SHOGUN  4.1.0

## Detailed Description

This class implements the quadratic time Maximum Mean Statistic as described in [1]. The MMD is the distance of two probability distributions $$p$$ and $$q$$ in a RKHS which we denote by

$\hat{\eta_k}=\text{MMD}[\mathcal{F},p,q]^2=\textbf{E}_{x,x'} \left[ k(x,x')\right]-2\textbf{E}_{x,y}\left[ k(x,y)\right] +\textbf{E}_{y,y'}\left[ k(y,y')\right]=||\mu_p - \mu_q||^2_\mathcal{F}$

.

Given two sets of samples $$\{x_i\}_{i=1}^{n_x}\sim p$$ and $$\{y_i\}_{i=1}^{n_y}\sim q$$, $$n_x+n_y=n$$, the unbiased estimate of the above statistic is computed as

$\hat{\eta}_{k,U}=\frac{1}{n_x(n_x-1)}\sum_{i=1}^{n_x}\sum_{j\neq i} k(x_i,x_j)+\frac{1}{n_y(n_y-1)}\sum_{i=1}^{n_y}\sum_{j\neq i}k(y_i,y_j) -\frac{2}{n_xn_y}\sum_{i=1}^{n_x}\sum_{j=1}^{n_y}k(x_i,y_j)$

A biased version is

$\hat{\eta}_{k,V}=\frac{1}{n_x^2}\sum_{i=1}^{n_x}\sum_{j=1}^{n_x} k(x_i,x_j)+\frac{1}{n_y^2}\sum_{i=1}^{n_y}\sum_{j=1}^{n_y}k(y_i,y_j) -\frac{2}{n_xn_y}\sum_{i=1}^{n_x}\sum_{j=1}^{n_y}k(x_i,y_j)$

When $$n_x=n_y=\frac{n}{2}$$, an incomplete version can also be computed as the following

$\hat{\eta}_{k,U^-}=\frac{1}{\frac{n}{2}(\frac{n}{2}-1)}\sum_{i\neq j} h(z_i,z_j)$

where for each pair $$z=(x,y)$$, $$h(z,z')=k(x,x')+k(y,y')-k(x,y')- k(x',y)$$.

The type (biased/unbiased/incomplete) can be selected via set_statistic_type(). Note that there are presently two setups for computing statistic. While using BIASED, UNBIASED or INCOMPLETE, the estimate returned by compute_statistic() is $$\frac{n_xn_y}{n_x+n_y}\hat{\eta}_k$$. If DEPRECATED ones are used, then this returns $$(n_x+n_y)\hat{\eta}_k$$ in general and $$(\frac{n}{2}) \hat{\eta}_k$$ when $$n_x=n_y=\frac{n}{2}$$. This holds for the null distribution samples as well.

Estimating variance of the asymptotic distribution of the statistic under null and alternative hypothesis can be done using compute_variance() method. This is internally done alongwise computing statistics to avoid recomputing the kernel.

Variance under null is computed as $$\sigma_{k,0}^2=2\hat{\kappa}_2=2(\kappa_2-2\kappa_1+\kappa_0)$$ where $$\kappa_0=\left(\mathbb{E}_{X,X'}k(X,X')\right )^2$$, $$\kappa_1=\mathbb{E}_X\left[(\mathbb{E}_{X'}k(X,X'))^2\right]$$, and $$\kappa_2=\mathbb{E}_{X,X'}k^2(X,X')$$ and variance under alternative is computed as

$\sigma_{k,A}^2=4\rho_y\left\{\mathbb{E}_X\left[\left(\mathbb{E}_{X'} k(X,X')-\mathbb{E}_Yk(X,Y)\right)^2 \right ] -\left(\mathbb{E}_{X,X'} k(X,X')-\mathbb{E}_{X,Y}k(X,Y) \right)^2\right \}+4\rho_x\left\{ \mathbb{E}_Y\left[\left(\mathbb{E}_{Y'}k(Y,Y')-\mathbb{E}_Xk(X,Y) \right)^2\right ] -\left(\mathbb{E}_{Y,Y'}k(Y,Y')-\mathbb{E}_{X,Y} k(X,Y) \right)^2\right \}$

where $$\rho_x=\frac{n_x}{n}$$ and $$\rho_y=\frac{n_y}{n}$$.

Note that statistic and variance estimation can be done for multiple kernels at once as well.

Along with the statistic comes a method to compute a p-value based on different methods. Permutation test 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.

MMD2_SPECTRUM_DEPRECATED: For a fast, consistent test based on the spectrum of the kernel matrix, as described in [2]. Only supported if Eigen3 is installed.

MMD2_SPECTRUM: Similar to the deprecated version except it estimates the statistic under null as $$\frac{n_xn_y}{n_x+n_y}\hat{\eta}_{k,U}\rightarrow \sum_r\lambda_r(Z_r^2-1)$$ instead (see method description for more details).

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

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.

For kernel selection see CMMDKernelSelection.

NOTE: $$n_x$$ and $$n_y$$ are represented by $$m$$ and $$n$$, respectively in the implementation.

[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.

Definition at line 158 of file QuadraticTimeMMD.h.

[legend]

## Public Member Functions

CQuadraticTimeMMD (CKernel *kernel, CFeatures *p_and_q, index_t m)

CQuadraticTimeMMD (CKernel *kernel, CFeatures *p, CFeatures *q)

virtual float64_t compute_statistic ()

SGVector< float64_tcompute_statistic (bool multiple_kernels)

virtual SGVector< float64_tcompute_variance ()

SGMatrix< float64_tcompute_variance (bool multiple_kernels)

float64_t compute_variance_under_null ()

float64_t compute_variance_under_alternative ()

virtual float64_t compute_p_value (float64_t statistic)

virtual float64_t compute_threshold (float64_t alpha)

virtual const char * get_name () const

virtual EStatisticType get_statistic_type () const

SGVector< float64_tsample_null_spectrum (index_t num_samples, index_t num_eigenvalues)

SGVector< float64_tsample_null_spectrum_DEPRECATED (index_t num_samples, index_t num_eigenvalues)

void set_num_samples_spectrum (index_t num_samples_spectrum)

void set_num_eigenvalues_spectrum (index_t num_eigenvalues_spectrum)

SGVector< float64_tfit_null_gamma ()

virtual void set_kernel (CKernel *kernel)

virtual CKernelget_kernel ()

virtual SGVector< float64_tsample_null ()

virtual void set_p_and_q (CFeatures *p_and_q)

virtual CFeaturesget_p_and_q ()

void set_m (index_t m)

index_t get_m ()

virtual float64_t perform_test ()

bool perform_test (float64_t alpha)

virtual void set_num_null_samples (index_t num_null_samples)

virtual void set_null_approximation_method (ENullApproximationMethod null_approximation_method)

virtual CSGObjectshallow_copy () const

virtual CSGObjectdeep_copy () const

virtual bool is_generic (EPrimitiveType *generic) const

template<class T >
void set_generic ()

template<>
void set_generic ()

template<>
void set_generic ()

template<>
void set_generic ()

template<>
void set_generic ()

template<>
void set_generic ()

template<>
void set_generic ()

template<>
void set_generic ()

template<>
void set_generic ()

template<>
void set_generic ()

template<>
void set_generic ()

template<>
void set_generic ()

template<>
void set_generic ()

template<>
void set_generic ()

template<>
void set_generic ()

template<>
void set_generic ()

void unset_generic ()

virtual void print_serializable (const char *prefix="")

virtual bool save_serializable (CSerializableFile *file, const char *prefix="")

virtual bool load_serializable (CSerializableFile *file, const char *prefix="")

void set_global_io (SGIO *io)

SGIOget_global_io ()

void set_global_parallel (Parallel *parallel)

Parallelget_global_parallel ()

void set_global_version (Version *version)

Versionget_global_version ()

SGStringList< char > get_modelsel_names ()

void print_modsel_params ()

char * get_modsel_param_descr (const char *param_name)

index_t get_modsel_param_index (const char *param_name)

void build_gradient_parameter_dictionary (CMap< TParameter *, CSGObject * > *dict)

virtual void update_parameter_hash ()

virtual bool parameter_hash_changed ()

virtual bool equals (CSGObject *other, float64_t accuracy=0.0, bool tolerant=false)

virtual CSGObjectclone ()

## Public Attributes

SGIOio

Parallelparallel

Versionversion

Parameterm_parameters

Parameterm_model_selection_parameters

uint32_t m_hash

## Protected Member Functions

SGVector< float64_tcompute_unbiased_statistic_variance (int m, int n)

SGVector< float64_tcompute_biased_statistic_variance (int m, int n)

SGVector< float64_tcompute_incomplete_statistic_variance (int n)

float64_t compute_unbiased_statistic (int m, int n)

float64_t compute_biased_statistic (int m, int n)

float64_t compute_incomplete_statistic (int n)

virtual void load_serializable_pre () throw (ShogunException)

virtual void load_serializable_post () throw (ShogunException)

virtual void save_serializable_pre () throw (ShogunException)

virtual void save_serializable_post () throw (ShogunException)

## Protected Attributes

index_t m_num_samples_spectrum

index_t m_num_eigenvalues_spectrum

CKernelm_kernel

CFeaturesm_p_and_q

index_t m_m

index_t m_num_null_samples

ENullApproximationMethod m_null_approximation_method

## Constructor & Destructor Documentation

default constructor

Definition at line 48 of file QuadraticTimeMMD.cpp.

 CQuadraticTimeMMD ( CKernel * kernel, CFeatures * p_and_q, index_t m )

Constructor

Parameters
 p_and_q feature data. Is assumed to contain samples from both p and q. First m samples from p, then from index m all samples from q kernel kernel to use p_and_q samples from p and q, appended m index of first sample of q

Definition at line 53 of file QuadraticTimeMMD.cpp.

 CQuadraticTimeMMD ( CKernel * kernel, CFeatures * p, CFeatures * q )

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

Parameters
 kernel kernel for MMD p samples from distribution p, will be copied and NOT SG_REF'ed q samples from distribution q, will be copied and NOT SG_REF'ed

Definition at line 60 of file QuadraticTimeMMD.cpp.

 CQuadraticTimeMMD ( CCustomKernel * custom_kernel, index_t m )

Constructor. This is a convienience constructor which 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

Parameters
 custom_kernel custom kernel for MMD, which is a kernel between the appended features p and q m index of first sample of q

Definition at line 66 of file QuadraticTimeMMD.cpp.

virtual

destructor

Definition at line 72 of file QuadraticTimeMMD.cpp.

## Member Function Documentation

 void build_gradient_parameter_dictionary ( CMap< TParameter *, CSGObject * > * dict )
inherited

Builds a dictionary of all parameters in SGObject as well of those of SGObjects that are parameters of this object. Dictionary maps parameters to the objects that own them.

Parameters
 dict dictionary of parameters to be built.

Definition at line 597 of file SGObject.cpp.

 CSGObject * clone ( )
virtualinherited

Creates a clone of the current object. This is done via recursively traversing all parameters, which corresponds to a deep copy. Calling equals on the cloned object always returns true although none of the memory of both objects overlaps.

Returns
an identical copy of the given object, which is disjoint in memory. NULL if the clone fails. Note that the returned object is SG_REF'ed

Definition at line 714 of file SGObject.cpp.

 float64_t compute_biased_statistic ( int m, int n )
protected

Wrapper method for computing biased estimate of MMD^2

Parameters
 m number of samples from p n number of samples from q
Returns
biased $$\text{MMD}^2$$ estimate $$\hat{\eta}_{k,V}$$

Definition at line 536 of file QuadraticTimeMMD.cpp.

 SGVector< float64_t > compute_biased_statistic_variance ( int m, int n )
protected

Helper method to compute biased estimate of squared quadratic time MMD and variance estimate under null and alternative hypothesis

Parameters
 m number of samples from p n number of samples from q
Returns
a vector of three values first - biased $$\text{MMD}^2$$ estimate $$\hat{\eta}_{k,V}$$ second - variance under null hypothesis (see class documentation) third - variance under alternative hypothesis (see class documentation)

Definition at line 239 of file QuadraticTimeMMD.cpp.

 float64_t compute_incomplete_statistic ( int n )
protected

Wrapper method for computing incomplete estimate of MMD^2

Parameters
 n number of samples from p and q
Returns
incomplete $$\text{MMD}^2$$ estimate $$\hat{\eta}_{k,U^-}$$

Definition at line 541 of file QuadraticTimeMMD.cpp.

 SGVector< float64_t > compute_incomplete_statistic_variance ( int n )
protected

Helper method to compute incomplete estimate of squared quadratic time MMD and variance estimate under null and alternative hypothesis

Parameters
 n number of samples from p and q
Returns
a vector of three values first - incomplete $$\text{MMD}^2$$ estimate $$\hat{\eta}_{k,U^-}$$ second - variance under null hypothesis (see class documentation) third - variance under alternative hypothesis (see class documentation)

Definition at line 385 of file QuadraticTimeMMD.cpp.

 float64_t compute_p_value ( float64_t statistic )
virtual

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/incomplete).

Parameters
 statistic statistic value to compute the p-value for
Returns
p-value parameter statistic is the (1-p) percentile of the null distribution

Reimplemented from CTwoSampleTest.

Definition at line 749 of file QuadraticTimeMMD.cpp.

 float64_t compute_statistic ( )
virtual

Computes the squared quadratic time MMD for the current data. Note that the type (biased/unbiased/incomplete) can be specified with set_statistic_type() method.

Returns
(biased, unbiased or incomplete) $$\frac{mn}{m+n}\hat{\eta}_k$$. If DEPRECATED types are used, then it returns $$(m+m)\hat{\eta}_k$$ in general and $$m\hat{\eta}_k$$ when $$m=n$$.

Implements CKernelTwoSampleTest.

Definition at line 546 of file QuadraticTimeMMD.cpp.

 SGVector< float64_t > compute_statistic ( bool multiple_kernels )
virtual

Same as compute_statistic(), but with the possibility to perform on multiple kernels at once

Parameters
 multiple_kernels if true, and underlying kernel is K_COMBINED, method will be executed on all subkernels on the same data
Returns
vector of results for subkernels

Implements CKernelTwoSampleTest.

Definition at line 663 of file QuadraticTimeMMD.cpp.

 float64_t compute_threshold ( float64_t alpha )
virtual

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/incomplete).

Parameters
 alpha test level to reject null-hypothesis
Returns
threshold for statistics to reject null-hypothesis

Reimplemented from CTwoSampleTest.

Definition at line 801 of file QuadraticTimeMMD.cpp.

 float64_t compute_unbiased_statistic ( int m, int n )
protected

Wrapper method for computing unbiased estimate of MMD^2

Parameters
 m number of samples from p n number of samples from q
Returns
unbiased $$\text{MMD}^2$$ estimate $$\hat{\eta}_{k,U}$$

Definition at line 531 of file QuadraticTimeMMD.cpp.

 SGVector< float64_t > compute_unbiased_statistic_variance ( int m, int n )
protected

Helper method to compute unbiased estimate of squared quadratic time MMD and variance estimate under null and alternative hypothesis

Parameters
 m number of samples from p n number of samples from q
Returns
a vector of three values first - unbiased $$\text{MMD}^2$$ estimate $$\hat{\eta}_{k,U}$$ second - variance under null hypothesis (see class documentation) third - variance under alternative hypothesis (see class documentation)

Definition at line 92 of file QuadraticTimeMMD.cpp.

 SGVector< float64_t > compute_variance ( )
virtual

Wrapper for computing variance estimate of the asymptotic distribution of the statistic (unbisaed/biased/incomplete) under null and alternative hypothesis (see class description for details)

Returns
a vector of two values containing asymptotic variance estimate under null and alternative, respectively

Definition at line 598 of file QuadraticTimeMMD.cpp.

 SGMatrix< float64_t > compute_variance ( bool multiple_kernels )

Same as compute_variance(), but with the possibility to perform on multiple kernels at once

Parameters
 multiple_kernels if true, and underlying kernel is K_COMBINED, method will be executed on all subkernels on the same data
Returns
matrix of results for subkernels, one row for each subkernel

Definition at line 704 of file QuadraticTimeMMD.cpp.

 float64_t compute_variance_under_alternative ( )

Wrapper method for compute_variance()

Returns
variance estimation of asymptotic distribution of statistic under alternative hypothesis

Definition at line 658 of file QuadraticTimeMMD.cpp.

 float64_t compute_variance_under_null ( )

Wrapper method for compute_variance()

Returns
variance estimation of asymptotic distribution of statistic under null hypothesis

Definition at line 653 of file QuadraticTimeMMD.cpp.

 CSGObject * deep_copy ( ) const
virtualinherited

A deep copy. All the instance variables will also be copied.

Definition at line 198 of file SGObject.cpp.

 bool equals ( CSGObject * other, float64_t accuracy = 0.0, bool tolerant = false )
virtualinherited

Recursively compares the current SGObject to another one. Compares all registered numerical parameters, recursion upon complex (SGObject) parameters. Does not compare pointers!

May be overwritten but please do with care! Should not be necessary in most cases.

Parameters
 other object to compare with accuracy accuracy to use for comparison (optional) tolerant allows linient check on float equality (within accuracy)
Returns
true if all parameters were equal, false if not

Definition at line 618 of file SGObject.cpp.

 SGVector< float64_t > fit_null_gamma ( )

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). To use, set BIASED_DEPRECATED as statistic type. 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.

Returns
vector with two parameter for gamma distribution. To use: call gamma_cdf(statistic, a, b).

Definition at line 1032 of file QuadraticTimeMMD.cpp.

 SGIO * get_global_io ( )
inherited

get the io object

Returns
io object

Definition at line 235 of file SGObject.cpp.

 Parallel * get_global_parallel ( )
inherited

get the parallel object

Returns
parallel object

Definition at line 277 of file SGObject.cpp.

 Version * get_global_version ( )
inherited

get the version object

Returns
version object

Definition at line 290 of file SGObject.cpp.

 virtual CKernel* get_kernel ( )
virtualinherited
Returns
underlying kernel, is SG_REF'ed

Definition at line 86 of file KernelTwoSampleTest.h.

 index_t get_m ( )
inherited
Returns
number of to be used samples m

Definition at line 127 of file TwoSampleTest.h.

 SGStringList< char > get_modelsel_names ( )
inherited
Returns
vector of names of all parameters which are registered for model selection

Definition at line 498 of file SGObject.cpp.

 char * get_modsel_param_descr ( const char * param_name )
inherited

Returns description of a given parameter string, if it exists. SG_ERROR otherwise

Parameters
 param_name name of the parameter
Returns
description of the parameter

Definition at line 522 of file SGObject.cpp.

 index_t get_modsel_param_index ( const char * param_name )
inherited

Returns index of model selection parameter with provided index

Parameters
 param_name name of model selection parameter
Returns
index of model selection parameter with provided name, -1 if there is no such

Definition at line 535 of file SGObject.cpp.

 virtual const char* get_name ( ) const
virtual
Returns
the class name

Implements CKernelTwoSampleTest.

Definition at line 280 of file QuadraticTimeMMD.h.

 CFeatures * get_p_and_q ( )
virtualinherited

Getter for joint features, SG_REF'ed

Returns
joint feature object

Reimplemented in CStreamingMMD.

Definition at line 171 of file TwoSampleTest.cpp.

 virtual EStatisticType get_statistic_type ( ) const
virtual

returns the statistic type of this test statistic

Implements CHypothesisTest.

Definition at line 286 of file QuadraticTimeMMD.h.

 bool is_generic ( EPrimitiveType * generic ) const
virtualinherited

If the SGSerializable is a class template then TRUE will be returned and GENERIC is set to the type of the generic.

Parameters
 generic set to the type of the generic if returning TRUE
Returns
TRUE if a class template.

Definition at line 296 of file SGObject.cpp.

 bool load_serializable ( CSerializableFile * file, const char * prefix = "" )
virtualinherited

Load this object from file. If it will fail (returning FALSE) then this object will contain inconsistent data and should not be used!

Parameters
 file where to load from prefix prefix for members
Returns
TRUE if done, otherwise FALSE

Definition at line 369 of file SGObject.cpp.

 void load_serializable_post ( ) throw ( ShogunException )
protectedvirtualinherited

Can (optionally) be overridden to post-initialize some member variables which are not PARAMETER::ADD'ed. Make sure that at first the overridden method BASE_CLASS::LOAD_SERIALIZABLE_POST is called.

Exceptions
 ShogunException will be thrown if an error occurs.

Definition at line 426 of file SGObject.cpp.

 void load_serializable_pre ( ) throw ( ShogunException )
protectedvirtualinherited

Can (optionally) be overridden to pre-initialize some member variables which are not PARAMETER::ADD'ed. Make sure that at first the overridden method BASE_CLASS::LOAD_SERIALIZABLE_PRE is called.

Exceptions
 ShogunException will be thrown if an error occurs.

Definition at line 421 of file SGObject.cpp.

 bool parameter_hash_changed ( )
virtualinherited
Returns
whether parameter combination has changed since last update

Definition at line 262 of file SGObject.cpp.

 float64_t perform_test ( )
virtualinherited

Performs the complete two-sample test on current data and returns a p-value.

This is a wrapper that calls compute_statistic first and then calls compute_p_value using the obtained statistic. In some statistic classes, it might be possible to compute statistic and p-value in one single run which is more efficient. Therefore, this method might be overwritten in subclasses.

The method for computing the p-value can be set via set_null_approximation_method().

Returns
p-value such that computed statistic is the (1-p) quantile of the estimated null distribution

Reimplemented in CStreamingMMD.

Definition at line 113 of file HypothesisTest.cpp.

 bool perform_test ( float64_t alpha )
inherited

Performs the complete two-sample test on current data and returns a binary answer wheter null hypothesis is rejected or not.

This is just a wrapper for the above perform_test() method that returns a p-value. If this p-value lies below the test level alpha, the null hypothesis is rejected.

Should not be overwritten in subclasses. (Therefore not virtual)

Parameters
 alpha test level alpha.
Returns
true if null hypothesis is rejected and false otherwise

Definition at line 121 of file HypothesisTest.cpp.

 void print_modsel_params ( )
inherited

prints all parameter registered for model selection and their type

Definition at line 474 of file SGObject.cpp.

 void print_serializable ( const char * prefix = "" )
virtualinherited

prints registered parameters out

Parameters
 prefix prefix for members

Definition at line 308 of file SGObject.cpp.

 SGVector< float64_t > sample_null ( )
virtualinherited

merges both sets of samples and computes the test statistic m_num_null_samples times. This version checks if a precomputed custom kernel is used, and, if so, just permutes it instead of re- computing it in every iteration.

Returns
vector of all statistics

Reimplemented from CTwoSampleTest.

Reimplemented in CStreamingMMD.

Definition at line 55 of file KernelTwoSampleTest.cpp.

 SGVector< float64_t > sample_null_spectrum ( index_t num_samples, index_t num_eigenvalues )

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 estimate is computed as

$\frac{n_xn_y}{n_x+n_y}\hat{\eta}_{k,U}\rightarrow\sum_{l=1}^\infty \lambda_l\left(Z^2_l-1 \right)$

where $${Z_l}\stackrel{i.i.d.}{\sim}\mathcal{N}(0,1)$$ and $$\lambda_l$$ are the eigenvalues of centered kernel matrix HKH.

kernel matrix needs to be stored in memory

Note that m*n/(m+n)*Null-distribution is returned, which is fine since the statistic is also m*n/(m+n)*MMD^2

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.

Parameters
 num_samples number of samples to draw 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.
Returns
samples from the estimated null distribution

Definition at line 854 of file QuadraticTimeMMD.cpp.

 SGVector< float64_t > sample_null_spectrum_DEPRECATED ( index_t num_samples, index_t num_eigenvalues )

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

$t\text{MMD}_u^2[\mathcal{F},X,Y]\rightarrow\sum_{l=1}^\infty \lambda_l\left((a_l\rho_x^{-\frac{1}{{2}}} -b_l\rho_y^{-\frac{1}{{2}}})^2-(\rho_x\rho_y)^{-1} \right)$

where $$t=m+n$$, $$\lim_{m,n\rightarrow\infty}m/t\rightarrow \rho_x$$ and $$\rho_y$$ likewise (equation 10 from [1]) and $$\lambda_l$$ are estimated as $$\frac{\nu_l}{(m+n)}$$, where $$\nu_l$$ are the eigenvalues of centered kernel matrix HKH.

The biased version uses

$t\text{MMD}_b^2[\mathcal{F},X,Y]\rightarrow\sum_{l=1}^\infty \lambda_l\left((a_l\rho_x^{-\frac{1}{{2}}}- b_l\rho_y^{-\frac{1}{{2}}})^2\right)$

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.

Parameters
 num_samples number of samples to draw 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.
Returns
samples from the estimated null distribution

Definition at line 933 of file QuadraticTimeMMD.cpp.

 bool save_serializable ( CSerializableFile * file, const char * prefix = "" )
virtualinherited

Save this object to file.

Parameters
 file where to save the object; will be closed during returning if PREFIX is an empty string. prefix prefix for members
Returns
TRUE if done, otherwise FALSE

Definition at line 314 of file SGObject.cpp.

 void save_serializable_post ( ) throw ( ShogunException )
protectedvirtualinherited

Can (optionally) be overridden to post-initialize some member variables which are not PARAMETER::ADD'ed. Make sure that at first the overridden method BASE_CLASS::SAVE_SERIALIZABLE_POST is called.

Exceptions
 ShogunException will be thrown if an error occurs.

Reimplemented in CKernel.

Definition at line 436 of file SGObject.cpp.

 void save_serializable_pre ( ) throw ( ShogunException )
protectedvirtualinherited

Can (optionally) be overridden to pre-initialize some member variables which are not PARAMETER::ADD'ed. Make sure that at first the overridden method BASE_CLASS::SAVE_SERIALIZABLE_PRE is called.

Exceptions
 ShogunException will be thrown if an error occurs.

Definition at line 431 of file SGObject.cpp.

 void set_generic ( )
inherited

Definition at line 41 of file SGObject.cpp.

 void set_generic ( )
inherited

Definition at line 46 of file SGObject.cpp.

 void set_generic ( )
inherited

Definition at line 51 of file SGObject.cpp.

 void set_generic ( )
inherited

Definition at line 56 of file SGObject.cpp.

 void set_generic ( )
inherited

Definition at line 61 of file SGObject.cpp.

 void set_generic ( )
inherited

Definition at line 66 of file SGObject.cpp.

 void set_generic ( )
inherited

Definition at line 71 of file SGObject.cpp.

 void set_generic ( )
inherited

Definition at line 76 of file SGObject.cpp.

 void set_generic ( )
inherited

Definition at line 81 of file SGObject.cpp.

 void set_generic ( )
inherited

Definition at line 86 of file SGObject.cpp.

 void set_generic ( )
inherited

Definition at line 91 of file SGObject.cpp.

 void set_generic ( )
inherited

Definition at line 96 of file SGObject.cpp.

 void set_generic ( )
inherited

Definition at line 101 of file SGObject.cpp.

 void set_generic ( )
inherited

Definition at line 106 of file SGObject.cpp.

 void set_generic ( )
inherited

Definition at line 111 of file SGObject.cpp.

 void set_generic ( )
inherited

set generic type to T

 void set_global_io ( SGIO * io )
inherited

set the io object

Parameters
 io io object to use

Definition at line 228 of file SGObject.cpp.

 void set_global_parallel ( Parallel * parallel )
inherited

set the parallel object

Parameters
 parallel parallel object to use

Definition at line 241 of file SGObject.cpp.

 void set_global_version ( Version * version )
inherited

set the version object

Parameters
 version version object to use

Definition at line 283 of file SGObject.cpp.

 virtual void set_kernel ( CKernel * kernel )
virtualinherited

Setter for the underlying kernel

Parameters
 kernel new kernel to use

Definition at line 77 of file KernelTwoSampleTest.h.

 void set_m ( index_t m )
inherited
Parameters
 m number of samples from first distribution p

Definition at line 162 of file TwoSampleTest.cpp.

 void set_null_approximation_method ( ENullApproximationMethod null_approximation_method )
virtualinherited

sets the method how to approximate the null-distribution

Parameters
 null_approximation_method method to use

Definition at line 61 of file HypothesisTest.cpp.

 void set_num_eigenvalues_spectrum ( index_t num_eigenvalues_spectrum )

setter for number of eigenvalues to use in spectrum based p-value computation. Maximum is m_m+m_n-1

Parameters
 num_eigenvalues_spectrum number of eigenvalues to use to approximate null-distributrion

Definition at line 1124 of file QuadraticTimeMMD.cpp.

 void set_num_null_samples ( index_t num_null_samples )
virtualinherited

sets the number of permutation iterations for sample_null()

Parameters
 num_null_samples how often permutation shall be done

Definition at line 67 of file HypothesisTest.cpp.

 void set_num_samples_spectrum ( index_t num_samples_spectrum )

setter for number of samples to use in spectrum based p-value computation.

Parameters
 num_samples_spectrum number of samples to draw from approximate null-distributrion

Definition at line 1118 of file QuadraticTimeMMD.cpp.

 void set_p_and_q ( CFeatures * p_and_q )
virtualinherited

Setter for joint features

Parameters
 p_and_q joint features from p and q to set

Reimplemented in CStreamingMMD.

Definition at line 154 of file TwoSampleTest.cpp.

 void set_statistic_type ( EQuadraticMMDType statistic_type )
Parameters
 statistic_type statistic type (biased/unbiased/incomplete) to use

Definition at line 1130 of file QuadraticTimeMMD.cpp.

 CSGObject * shallow_copy ( ) const
virtualinherited

A shallow copy. All the SGObject instance variables will be simply assigned and SG_REF-ed.

Reimplemented in CGaussianKernel.

Definition at line 192 of file SGObject.cpp.

 void unset_generic ( )
inherited

unset generic type

this has to be called in classes specializing a template class

Definition at line 303 of file SGObject.cpp.

 void update_parameter_hash ( )
virtualinherited

Updates the hash of current parameter combination

Definition at line 248 of file SGObject.cpp.

## Member Data Documentation

 SGIO* io
inherited

io

Definition at line 369 of file SGObject.h.

inherited

parameters wrt which we can compute gradients

Definition at line 384 of file SGObject.h.

 uint32_t m_hash
inherited

Hash of parameter values

Definition at line 387 of file SGObject.h.

 CKernel* m_kernel
protectedinherited

underlying kernel

Definition at line 121 of file KernelTwoSampleTest.h.

 index_t m_m
protectedinherited

defines the first index of samples of q

Definition at line 139 of file TwoSampleTest.h.

 Parameter* m_model_selection_parameters
inherited

model selection parameters

Definition at line 381 of file SGObject.h.

 ENullApproximationMethod m_null_approximation_method
protectedinherited

Defines how the the null distribution is approximated

Definition at line 177 of file HypothesisTest.h.

 index_t m_num_eigenvalues_spectrum
protected

number of Eigenvalues for spectrum null-dstribution-approximation

Definition at line 479 of file QuadraticTimeMMD.h.

 index_t m_num_null_samples
protectedinherited

number of iterations for sampling from null-distributions

Definition at line 174 of file HypothesisTest.h.

 index_t m_num_samples_spectrum
protected

number of samples for spectrum null-dstribution-approximation

Definition at line 476 of file QuadraticTimeMMD.h.

 CFeatures* m_p_and_q
protectedinherited

concatenated samples of the two distributions (two blocks)

Definition at line 136 of file TwoSampleTest.h.

 Parameter* m_parameters
inherited

parameters

Definition at line 378 of file SGObject.h.

protected

type of statistic (biased/unbiased/incomplete as well as deprecated versions of biased/unbiased)

Definition at line 484 of file QuadraticTimeMMD.h.

 Parallel* parallel
inherited

parallel

Definition at line 372 of file SGObject.h.

 Version* version
inherited

version

Definition at line 375 of file SGObject.h.

The documentation for this class was generated from the following files:

SHOGUN Machine Learning Toolbox - Documentation