_idx = i;
                for(i=0;i<nr_class;i++)
                        free(pairwise_prob[i]);
                free(dec_values);
                free(pairwise_prob);         
                return model->label[prob_max_idx];
        }
        else 
                return svm_predict(model, x);
}

static const char *svm_type_table[] =
{
        "c_svc","nu_svc","one_class","epsilon_svr","nu_svr",NULL
};

static const char *kernel_type_table[]=
{
        "linear","polynomial","rbf","sigmoid","precomputed",NULL
};

int svm_save_model(const char *model_file_name, const svm_model *model)
{
        FILE *fp = fopen(model_file_name,"w");
        if(fp==NULL) return -1;

        char *old_locale = strdup(setlocale(LC_ALL, NULL));
        setlocale(LC_ALL, "C");

        const svm_parameter& param = model->param;

        fprintf(fp,"svm_type %s\n", svm_type_table[param.svm_type]);
        fprintf(fp,"kernel_type %s\n", kernel_type_table[param.kernel_type]);

        if(param.kernel_type == POLY)
                fprintf(fp,"degree %d\n", param.degree);

        if(param.kernel_type == POLY || param.kernel_type == RBF || param.kernel_type == SIGMOID)
                fprintf(fp,"gamma %g\n", param.gamma);

        if(param.kernel_type == POLY || param.kernel_type == SIGMOID)
                fprintf(fp,"coef0 %g\n", param.coef0);

        int nr_class = model->nr_class;
        int l = model->l;
        fprintf(fp, "nr_class %d\n", nr_class);
        fprintf(fp, "total_sv %d\n",l);
        
        {
                fprintf(fp, "rho");
                for(int i=0;i<nr_class*(nr_class-1)/2;i++)
                        fprintf(fp," %g",model->rho[i]);
                fprintf(fp, "\n");
        }
        
        if(model->label)
        {
                fprintf(fp, "label");
                for(int i=0;i<nr_class;i++)
                        fprintf(fp," %d",model->label[i]);
                fprintf(fp, "\n");
        }

        if(model->probA) // regression has probA only
        {
                fprintf(fp, "probA");
                for(int i=0;i<nr_class*(nr_class-1)/2;i++)
                        fprintf(fp," %g",model->probA[i]);
                fprintf(fp, "\n");
        }
        if(model->probB)
        {
                fprintf(fp, "probB");
                for(int i=0;i<nr_class*(nr_class-1)/2;i++)
                        fprintf(fp," %g",model->probB[i]);
                fprintf(fp, "\n");
        }

        if(model->nSV)
        {
                fprintf(fp, "nr_sv");
                for(int i=0;i<nr_class;i++)
                        fprintf(fp," %d",model->nSV[i]);
                fprintf(fp, "\n");
        }

        fprintf(fp, "SV\n");
        const double * const *sv_coef = model->sv_coef;
        const svm_node * const *SV = model->SV;

        for(int i=0;i<l;i++)
        {
                for(int j=0;j<nr_class-1;j++)
                        fprintf(fp, "%.16g ",sv_coef[j][i]);

                const svm_node *p = SV[i];

                if(param.kernel_type == PRECOMPUTED)
                        fprintf(fp,"0:%d ",(int)(p->value));
                else
                        while(p->index != -1)
                        {
                                fprintf(fp,"%d:%.8g ",p->index,p->value);
                                p++;
                        }
                fprintf(fp, "\n");
        }

        setlocale(LC_ALL, old_locale);
        free(old_locale);

        if (ferror(fp) != 0 || fclose(fp) != 0) return -1;
        else return 0;
}

static char *line = NULL;
static int max_line_len;

static char* readline(FILE *input)
{
        int len;

        if(fgets(line,max_line_len,input) == NULL)
                return NULL;

        while(strrchr(line,'\n') == NULL)
        {
                max_line_len *= 2;
                line = (char *) realloc(line,max_line_len);
                len = (int) strlen(line);
                if(fgets(line+len,max_line_len-len,input) == NULL)
                        break;
        }
        return line;
}

svm_model *svm_load_model(const char *model_file_name)
{
        FILE *fp = fopen(model_file_name,"rb");
        if(fp==NULL) return NULL;

        char *old_locale = strdup(setlocale(LC_ALL, NULL));
        setlocale(LC_ALL, "C");

        // read parameters

        svm_model *model = Malloc(svm_model,1);
        svm_parameter& param = model->param;
        model->rho = NULL;
        model->probA = NULL;
        model->probB = NULL;
        model->label = NULL;
        model->nSV = NULL;

        char cmd[81];
        while(1)
        {
                fscanf(fp,"%80s",cmd);

                if(strcmp(cmd,"svm_type")==0)
                {
                        fscanf(fp,"%80s",cmd);
                        int i;
                        for(i=0;svm_type_table[i];i++)
                        {
                                if(strcmp(svm_type_table[i],cmd)==0)
                                {
                                        param.svm_type=i;
                                        break;
                                }
                        }
                        if(svm_type_table[i] == NULL)
                        {
                                fprintf(stderr,"unknown svm type.\n");
                                
                                setlocale(LC_ALL, old_locale);
                                free(old_locale);
                                free(model->rho);
                                free(model->label);
                                free(model->nSV);
                                free(model);
                                return NULL;
                        }
                }
                else if(strcmp(cmd,"kernel_type")==0)
                {               
                        fscanf(fp,"%80s",cmd);
                        int i;
                        for(i=0;kernel_type_table[i];i++)
                        {
                                if(strcmp(kernel_type_table[i],cmd)==0)
                                {
                                        param.kernel_type=i;
                                        break;
                                }
                        }
                        if(kernel_type_table[i] == NULL)
                        {
                                fprintf(stderr,"unknown kernel function.\n");
                                
                                setlocale(LC_ALL, old_locale);
                                free(old_locale);
                                free(model->rho);
                                free(model->label);
                                free(model->nSV);
                                free(model);
                                return NULL;
                        }
                }
                else if(strcmp(cmd,"degree")==0)
                        fscanf(fp,"%d",&param.degree);
                else if(strcmp(cmd,"gamma")==0)
                        fscanf(fp,"%lf",&param.gamma);
                else if(strcmp(cmd,"coef0")==0)
                        fscanf(fp,"%lf",&param.coef0);
                else if(strcmp(cmd,"nr_class")==0)
                        fscanf(fp,"%d",&model->nr_class);
                else if(strcmp(cmd,"total_sv")==0)
                        fscanf(fp,"%d",&model->l);
                else if(strcmp(cmd,"rho")==0)
                {
                        int n = model->nr_class * (model->nr_class-1)/2;
                        model->rho = Malloc(double,n);
                        for(int i=0;i<n;i++)
                                fscanf(fp,"%lf",&model->rho[i]);
                }
                else if(strcmp(cmd,"label")==0)
                {
                        int n = model->nr_class;
                        model->label = Malloc(int,n);
                        for(int i=0;i<n;i++)
                                fscanf(fp,"%d",&model->label[i]);
                }
                else if(strcmp(cmd,"probA")==0)
                {
                        int n = model->nr_class * (model->nr_class-1)/2;
                        model->probA = Malloc(double,n);
                        for(int i=0;i<n;i++)
                                fscanf(fp,"%lf",&model->probA[i]);
                }
                else if(strcmp(cmd,"probB")==0)
                {
                        int n = model->nr_class * (model->nr_class-1)/2;
                        model->probB = Malloc(double,n);
                        for(int i=0;i<n;i++)
                                fscanf(fp,"%lf",&model->probB[i]);
                }
                else if(strcmp(cmd,"nr_sv")==0)
                {
                        int n = model->nr_class;
                        model->nSV = Malloc(int,n);
                        for(int i=0;i<n;i++)
                                fscanf(fp,"%d",&model->nSV[i]);
                }
                else if(strcmp(cmd,"SV")==0)
                {
                        while(1)
                        {
                                int c = getc(fp);
                                if(c==EOF || c=='\n') break;    
                        }
                        break;
                }
                else
                {
                        fprintf(stderr,"unknown text in model file: [%s]\n",cmd);
                        
                        setlocale(LC_ALL, old_locale);
                        free(old_locale);
                        free(model->rho);
                        free(model->label);
                        free(model->nSV);
                        free(model);
                        return NULL;
                }
        }

        // read sv_coef and SV

        int elements = 0;
        long pos = ftell(fp);

        max_line_len = 1024;
        line = Malloc(char,max_line_len);
        char *p,*endptr,*idx,*val;

        while(readline(fp)!=NULL)
        {
                p = strtok(line,":");
                while(1)
                {
                        p = strtok(NULL,":");
                        if(p == NULL)
                                break;
                        ++elements;
                }
        }
        elements += model->l;

        fseek(fp,pos,SEEK_SET);

        int m = model->nr_class - 1;
        int l = model->l;
        model->sv_coef = Malloc(double *,m);
        int i;
        for(i=0;i<m;i++)
                model->sv_coef[i] = Malloc(double,l);
        model->SV = Malloc(svm_node*,l);
        svm_node *x_space = NULL;
        if(l>0) x_space = Malloc(svm_node,elements);

        int j=0;
        for(i=0;i<l;i++)
        {
                readline(fp);
                model->SV[i] = &x_space[j];

                p = strtok(line, " \t");
                model->sv_coef[0][i] = strtod(p,&endptr);
                for(int k=1;k<m;k++)
                {
                        p = strtok(NULL, " \t");
                        model->sv_coef[k][i] = strtod(p,&endptr);
                }

                while(1)
                {
                        idx = strtok(NULL, ":");
                        val = strtok(NULL, " \t");

                        if(val == NULL)
                                break;
                        x_space[j].index = (int) strtol(idx,&endptr,10);
                        x_space[j].value = strtod(val,&endptr);

                        ++j;
                }
                x_space[j++].index = -1;
        }
        free(line);

        setlocale(LC_ALL, old_locale);
        free(old_locale);

        if (ferror(fp) != 0 || fclose(fp) != 0)
                return NULL;

        model->free_sv = 1;     // XXX
        return model;
}

void svm_free_model_content(svm_model* model_ptr)
{
        if(model_ptr->free_sv && model_ptr->l > 0 && model_ptr->SV != NULL)
                free((void *)(model_ptr->SV[0]));
        if(model_ptr->sv_coef)
        {
                for(int i=0;i<model_ptr->nr_class-1;i++)
                        free(model_ptr->sv_coef[i]);
        }

        free(model_ptr->SV);
        model_ptr->SV = NULL;

        free(model_ptr->sv_coef);
        model_ptr->sv_coef = NULL;

        free(model_ptr->rho);
        model_ptr->rho = NULL;

        free(model_ptr->label);
        model_ptr->label= NULL;

        free(model_ptr->probA);
        model_ptr->probA = NULL;

        free(model_ptr->probB);
        model_ptr->probB= NULL;

        free(model_ptr->nSV);
        model_ptr->nSV = NULL;
}

void svm_free_and_destroy_model(svm_model** model_ptr_ptr)
{
        if(model_ptr_ptr != NULL && *model_ptr_ptr != NULL)
        {
                svm_free_model_content(*model_ptr_ptr);
                free(*model_ptr_ptr);
                *model_ptr_ptr = NULL;
        }
}

void svm_destroy_param(svm_parameter* param)
{
        free(param->weight_label);
        free(param->weight);
}

const char *svm_check_parameter(const svm_problem *prob, const svm_parameter *param)
{
        // svm_type

        int svm_type = param->svm_type;
        if(svm_type != C_SVC &&
           svm_type != NU_SVC &&
           svm_type != ONE_CLASS &&
           svm_type != EPSILON_SVR &&
           svm_type != NU_SVR)
                return "unknown svm type";
        
        // kernel_type, degree
        
        int kernel_type = param->kernel_type;
        if(kernel_type != LINEAR &&
           kernel_type != POLY &&
           kernel_type != RBF &&
           kernel_type != SIGMOID &&
           kernel_type != PRECOMPUTED)
                return "unknown kernel type";

        if(param->gamma < 0)
                return "gamma < 0";

        if(param->degree < 0)
                return "degree of polynomial kernel < 0";

        // cache_size,eps,C,nu,p,shrinking

        if(param->cache_size <= 0)
                return "cache_size <= 0";

        if(param->eps <= 0)
                return "eps <= 0";

        if(svm_type == C_SVC ||
           svm_type == EPSILON_SVR ||
           svm_type == NU_SVR)
                if(param->C <= 0)
                        return "C <= 0";

        if(svm_type == NU_SVC ||
           svm_type == ONE_CLASS ||
           svm_type == NU_SVR)
                if(param->nu <= 0 || param->nu > 1)
                        return "nu <= 0 or nu > 1";

        if(svm_type == EPSILON_SVR)
                if(param->p < 0)
                        return "p < 0";

        if(param->shrinking != 0 &&
           param->shrinking != 1)
                return "shrinking != 0 and shrinking != 1";

        if(param->probability != 0 &&
           param->probability != 1)
                return "probability != 0 and probability != 1";

        if(param->probability == 1 &&
           svm_type == ONE_CLASS)
                return "one-class SVM probability output not supported yet";


        // check whether nu-svc is feasible
        
        if(svm_type == NU_SVC)
        {
                int l = prob->l;
                int max_nr_class = 16;
                int nr_class = 0;
                int *label = Malloc(int,max_nr_class);
                int *count = Malloc(int,max_nr_class);

                int i;
                for(i=0;i<l;i++)
                {
                        int this_label = (int)prob->y[i];
                        int j;
                        for(j=0;j<nr_class;j++)
                                if(this_label == label[j])
                                {
                                        ++count[j];
                                        break;
                                }
                        if(j == nr_class)
                        {
                                if(nr_class == max_nr_class)
                                {
                                        max_nr_class *= 2;
                                        label = (int *)realloc(label,max_nr_class*sizeof(int));
                                        count = (int *)realloc(count,max_nr_class*sizeof(int));
                                }
                                label[nr_class] = this_label;
                                count[nr_class] = 1;
                                ++nr_class;
                        }
                }
        
                for(i=0;i<nr_class;i++)
                {
                        int n1 = count[i];
                        for(int j=i+1;j<nr_class;j++)
                        {
                                int n2 = count[j];
                                if(param->nu*(n1+n2)/2 > min(n1,n2))
                                {
                                        free(label);
                                        free(count);
                                        return "specified nu is infeasible";
                                }
                        }
                }
                free(label);
                free(count);
        }

        return NULL;
}

int svm_check_probability_model(const svm_model *model)
{
        return ((model->param.svm_type == C_SVC || model->param.svm_type == NU_SVC) &&
                model->probA!=NULL && model->probB!=NULL) ||
                ((model->param.svm_type == EPSILON_SVR || model->param.svm_type == NU_SVR) &&
                 model->probA!=NULL);
}

void svm_set_print_string_function(void (*print_func)(const char *))
{
        if(print_func == NULL)
                svm_print_string = &print_string_stdout;
        else
                svm_print_string = print_func;
}