SHOGUN
3.2.1
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This class implements the NOrmalized Cross Covariance Operator (NOCCO) based independence test as described in [1].
The test of independence is performed as follows: Given samples \(Z=\{(x_i, y_i)\}_{i=1}^n\) from the joint distribution \(\textbf{P}_{XY}\), does the joint distribution factorize as \(\textbf{P}_{XY}=\textbf{P}_X \textbf{P}_Y\)? The null hypothesis says yes and the alternative hypothesis says no.
The dependence of the random variables \(\mathbf X=\{x_i\}\) and \( \mathbf Y=\{y_i\}\) can be measured via the cross-covariance operator \(\boldsymbol\Sigma_{YX}\) which becomes \(\mathbf{0}\) if and only if \(\mathbf X\) and \(\mathbf Y\) are independent. This term factorizes as \(\boldsymbol\Sigma_{YX}=\boldsymbol\Sigma_{YY}^{\frac{1}{2}}\mathbf{V}_ {YX}\boldsymbol\Sigma_{XX}^{\frac{1}{2}}\), where \(\boldsymbol\Sigma_ {XX}\) and \(\boldsymbol\Sigma_{YY}\) are known as covariance operator and \(\mathbf{V}_{YX}\) is known as normalized cross-covariance operator. The paper uses the Hilbert-Schmidt norm of \(\mathbf V_{YX}\) as a dependence measure of the independence test (see paper for theroretical details).
This class overrides the compute_statistic() method of the superclass which computes an unbiased estimate of the normalized cross covariance operator norm. Given the kernels \(K\) (for \(\mathbf X\)) and \(L\) (for \(\mathbf Y\)), if we denote the doubly centered Gram matrices as \(\mathbf{G}_X=\mathbf{HKH}\) and \(\mathbf{G}_Y=\mathbf{HLH}\) (where \(\mathbf H=\mathbf I-\frac{1}{n}\mathbf{1}\)), then the operator norm is estimated as
\[ \hat{I}^{\text{NOCCO}}=\text{Trace}\left[\mathbf{R_X R_Y}\right] \]
where \(\mathbf{R}_X=\mathbf{G}_X(\mathbf{G}_X+n\varepsilon_n\mathbf{I}) ^{-1}\) and \(\mathbf{R}_Y=\mathbf{G}_Y(\mathbf{G}_Y+n\varepsilon_n \mathbf{I})^{-1}\) and \(\varepsilon_n\gt 0\) is a regularization constant.
In order to avoid computing direct inverse in the above terms for avoiding numerical issues, this class uses Cholesky decomposition of matrices \(\mathbf{GG}_*=\mathbf{LL}^\top\) (where \(\mathbf{GG}_*=(\mathbf{G}_*+ n\varepsilon_n\mathbf{I})^{-1}\)) and solve systems \(\mathbf{GG}_* \mathbf x_i=\mathbf{LL}^\top\mathbf x_i=\mathbf e_i\) ( \(\mathbf e_i\) being the \(i^{\text{th}}\) column of \(\mathbf I_n\)) one by one. On the fly it then uses the solution vectors \(\mathbf x_i\) to compute the matrix-matrix product \(\mathbf C_*=\mathbf G_*\mathbf{GG}_*^{-1}\) using \(\mathbf C_{*,(j,i)}=\mathbf G_{*,j}\cdot \mathbf x_i\), where $ G_{*,j}$ is the \(j^{\text{th}}\) row of \(\mathbf G_*$ (or column, since it is symmetric) and then discarding the vector. The final trace computation is also simplified using the symmetry of the matrices \) R_X \( and \) R_Y
Public Member Functions | |
CNOCCO () | |
CNOCCO (CKernel *kernel_p, CKernel *kernel_q, CFeatures *p, CFeatures *q) | |
virtual | ~CNOCCO () |
virtual float64_t | compute_statistic () |
virtual float64_t | compute_p_value (float64_t statistic) |
virtual float64_t | compute_threshold (float64_t alpha) |
virtual const char * | get_name () const |
virtual EStatisticType | get_statistic_type () const |
virtual void | set_p (CFeatures *p) |
virtual void | set_q (CFeatures *q) |
void | set_epsilon (float64_t epsilon) |
float64_t | get_epsilon () const |
virtual SGVector< float64_t > | sample_null () |
virtual void | set_kernel_p (CKernel *kernel_p) |
virtual void | set_kernel_q (CKernel *kernel_q) |
virtual CKernel * | get_kernel_p () |
virtual CKernel * | get_kernel_q () |
virtual CFeatures * | get_p () |
virtual CFeatures * | get_q () |
virtual float64_t | perform_test () |
bool | perform_test (float64_t alpha) |
virtual void | set_num_null_samples (index_t num_null_samples) |
virtual void | set_null_approximation_method (ENullApproximationMethod null_approximation_method) |
virtual CSGObject * | shallow_copy () const |
virtual CSGObject * | deep_copy () const |
virtual bool | is_generic (EPrimitiveType *generic) const |
template<class T > | |
void | set_generic () |
template<> | |
void | set_generic () |
template<> | |
void | set_generic () |
template<> | |
void | set_generic () |
template<> | |
void | set_generic () |
template<> | |
void | set_generic () |
template<> | |
void | set_generic () |
template<> | |
void | set_generic () |
template<> | |
void | set_generic () |
template<> | |
void | set_generic () |
template<> | |
void | set_generic () |
template<> | |
void | set_generic () |
template<> | |
void | set_generic () |
template<> | |
void | set_generic () |
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void | set_generic () |
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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::get_version_parameter()) |
virtual bool | load_serializable (CSerializableFile *file, const char *prefix="", int32_t param_version=Version::get_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_gradient_parameter_dictionary (CMap< TParameter *, CSGObject * > *dict) |
virtual void | update_parameter_hash () |
virtual bool | parameter_hash_changed () |
virtual bool | equals (CSGObject *other, float64_t accuracy=0.0, bool tolerant=false) |
virtual CSGObject * | clone () |
Public Attributes | |
SGIO * | io |
Parallel * | parallel |
Version * | version |
Parameter * | m_parameters |
Parameter * | m_model_selection_parameters |
Parameter * | m_gradient_parameters |
ParameterMap * | m_parameter_map |
uint32_t | m_hash |
Protected Member Functions | |
SGMatrix< float64_t > | compute_helper (SGMatrix< float64_t > m) |
SGMatrix< float64_t > | get_kernel_matrix_K () |
SGMatrix< float64_t > | get_kernel_matrix_L () |
virtual 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) |
Protected Attributes | |
CKernel * | m_kernel_p |
CKernel * | m_kernel_q |
CFeatures * | m_p |
CFeatures * | m_q |
index_t | m_num_null_samples |
ENullApproximationMethod | m_null_approximation_method |
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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.
dict | dictionary of parameters to be built. |
Definition at line 1243 of file SGObject.cpp.
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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.
Definition at line 1360 of file SGObject.cpp.
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.
statistic | statistic value to compute the p-value for |
Reimplemented from CHypothesisTest.
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Computes the NOCCO statistic (see class description) for underlying kernels and data.
Note that since kernel matrices have to be stored, it has quadratic space costs.
Implements CHypothesisTest.
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.
alpha | test level to reject null-hypothesis |
Reimplemented from CHypothesisTest.
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virtualinherited |
A deep copy. All the instance variables will also be copied.
Definition at line 200 of file SGObject.cpp.
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.
other | object to compare with |
accuracy | accuracy to use for comparison (optional) |
tolerant | allows linient check on float equality (within accuracy) |
Definition at line 1264 of file SGObject.cpp.
float64_t get_epsilon | ( | ) | const |
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Definition at line 158 of file KernelIndependenceTest.cpp.
Definition at line 184 of file KernelIndependenceTest.cpp.
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Getter for kernel for features from p, SG_REF'ed
Definition at line 146 of file KernelIndependenceTest.cpp.
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Getter for kernel for features from q, SG_REF'ed
Definition at line 152 of file KernelIndependenceTest.cpp.
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Definition at line 1135 of file SGObject.cpp.
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Returns description of a given parameter string, if it exists. SG_ERROR otherwise
param_name | name of the parameter |
Definition at line 1159 of file SGObject.cpp.
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Returns index of model selection parameter with provided index
param_name | name of model selection parameter |
Definition at line 1172 of file SGObject.cpp.
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Getter for features from p, SG_REF'ed
Definition at line 121 of file IndependenceTest.cpp.
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Getter for features from q, SG_REF'ed
Definition at line 127 of file IndependenceTest.cpp.
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Implements CHypothesisTest.
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If the SGSerializable is a class template then TRUE will be returned and GENERIC is set to the type of the generic.
generic | set to the type of the generic if returning TRUE |
Definition at line 297 of file SGObject.cpp.
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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)
file_version | parameter version of the file |
current_version | version from which mapping begins (you want to use Version::get_version_parameter() for this in most cases) |
file | file to load from |
prefix | prefix for members |
Definition at line 704 of file SGObject.cpp.
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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
param_info | information of parameter |
file_version | parameter version of the file, must be <= provided parameter version |
file | file to load from |
prefix | prefix for members |
Definition at line 545 of file SGObject.cpp.
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Load this object from file. If it will fail (returning FALSE) then this object will contain inconsistent data and should not be used!
file | where to load from |
prefix | prefix for members |
param_version | (optional) a parameter version different to (this is mainly for testing, better do not use) |
Definition at line 374 of file SGObject.cpp.
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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.
ShogunException | will be thrown if an error occurs. |
Reimplemented in CKernel, CWeightedDegreePositionStringKernel, CList, CAlphabet, CLinearHMM, CGaussianKernel, CInverseMultiQuadricKernel, CCircularKernel, and CExponentialKernel.
Definition at line 1062 of file SGObject.cpp.
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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.
ShogunException | will be thrown if an error occurs. |
Reimplemented in CDynamicArray< T >, CDynamicArray< float64_t >, CDynamicArray< float32_t >, CDynamicArray< int32_t >, CDynamicArray< char >, CDynamicArray< bool >, and CDynamicObjectArray.
Definition at line 1057 of file SGObject.cpp.
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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
param_base | set of TParameter instances that are mapped to the provided target parameter infos |
base_version | version of the parameter base |
target_param_infos | set of SGParamInfo instances that specify the target parameter base |
Definition at line 742 of file SGObject.cpp.
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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
param_base | set of TParameter instances to use for migration |
target | parameter info for the resulting TParameter |
Definition at line 949 of file SGObject.cpp.
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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)
param_base | set of TParameter instances to use for migration |
target | parameter info for the resulting TParameter |
replacement | (used as output) here the TParameter instance which is returned by migration is created into |
to_migrate | the only source that is used for migration |
old_name | with this parameter, a name change may be specified |
Definition at line 889 of file SGObject.cpp.
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Definition at line 263 of file SGObject.cpp.
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Performs the complete two-sample test on current data and returns a p-value.
This is a wrapper that calls compute_statistic first and then calls compute_p_value using the obtained statistic. In some statistic classes, it might be possible to compute statistic and p-value in one single run which is more efficient. Therefore, this method might be overwritten in subclasses.
The method for computing the p-value can be set via set_null_approximation_method().
Reimplemented in CStreamingMMD.
Definition at line 112 of file HypothesisTest.cpp.
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Performs the complete two-sample test on current data and returns a binary answer wheter null hypothesis is rejected or not.
This is just a wrapper for the above perform_test() method that returns a p-value. If this p-value lies below the test level alpha, the null hypothesis is rejected.
Should not be overwritten in subclasses. (Therefore not virtual)
alpha | test level alpha. |
Definition at line 120 of file HypothesisTest.cpp.
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prints all parameter registered for model selection and their type
Definition at line 1111 of file SGObject.cpp.
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prints registered parameters out
prefix | prefix for members |
Definition at line 309 of file SGObject.cpp.
Merges both sets of samples and computes the test statistic m_num_null_sample times. This version precomputes the kenrel matrix once by hand, then samples using this one. The matrix has to be stored anyway when statistic is computed.
Reimplemented from CKernelIndependenceTest.
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Save this object to file.
file | where to save the object; will be closed during returning if PREFIX is an empty string. |
prefix | prefix for members |
param_version | (optional) a parameter version different to (this is mainly for testing, better do not use) |
Definition at line 315 of file SGObject.cpp.
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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.
ShogunException | will be thrown if an error occurs. |
Reimplemented in CKernel.
Definition at line 1072 of file SGObject.cpp.
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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.
ShogunException | will be thrown if an error occurs. |
Reimplemented in CKernel, CDynamicArray< T >, CDynamicArray< float64_t >, CDynamicArray< float32_t >, CDynamicArray< int32_t >, CDynamicArray< char >, CDynamicArray< bool >, and CDynamicObjectArray.
Definition at line 1067 of file SGObject.cpp.
void set_epsilon | ( | float64_t | epsilon | ) |
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Definition at line 42 of file SGObject.cpp.
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Definition at line 47 of file SGObject.cpp.
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Definition at line 52 of file SGObject.cpp.
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Definition at line 57 of file SGObject.cpp.
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Definition at line 62 of file SGObject.cpp.
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Definition at line 67 of file SGObject.cpp.
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Definition at line 72 of file SGObject.cpp.
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Definition at line 77 of file SGObject.cpp.
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Definition at line 82 of file SGObject.cpp.
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Definition at line 87 of file SGObject.cpp.
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Definition at line 92 of file SGObject.cpp.
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Definition at line 97 of file SGObject.cpp.
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Definition at line 102 of file SGObject.cpp.
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Definition at line 107 of file SGObject.cpp.
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Definition at line 112 of file SGObject.cpp.
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set generic type to T
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set the parallel object
parallel | parallel object to use |
Definition at line 243 of file SGObject.cpp.
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set the version object
version | version object to use |
Definition at line 284 of file SGObject.cpp.
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Setter for kernel for features from distribution p, SG_REFs it
kernel_p | kernel for features from p |
Definition at line 130 of file KernelIndependenceTest.cpp.
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Setter for kernel for features from distribution q, SG_REFs it
kernel_q | kernel for features from q |
Definition at line 138 of file KernelIndependenceTest.cpp.
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sets the method how to approximate the null-distribution
null_approximation_method | method to use |
Definition at line 60 of file HypothesisTest.cpp.
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sets the number of permutation iterations for sample_null()
num_null_samples | how often permutation shall be done |
Definition at line 66 of file HypothesisTest.cpp.
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Setter for features from distribution p, SG_REFs it
p | features from p |
Reimplemented from CIndependenceTest.
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Setter for features from distribution q, SG_REFs it
q | features from q |
Reimplemented from CIndependenceTest.
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A shallow copy. All the SGObject instance variables will be simply assigned and SG_REF-ed.
Reimplemented in CGaussianKernel.
Definition at line 194 of file SGObject.cpp.
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unset generic type
this has to be called in classes specializing a template class
Definition at line 304 of file SGObject.cpp.
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Updates the hash of current parameter combination
Definition at line 250 of file SGObject.cpp.
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io
Definition at line 496 of file SGObject.h.
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parameters wrt which we can compute gradients
Definition at line 511 of file SGObject.h.
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Hash of parameter values
Definition at line 517 of file SGObject.h.
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underlying kernel for p
Definition at line 137 of file KernelIndependenceTest.h.
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underlying kernel for q
Definition at line 140 of file KernelIndependenceTest.h.
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model selection parameters
Definition at line 508 of file SGObject.h.
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Defines how the the null distribution is approximated
Definition at line 177 of file HypothesisTest.h.
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number of iterations for sampling from null-distributions
Definition at line 174 of file HypothesisTest.h.
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samples of the distribution p
Definition at line 116 of file IndependenceTest.h.
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map for different parameter versions
Definition at line 514 of file SGObject.h.
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parameters
Definition at line 505 of file SGObject.h.
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samples of the distribution q
Definition at line 119 of file IndependenceTest.h.
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parallel
Definition at line 499 of file SGObject.h.
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version
Definition at line 502 of file SGObject.h.