QuadraticTimeMMD.cpp

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/*
 * This program is free software; you can redistribute it and/or modify
 * it under the terms of the GNU General Public License as published by
 * the Free Software Foundation; either version 3 of the License, or
 * (at your option) any later version.
 *
 * Written (W) 2012 Heiko Strathmann
 */

#include <shogun/statistics/QuadraticTimeMMD.h>
#include <shogun/features/Features.h>
#include <shogun/mathematics/Statistics.h>
#include <shogun/kernel/Kernel.h>

using namespace shogun;

CQuadraticTimeMMD::CQuadraticTimeMMD() : CKernelTwoSampleTestStatistic()
{
    init();
}

CQuadraticTimeMMD::CQuadraticTimeMMD(CKernel* kernel, CFeatures* p_and_q,
        index_t m) :
        CKernelTwoSampleTestStatistic(kernel, p_and_q, m)
{
    init();

    if (p_and_q && m!=p_and_q->get_num_vectors()/2)
    {
        SG_ERROR("CQuadraticTimeMMD: Only features with equal number of vectors "
                "are currently possible\n");
    }
}

CQuadraticTimeMMD::CQuadraticTimeMMD(CKernel* kernel, CFeatures* p,
        CFeatures* q) : CKernelTwoSampleTestStatistic(kernel, p, q)
{
    init();

    if (p && q && p->get_num_vectors()!=q->get_num_vectors())
    {
        SG_ERROR("CQuadraticTimeMMD: Only features with equal number of vectors "
                "are currently possible\n");
    }
}

CQuadraticTimeMMD::~CQuadraticTimeMMD()
{

}

void CQuadraticTimeMMD::init()
{
    SG_ADD(&m_num_samples_spectrum, "num_samples_spectrum", "Number of samples"
            " for spectrum method null-distribution approximation",
            MS_NOT_AVAILABLE);
    SG_ADD(&m_num_eigenvalues_spectrum, "num_eigenvalues_spectrum", "Number of "
            " Eigenvalues for spectrum method null-distribution approximation",
            MS_NOT_AVAILABLE);
    SG_ADD((machine_int_t*)&m_statistic_type, "statistic_type",
            "Biased or unbiased MMD", MS_NOT_AVAILABLE);

    m_num_samples_spectrum=0;
    m_num_eigenvalues_spectrum=0;
    m_statistic_type=UNBIASED;
}

float64_t CQuadraticTimeMMD::compute_unbiased_statistic()
{
    /* split computations into three terms from JLMR paper (see documentation )*/
    index_t m=m_m;

    /* init kernel with features */
    m_kernel->init(m_p_and_q, m_p_and_q);

    /* first term */
    float64_t first=0;
    for (index_t i=0; i<m; ++i)
    {
        /* ensure i!=j while adding up */
        for (index_t j=0; j<m; ++j)
            first+=i==j ? 0 : m_kernel->kernel(i,j);
    }
    first/=(m-1);

    /* second term */
    float64_t second=0;
    for (index_t i=m_m; i<m_m+m; ++i)
    {
        /* ensure i!=j while adding up */
        for (index_t j=m_m; j<m_m+m; ++j)
            second+=i==j ? 0 : m_kernel->kernel(i,j);
    }
    second/=(m-1);

    /* third term */
    float64_t third=0;
    for (index_t i=0; i<m; ++i)
    {
        for (index_t j=m_m; j<m_m+m; ++j)
            third+=m_kernel->kernel(i,j);
    }
    third*=2.0/m;

    return first+second-third;
}

float64_t CQuadraticTimeMMD::compute_biased_statistic()
{
    /* split computations into three terms from JLMR paper (see documentation )*/
    index_t m=m_m;

    /* init kernel with features */
    m_kernel->init(m_p_and_q, m_p_and_q);

    /* first term */
    float64_t first=0;
    for (index_t i=0; i<m; ++i)
    {
        for (index_t j=0; j<m; ++j)
            first+=m_kernel->kernel(i,j);
    }
    first/=m;

    /* second term */
    float64_t second=0;
    for (index_t i=m_m; i<m_m+m; ++i)
    {
        for (index_t j=m_m; j<m_m+m; ++j)
            second+=m_kernel->kernel(i,j);
    }
    second/=m;

    /* third term */
    float64_t third=0;
    for (index_t i=0; i<m; ++i)
    {
        for (index_t j=m_m; j<m_m+m; ++j)
            third+=m_kernel->kernel(i,j);
    }
    third*=2.0/m;

    return first+second-third;
}

float64_t CQuadraticTimeMMD::compute_statistic()
{
    if (!m_kernel)
        SG_ERROR("%s::compute_statistic(): No kernel specified!\n", get_name());

    float64_t result=0;
    switch (m_statistic_type)
    {
    case UNBIASED:
        result=compute_unbiased_statistic();
        break;
    case BIASED:
        result=compute_biased_statistic();
        break;
    default:
        SG_ERROR("CQuadraticTimeMMD::compute_statistic(): Unknown statistic "
                "type!\n");
        break;
    }

    return result;
}

float64_t CQuadraticTimeMMD::compute_p_value(float64_t statistic)
{
    float64_t result=0;

    switch (m_null_approximation_method)
    {
    case MMD2_SPECTRUM:
    {
#ifdef HAVE_LAPACK
        /* get samples from null-distribution and compute p-value of statistic */
        SGVector<float64_t> null_samples=sample_null_spectrum(
                m_num_samples_spectrum, m_num_eigenvalues_spectrum);
        CMath::qsort(null_samples);
        index_t pos=CMath::find_position_to_insert(null_samples, statistic);
        result=1.0-((float64_t)pos)/null_samples.vlen;
#else // HAVE_LAPACK
        SG_ERROR("CQuadraticTimeMMD::compute_p_value(): Only possible if "
                "shogun is compiled with LAPACK enabled\n");
#endif // HAVE_LAPACK
        break;
    }

    case MMD2_GAMMA:
    {
        /* fit gamma and return cdf at statistic */
        SGVector<float64_t> params=fit_null_gamma();
        result=CStatistics::gamma_cdf(statistic, params[0], params[1]);
        break;
    }

    default:
        result=CKernelTwoSampleTestStatistic::compute_p_value(statistic);
        break;
    }

    return result;
}

float64_t CQuadraticTimeMMD::compute_threshold(float64_t alpha)
{
    float64_t result=0;

    switch (m_null_approximation_method)
    {
    case MMD2_SPECTRUM:
    {
#ifdef HAVE_LAPACK
        /* get samples from null-distribution and compute threshold */
        SGVector<float64_t> null_samples=sample_null_spectrum(
                m_num_samples_spectrum, m_num_eigenvalues_spectrum);
        CMath::qsort(null_samples);
        result=null_samples[CMath::floor(null_samples.vlen*(1-alpha))];
#else // HAVE_LAPACK
        SG_ERROR("CQuadraticTimeMMD::compute_threshold(): Only possible if "
                "shogun is compiled with LAPACK enabled\n");
#endif // HAVE_LAPACK
        break;
    }

    case MMD2_GAMMA:
    {
        /* fit gamma and return inverse cdf at alpha */
        SGVector<float64_t> params=fit_null_gamma();
        result=CStatistics::inverse_gamma_cdf(alpha, params[0], params[1]);
        break;
    }

    default:
        /* bootstrapping is handled here */
        result=CKernelTwoSampleTestStatistic::compute_threshold(alpha);
        break;
    }

    return result;
}


#ifdef HAVE_LAPACK
SGVector<float64_t> CQuadraticTimeMMD::sample_null_spectrum(index_t num_samples,
        index_t num_eigenvalues)
{
    if (m_m!=m_p_and_q->get_num_vectors()/2)
    {
        SG_ERROR("%s::sample_null_spectrum(): Currently, only equal "
                "sample sizes are supported\n", get_name());
    }

    if (num_samples<=2)
    {
        SG_ERROR("%s::sample_null_spectrum(): Number of samples has to be at"
                " least 2, better in the hundrets", get_name());
    }

    if (num_eigenvalues>2*m_m-1)
    {
        SG_ERROR("%s::sample_null_spectrum(): Number of Eigenvalues too large\n",
                get_name());
    }

    if (num_eigenvalues<1)
    {
        SG_ERROR("%s::sample_null_spectrum(): Number of Eigenvalues too small\n",
                get_name());
    }

    /* evtl. warn user not to use wrong statistic type */
    if (m_statistic_type!=BIASED)
    {
        SG_WARNING("%s::sample_null_spectrum(): Note: provided statistic has "
                "to be BIASED. Please ensure that! To get rid of warning,"
                "call %s::set_statistic_type(BIASED)\n", get_name(),
                get_name());
    }

    /* imaginary matrix K=[K KL; KL' L] (MATLAB notation)
     * K is matrix for XX, L is matrix for YY, KL is XY, LK is YX
     * works since X and Y are concatenated here */
    m_kernel->init(m_p_and_q, m_p_and_q);
    SGMatrix<float64_t> K=m_kernel->get_kernel_matrix();

    /* center matrix K=H*K*H */
    K.center();

    /* compute eigenvalues and select num_eigenvalues largest ones */
    SGVector<float64_t> eigenvalues=
            SGMatrix<float64_t>::compute_eigenvectors(K);
    SGVector<float64_t> largest_ev(num_eigenvalues);

    /* take largest EV, scale by 1/2/m on the fly and take abs value*/
    for (index_t i=0; i<num_eigenvalues; ++i)
        largest_ev[i]=CMath::abs(
                1.0/2/m_m*eigenvalues[eigenvalues.vlen-1-i]);

    /* finally, sample from null distribution */
    SGVector<float64_t> null_samples(num_samples);
    for (index_t i=0; i<num_samples; ++i)
    {
        /* 2*sum(kEigs.*(randn(length(kEigs),1)).^2); */
        null_samples[i]=0;
        for (index_t j=0; j<largest_ev.vlen; ++j)
            null_samples[i]+=largest_ev[j]*CMath::pow(CMath::randn_double(), 2);

        null_samples[i]*=2;
    }

    return null_samples;
}
#endif // HAVE_LAPACK

SGVector<float64_t> CQuadraticTimeMMD::fit_null_gamma()
{
    SG_WARNING("CQuadraticTimeMMD::fit_null_gamma(): TODO: combine with "
            "compute_statistic to be more efficient!\n");

    if (m_m!=m_p_and_q->get_num_vectors()/2)
    {
        SG_ERROR("%s::compute_p_value_gamma(): Currently, only equal "
                "sample sizes are supported\n", get_name());
    }

    /* evtl. warn user not to use wrong statistic type */
    if (m_statistic_type!=BIASED)
    {
        SG_WARNING("%s::compute_p_value(): Note: provided statistic has "
                "to be BIASED. Please ensure that! To get rid of warning,"
                "call %s::set_statistic_type(BIASED)\n", get_name(),
                get_name());
    }

    /* imaginary matrix K=[K KL; KL' L] (MATLAB notation)
     * K is matrix for XX, L is matrix for YY, KL is XY, LK is YX
     * works since X and Y are concatenated here */
    m_kernel->init(m_p_and_q, m_p_and_q);

    /* compute mean under H0 of MMD, which is
     * meanMMD  = 2/m * ( 1  - 1/m*sum(diag(KL))  );
     * in MATLAB.
     * Remove diagonals on the fly */
    float64_t mean_mmd=0;
    for (index_t i=0; i<m_m; ++i)
    {
        /* virtual KL matrix is in upper right corner of SHOGUN K matrix
         * so this sums the diagonal of the matrix between X and Y*/
        mean_mmd+=m_kernel->kernel(i, m_m+i);
    }
    mean_mmd=2.0/m_m*(1.0-1.0/m_m*mean_mmd);

    /* compute variance under H0 of MMD, which is
     * varMMD = 2/m/(m-1) * 1/m/(m-1) * sum(sum( (K + L - KL - KL').^2 ));
     * in MATLAB, so sum up all elements */
    float64_t var_mmd=0;
    for (index_t i=0; i<m_m; ++i)
    {
        for (index_t j=0; j<m_m; ++j)
        {
            /* dont add diagonal of all pairs of imaginary kernel matrices */
            if (i==j || m_m+i==j || m_m+j==i)
                continue;

            float64_t to_add=m_kernel->kernel(i, j);
            to_add+=m_kernel->kernel(m_m+i, m_m+j);
            to_add-=m_kernel->kernel(i, m_m+j);
            to_add-=m_kernel->kernel(m_m+i, j);
            var_mmd+=CMath::pow(to_add, 2);
        }
    }
    var_mmd*=2.0/m_m/(m_m-1)*1.0/m_m/(m_m-1);

    /* parameters for gamma distribution */
    float64_t a=CMath::pow(mean_mmd, 2)/var_mmd;
    float64_t b=var_mmd*m_m / mean_mmd;

    SGVector<float64_t> result(2);
    result[0]=a;
    result[1]=b;

    return result;
}

void CQuadraticTimeMMD::set_num_samples_sepctrum(index_t
        num_samples_spectrum)
{
    m_num_samples_spectrum=num_samples_spectrum;
}

void CQuadraticTimeMMD::set_num_eigenvalues_spectrum(
        index_t num_eigenvalues_spectrum)
{
    m_num_eigenvalues_spectrum=num_eigenvalues_spectrum;
}

void CQuadraticTimeMMD::set_statistic_type(EQuadraticMMDType
        statistic_type)
{
    m_statistic_type=statistic_type;
}