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CLinearTimeMMD Class Reference

Detailed Description

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

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

.

Given two sets of samples $\{x_i\}_{i=1}^m\sim p$ and $\{y_i\}_{i=1}^n\sim q$ the (unbiased) statistic is computed as

\[ \text{MMD}_l^2[\mathcal{F},X,Y]=\frac{1}{m_2}\sum_{i=1}^{m_2} h(z_{2i},z_{2i+1}) \]

where

\[ h(z_{2i},z_{2i+1})=k(x_{2i},x_{2i+1})+k(y_{2i},y_{2i+1})-k(x_{2i},y_{2i+1})- k(x_{2i+1},y_{2i}) \]

and $ m_2=\lfloor\frac{m}{2} \rfloor$.

Along with the statistic comes a method to compute a p-value based on a Gaussian approximation of the null-distribution which is also possible in linear time and constant space. Bootstrapping, is also possible. If unsure which one to use, bootstrapping with 250 iterations always is correct (but slow). When the sample size is large (>1000) at least, the Gaussian approximation is an accurate and much faster choice than bootstrapping.

To choose, use set_null_approximation_method() and choose from

MMD1_GAUSSIAN: Approximates the null-distribution with a Gaussian. Only use from at least 1000 samples.

BOOTSTRAPPING: For permuting available samples to sample null-distribution

Comes with a method for selecting kernel weights, if a combined kernel on combined features is used. See optimize_kernel_weights(). See [2]

A very basic method for kernel selection when using CGaussianKernel is to use the median distance of the underlying data. See examples how to do that. More advanced methods will follow in the near future. However, the median heuristic works in quite some cases. See [1].

[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]: TODO, not yet published

Definition at line 74 of file LinearTimeMMD.h.

Inheritance diagram for CLinearTimeMMD:
Inheritance graph
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Public Member Functions

 CLinearTimeMMD ()
 CLinearTimeMMD (CKernel *kernel, CFeatures *p_and_q, index_t q_start)
 CLinearTimeMMD (CKernel *kernel, CFeatures *p, CFeatures *q)
virtual ~CLinearTimeMMD ()
virtual float64_t compute_statistic ()
virtual float64_t compute_p_value (float64_t statistic)
virtual float64_t compute_threshold (float64_t alpha)
virtual float64_t compute_variance_estimate ()
virtual void optimize_kernel_weights ()
void set_opt_max_iterations (index_t opt_max_iterations)
void set_opt_epsilon (float64_t opt_epsilon)
void set_opt_low_cut (float64_t opt_low_cut)
void set_opt_regularization_eps (float64_t opt_regularization_eps)
virtual const char * get_name () const
virtual SGVector< float64_t > bootstrap_null ()
virtual void set_bootstrap_iterations (index_t bootstrap_iterations)
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 ()
void unset_generic ()
virtual void print_serializable (const char *prefix="")
virtual bool save_serializable (CSerializableFile *file, const char *prefix="", int32_t param_version=VERSION_PARAMETER)
virtual bool load_serializable (CSerializableFile *file, const char *prefix="", int32_t param_version=VERSION_PARAMETER)
DynArray< TParameter * > * load_file_parameters (const SGParamInfo *param_info, int32_t file_version, CSerializableFile *file, const char *prefix="")
DynArray< TParameter * > * load_all_file_parameters (int32_t file_version, int32_t current_version, CSerializableFile *file, const char *prefix="")
void map_parameters (DynArray< TParameter * > *param_base, int32_t &base_version, DynArray< const SGParamInfo * > *target_param_infos)
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_parameter_dictionary (CMap< TParameter *, CSGObject * > &dict)

Static Public Member Functions

static const float64_t * get_Q_col (uint32_t i)
static void print_state (libqp_state_T state)

Public Attributes

SGIO * io
Parallel * parallel
Version * version
Parameter * m_parameters
Parameter * m_model_selection_parameters
ParameterMap * m_parameter_map
uint32_t m_hash

Protected Member Functions

virtual TParameter * migrate (DynArray< TParameter * > *param_base, const SGParamInfo *target)
virtual void one_to_one_migration_prepare (DynArray< TParameter * > *param_base, const SGParamInfo *target, TParameter *&replacement, TParameter *&to_migrate, char *old_name=NULL)
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)
virtual bool update_parameter_hash ()

Protected Attributes

index_t m_opt_max_iterations
float64_t m_opt_epsilon
float64_t m_opt_low_cut
float64_t m_opt_regularization_eps
CKernel * m_kernel
CFeatures * m_p_and_q
index_t m_q_start
index_t m_bootstrap_iterations
ENullApproximationMethod m_null_approximation_method

Static Protected Attributes

static SGMatrix< float64_t > m_Q = SGMatrix<float64_t>()

Constructor & Destructor Documentation

CLinearTimeMMD ( )

Definition at line 20 of file LinearTimeMMD.cpp.

CLinearTimeMMD ( CKernel *  kernel,
CFeatures *  p_and_q,
index_t  q_start 
)

Constructor

Parameters
p_and_qfeature data. Is assumed to contain samples from both p and q. First all samples from p, then from index q_start all samples from q
kernelkernel to use
p_and_qsamples from p and q, appended
q_startindex of first sample of q

Definition at line 26 of file LinearTimeMMD.cpp.

CLinearTimeMMD ( 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
kernelkernel for MMD
psamples from distribution p, will be copied and NOT SG_REF'ed
qsamples from distribution q, will be copied and NOT SG_REF'ed

Definition at line 39 of file LinearTimeMMD.cpp.

~CLinearTimeMMD ( )
virtual

Definition at line 51 of file LinearTimeMMD.cpp.

Member Function Documentation

SGVector< float64_t > bootstrap_null ( )
virtualinherited

merges both sets of samples and computes the test statistic m_bootstrap_iteration 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 CTwoDistributionsTestStatistic.

Definition at line 60 of file KernelTwoSampleTestStatistic.cpp.

void build_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
dictdictionary of parameters to be built.

Definition at line 1204 of file SGObject.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.

The method for computing the p-value can be set via set_null_approximation_method(). Since the null- distribution is normal, a Gaussian approximation is available.

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

Reimplemented from CTwoDistributionsTestStatistic.

Definition at line 112 of file LinearTimeMMD.cpp.

float64_t compute_statistic ( )
virtual

Computes the squared linear time MMD for the current data. This is an unbiased estimate

Returns
squared linear time MMD

Implements CTestStatistic.

Definition at line 73 of file LinearTimeMMD.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.

The method for computing the p-value can be set via set_null_approximation_method(). Since the null- distribution is normal, a Gaussian approximation is available.

Parameters
alphatest level to reject null-hypothesis
Returns
threshold for statistics to reject null-hypothesis

Reimplemented from CTwoDistributionsTestStatistic.

Definition at line 135 of file LinearTimeMMD.cpp.

float64_t compute_variance_estimate ( )
virtual

computes a linear time estimate of the variance of the squared linear time mmd, which may be used for an approximation of the null-distribution The value is the variance of the vector of which the linear time MMD is the mean.

Returns
variance estimate

Definition at line 158 of file LinearTimeMMD.cpp.

virtual CSGObject* deep_copy ( ) const
virtualinherited

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

Definition at line 131 of file SGObject.h.

SGIO * get_global_io ( )
inherited

get the io object

Returns
io object

Definition at line 224 of file SGObject.cpp.

Parallel * get_global_parallel ( )
inherited

get the parallel object

Returns
parallel object

Definition at line 259 of file SGObject.cpp.

Version * get_global_version ( )
inherited

get the version object

Returns
version object

Definition at line 272 of file SGObject.cpp.

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

Definition at line 1108 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_namename of the parameter
Returns
description of the parameter

Definition at line 1132 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_namename of model selection parameter
Returns
index of model selection parameter with provided name, -1 if there is no such

Definition at line 1145 of file SGObject.cpp.

virtual const char* get_name ( ) const
virtual

Returns the name of the SGSerializable instance. It MUST BE the CLASS NAME without the prefixed `C'.

Returns
name of the SGSerializable

Implements CKernelTwoSampleTestStatistic.

Definition at line 197 of file LinearTimeMMD.h.

const float64_t * get_Q_col ( uint32_t  i)
static

return pointer to i-th column of m_Q. Helper for libqp

Definition at line 385 of file LinearTimeMMD.cpp.

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

Definition at line 278 of file SGObject.cpp.

DynArray< TParameter * > * load_all_file_parameters ( int32_t  file_version,
int32_t  current_version,
CSerializableFile *  file,
const char *  prefix = "" 
)
inherited

maps all parameters of this instance to the provided file version and loads all parameter data from the file into an array, which is sorted (basically calls load_file_parameter(...) for all parameters and puts all results into a sorted array)

Parameters
file_versionparameter version of the file
current_versionversion from which mapping begins (you want to use VERSION_PARAMETER for this in most cases)
filefile to load from
prefixprefix for members
Returns
(sorted) array of created TParameter instances with file data

Definition at line 679 of file SGObject.cpp.

DynArray< TParameter * > * load_file_parameters ( const SGParamInfo *  param_info,
int32_t  file_version,
CSerializableFile *  file,
const char *  prefix = "" 
)
inherited

loads some specified parameters from a file with a specified version The provided parameter info has a version which is recursively mapped until the file parameter version is reached. Note that there may be possibly multiple parameters in the mapping, therefore, a set of TParameter instances is returned

Parameters
param_infoinformation of parameter
file_versionparameter version of the file, must be <= provided parameter version
filefile to load from
prefixprefix for members
Returns
new array with TParameter instances with the attached data

Definition at line 523 of file SGObject.cpp.

bool load_serializable ( CSerializableFile *  file,
const char *  prefix = "",
int32_t  param_version = VERSION_PARAMETER 
)
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
filewhere to load from
prefixprefix for members
param_version(optional) a parameter version different to (this is mainly for testing, better do not use)
Returns
TRUE if done, otherwise FALSE

Reimplemented in CModelSelectionParameters.

Definition at line 354 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
ShogunExceptionWill be thrown if an error occurres.

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

Definition at line 1033 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
ShogunExceptionWill be thrown if an error occurres.

Definition at line 1028 of file SGObject.cpp.

void map_parameters ( DynArray< TParameter * > *  param_base,
int32_t &  base_version,
DynArray< const SGParamInfo * > *  target_param_infos 
)
inherited

Takes a set of TParameter instances (base) with a certain version and a set of target parameter infos and recursively maps the base level wise to the current version using CSGObject::migrate(...). The base is replaced. After this call, the base version containing parameters should be of same version/type as the initial target parameter infos. Note for this to work, the migrate methods and all the internal parameter mappings have to match

Parameters
param_baseset of TParameter instances that are mapped to the provided target parameter infos
base_versionversion of the parameter base
target_param_infosset of SGParamInfo instances that specify the target parameter base

Definition at line 717 of file SGObject.cpp.

TParameter * migrate ( DynArray< TParameter * > *  param_base,
const SGParamInfo *  target 
)
protectedvirtualinherited

creates a new TParameter instance, which contains migrated data from the version that is provided. The provided parameter data base is used for migration, this base is a collection of all parameter data of the previous version. Migration is done FROM the data in param_base TO the provided param info Migration is always one version step. Method has to be implemented in subclasses, if no match is found, base method has to be called.

If there is an element in the param_base which equals the target, a copy of the element is returned. This represents the case when nothing has changed and therefore, the migrate method is not overloaded in a subclass

Parameters
param_baseset of TParameter instances to use for migration
targetparameter info for the resulting TParameter
Returns
a new TParameter instance with migrated data from the base of the type which is specified by the target parameter

Definition at line 923 of file SGObject.cpp.

void one_to_one_migration_prepare ( DynArray< TParameter * > *  param_base,
const SGParamInfo *  target,
TParameter *&  replacement,
TParameter *&  to_migrate,
char *  old_name = NULL 
)
protectedvirtualinherited

This method prepares everything for a one-to-one parameter migration. One to one here means that only ONE element of the parameter base is needed for the migration (the one with the same name as the target). Data is allocated for the target (in the type as provided in the target SGParamInfo), and a corresponding new TParameter instance is written to replacement. The to_migrate pointer points to the single needed TParameter instance needed for migration. If a name change happened, the old name may be specified by old_name. In addition, the m_delete_data flag of to_migrate is set to true. So if you want to migrate data, the only thing to do after this call is converting the data in the m_parameter fields. If unsure how to use - have a look into an example for this. (base_migration_type_conversion.cpp for example)

Parameters
param_baseset of TParameter instances to use for migration
targetparameter info for the resulting TParameter
replacement(used as output) here the TParameter instance which is returned by migration is created into
to_migratethe only source that is used for migration
old_namewith this parameter, a name change may be specified

Definition at line 864 of file SGObject.cpp.

void optimize_kernel_weights ( )
virtual

Selects optimal kernel weights (if the underlying kernel and features are combined ones) using the ratio of the squared MMD by its standard deviation as a criterion, i.e.

\[ \frac{\text{MMD}_l^2[\mathcal{F},X,Y]}{\sigma_l} \]

where both expressions are estimated in linear time. This comes down to solving a convex program which is quadratic in the number of kernels.

SHOGUN has to be compiled with LAPACK to make this available. See set_opt* methods for optimization parameters.

IMPORTANT: Kernel weights have to be learned on different data than is used for testing/evaluation!

TODO check whether other types of combined kernels/features might be allowed

Definition at line 208 of file LinearTimeMMD.cpp.

void print_modsel_params ( )
inherited

prints all parameter registered for model selection and their type

Definition at line 1084 of file SGObject.cpp.

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

prints registered parameters out

Parameters
prefixprefix for members

Definition at line 290 of file SGObject.cpp.

void print_state ( libqp_state_T  state)
static

helper functions that prints current state

Definition at line 390 of file LinearTimeMMD.cpp.

bool save_serializable ( CSerializableFile *  file,
const char *  prefix = "",
int32_t  param_version = VERSION_PARAMETER 
)
virtualinherited

Save this object to file.

Parameters
filewhere to save the object; will be closed during returning if PREFIX is an empty string.
prefixprefix for members
param_version(optional) a parameter version different to (this is mainly for testing, better do not use)
Returns
TRUE if done, otherwise FALSE

Reimplemented in CModelSelectionParameters.

Definition at line 296 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
ShogunExceptionWill be thrown if an error occurres.

Reimplemented in CKernel.

Definition at line 1043 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
ShogunExceptionWill be thrown if an error occurres.

Reimplemented in CKernel.

Definition at line 1038 of file SGObject.cpp.

void set_bootstrap_iterations ( index_t  bootstrap_iterations)
virtualinherited

sets the number of bootstrap iterations for bootstrap_null()

Parameters
bootstrap_iterationshow often bootstrapping shall be done

Definition at line 44 of file TestStatistic.cpp.

void set_generic< floatmax_t > ( )
inherited

set generic type to T

Definition at line 41 of file SGObject.cpp.

void set_global_io ( SGIO *  io)
inherited

set the io object

Parameters
ioio object to use

Definition at line 217 of file SGObject.cpp.

void set_global_parallel ( Parallel *  parallel)
inherited

set the parallel object

Parameters
parallelparallel object to use

Definition at line 230 of file SGObject.cpp.

void set_global_version ( Version *  version)
inherited

set the version object

Parameters
versionversion object to use

Definition at line 265 of file SGObject.cpp.

void set_null_approximation_method ( ENullApproximationMethod  null_approximation_method)
virtualinherited

sets the method how to approximate the null-distribution

Parameters
null_approximation_methodmethod to use

Definition at line 38 of file TestStatistic.cpp.

void set_opt_epsilon ( float64_t  opt_epsilon)

Sets the stopping criterion epsilon for optimizing kernel weights

Definition at line 177 of file LinearTimeMMD.h.

void set_opt_low_cut ( float64_t  opt_low_cut)

Sets the low cut for optimizing kernel weights (weight below are set to zero

Definition at line 183 of file LinearTimeMMD.h.

void set_opt_max_iterations ( index_t  opt_max_iterations)

Sets the max. number of iterations for optimizing kernel weights

Definition at line 171 of file LinearTimeMMD.h.

void set_opt_regularization_eps ( float64_t  opt_regularization_eps)

Sets regularization constant. This value is added on diagonal of matrix for optimizing kernel weights

Definition at line 190 of file LinearTimeMMD.h.

virtual 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 122 of file SGObject.h.

void unset_generic ( )
inherited

unset generic type

this has to be called in classes specializing a template class

Definition at line 285 of file SGObject.cpp.

bool update_parameter_hash ( )
protectedvirtualinherited

Updates the hash of current parameter combination.

Returns
bool if parameter combination has changed since last update.

Definition at line 237 of file SGObject.cpp.

Member Data Documentation

SGIO* io
inherited

io

Definition at line 462 of file SGObject.h.

index_t m_bootstrap_iterations
protectedinherited

number of iterations for bootstrapping null-distributions

Definition at line 98 of file TestStatistic.h.

uint32_t m_hash
inherited

Hash of parameter values

Definition at line 480 of file SGObject.h.

CKernel* m_kernel
protectedinherited

underlying kernel

Definition at line 83 of file KernelTwoSampleTestStatistic.h.

Parameter* m_model_selection_parameters
inherited

model selection parameters

Definition at line 474 of file SGObject.h.

ENullApproximationMethod m_null_approximation_method
protectedinherited

Defines how the the null distribution is approximated

Definition at line 101 of file TestStatistic.h.

float64_t m_opt_epsilon
protected

stopping accuracy of qp solver

Definition at line 219 of file LinearTimeMMD.h.

float64_t m_opt_low_cut
protected

low cut for weights, if weights are under this value, are set to zero

Definition at line 222 of file LinearTimeMMD.h.

index_t m_opt_max_iterations
protected

maximum number of iterations of qp solver

Definition at line 216 of file LinearTimeMMD.h.

float64_t m_opt_regularization_eps
protected

regularization epsilon that is added to diagonal of Q matrix

Definition at line 225 of file LinearTimeMMD.h.

CFeatures* m_p_and_q
protectedinherited

concatenated samples of the two distributions (two blocks)

Definition at line 94 of file TwoDistributionsTestStatistic.h.

ParameterMap* m_parameter_map
inherited

map for different parameter versions

Definition at line 477 of file SGObject.h.

Parameter* m_parameters
inherited

parameters

Definition at line 471 of file SGObject.h.

SGMatrix< float64_t > m_Q = SGMatrix<float64_t>()
staticprotected

matrix for selection of kernel weights (static because of libqp)

Definition at line 228 of file LinearTimeMMD.h.

index_t m_q_start
protectedinherited

defines the first index of samples of q

Definition at line 97 of file TwoDistributionsTestStatistic.h.

Parallel* parallel
inherited

parallel

Definition at line 465 of file SGObject.h.

Version* version
inherited

version

Definition at line 468 of file SGObject.h.

The documentation for this class was generated from the following files:
SHOGUN Machine Learning Toolbox - Documentation