Description of the class THSIC. More...

#include <THSIC.h>

Inheritance diagram for URANIE::Sensitivity::THSIC:

Collaboration diagram for URANIE::Sensitivity::THSIC:

Public Types
enum	EEstimatedVariance { kUnknown, kStdEmpirical, kMedianeDelta }

Public Types inherited from URANIE::Sensitivity::TSensitivity
enum	ELauncher { kCode, kCodeRemote, kFunction, kRun, kUnknown }

Public Member Functions
Constructors and destructor
	THSIC ()
	Default constructor. More...

	THSIC (URANIE::DataServer::TDataServer tds, const char sInputs, const char sOutputs, Option_t option="")
	THSIC constructor with data contined in the TDS and the inputs and outputs attributes. More...

virtual	~THSIC ()
	Default destructor. More...

Setting and Getting attributs
EEstimatedVariance	getEstimatedVariance ()
	Which method used to estimate the variance of the gaussian kernel ( kUnknown \| kStdEmpirical \| kMedianeDelta ) More...

void	setEstimatedVariance (EEstimatedVariance nMethod)
	Set the method used to estimate the variance of the gaussian kernel ( kUnknown \| kStdEmpirical \| kMedianeDelta ) More...

Int_t	getNPermutationSample ()
	getNPermutationSample More...

void	setNPermutationSample (Int_t ns)
	Set the Permutation sample permutation Size (default NDefaultPermutation) More...

void	setThresholdPermutation (Int_t n)

Int_t	getThresholdPermutation ()
	Get the Threshold (nS/nX) to use the permutation permutation test to compute the p-value (default NDefaultThresholdPermutation) More...

void	setThresholdGamma (Int_t n)
	Set the Threshold (nS/nX) to use the Gamma approximation to cumpute the p-value (default NDefaultThresholdGamma) More...

Int_t	getThresholdGamma ()
	Get the Threshold (nS/nX) to use the Gamma approximation to cumpute the p-value (default NDefaultThresholdGamma) More...

Parse the option
The options treated by the THSIC object were : "empiri" : Choose the Standard deviation of each attributes to compute the $ \sigma$ parameter "median" : Choose the median distance between points of each attributes to compute the $ \sigma$ parameter See also TSensitivity::parseOption()
virtual void	parseOption (Option_t *option="")
	Parse the option given by the user. More...

Printing Log
void	printSummary ()
	Prints a concise summary of results to the terminal. More...

virtual void	printLog (Option_t *option="")
	Prints all attributes of the class for the current instance. More...

All methods used for elementary tasks
void	generateHMatrix (Option_t *option="")
	generate the $ H $ matrix More...

TMatrixD	doubleCenteringRect (const TMatrixD &matK)
	Centers the rows and columns of the provided matrix. More...

TMatrixDSym	doubleCentering (const TMatrixDSym &matK)
	Centers the rows and columns of the provided matrix. More...

Double_t	estimateSigma (const char sAtt, Option_t option="")
	Computes the bandwidth parameter of the specified attribute. More...

TMatrixDSym	getMatrixKGaussian (const char *sAtt, Double_t dTheta)
	Constructs the Gram matrix associated with the specified attribute. More...

Double_t	biasedHSIC (const TMatrixDSym &matKx, const TMatrixDSym &matKy)
	Computes the V-statistic estimate of $ HSIC(X, Y) $ where $ X $. The computation is based on the Gram matrices $ K_{x} $ and $ K_{y} $ which are constructed beforehand from the observations of $ X $ and $ Y $. More...

Double_t	unbiasedHSIC (const TMatrixDSym &matKx, const TMatrixDSym &matKy)
	Computes the U-statistic estimate of $ HSIC(X, Y) $. The computation is based on the Gram matrices $ K_{x} $ and $ K_{y} $ which are constructed beforehand from the observations of $ X $ and $ Y $. More...

Double_t	getProductOfNorms (const TMatrixDSym &matKx, const TMatrixDSym &matKy)
	Estimates the product of the squared Hilbert-Schmidt (HS) norms of the cross-covariance operators $ C_{xx} $ and $ C_{yy} $. More...

void	permuteKMatrix (const TMatrixDSym &theMatK, TList tlstPerm, TList tlstPermMat)
	Given a Gram matrix and a list of permutations, constructs the permuted matrices and store them in a list. More...

Overloaded methods
virtual void	generateSample (Option_t *option="")
	No action is performed, but a warning is returned. More...

void	computeIndexes (Option_t *option="")
	Calls the method evaluateIndexes. More...

void	evaluateIndexes (Option_t *option)
	Applies the HSIC methodology. More...

void	drawIndexes (TString sTitle, const char select="", Option_t option="")
	Draw the indexes computed by the method. More...

Public Member Functions inherited from URANIE::Sensitivity::TSensitivityTest
	TSensitivityTest ()
	Default constructor. More...

	TSensitivityTest (URANIE::DataServer::TDataServer tds, const char varexpinput, const char varexpoutput, Option_t option="")
	Default constructor. More...

virtual	~TSensitivityTest ()
	Default destructor. More...

Public Member Functions inherited from URANIE::Sensitivity::TSensitivity
	TSensitivity ()
	Default constructor. More...

	TSensitivity (URANIE::DataServer::TDataServer tds, const char fcn, Int_t ns, const char varexpinput="", const char varexpoutput="", Option_t *option="")
	Default constructor with the name of a function. More...

	TSensitivity (URANIE::DataServer::TDataServer tds, void(fcn)(Double_t , Double_t ), const char varexpinput, const char varexpoutput, Int_t ns, Option_t *option="")

	TSensitivity (URANIE::DataServer::TDataServer tds, const char varexpinput, const char varexpoutput, Option_t option="")
	Default constructor. More...

	TSensitivity (URANIE::DataServer::TDataServer tds, URANIE::Launcher::TCode code, Int_t ns, Option_t *option="")
	Default constructor with TCode arg. More...

	TSensitivity (URANIE::DataServer::TDataServer tds, URANIE::Relauncher::TRun run, Int_t ns, Option_t *option="")
	Default constructor with TRun arg. More...

virtual	~TSensitivity ()
	Default destructor. More...

Int_t	getID ()

virtual TTree *	getResultTuple (bool commonresulttuple=true)
	Get the result ntuple (default parameter unused but for Morris method) More...

double	getValue (const char sorder="", const char sinputname="", const char *sselect="")

vector< int > *	getAttributeElements (string str)

void	setFunction (void(fct)(Double_t , Double_t *), Int_t nx=-1, Int_t ny=1)

TString	getFunctionName ()

void	setSeed (UInt_t nval)

UInt_t	getSeed ()

virtual void	setMethodName (const char *str)
	Set the Method name. More...

const char *	getMethodName ()
	Get the method name. More...

Bool_t	getNoIntermediateSaved ()
	Get the noIntermediateSaved flag. More...

const char *	getIteratorName ()
	Get the name of the iterator attribut of the method. More...

void	setSensitivityIteratorName (const char *str)
	Set the iterator name devoted to compute sensitivity indexes. More...

void	setTimeName (TString sname)
	Set the name of the time attribute (only one) More...

TString	getTimeName ()
	Get the name of the time attribute. More...

virtual void	setDrawProgressBar (Bool_t bbool=kTRUE)
	Set the "draw progress bar" flag. More...

void	setUsingErrors (bool thebool)
	Set the "using error results anyway" option. More...

Bool_t	getDrawProgressBar ()
	Get the clean flag. More...

Bool_t	isInputCorrelated ()

TMatrixD	getMatrixInputCorrelation ()

Int_t	getNInput ()
	Get the number of input attributes. More...

Int_t	getNOutput ()
	Get the number of output attributes. More...

void	setInputCorrelationMatrix (TMatrixD Corr)

void	checkOutputRequested (string attlist, bool fromoption=false)
	Check the output list requested by the user. More...

void	computeIndexes (Option_t *option="")
	Compute the Sensitivity Indexes. More...

void	fillIndex (const char sinputname, const char sorder, Double_t dval, const char *algo="", Double_t dvalCILower=-1.0, Double_t dvalCIUpper=-1.0)
	Method to fill in the tree the value of Sensitivity indexes for an input attribute. More...

virtual void	createTuple (Option_t *option="")

virtual void	setLog ()

void	unsetLog ()

void	changeLog ()

Bool_t	getLog ()

void	setNLauncher (ELauncher codeLauncher)

Static Public Member Functions
Static methods
static TString	convertRule (EEstimatedVariance evRule)
	Outputs the string representation of any enumerator of EEstimatedVariance. More...

Private Attributes
Private and Public attributes
EEstimatedVariance	_nEstimatedVariance
	< The H matrix defined by H := ( Kronecker(i,j) - 1/nS) More...

Int_t	_nPermutationSample
	the method to compute the variance for the gaussian kernel More...

Int_t	_nThresholdPermutation
	The number of permutation sample ( permutation ) (default NDefaultPermutation) More...

Int_t	_nThresholdGamma
	The Threshold (nS/nX) to use the permutation permutation test to cumpute the p-value (default 5) More...

Additional Inherited Members
Public Attributes inherited from URANIE::Sensitivity::TSensitivity
URANIE::DataServer::TDataServer *	_tds
	Pointeur vers un TDS. More...

TList *	_listOfInputAttributes
	List of the input branches. More...

TList *	_listOfOutputAttributes
	List of the input branches. More...

TString	_sTimeAttribute
	The name of the Time attribute. More...

Int_t	_nS
	The number of simulation or other information depend on the used method. More...

Int_t	_nX
	Dimension of the input. More...

UInt_t	_nY
	Dimension of the target. More...

UInt_t	_nElY
	Number of element for one selected output. More...

Int_t	_nbOut
	Total number of Output to be considered. More...

Int_t	_iOut
	counter for output More...

unsigned int	_iy
	iterator over number of element More...

unsigned int	_iely
	iterator over number of element More...

ELauncher	_nLauncher
	The type of launcher. More...

TString	_sFunctionName
	The Name of the evaluatuor. More...

URANIE::Launcher::TCode *	_code
	The tcode. More...

URANIE::Relauncher::TRun *	_run

TObjArray *	_drawingGarbageCollector
	Garbage collector for prints. More...

Int_t	_nSeed
	The seed of the random generator. More...

Bool_t	_bChosenOutputs
	Fact that the input list is provided or not. More...

Bool_t	_blog
	Boolean for edit the log. More...

Bool_t	_bquiet
	Quiet mode. More...

Bool_t	_bdrawProgressBar
	Boolean to know if the progress bar has to be drawn. More...

Bool_t	_bnoIntermediateSaved
	Boolean to know if the progress bar has to be drawn. More...

TString	_sIteratorName
	The specific iterator attribute for the method. More...

TString	_sMethodName
	The method name. More...

TString	_sSelectedOutput
	The output. More...

TString	_sSelectedInput
	The input. More...

map< string, unsigned int >	_mAttributeSize
	Map of size of element for attribute;. More...

map< string, vector< int > >	_mAttributeElements
	Map of Elements number to run (if vector subselection is requested) More...

vector< string >	_vOutputNames
	Name of the output. More...

TCanvas *	_canvas
	Canvas object to deal with. More...

void(*	_pFunction )(double , double )

TTree *	_ntresult
	The TTree of results. More...

Protected Member Functions inherited from URANIE::Sensitivity::TSensitivity
virtual void	preTreatment ()
	PreTreatment for every output. More...

virtual void	postTreatment ()
	PostTreatment for every output. More...

void	setNoIntermediateSaved (Bool_t bbool=kTRUE)
	Set the "only final file" flag. More...

void	checkCanvasCreation (bool newcan)
	Create a canvas if needed. More...

void	drawIndexesHistogram (TString sTitre, const char select="", Option_t option="")
	Draws indexes with an histogram. More...

void	drawIndexesPie (TString sTitre, const char select="", Option_t option="")
	Draws indexes with an pie chart. More...

Protected Attributes inherited from URANIE::Sensitivity::TSensitivity
Char_t	_sOutputAttribute [MAXLENGTHSTRING]
	The name of the output attribute. More...

Char_t	_sInputAttribute [MAXLENGTHSTRING]
	The name of the input attribute. More...

Char_t	_sOrder [MAXLENGTHSTRING]
	The order of sensitivity indexes. More...

Char_t	_sMethod [MAXLENGTHSTRING]
	The name of the method. More...

Char_t	_sAlgorithm [MAXLENGTHSTRING]
	The name of the algorithm to compute the index. More...

Double_t	_valSobolCrt
	The value of sensitivity indexes. More...

Double_t	_valSobolCILower
	The value of lower Condidence Interval (95) More...

Double_t	_valSobolCIUpper
	The value of upper Condidence Interval (95) More...

TMatrixD	_minputCorr
	Input correlation matrix if sample needs to be correlated. More...

Bool_t	_bisInputCorrelated
	State whether the input correlation matrix is set. More...

Bool_t	_bgoingThroughError
	State whether the error must not block the computation. More...

Detailed Description

Description of the class THSIC.

Introduction

The goal of this class is to find the list of input attributes which are dependent with the output attributes using the "HSIC" values (for "Hilbert-Schmidt Independence Criterion"). This approach was introduced by Gretton (2005, 2007).

Moreover, in the same time, we compute the $ R^{2}_{HSIC} $ proposed by Da Veiga (2015) given by the formula

\[ R^{2}_{HSIC} ( X, Y) \; = \; \frac{ HSIC ( X, Y) }{\sqrt{ HSIC ( X, X) \, \times \, HSIC ( Y, Y)}} \]

Draw Indexes

We use a "Pareto chart" to draw the "p-Values" and the $ R^{2}_{HSIC} $.

Bibliography

Da Veiga, S. " Global Sensitivity Analysis with Dependence Measures ", https://arxiv.org/abs/1311.2483, 2013
Zhang, Q., Filippi, S., Gretton, A., Sejdinovic, D., "Large-Scale Kernel Methods for Independence Testing", 2016
Gretton, A., Fukumizu, K., Teo, C.H., Song, L., Schoelkopf, B., et Smola, A., "A Kernel Statistical Test of Independence,", NIPS 2007
Gretton, A., Herbrich, R., Smola, A., Bousquet, O., et Schoelkopf, B., "Kernel Methods for Measuring Independence", Journal of Machine Learning Research, 6 , pp.2075–2129, 2005
The Pareto chart, https://en.wikipedia.org/wiki/Pareto_chart

Member Enumeration Documentation

◆ EEstimatedVariance

enum URANIE::Sensitivity::THSIC::EEstimatedVariance

Enumerator
kUnknown
kStdEmpirical
kMedianeDelta

Constructor & Destructor Documentation

◆ THSIC() [1/2]

URANIE::Sensitivity::THSIC::THSIC ( )

Default constructor.

Referenced by ClassImp().

◆ THSIC() [2/2]

URANIE::Sensitivity::THSIC::THSIC	(	URANIE::DataServer::TDataServer *	tds,
		const char *	sInputs,
		const char *	sOutputs,
		Option_t *	option = `""`
	)

THSIC constructor with data contined in the TDS and the inputs and outputs attributes.

Parameters

tds
sInputs
sOutputs
option	(Option_t*) option to pass [""]

◆ ~THSIC()

virtual URANIE::Sensitivity::THSIC::~THSIC ( )

virtual

Default destructor.

Referenced by ClassImp().

Member Function Documentation

◆ biasedHSIC()

Double_t URANIE::Sensitivity::THSIC::biasedHSIC	(	const TMatrixDSym &	matKx,
		const TMatrixDSym &	matKy
	)

Computes the V-statistic estimate of $ HSIC(X, Y) $ where $ X $. The computation is based on the Gram matrices $ K_{x} $ and $ K_{y} $ which are constructed beforehand from the observations of $ X $ and $ Y $.

The formula of the V-statistic estimator can be found in: [Zhang et al., 2018] "Large-scale kernel methods for independence testing", Statistics and Computing", page 117, Eq (9).

Let $ \widetilde{K}_{x} $ be the double-centered Gram matrix built from the observations of $ X $. Let $ \widetilde{K}_{y} $ be the double-centered Gram matrix built from the observations of $ Y $.

The formula of the V-statistic estimator is given by:

\[ \widehat{HSIC}_{V}(X, Y) \, = \, \frac{1}{nS^2} \, Tr( \widetilde{K}_{x} \widetilde{K}_{y} ) \]

Parameters

matKx	(TMatrixDSym) The Gram matrix built from the observations of $ X $.
matKy	(TMatrixDSym) The Gram matrix built from the observations of $ Y $.

Returns: (Double_t) The value of the V-statistic estimate of $ HSIC(X, Y) $.

Referenced by ClassImp().

◆ computeIndexes()

void URANIE::Sensitivity::THSIC::computeIndexes ( Option_t * option = "" )

Calls the method evaluateIndexes.

Parameters

option (Option_t*) Specifies the option to pass to the method evaluateIndexes.

Referenced by ClassImp().

◆ convertRule()

static TString URANIE::Sensitivity::THSIC::convertRule ( EEstimatedVariance evRule )

static

Outputs the string representation of any enumerator of EEstimatedVariance.

Parameters

evRule (EEstimatedVariance) enumerator of EEstimatedVariance that must be represented by a string

Referenced by ClassImp().

◆ doubleCentering()

TMatrixDSym URANIE::Sensitivity::THSIC::doubleCentering ( const TMatrixDSym & matK )

Centers the rows and columns of the provided matrix.

Parameters

matK	(TMatrixDSym) The $ n $-by- $ n $ matrix $ K $ that needs to be centered.

Returns: (TMatrixDSym) The double-centered matrix produced from $ $ K $.

It is defined as $ \widetilde{K} := H \, K \, H $ where: $ H $ is the centering matrix of dimension $ n $.

Note that the sum of each row (resp. column) of $ \widetilde{K} $ is equal to zero.

Referenced by ClassImp().

◆ doubleCenteringRect()

TMatrixD URANIE::Sensitivity::THSIC::doubleCenteringRect ( const TMatrixD & matK )

Centers the rows and columns of the provided matrix.

Parameters

matK	(TMatrixD) The $ n $-by- $ m $ matrix $ K $ that needs to be centered.

Returns: (TMatrixD) The double-centered matrix produced from $ $ K $.

It is defined as $ \widetilde{K} := H_n \, K \, H_m $ where: $ \forall \, r \geq 1 $, $ H_r $ is the centering matrix of dimension $ r $.

Note that the sum of each row (resp. column) of $ \widetilde{K} $ is equal to zero.

Referenced by ClassImp().

◆ drawIndexes()

void URANIE::Sensitivity::THSIC::drawIndexes	(	TString	sTitle,
		const char *	select = `""`,
		Option_t *	option = `""`
	)

virtual

Draw the indexes computed by the method.

Parameters

sTitle	(TString) the title of the graph
select	(const char*) the select argument
option	(Option_t*) option to pass [""]

Reimplemented from URANIE::Sensitivity::TSensitivity.

Referenced by ClassImp().

◆ estimateSigma()

Double_t URANIE::Sensitivity::THSIC::estimateSigma	(	const char *	sAtt,
		Option_t *	option = `""`
	)

Computes the bandwidth parameter of the specified attribute.

This function calculates the bandwidth parameter based on the observations of the specified attribute, which are stored in the attached TDataServer.

Parameters

sAtt	(const char*) The name of the attribute for which the bandwith parameter is to be computed.
option	(Option_t*) The rule of thumb used to compute the bandwidth parameter from the data. The user can choose between: "empiri": empirical standard deviation, "median": median of pair distances.

Returns: (Double_t) The computed value of the bandwidth parameter.

Referenced by ClassImp().

◆ evaluateIndexes()

void URANIE::Sensitivity::THSIC::evaluateIndexes ( Option_t * option )

virtual

Applies the HSIC methodology.

For each input-output pair $ (X_i, Y_j) $, this function estimates three sensitivity indices:

the (standard) HSIC index,
the normalized HSIC index (also called R2-HSIC index),
the p-value of the independence test.

Parameters

option (Option_t*) Some possible options (level of verbosity, estimator type, test procedure).

By default:

The function operates in verbose mode, providing many details.
HSIC indices (and R2-HSIC indices) are estimated with V-statistics, which yields (slightly) biased but non-negative estimates.
The p-values are estimated with the asymptotic test procedure (based on asymptotic formulas).

The option argument allows customization. It should be a concatenation of substrings, such as: "[option_1]" or "[option_1],[option_2]" or "[option_1],[option_2],[option_3]".

Valid substrings include:

"quiet" to enable the silent mode (instead of the verbose mode),
"unbiased" to estimate HSIC indices with U-statistics (instead of V-statistics),
"nperm=500" to estimate the p-values with the permutation-based test procedure (instead of the asymptotic test procedure). This procedure will apply $ B = 500 $ random permutations on the output data before estimating the p-value. Note that the number of permutations can be set to any integer greater than $ 7 $.

Implements URANIE::Sensitivity::TSensitivity.

Referenced by ClassImp().

◆ generateHMatrix()

void URANIE::Sensitivity::THSIC::generateHMatrix ( Option_t * option = "" )

generate the $ H $ matrix

The $ H $ matrix is defined by the formula

\[ H (i, j) \, = \, 1\!\!I_{nS, nS} \: - \: \frac{1}{nS} \: \delta_{i=j} \]

Parameters

option (Option_t *) option to pass [""]

Referenced by ClassImp().

◆ generateSample()

virtual void URANIE::Sensitivity::THSIC::generateSample ( Option_t * option = "" )

virtual

No action is performed, but a warning is returned.

Remember that any THSIC instance is created from an already filled TDataServer.

Parameters

option (Option_t*) Useless here.

Implements URANIE::Sensitivity::TSensitivity.

Referenced by ClassImp().

◆ getEstimatedVariance()

EEstimatedVariance URANIE::Sensitivity::THSIC::getEstimatedVariance ( )

inline

Which method used to estimate the variance of the gaussian kernel ( kUnknown | kStdEmpirical | kMedianeDelta )

kStdEmpirical : the $ \sigma $ parameters is estimated by the emperical standatd deviation of the attributes
kMedianeDelta : use the median distance between points
Returns
the method used to estimated the variance of the gaussian kernel ( kStdEmpirical | kMedianeDelta )

References _nEstimatedVariance.

◆ getMatrixKGaussian()

TMatrixDSym URANIE::Sensitivity::THSIC::getMatrixKGaussian	(	const char *	sAtt,
		Double_t	dTheta
	)

Constructs the Gram matrix associated with the specified attribute.

This function uses the observations of the specified attribute, stored in the attached TDataServer, for kernelization. The kernel function applied to the data is a Gaussian kernel, whose parameter is provided as second argument.

Parameters

sAtt	(const char*) The name of the attribute for which the Gram matrix is to be constructed.
dTheta	(Double_t) The parameter of the Gaussian kernel.

Returns: (TMatrixDSym) The constructed Gram matrix.

Referenced by ClassImp().

◆ getNPermutationSample()

Int_t URANIE::Sensitivity::THSIC::getNPermutationSample ( )

inline

getNPermutationSample

Returns: The number of Permutation sample (permutation zize) to estimate the pValues

References _nPermutationSample.

◆ getProductOfNorms()

Double_t URANIE::Sensitivity::THSIC::getProductOfNorms	(	const TMatrixDSym &	matKx,
		const TMatrixDSym &	matKy
	)

Estimates the product of the squared Hilbert-Schmidt (HS) norms of the cross-covariance operators $ C_{xx} $ and $ C_{yy} $.

This quantity appears in the asymptotic variance of the test statistic (either the U-statistic or V-statistic estimator of the HSIC) under the null hypothesis. The estimator is equal to the mean of all the off-diagonal coefficients of the following matrix: $ \left( \widetilde{K}_{x} \odot \widetilde{K}_{y} \right)^{ \odot 2} $ where:

$ \widetilde{K}_{x} $ and $ \widetilde{K}_{y} $ are the double-centered matrices derived from $ K_{x} $ and $ K_{y} $,
$ \odot $ denotes the Hadamard product (element-wise product).

Further details can be found in [Gretton et al., 2007]. Note that the product of the squared HS norms appears in Theorem 4 and Appendix A.2.

Parameters

matKx	(TMatrixDSym) The input Gram matrix $ K_{x} $.
matKy	(TMatrixDSym) The output Gram matrix $ K_{y} $.

Returns: (Double_t) An estimate of the product of the two squared HS norms.

Referenced by ClassImp().

◆ getThresholdGamma()

Int_t URANIE::Sensitivity::THSIC::getThresholdGamma ( )

inline

Get the Threshold (nS/nX) to use the Gamma approximation to cumpute the p-value (default NDefaultThresholdGamma)

Returns: The current Threshold (nS/nX) for using the Gamma approximation

References _nThresholdGamma.

◆ getThresholdPermutation()

Int_t URANIE::Sensitivity::THSIC::getThresholdPermutation ( )

inline

Get the Threshold (nS/nX) to use the permutation permutation test to compute the p-value (default NDefaultThresholdPermutation)

Returns: The current Threshold (nS/nX) for using the permutation permutation test

References _nThresholdPermutation.

◆ parseOption()

virtual void URANIE::Sensitivity::THSIC::parseOption ( Option_t * option = "" )

virtual

Parse the option given by the user.

Parameters

option (Option_t) the string of option

Reimplemented from URANIE::Sensitivity::TSensitivity.

Referenced by ClassImp().

◆ permuteKMatrix()

void URANIE::Sensitivity::THSIC::permuteKMatrix	(	const TMatrixDSym &	theMatK,
		TList *	tlstPerm,
		TList *	tlstPermMat
	)

Given a Gram matrix and a list of permutations, constructs the permuted matrices and store them in a list.

Let us recall that a permutation is a one-to-one map $ \sigma: \{ 1, \ldots, nS \} \rightarrow \{ 1, \ldots, nS \} $. For a given Gram matrix $ K = \left[ K(i,j) \right]_{1 \leq i, j \leq nS} $ and a given permutation $ \sigma $, the permuted Gram matrix is defined as:

\[ K_{\sigma} := \left[ K \left( \sigma(i), \sigma(j) \right) \right]_{1 \leq i, j \leq nS} \]

.

Parameters

theMatK	(TMatrixDSym) Gram matrix that is to be permuted.
tlstPerm	(TList) Pointer to the list of permutations. Each element in tlstPerm is a pointer to a TPatternsEventList (class defined in the DataServer module). The first element in *tlstPerm must be a pointer to a TPatternsEventList representing the identity matrix.
tlstPermMat	(TList) Pointer to the list of permuted Gram matrices. An empty list must be passed as argument. For each new permutation found in tlstPerm, theMatK is permuted accordingly, and a pointer to the resulting TMatrixDSym is stored in *tlstKPermMat.

Referenced by ClassImp().

◆ printLog()

virtual void URANIE::Sensitivity::THSIC::printLog ( Option_t * option = "" )

virtual

Prints all attributes of the class for the current instance.

Parameters

option (Option_t*) Directly forwarded to the TSensitivityTest::printLog method.

Reimplemented from URANIE::Sensitivity::TSensitivityTest.

Referenced by ClassImp().

◆ printSummary()

void URANIE::Sensitivity::THSIC::printSummary ( )

Prints a concise summary of results to the terminal.

Specifically, for each input-output pair $ (X_i, Y_j) $, the following results are displayed:

the HSIC index,
the normalized HSIC index (or R2-HSIC index),
the p-value of the independence test.

Referenced by ClassImp().

◆ setEstimatedVariance()

void URANIE::Sensitivity::THSIC::setEstimatedVariance ( EEstimatedVariance nMethod )

Set the method used to estimate the variance of the gaussian kernel ( kUnknown | kStdEmpirical | kMedianeDelta )

Parameters

nMethod (EEstimatedVariance) One of the ( kStdEmpirical | kMedianeDelta ) choice

Exceptions

UErrorExceptions for the case (kUnknown)

Referenced by ClassImp().

◆ setNPermutationSample()

void URANIE::Sensitivity::THSIC::setNPermutationSample ( Int_t ns )

Set the Permutation sample permutation Size (default NDefaultPermutation)

Parameters

ns	size of the permutation permutation size

Referenced by ClassImp().

◆ setThresholdGamma()

void URANIE::Sensitivity::THSIC::setThresholdGamma ( Int_t n )

Set the Threshold (nS/nX) to use the Gamma approximation to cumpute the p-value (default NDefaultThresholdGamma)

Parameters

n	(Int_t) the Threshold (nS/nX) for using the Gamma approximation

Referenced by ClassImp().

◆ setThresholdPermutation()

void URANIE::Sensitivity::THSIC::setThresholdPermutation ( Int_t n )

Set the threshold (nS/nX) to use the permutation permutation test to cumpute the p-value (default NDefaultThresholdPermutation)

Referenced by ClassImp().

◆ unbiasedHSIC()

Double_t URANIE::Sensitivity::THSIC::unbiasedHSIC	(	const TMatrixDSym &	matKx,
		const TMatrixDSym &	matKy
	)

Computes the U-statistic estimate of $ HSIC(X, Y) $. The computation is based on the Gram matrices $ K_{x} $ and $ K_{y} $ which are constructed beforehand from the observations of $ X $ and $ Y $.

The formula of the U-statistic estimator can be found in: [Zhang et al., 2018] "Large-scale kernel methods for independence testing", Statistics and Computing", page 117, Eq (8).

Let $ L_{x} $ be the matrix obtained from the Gram matrix $ K_{x} $ after setting the diagonal elements to zero. Let $ L_{y} $ be the matrix obtained from the Gram matrix $ K_{y} $ after setting the diagonal elements to zero. Let $ 1 $ be the vector of $ \mathbb{R}^{nS} $ filled with ones.

The formula of the U-statistic estimator is given by:

\[ \widehat{HSIC}_{U}(X, Y) \, = \, \frac{1}{ nS (nS-3) } \, \left[ Tr( L_{x} L_{y} ) + \frac{1}{ (nS-1)(nS-2) } \, 1^T L_{x} 1 \times 1^T L_{y} 1 - \frac{2}{ nS-2 } \, 1^T L_{x} L_{y} 1 \right] \]

Parameters

matKx	(TMatrixDSym) The Gram matrix built from the observations of $ X $.
matKy	(TMatrixDSym) The Gram matrix built from the observations of $ Y $.

Returns: (Double_t) The value of the U-statistic estimate of $ HSIC(X, Y) $.

Referenced by ClassImp().

Member Data Documentation

◆ _nEstimatedVariance

EEstimatedVariance URANIE::Sensitivity::THSIC::_nEstimatedVariance

private

< The H matrix defined by H := ( Kronecker(i,j) - 1/nS)

Referenced by ClassImp(), and getEstimatedVariance().

◆ _nPermutationSample

Int_t URANIE::Sensitivity::THSIC::_nPermutationSample

private

the method to compute the variance for the gaussian kernel

Referenced by ClassImp(), and getNPermutationSample().

◆ _nThresholdGamma

Int_t URANIE::Sensitivity::THSIC::_nThresholdGamma

private

The Threshold (nS/nX) to use the permutation permutation test to cumpute the p-value (default 5)

Referenced by ClassImp(), and getThresholdGamma().

◆ _nThresholdPermutation

Int_t URANIE::Sensitivity::THSIC::_nThresholdPermutation

private

The number of permutation sample ( permutation ) (default NDefaultPermutation)

Referenced by ClassImp(), and getThresholdPermutation().

matKx	(TMatrixDSym) The input Gram matrix \( K_{x} \).
matKy	(TMatrixDSym) The output Gram matrix \( K_{y} \).

Documentation / Developer's manual

Available modules

Public Types

Public Member Functions

Static Public Member Functions

Private Attributes

Additional Inherited Members

Detailed Description

Description of the class THSIC.

Introduction

Draw Indexes

Bibliography

Member Enumeration Documentation

◆ EEstimatedVariance

Constructor & Destructor Documentation

◆ THSIC() [1/2]

◆ THSIC() [2/2]

◆ ~THSIC()

Member Function Documentation

◆ biasedHSIC()

◆ computeIndexes()

◆ convertRule()

◆ doubleCentering()

◆ doubleCenteringRect()

◆ drawIndexes()

◆ estimateSigma()

◆ evaluateIndexes()

◆ generateHMatrix()

◆ generateSample()

◆ getEstimatedVariance()

◆ getMatrixKGaussian()

◆ getNPermutationSample()

◆ getProductOfNorms()

◆ getThresholdGamma()

◆ getThresholdPermutation()

◆ parseOption()

◆ permuteKMatrix()

◆ printLog()

◆ printSummary()

◆ setEstimatedVariance()

◆ setNPermutationSample()

◆ setThresholdGamma()

◆ setThresholdPermutation()

◆ unbiasedHSIC()

Member Data Documentation

◆ _nEstimatedVariance

◆ _nPermutationSample

◆ _nThresholdGamma

◆ _nThresholdPermutation