QuadraticTimeMMD.h

/*
 * Copyright (c) The Shogun Machine Learning Toolbox
 * Written (w) 2012-2013 Heiko Strathmann
 * All rights reserved.
 *
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 * modification, are permitted provided that the following conditions are met:
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 *    list of conditions and the following disclaimer.
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#ifndef __QUADRACTIMEMMD_H_
#define __QUADRACTIMEMMD_H_

#include <shogun/statistics/KernelTwoSampleTest.h>

namespace shogun
{

class CFeatures;
class CKernel;
class CCustomKernel;

/** Enum to select which statistic type of quadratic time MMD should be computed */
enum EQuadraticMMDType
{
	BIASED, UNBIASED
};

/** @brief This class implements the quadratic time Maximum Mean Statistic as
 * described in [1].
 * The MMD is the distance of two probability distributions \f$p\f$ and \f$q\f$
 * in a RKHS
 * \f[
 * \text{MMD}[\mathcal{F},p,q]^2=\textbf{E}_{x,x'}\left[ k(x,x')\right]-
 * 2\textbf{E}_{x,y}\left[ k(x,y)\right]
 * +\textbf{E}_{y,y'}\left[ k(y,y')\right]=||\mu_p - \mu_q||^2_\mathcal{F}
 * \f]
 *
 * Given two sets of samples \f$\{x_i\}_{i=1}^m\sim p\f$ and
 * \f$\{y_i\}_{i=1}^n\sim q\f$
 * the (unbiased) statistic is computed as
 *
 * \f[
 * \text{MMD}_u^2[\mathcal{F},X,Y]=\frac{1}{m(m-1)}\sum_{i=1}^m\sum_{j\neq i}^m
 * k(x_i,x_j) + \frac{1}{n(n-1)}\sum_{i=1}^n\sum_{j\neq i}^nk(y_i,y_j)
 * - \frac{2}{mn}\sum_{i=1}^m\sum_{j=1}^nk(x_i,y_j)
 * \f]
 *
 * A biased version is
 *
 * \f[
 * \text{MMD}_b^2[\mathcal{F},X,Y]=\frac{1}{m^2}\sum_{i=1}^m\sum_{j=1}^m
 * k(x_i,x_j) + \frac{1}{n^2}\sum_{i=1}^n\sum_{j=1}^nk(y_i,y_j) -
 * \frac{2}{mn}\sum_{i=1}^m\sum_{j=1}^nk(x_i,y_j)
 * \f]
 *
 * The type (biased/unbiased) can be selected via set_statistic_type().
 * Note that computing the statistic returns \f$m\text{MMD}^2\f$ when \f$m\f$
 * equals \f$n\f$, i.e. number of samples from both distributions are same,
 * and \f$(m+n)\text{MMD}^2\f$ in general case; same holds for the null
 * distribution samples.
 *
 * Along with the statistic comes a method to compute a p-value based on
 * different methods. Sampling from null is also possible. If unsure which one to
 * use, sampling with 250 permutation iterations always is correct (but slow).
 *
 * To choose, use set_null_approximation_method() and choose from.
 *
 * If you do not know about your data, but want to use the MMD from a kernel
 * matrix, just use the custom kernel constructor. Everything else will work as
 * usual.
 *
 * MMD2_SPECTRUM: for a fast, consistent test based on the spectrum of the kernel
 * matrix, as described in [2]. Only supported if LAPACK is installed.
 *
 * MMD2_GAMMA: for a very fast, but not consistent test based on moment matching
 * of a Gamma distribution, as described in [2].
 *
 * PERMUTATION: For permuting available samples to sample null-distribution
 *
 * For kernel selection see CMMDKernelSelection.
 *
 * [1]: Gretton, A., Borgwardt, K. M., Rasch, M. J., Schoelkopf, B., & Smola, A. (2012).
 * A Kernel Two-Sample Test. Journal of Machine Learning Research, 13, 671-721.
 *
 * [2]: Gretton, A., Fukumizu, K., & Harchaoui, Z. (2011).
 * A fast, consistent kernel two-sample test.
 *
 */
class CQuadraticTimeMMD : public CKernelTwoSampleTest
{
public:
	/** default constructor */
	CQuadraticTimeMMD();

	/** Constructor
	 *
	 * @param p_and_q feature data. Is assumed to contain samples from both
	 * p and q. First all samples from p, then from index m all
	 * samples from q
	 *
	 * @param kernel kernel to use
	 * @param p_and_q samples from p and q, appended
	 * @param m index of first sample of q
	 */
	CQuadraticTimeMMD(CKernel* kernel, CFeatures* p_and_q, index_t m);

	/** Constructor.
	 * This is a convienience constructor which copies both features to one
	 * element and then calls the other constructor. Needs twice the memory
	 * for a short time
	 *
	 * @param kernel kernel for MMD
	 * @param p samples from distribution p, will be copied and NOT
	 * SG_REF'ed
	 * @param q samples from distribution q, will be copied and NOT
	 * SG_REF'ed
	 */
	CQuadraticTimeMMD(CKernel* kernel, CFeatures* p, CFeatures* q);

	/** Constructor.
	 * This is a convienience constructor which copies allows to only specify
	 * a custom kernel. In this case, the features are completely ignored
	 * and all computations will be done on the custom kernel
	 *
	 * @param custom_kernel custom kernel for MMD, which is a kernel between
	 * the appended features p and q
	 * @param m index of first sample of q
	 */
	CQuadraticTimeMMD(CCustomKernel* custom_kernel, index_t m);

	/** destructor */
	virtual ~CQuadraticTimeMMD();

	/** Computes the squared quadratic time MMD for the current data. Note
	 * that the type (biased/unbiased) can be specified with
	 * set_statistic_type() method. Note that it returns m*MMD.
	 *
	 * @return (biased or unbiased) squared quadratic time MMD
	 */
	virtual float64_t compute_statistic();

	/** Same as compute_statistic(), but with the possibility to perform on
	 * multiple kernels at once
	 *
	 * @param multiple_kernels if true, and underlying kernel is K_COMBINED,
	 * method will be executed on all subkernels on the same data
	 * @return vector of results for subkernels
	 */
	virtual SGVector<float64_t> compute_statistic(bool multiple_kernels);

	/** computes a p-value based on current method for approximating the
	 * null-distribution. The p-value is the 1-p quantile of the null-
	 * distribution where the given statistic lies in.
	 *
	 * Not all methods for computing the p-value are compatible with all
	 * methods of computing the statistic (biased/unbiased).
	 *
	 * @param statistic statistic value to compute the p-value for
	 * @return p-value parameter statistic is the (1-p) percentile of the
	 * null distribution
	 */
	virtual float64_t compute_p_value(float64_t statistic);

	/** computes a threshold based on current method for approximating the
	 * null-distribution. The threshold is the value that a statistic has
	 * to have in ordner to reject the null-hypothesis.
	 *
	 * Not all methods for computing the p-value are compatible with all
	 * methods of computing the statistic (biased/unbiased).
	 *
	 * @param alpha test level to reject null-hypothesis
	 * @return threshold for statistics to reject null-hypothesis
	 */
	virtual float64_t compute_threshold(float64_t alpha);

	/** @return the class name */
	virtual const char* get_name() const
	{
		return "QuadraticTimeMMD";
	};

	/** returns the statistic type of this test statistic */
	virtual EStatisticType get_statistic_type() const
	{
		return S_QUADRATIC_TIME_MMD;
	}

#ifdef HAVE_EIGEN3
	/** Returns a set of samples of an estimate of the null distribution
	 * using the Eigen-spectrum of the centered kernel matrix of the merged
	 * samples of p and q. May be used to compute p_value (easy).
	 *
	 * The unbiased version uses
	 * \f[
	 *	t\text{MMD}_u^2[\mathcal{F},X,Y]\xrightarrow{D}\sum_{l=1}^\infty
	 *	\lambda_l\left((a_l\rho_x^{-\frac{1}{{2}}}
	 *	-b_l\rho_y^{-\frac{1}{{2}}})^2-(\rho_x\rho_y)^{-1} \right)
	 * \f]
	 * where \f$t=m+n\f$, \f$\lim_{m,n\rightarrow\infty}m/t\rightarrow
	 * \rho_x\f$ and \f$\rho_y\f$ likewise (equation 10 from [1]) and
	 * \f$\lambda_l\f$ are estimated as \f$\frac{\nu_l}{(m+n)}\f$, where
	 * \f$\nu_l\f$ are the eigenvalues of centered kernel matrix HKH.
	 *
	 * The biased version uses
	 * \f[
	 * 	t\text{MMD}_b^2[\mathcal{F},X,Y]\xrightarrow{D}\sum_{l=1}^\infty
	 *	\lambda_l\left((a_l\rho_x^{-\frac{1}{{2}}}-
	 *	b_l\rho_y^{-\frac{1}{{2}}})^2\right)
	 * \f]
	 *
	 * kernel matrix needs to be stored in memory
	 *
	 * Note that (m+n)*Null-distribution is returned,
	 * which is fine since the statistic is also (m+n)*MMD:
	 * except when m and n are equal, then m*MMD^2 is returned
	 *
	 * Works well if the kernel matrix is NOT diagonal dominant.
	 * See Gretton, A., Fukumizu, K., & Harchaoui, Z. (2011).
	 * A fast, consistent kernel two-sample test.
	 *
	 * @param num_samples number of samples to draw
	 * @param num_eigenvalues number of eigenvalues to use to draw samples
	 * Maximum number of m+n-1 where m and n are the sizes of samples from
	 * p and q respectively.
	 * It is usually safe to use a smaller number since they decay very
	 * fast, however, a conservative approach would be to use all (-1 does
	 * this). See paper for details.
	 * @return samples from the estimated null distribution
	 */
	SGVector<float64_t> sample_null_spectrum(index_t num_samples,
			index_t num_eigenvalues);
#endif // HAVE_EIGEN3

	/** setter for number of samples to use in spectrum based p-value
	 * computation.
	 *
	 * @param num_samples_spectrum number of samples to draw from
	 * approximate null-distributrion
	 */
	void set_num_samples_sepctrum(index_t num_samples_spectrum);

	/** setter for number of eigenvalues to use in spectrum based p-value
	 * computation. Maximum is m_m+m_n-1
	 *
	 * @param num_eigenvalues_spectrum number of eigenvalues to use to
	 * approximate null-distributrion
	 */
	void set_num_eigenvalues_spectrum(index_t num_eigenvalues_spectrum);

	/** @param statistic_type statistic type (biased/unbiased) to use */
	void set_statistic_type(EQuadraticMMDType statistic_type);

	/** Approximates the null-distribution by the two parameter gamma
	 * distribution. It works in O(m^2) where m is the number of samples
	 * from each distribution. Its very fast, but may be inaccurate.
	 * However, there are cases where it performs very well.
	 * Returns parameters of gamma distribution that is fitted.
	 *
	 * Called by compute_p_value() if null approximation method is set to
	 * MMD2_GAMMA.
	 *
	 * Note that when being used for constructing a test, the provided
	 * statistic HAS to be the biased version
	 * (see paper for details). Note that m*Null-distribution is fitted,
	 * which is fine since the statistic is also m*MMD.
	 *
	 * See Gretton, A., Fukumizu, K., & Harchaoui, Z. (2011).
	 * A fast, consistent kernel two-sample test.
	 *
	 * @return vector with two parameter for gamma distribution. To use:
	 * call gamma_cdf(statistic, a, b).
	 */
	SGVector<float64_t> fit_null_gamma();

protected:
	/** helper method to compute unbiased squared quadratic time MMD */
	virtual float64_t compute_unbiased_statistic();

	/** helper method to compute biased squared quadratic time MMD */
	virtual float64_t compute_biased_statistic();

private:
	/** register parameters and initialize with defaults */
	void init();

protected:
	/** number of samples for spectrum null-dstribution-approximation */
	index_t m_num_samples_spectrum;

	/** number of Eigenvalues for spectrum null-dstribution-approximation */
	index_t m_num_eigenvalues_spectrum;

	/** type of statistic (biased/unbiased) */
	EQuadraticMMDType m_statistic_type;
};

}

#endif /* __QUADRACTIMEMMD_H_ */