Detailed Description

This class implements the NOrmalized Cross Covariance Operator (NOCCO) based independence test as described in [1].

The test of independence is performed as follows: Given samples $Z=\{(x_i, y_i)\}_{i=1}^n$ from the joint distribution $\textbf{P}_{XY}$, does the joint distribution factorize as $\textbf{P}_{XY}=\textbf{P}_X \textbf{P}_Y$? The null hypothesis says yes and the alternative hypothesis says no.

The dependence of the random variables $\mathbf X=\{x_i\}$ and $ \mathbf Y=\{y_i\}$ can be measured via the cross-covariance operator $\boldsymbol\Sigma_{YX}$ which becomes $\mathbf{0}$ if and only if $\mathbf X$ and $\mathbf Y$ are independent. This term factorizes as $\boldsymbol\Sigma_{YX}=\boldsymbol\Sigma_{YY}^{\frac{1}{2}}\mathbf{V}_ {YX}\boldsymbol\Sigma_{XX}^{\frac{1}{2}}$, where $\boldsymbol\Sigma_ {XX}$ and $\boldsymbol\Sigma_{YY}$ are known as covariance operator and $\mathbf{V}_{YX}$ is known as normalized cross-covariance operator. The paper uses the Hilbert-Schmidt norm of $\mathbf V_{YX}$ as a dependence measure of the independence test (see paper for theroretical details).

This class overrides the compute_statistic() method of the superclass which computes an unbiased estimate of the normalized cross covariance operator norm. Given the kernels $K$ (for $\mathbf X$) and $L$ (for $\mathbf Y$), if we denote the doubly centered Gram matrices as $\mathbf{G}_X=\mathbf{HKH}$ and $\mathbf{G}_Y=\mathbf{HLH}$ (where $\mathbf H=\mathbf I-\frac{1}{n}\mathbf{1}$), then the operator norm is estimated as

\[ \hat{I}^{\text{NOCCO}}=\text{Trace}\left[\mathbf{R_X R_Y}\right] \]

where $\mathbf{R}_X=\mathbf{G}_X(\mathbf{G}_X+n\varepsilon_n\mathbf{I}) ^{-1}$ and $\mathbf{R}_Y=\mathbf{G}_Y(\mathbf{G}_Y+n\varepsilon_n \mathbf{I})^{-1}$ and $\varepsilon_n\gt 0$ is a regularization constant.

In order to avoid computing direct inverse in the above terms for avoiding numerical issues, this class uses Cholesky decomposition of matrices $\mathbf{GG}_*=\mathbf{LL}^\top$ (where $\mathbf{GG}_*=(\mathbf{G}_*+ n\varepsilon_n\mathbf{I})^{-1}$) and solve systems $\mathbf{GG}_* \mathbf x_i=\mathbf{LL}^\top\mathbf x_i=\mathbf e_i$ ( $\mathbf e_i$ being the $i^{\text{th}}$ column of $\mathbf I_n$) one by one. On the fly it then uses the solution vectors $\mathbf x_i$ to compute the matrix-matrix product $\mathbf C_*=\mathbf G_*\mathbf{GG}_*^{-1}$ using $\mathbf C_{*,(j,i)}=\mathbf G_{*,j}\cdot \mathbf x_i$, where $ G_{*,j}$ is the $j^{\text{th}}$ row of $\mathbf G_*$ (or column, since it is symmetric) and then discarding the vector. The final trace computation is also simplified using the symmetry of the matrices $ R_X $ and $ R_Y

Definition at line 109 of file NOCCO.h.

Inheritance diagram for CNOCCO:

[legend]

Public Member Functions
	CNOCCO ()

	CNOCCO (CKernel kernel_p, CKernel kernel_q, CFeatures p, CFeatures q)

virtual	~CNOCCO ()

virtual float64_t	compute_statistic ()

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

virtual void	set_p (CFeatures *p)

virtual void	set_q (CFeatures *q)

void	set_epsilon (float64_t epsilon)

float64_t	get_epsilon () const

virtual SGVector< float64_t >	sample_null ()

virtual void	set_kernel_p (CKernel *kernel_p)

virtual void	set_kernel_q (CKernel *kernel_q)

virtual CKernel *	get_kernel_p ()

virtual CKernel *	get_kernel_q ()

virtual CFeatures *	get_p ()

virtual CFeatures *	get_q ()

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 CSGObject *	shallow_copy () const

virtual CSGObject *	deep_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)

SGIO *	get_global_io ()

void	set_global_parallel (Parallel *parallel)

Parallel *	get_global_parallel ()

void	set_global_version (Version *version)

Version *	get_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 CSGObject *	clone ()

Public Attributes
SGIO *	io

Parallel *	parallel

Version *	version

Parameter *	m_parameters

Parameter *	m_model_selection_parameters

Parameter *	m_gradient_parameters

uint32_t	m_hash

Protected Member Functions
SGMatrix< float64_t >	compute_helper (SGMatrix< float64_t > m)

SGMatrix< float64_t >	get_kernel_matrix_K ()

SGMatrix< float64_t >	get_kernel_matrix_L ()

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
CKernel *	m_kernel_p

CKernel *	m_kernel_q

CFeatures *	m_p

CFeatures *	m_q

index_t	m_num_null_samples

ENullApproximationMethod	m_null_approximation_method

Constructor & Destructor Documentation

CNOCCO ( )

Constructor

Definition at line 44 of file NOCCO.cpp.

CNOCCO	(	CKernel *	kernel_p,
		CKernel *	kernel_q,
		CFeatures *	p,
		CFeatures *	q
	)

Constructor.

Initializes the kernels and features from the two distributions and SG_REFs them

Parameters

kernel_p	kernel to use on samples from p
kernel_q	kernel to use on samples from q
p	samples from distribution p
q	samples from distribution q

Definition at line 49 of file NOCCO.cpp.

~CNOCCO ( )

virtual

Destructor

Definition at line 66 of file NOCCO.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.

SGMatrix< float64_t > compute_helper ( SGMatrix< float64_t > m )

protected

Helper method which computes the matrix times matrix inverse using LLT solve (Cholesky) withoout storing the inverse (see class documentation).

Parameters

m	the centered Gram matrix

Returns: the result matrix of the multiplication

Definition at line 109 of file NOCCO.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.

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 CHypothesisTest.

Definition at line 189 of file NOCCO.cpp.

float64_t compute_statistic ( )

virtual

Computes the NOCCO statistic (see class description) for underlying kernels and data.

Note that since kernel matrices have to be stored, it has quadratic space costs.

Returns: unbiased estimate of NOCCO

Implements CHypothesisTest.

Definition at line 148 of file NOCCO.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.

Parameters

alpha test level to reject null-hypothesis

Returns: threshold for statistics to reject null-hypothesis

Reimplemented from CHypothesisTest.

Definition at line 208 of file NOCCO.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.

float64_t get_epsilon ( ) const

Returns: epsilon the regularization parameter

Definition at line 104 of file NOCCO.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.

SGMatrix< float64_t > get_kernel_matrix_K ( )

protectedinherited

Returns: kernel matrix on samples from p. Distinguishes CustomKernels

Definition at line 158 of file KernelIndependenceTest.cpp.

SGMatrix< float64_t > get_kernel_matrix_L ( )

protectedinherited

Returns: kernel matrix on samples from q. Distinguishes CustomKernels

Definition at line 184 of file KernelIndependenceTest.cpp.

CKernel * get_kernel_p ( )

virtualinherited

Getter for kernel for features from p, SG_REF'ed

Returns: kernel for features from p

Definition at line 146 of file KernelIndependenceTest.cpp.

CKernel * get_kernel_q ( )

virtualinherited

Getter for kernel for features from q, SG_REF'ed

Returns: kernel for features from q

Definition at line 152 of file KernelIndependenceTest.cpp.

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 CKernelIndependenceTest.

Definition at line 160 of file NOCCO.h.

CFeatures * get_p ( )

virtualinherited

Getter for features from p, SG_REF'ed

Returns: feature object from p

Definition at line 121 of file IndependenceTest.cpp.

CFeatures * get_q ( )

virtualinherited

Getter for features from q, SG_REF'ed

Returns: feature object from q

Definition at line 127 of file IndependenceTest.cpp.

virtual EStatisticType get_statistic_type ( ) const

virtual

Returns: the statistic type of this test statistic

Implements CHypothesisTest.

Definition at line 166 of file NOCCO.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.

Reimplemented in CKernel, CWeightedDegreePositionStringKernel, CList, CAlphabet, CLinearHMM, CGaussianKernel, CInverseMultiQuadricKernel, CCircularKernel, and CExponentialKernel.

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.

Reimplemented in CDynamicArray< T >, CDynamicArray< float64_t >, CDynamicArray< float32_t >, CDynamicArray< int32_t >, CDynamicArray< char >, CDynamicArray< bool >, and CDynamicObjectArray.

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 ( )

virtual

Merges both sets of samples and computes the test statistic m_num_null_sample times. This version precomputes the kenrel matrix once by hand, then samples using this one. The matrix has to be stored anyway when statistic is computed.

Returns: vector of all statistics

Reimplemented from CKernelIndependenceTest.

Definition at line 227 of file NOCCO.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.

Reimplemented in CKernel, CDynamicArray< T >, CDynamicArray< float64_t >, CDynamicArray< float32_t >, CDynamicArray< int32_t >, CDynamicArray< char >, CDynamicArray< bool >, and CDynamicObjectArray.

Definition at line 431 of file SGObject.cpp.

void set_epsilon ( float64_t epsilon )

Setter for regularization parameter epsilon

Parameters

epsilon the regularization parameter

Definition at line 99 of file NOCCO.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.

void set_kernel_p ( CKernel * kernel_p )

virtualinherited

Setter for kernel for features from distribution p, SG_REFs it

Parameters

kernel_p kernel for features from p

Definition at line 130 of file KernelIndependenceTest.cpp.

void set_kernel_q ( CKernel * kernel_q )

virtualinherited

Setter for kernel for features from distribution q, SG_REFs it

Parameters

kernel_q kernel for features from q

Definition at line 138 of file KernelIndependenceTest.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_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_p ( CFeatures * p )

virtual

Setter for features from distribution p, SG_REFs it

Parameters

p	features from p

Reimplemented from CIndependenceTest.

Definition at line 85 of file NOCCO.cpp.

void set_q ( CFeatures * q )

virtual

Setter for features from distribution q, SG_REFs it

Parameters

q	features from q

Reimplemented from CIndependenceTest.

Definition at line 92 of file NOCCO.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.

Parameter* m_gradient_parameters

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_p

protectedinherited

underlying kernel for p

Definition at line 137 of file KernelIndependenceTest.h.

CKernel* m_kernel_q

protectedinherited

underlying kernel for q

Definition at line 140 of file KernelIndependenceTest.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_null_samples

protectedinherited

number of iterations for sampling from null-distributions

Definition at line 174 of file HypothesisTest.h.

CFeatures* m_p

protectedinherited

samples of the distribution p

Definition at line 116 of file IndependenceTest.h.

Parameter* m_parameters

inherited

parameters

Definition at line 378 of file SGObject.h.

CFeatures* m_q

protectedinherited

samples of the distribution q

Definition at line 119 of file IndependenceTest.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:

Detailed Description

Public Member Functions

Public Attributes

Protected Member Functions

Protected Attributes

Constructor & Destructor Documentation

Member Function Documentation

Member Data Documentation