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CStatistics Class Reference

Detailed Description

Class that contains certain functions related to statistics, such as probability/cumulative distribution functions, different statistics, etc.

Definition at line 32 of file Statistics.h.

Inheritance diagram for CStatistics:
Inheritance graph
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Classes

struct  SigmoidParamters
 

Public Member Functions

virtual const char * get_name () const
 
template<>
floatmax_t mean (SGVector< complex128_t > vec)
 mean not implemented for complex128_t, returns 0.0 instead More...
 
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 ()
 

Static Public Member Functions

template<class T >
static floatmax_t mean (SGVector< T > vec)
 
static float64_t median (SGVector< float64_t > values, bool modify=false, bool in_place=false)
 
static float64_t matrix_median (SGMatrix< float64_t > values, bool modify=false, bool in_place=false)
 
static float64_t variance (SGVector< float64_t > values)
 
static float64_t std_deviation (SGVector< float64_t > values)
 
static SGVector< float64_tmatrix_mean (SGMatrix< float64_t > values, bool col_wise=true)
 
static SGVector< float64_tmatrix_variance (SGMatrix< float64_t > values, bool col_wise=true)
 
static SGVector< float64_tmatrix_std_deviation (SGMatrix< float64_t > values, bool col_wise=true)
 
static SGMatrix< float64_tcovariance_matrix (SGMatrix< float64_t > observations, bool in_place=false)
 
static float64_t confidence_intervals_mean (SGVector< float64_t > values, float64_t alpha, float64_t &conf_int_low, float64_t &conf_int_up)
 
static float64_t inverse_student_t (int32_t k, float64_t p)
 
static float64_t inverse_incomplete_beta (float64_t a, float64_t b, float64_t y)
 
static float64_t incomplete_beta (float64_t a, float64_t b, float64_t x)
 
static float64_t inverse_normal_cdf (float64_t y0)
 
static float64_t inverse_normal_cdf (float64_t y0, float64_t mean, float64_t std_dev)
 
static float64_t lgamma (float64_t x)
 
static floatmax_t lgammal (floatmax_t x)
 
static float64_t tgamma (float64_t x)
 
static float64_t incomplete_gamma (float64_t a, float64_t x)
 
static float64_t incomplete_gamma_completed (float64_t a, float64_t x)
 
static float64_t gamma_cdf (float64_t x, float64_t a, float64_t b)
 
static float64_t inverse_gamma_cdf (float64_t p, float64_t a, float64_t b)
 
static float64_t inverse_incomplete_gamma_completed (float64_t a, float64_t y0)
 
static float64_t normal_cdf (float64_t x, float64_t std_dev=1)
 
static float64_t lnormal_cdf (float64_t x)
 
static float64_t chi2_cdf (float64_t x, float64_t k)
 
static float64_t fdistribution_cdf (float64_t x, float64_t d1, float64_t d2)
 
static float64_t erfc8_weighted_sum (float64_t x)
 
static float64_t error_function (float64_t x)
 
static float64_t error_function_complement (float64_t x)
 
static float64_t mutual_info (float64_t *p1, float64_t *p2, int32_t len)
 
static float64_t relative_entropy (float64_t *p, float64_t *q, int32_t len)
 
static float64_t entropy (float64_t *p, int32_t len)
 
static SGVector< float64_tfishers_exact_test_for_multiple_2x3_tables (SGMatrix< float64_t > tables)
 
static float64_t fishers_exact_test_for_2x3_table (SGMatrix< float64_t > table)
 
static SGVector< int32_t > sample_indices (int32_t sample_size, int32_t N)
 
static float64_t dlgamma (float64_t x)
 
static SigmoidParamters fit_sigmoid (SGVector< float64_t > scores)
 
static float64_t log_det_general (const SGMatrix< float64_t > A)
 
static float64_t log_det (SGMatrix< float64_t > m)
 
static float64_t log_det (const SGSparseMatrix< float64_t > m)
 
static SGMatrix< float64_tsample_from_gaussian (SGVector< float64_t > mean, SGMatrix< float64_t > cov, int32_t N=1, bool precision_matrix=false)
 
static SGMatrix< float64_tsample_from_gaussian (SGVector< float64_t > mean, SGSparseMatrix< float64_t > cov, int32_t N=1, bool precision_matrix=false)
 

Public Attributes

SGIOio
 
Parallelparallel
 
Versionversion
 
Parameterm_parameters
 
Parameterm_model_selection_parameters
 
Parameterm_gradient_parameters
 
uint32_t m_hash
 

Static Public Attributes

static const float64_t ERFC_CASE1 =0.0492
 
static const float64_t ERFC_CASE2 =-11.3137
 

Protected Member Functions

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)
 

Static Protected Member Functions

static float64_t ibetaf_incompletebetaps (float64_t a, float64_t b, float64_t x, float64_t maxgam)
 
static float64_t ibetaf_incompletebetafe (float64_t a, float64_t b, float64_t x, float64_t big, float64_t biginv)
 
static float64_t ibetaf_incompletebetafe2 (float64_t a, float64_t b, float64_t x, float64_t big, float64_t biginv)
 
static bool equal (float64_t a, float64_t b)
 
static bool not_equal (float64_t a, float64_t b)
 
static bool less (float64_t a, float64_t b)
 
static bool less_equal (float64_t a, float64_t b)
 
static bool greater (float64_t a, float64_t b)
 
static bool greater_equal (float64_t a, float64_t b)
 

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
dictdictionary of parameters to be built.

Definition at line 597 of file SGObject.cpp.

float64_t chi2_cdf ( float64_t  x,
float64_t  k 
)
static

Evaluates the CDF of the chi square distribution with parameter k at \(x\). Based on Wikipedia definition.

Parameters
xposition to evaluate
kparameter
Returns
chi square CDF at \(x\)

Definition at line 1751 of file Statistics.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 confidence_intervals_mean ( SGVector< float64_t values,
float64_t  alpha,
float64_t conf_int_low,
float64_t conf_int_up 
)
static

Calculates the sample mean of a given set of samples and also computes the confidence interval for the actual mean for a given p-value, assuming that the actual variance and mean are unknown (These are estimated by the samples). Based on Student's t-distribution.

Only for normally distributed data

Parameters
valuesvector of values that are used for calculations
alphaactual mean lies in confidence interval with (1-alpha)*100%
conf_int_lowlower confidence interval border is written here
conf_int_upupper confidence interval border is written here
Returns
sample mean

Definition at line 335 of file Statistics.cpp.

SGMatrix< float64_t > covariance_matrix ( SGMatrix< float64_t observations,
bool  in_place = false 
)
static

Computes the empirical estimate of the covariance matrix of the given data which is organized as num_cols variables with num_rows observations.

Data is centered before matrix is computed. May be done in place. In this case, the observation matrix is changed (centered).

Given sample matrix \(X\), first, column mean is removed to create \(\bar X\). Then \(\text{cov}(X)=(X-\bar X)^T(X - \bar X)\) is returned.

Needs SHOGUN to be compiled with LAPACK.

Parameters
observationsdata matrix organized as one variable per column
in_placeoptional, if set to true, observations matrix will be centered, if false, a copy will be created an centered.
Returns
covariance matrix empirical estimate

Definition at line 311 of file Statistics.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.

float64_t dlgamma ( float64_t  x)
static

Derivative of the log gamma function.

Parameters
xinput
Returns
derivative of the log gamma input

Definition at line 2071 of file Statistics.cpp.

float64_t entropy ( float64_t p,
int32_t  len 
)
static
Returns
entropy of \(p\) which is given in logspace

Definition at line 2035 of file Statistics.cpp.

static bool equal ( float64_t  a,
float64_t  b 
)
staticprotected

method to make ALGLIB integration easier

Definition at line 649 of file Statistics.h.

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
otherobject to compare with
accuracyaccuracy to use for comparison (optional)
tolerantallows 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 erfc8_weighted_sum ( float64_t  x)
static

Use to estimates erfc(x) valid for -100 < x < -8

Parameters
xreal value
Returns
weighted sum

Definition at line 1763 of file Statistics.cpp.

float64_t error_function ( float64_t  x)
static

Error function

The integral is

\[ \text{error\_function}(x)= \frac{2}{\sqrt{pi}}\int_0^x \exp (-t^2) dt \]

For \(0 \leq |x| < 1, \text{error\_function}(x) = x \frac{P4(x^2)}{Q5(x^2)}\) otherwise \(\text{error\_function}(x) = 1 - \text{error\_function\_complement}(x)\).

Taken from ALGLIB under gpl2+

Definition at line 1809 of file Statistics.cpp.

float64_t error_function_complement ( float64_t  x)
static

Complementary error function

\[ 1 - \text{error\_function}(x) = \text{error\_function\_complement}(x)= \frac{2}{\sqrt{\pi}}\int_x^\infty \exp\left(-t^2 \right)dt \]

For small \(x\), \(\text{error\_function\_complement}(x) = 1 - \text{error\_function}(x)\); otherwise rational approximations are computed.

Taken from ALGLIB under gpl2+

Definition at line 1848 of file Statistics.cpp.

float64_t fdistribution_cdf ( float64_t  x,
float64_t  d1,
float64_t  d2 
)
static

Evaluates the CDF of the F-distribution with parameters \(d1,d2\) at \(x\). Based on Wikipedia definition.

Parameters
xposition to evaluate
d1parameter 1
d2parameter 2
Returns
F-distribution CDF at \(x\)

Definition at line 1757 of file Statistics.cpp.

float64_t fishers_exact_test_for_2x3_table ( SGMatrix< float64_t table)
static

fisher's test for 2x3 table

Parameters
table

Definition at line 1906 of file Statistics.cpp.

SGVector< float64_t > fishers_exact_test_for_multiple_2x3_tables ( SGMatrix< float64_t tables)
static

fisher's test for multiple 2x3 tables

Parameters
tables

Definition at line 1891 of file Statistics.cpp.

CStatistics::SigmoidParamters fit_sigmoid ( SGVector< float64_t scores)
static

Converts a given vector of scores to calibrated probabilities by fitting a sigmoid function using the method described in Lin, H., Lin, C., and Weng, R. (2007). A note on Platt's probabilistic outputs for support vector machines.

This can be used to transform scores to probabilities as setting \(pf=x*a+b\) for a given score \(x\) and computing \(\frac{\exp(-f)}{1+}exp(-f)}\) if \(f\geq 0\) and \(\frac{1}{(1+\exp(f)}\) otherwise

Parameters
scoresscores to fit the sigmoid to
Returns
struct containing the sigmoid's shape parameters a and b

Definition at line 2336 of file Statistics.cpp.

float64_t gamma_cdf ( float64_t  x,
float64_t  a,
float64_t  b 
)
static

Evaluates the CDF of the gamma distribution with given parameters \(a, b\) at \(x\). Based on Wikipedia definition and ALGLIB routines.

Parameters
xposition to evaluate
ashape parameter
bscale parameter
Returns
gamma CDF at \(x\)

Definition at line 1508 of file Statistics.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.

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_namename 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_namename 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
object name

Implements CSGObject.

Definition at line 495 of file Statistics.h.

static bool greater ( float64_t  a,
float64_t  b 
)
staticprotected

method to make ALGLIB integration easier

Definition at line 661 of file Statistics.h.

static bool greater_equal ( float64_t  a,
float64_t  b 
)
staticprotected

method to make ALGLIB integration easier

Definition at line 664 of file Statistics.h.

float64_t ibetaf_incompletebetafe ( float64_t  a,
float64_t  b,
float64_t  x,
float64_t  big,
float64_t  biginv 
)
staticprotected

Continued fraction expansion #1 for incomplete beta integral

Taken from ALGLIB under gpl2+

Definition at line 1175 of file Statistics.cpp.

float64_t ibetaf_incompletebetafe2 ( float64_t  a,
float64_t  b,
float64_t  x,
float64_t  big,
float64_t  biginv 
)
staticprotected

Continued fraction expansion #2 for incomplete beta integral

Taken from ALGLIB under gpl2+

Definition at line 1278 of file Statistics.cpp.

float64_t ibetaf_incompletebetaps ( float64_t  a,
float64_t  b,
float64_t  x,
float64_t  maxgam 
)
staticprotected

Power series for incomplete beta integral. Use when \(bx\) is small and \(x\) not too close to \(1\).

Taken from ALGLIB under gpl2+

Definition at line 1121 of file Statistics.cpp.

float64_t incomplete_beta ( float64_t  a,
float64_t  b,
float64_t  x 
)
static

Incomplete beta integral

Returns incomplete beta integral of the arguments, evaluated from zero to \(x\). The function is defined as

\[ \frac{\Gamma(a+b)}{\Gamma(a)\Gamma(b)}\int_0^x t^{a-1} (1-t)^{b-1} dt. \]

The domain of definition is \(0 \leq x \leq 1\). In this implementation \(a\) and \(b\) are restricted to positive values. The integral from \(x\) to \(1\) may be obtained by the symmetry relation

\[ 1-\text{incomplete\_beta}(a,b,x)=\text{incomplete\_beta}(b,a,1-x). \]

The integral is evaluated by a continued fraction expansion or, when \(b\cdot x\) is small, by a power series.

Taken from ALGLIB under gpl2+

Definition at line 862 of file Statistics.cpp.

float64_t incomplete_gamma ( float64_t  a,
float64_t  x 
)
static

Incomplete gamma integral

Given \(p\), the function finds \(x\) such that

\[ \text{incomplete\_gamma}(a,x)=\frac{1}{\Gamma(a)}}\int_0^x e^{-t} t^{a-1} dt. \]

In this implementation both arguments must be positive. The integral is evaluated by either a power series or continued fraction expansion, depending on the relative values of \(a\) and \(x\).

Taken from ALGLIB under gpl2+

Definition at line 1383 of file Statistics.cpp.

float64_t incomplete_gamma_completed ( float64_t  a,
float64_t  x 
)
static

Complemented incomplete gamma integral

The function is defined by

\[ \text{incomplete\_gamma\_completed}(a,x)=1-\text{incomplete\_gamma}(a,x) = \frac{1}{\Gamma (a)}\int_x^\infty e^{-t} t^{a-1} dt \]

In this implementation both arguments must be positive. The integral is evaluated by either a power series or continued fraction expansion, depending on the relative values of \(a\) and \(x\).

Taken from ALGLIB under gpl2+

Definition at line 1424 of file Statistics.cpp.

float64_t inverse_gamma_cdf ( float64_t  p,
float64_t  a,
float64_t  b 
)
static

Evaluates the inverse CDF of the gamma distribution with given parameters \(a\), \(b\) at \(x\), such that result equals \(\text{gamma\_cdf}(x,a,b)\).

Parameters
pposition to evaluate
ashape parameter
bscale parameter
Returns
\(x\) such that result equals \(\text{gamma\_cdf}(x,a,b)\).

Definition at line 1514 of file Statistics.cpp.

float64_t inverse_incomplete_beta ( float64_t  a,
float64_t  b,
float64_t  y 
)
static

Inverse of incomplete beta integral

Given \(y\), the function finds \(x\) such that

\(\text{inverse\_incomplete\_beta}( a, b, x ) = y .\)

The routine performs interval halving or Newton iterations to find the root of \(\text{inverse\_incomplete\_beta}( a, b, x )-y=0.\)

Taken from ALGLIB under gpl2+

Definition at line 410 of file Statistics.cpp.

float64_t inverse_incomplete_gamma_completed ( float64_t  a,
float64_t  y0 
)
static

Inverse of complemented incomplete gamma integral

Given \(p\), the function finds \(x\) such that

\(\text{inverse\_incomplete\_gamma\_completed}( a, x ) = p.\)

Starting with the approximate value \( x=a t^3\), where \( t = 1 - d - \text{ndtri}(p) \sqrt{d} \) and \( d = \frac{1}{9}a \)

The routine performs up to 10 Newton iterations to find the root of \( \text{inverse\_incomplete\_gamma\_completed}( a, x )-p=0\)

Taken from ALGLIB under gpl2+

Definition at line 1522 of file Statistics.cpp.

float64_t inverse_normal_cdf ( float64_t  y0)
static

Inverse of Normal distribution function

Returns the argument, \(x\), for which the area under the Gaussian probability density function (integrated from minus infinity to \(x\)) is equal to \(y\).

For small arguments \(0 < y < \exp(-2)\), the program computes \(z = \sqrt{ -2.0 \log(y) }\); then the approximation is \(x = z - \frac{log(z)}{z} - \frac{1}{z} \frac{P(\frac{1}{z})}{ Q(\frac{1}{z}}\). There are two rational functions \(\frac{P}{Q}\), one for \(0 < y < \exp(-32)\) and the other for \(y\) up to \(\exp(-2)\). For larger arguments, \(w = y - 0.5\), and \(\frac{x}{\sqrt{2\pi}} = w + w^3 R(\frac{w^2)}{S(w^2)})\).

Taken from ALGLIB under gpl2+

Definition at line 1004 of file Statistics.cpp.

float64_t inverse_normal_cdf ( float64_t  y0,
float64_t  mean,
float64_t  std_dev 
)
static

same as other version, but with custom mean and variance

Definition at line 998 of file Statistics.cpp.

float64_t inverse_student_t ( int32_t  k,
float64_t  p 
)
static

Functional inverse of Student's t distribution

Given probability \(p\), finds the argument \(t\) such that \(\text{student\_t}(k,t)=p\)

Taken from ALGLIB under gpl2+

Definition at line 362 of file Statistics.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 296 of file SGObject.cpp.

static bool less ( float64_t  a,
float64_t  b 
)
staticprotected

method to make ALGLIB integration easier

Definition at line 655 of file Statistics.h.

static bool less_equal ( float64_t  a,
float64_t  b 
)
staticprotected

method to make ALGLIB integration easier

Definition at line 658 of file Statistics.h.

static float64_t lgamma ( float64_t  x)
static
Returns
natural logarithm of the gamma function of input

Definition at line 275 of file Statistics.h.

static floatmax_t lgammal ( floatmax_t  x)
static
Returns
natural logarithm of the gamma function of input for large numbers

Definition at line 282 of file Statistics.h.

float64_t lnormal_cdf ( float64_t  x)
static

returns logarithm of the cumulative distribution function (CDF) of Gaussian distribution \(N(0, 1)\):

\[ \text{lnormal\_cdf}(x)=log\left(\frac{1}{2}+ \frac{1}{2}\text{error\_function}(\frac{x}{\sqrt{2}})\right) \]

This method uses asymptotic expansion for \(x<-10.0\), otherwise it returns \(log(\text{normal\_cdf}(x))\).

Parameters
xreal value
Returns
\(log(\text{normal\_cdf}(x))\)

Definition at line 1691 of file Statistics.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
filewhere to load from
prefixprefix 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
ShogunExceptionwill 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
ShogunExceptionwill 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.

float64_t log_det ( SGMatrix< float64_t m)
static

The log determinant of a dense matrix

The log determinant of a positive definite symmetric real valued matrix is calculated as

\[ \text{log\_determinant}(M) = \text{log}(\text{determinant}(L)\times\text{determinant}(L')) = 2\times \sum_{i}\text{log}(L_{i,i}) \]

Where, \(M = L\times L'\) as per Cholesky decomposition.

Parameters
minput matrix
Returns
the log determinant value

Definition at line 2174 of file Statistics.cpp.

float64_t log_det ( const SGSparseMatrix< float64_t m)
static

The log determinant of a sparse matrix

The log determinant of symmetric positive definite sparse matrix is calculated in a similar way as the dense case. But using cholesky decomposition on sparse matrices may suffer from fill-in phenomenon, i.e. the factors may not be as sparse. The SimplicialCholesky module for sparse matrix in eigen3 library uses an approach called approximate minimum degree reordering, or amd, which permutes the matrix beforehand and results in much sparser factors. If \(P\) is the permutation matrix, it computes \(\text{LLT}(P\times M\times P^{-1}) = L\times L'\).

Parameters
minput sparse matrix
Returns
the log determinant value

Definition at line 2197 of file Statistics.cpp.

float64_t log_det_general ( const SGMatrix< float64_t A)
static

The log determinant of a dense matrix

If determinant of the input matrix is positive, it returns the logarithm of the value. If not, it returns CMath::INFTY Note that the input matrix is not required to be symmetric positive definite. This method is slower than log_det() if input matrix is known to be symmetric positive definite

It is adapted from Gaussian Process Machine Learning Toolbox http://www.gaussianprocess.org/gpml/code/matlab/doc/

Parameters
Ainput matrix
Returns
the log determinant value

Definition at line 2120 of file Statistics.cpp.

SGVector< float64_t > matrix_mean ( SGMatrix< float64_t values,
bool  col_wise = true 
)
static

Calculates mean of given values. Given \(\{x_1, ..., x_m\}\), this is \(\frac{1}{m}\sum_{i=1}^m x_i\)

Computes the mean for each row/col of matrix

Parameters
valuesvector of values
col_wiseif true, every column vector will be used, row vectors otherwise
Returns
mean of given values

Definition at line 218 of file Statistics.cpp.

float64_t matrix_median ( SGMatrix< float64_t values,
bool  modify = false,
bool  in_place = false 
)
static

Calculates median of given values. Matrix is seen as a long vector for this. The median is the value that one gets when the input vector is sorted and then selects the middle value.

This method is just a wrapper for median(). See this method for license of QuickSelect and Torben.

Parameters
valuesvector of values
modifyif false, array is modified while median is computed (Using QuickSelect). If true, median is computed without modifications, which is slower. There are two methods to choose from.
in_placeif set false, the vector is copied and then computed using QuickSelect. If set true, median is computed in-place using Torben method.
Returns
median of given values

Definition at line 192 of file Statistics.cpp.

SGVector< float64_t > matrix_std_deviation ( SGMatrix< float64_t values,
bool  col_wise = true 
)
static

Calculates unbiased empirical standard deviation estimator of given values. Given \(\{x_1, ..., x_m\}\), this is \(\sqrt{\frac{1}{m-1}\sum_{i=1}^m (x-\bar{x})^2}\) where \(\bar x=\frac{1}{m}\sum_{i=1}^m x_i\)

Computes the variance for each row/col of matrix

Parameters
valuesvector of values
col_wiseif true, every column vector will be used, row vectors otherwise
Returns
variance of given values

Definition at line 300 of file Statistics.cpp.

SGVector< float64_t > matrix_variance ( SGMatrix< float64_t values,
bool  col_wise = true 
)
static

Calculates unbiased empirical variance estimator of given values. Given \(\{x_1, ..., x_m\}\), this is \(\frac{1}{m-1}\sum_{i=1}^m (x-\bar{x})^2\) where \(\bar x=\frac{1}{m}\sum_{i=1}^m x_i\)

Computes the variance for each row/col of matrix

Parameters
valuesvector of values
col_wiseif true, every column vector will be used, row vectors otherwise
Returns
variance of given values

Definition at line 255 of file Statistics.cpp.

static floatmax_t mean ( SGVector< T >  vec)
static

Calculates mean of given values. Given \(\{x_1, ..., x_m\}\), this is \(\frac{1}{m}\sum_{i=1}^m x_i\)

Parameters
vecvector of values
Returns
mean of given values

Definition at line 44 of file Statistics.h.

floatmax_t mean ( SGVector< complex128_t vec)

mean not implemented for complex128_t, returns 0.0 instead

Definition at line 669 of file Statistics.h.

float64_t median ( SGVector< float64_t values,
bool  modify = false,
bool  in_place = false 
)
static

Calculates median of given values. The median is the value that one gets when the input vector is sorted and then selects the middle value.

QuickSelect method copyright: This Quickselect routine is based on the algorithm described in "Numerical recipes in C", Second Edition, Cambridge University Press, 1992, Section 8.5, ISBN 0-521-43108-5 This code by Nicolas Devillard - 1998. Public domain.

Torben method copyright: The following code is public domain. Algorithm by Torben Mogensen, implementation by N. Devillard. Public domain.

Both methods adapted to SHOGUN by Heiko Strathmann.

Parameters
valuesvector of values
modifyif false, array is modified while median is computed (Using QuickSelect). If true, median is computed without modifications, which is slower. There are two methods to choose from.
in_placeif set false, the vector is copied and then computed using QuickSelect. If set true, median is computed in-place using Torben method.
Returns
median of given values

Definition at line 40 of file Statistics.cpp.

float64_t mutual_info ( float64_t p1,
float64_t p2,
int32_t  len 
)
static
Returns
mutual information of \(p\) which is given in logspace where \(p,q\) are given in logspace

Definition at line 2014 of file Statistics.cpp.

float64_t normal_cdf ( float64_t  x,
float64_t  std_dev = 1 
)
static

Normal distribution function

Returns the area under the Gaussian probability density function, integrated from minus infinity to \(x\):

\[ \text{normal\_cdf}(x)=\frac{1}{\sqrt{2\pi}} \int_{-\infty}^x \exp \left( -\frac{t^2}{2} \right) dt = \frac{1+\text{error\_function}(z) }{2} \]

where \( z = \frac{x}{\sqrt{2} \sigma}\) and \( \sigma \) is the standard deviation. Computation is via the functions \(\text{error\_function}\) and \(\text{error\_function\_completement}\).

Taken from ALGLIB under gpl2+ Custom variance added by Heiko Strathmann

Definition at line 1681 of file Statistics.cpp.

static bool not_equal ( float64_t  a,
float64_t  b 
)
staticprotected

method to make ALGLIB integration easier

Definition at line 652 of file Statistics.h.

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

Definition at line 262 of file SGObject.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
prefixprefix for members

Definition at line 308 of file SGObject.cpp.

float64_t relative_entropy ( float64_t p,
float64_t q,
int32_t  len 
)
static
Returns
relative entropy \(H(P||Q)\) where \(p,q\) are given in logspace

Definition at line 2025 of file Statistics.cpp.

SGMatrix< float64_t > sample_from_gaussian ( SGVector< float64_t mean,
SGMatrix< float64_t cov,
int32_t  N = 1,
bool  precision_matrix = false 
)
static

Sampling from a multivariate Gaussian distribution with dense covariance matrix

Sampling is performed by taking samples from \(N(0, I)\), then using cholesky factor of the covariance matrix, \(\Sigma\) and performing

\[S_{N(\mu,\Sigma)}=S_{N(0,I)}*L^{T}+\mu\]

where \(\Sigma=L*L^{T}\) and \(\mu\) is the mean vector.

Parameters
meanthe mean vector
covthe covariance matrix
Nnumber of samples
precision_matrixif true, sample from N(mu,C^-1)
Returns
the sample matrix of size \(N\times dim\)

Definition at line 2217 of file Statistics.cpp.

SGMatrix< float64_t > sample_from_gaussian ( SGVector< float64_t mean,
SGSparseMatrix< float64_t cov,
int32_t  N = 1,
bool  precision_matrix = false 
)
static

Sampling from a multivariate Gaussian distribution with sparse covariance matrix

Sampling is performed in similar way as of dense covariance matrix, but direct cholesky factorization of sparse matrices could be inefficient. So, this method uses permutation matrix for factorization and then permutes back the final samples before adding the mean.

Parameters
meanthe mean vector
covthe covariance matrix
Nnumber of samples
precision_matrixif true, sample from N(mu,C^-1)
Returns
the sample matrix of size \(N\times dim\)

Definition at line 2267 of file Statistics.cpp.

SGVector< int32_t > sample_indices ( int32_t  sample_size,
int32_t  N 
)
static

sample indices

Parameters
sample_sizesize of sample to pick
Ntotal number of indices

Definition at line 2045 of file Statistics.cpp.

bool save_serializable ( CSerializableFile file,
const char *  prefix = "" 
)
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
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
ShogunExceptionwill 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
ShogunExceptionwill 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_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
ioio object to use

Definition at line 228 of file SGObject.cpp.

void set_global_parallel ( Parallel parallel)
inherited

set the parallel object

Parameters
parallelparallel object to use

Definition at line 241 of file SGObject.cpp.

void set_global_version ( Version version)
inherited

set the version object

Parameters
versionversion object to use

Definition at line 283 of file SGObject.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.

float64_t std_deviation ( SGVector< float64_t values)
static

Calculates unbiased empirical standard deviation estimator of given values. Given \(\{x_1, ..., x_m\}\), this is \(\sqrt{\frac{1}{m-1}\sum_{i=1}^m (x-\bar{x})^2}\) where \(\bar x=\frac{1}{m}\sum_{i=1}^m x_i\)

Parameters
valuesvector of values
Returns
variance of given values

Definition at line 295 of file Statistics.cpp.

static float64_t tgamma ( float64_t  x)
static
Returns
gamma function of input

Definition at line 292 of file Statistics.h.

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.

float64_t variance ( SGVector< float64_t values)
static

Calculates unbiased empirical variance estimator of given values. Given \(\{x_1, ..., x_m\}\), this is \(\frac{1}{m-1}\sum_{i=1}^m (x-\bar{x})^2\) where \(\bar x=\frac{1}{m}\sum_{i=1}^m x_i\)

Parameters
valuesvector of values
Returns
variance of given values

Definition at line 204 of file Statistics.cpp.

Member Data Documentation

const float64_t ERFC_CASE1 =0.0492
static

Magic number for computing lnormal_cdf

Definition at line 620 of file Statistics.h.

const float64_t ERFC_CASE2 =-11.3137
static

Magic number for computing lnormal_cdf

Definition at line 623 of file Statistics.h.

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.

Parameter* m_model_selection_parameters
inherited

model selection parameters

Definition at line 381 of file SGObject.h.

Parameter* m_parameters
inherited

parameters

Definition at line 378 of file SGObject.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