|
SHOGUN
4.1.0
|
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 成员函数 | |
| CNOCCO () | |
| CNOCCO (CKernel *kernel_p, CKernel *kernel_q, CFeatures *p, CFeatures *q) | |
| virtual | ~CNOCCO () |
| virtual float64_t | compute_statistic () |
| virtual float64_t | compute_p_value (float64_t statistic) |
| virtual float64_t | compute_threshold (float64_t alpha) |
| virtual const char * | get_name () const |
| virtual EStatisticType | get_statistic_type () const |
| virtual void | set_p (CFeatures *p) |
| virtual void | set_q (CFeatures *q) |
| void | set_epsilon (float64_t epsilon) |
| float64_t | get_epsilon () const |
| virtual SGVector< float64_t > | sample_null () |
| virtual void | set_kernel_p (CKernel *kernel_p) |
| virtual void | set_kernel_q (CKernel *kernel_q) |
| virtual CKernel * | get_kernel_p () |
| virtual CKernel * | get_kernel_q () |
| virtual CFeatures * | get_p () |
| virtual CFeatures * | get_q () |
| virtual float64_t | perform_test () |
| bool | perform_test (float64_t alpha) |
| virtual void | set_num_null_samples (index_t num_null_samples) |
| virtual void | set_null_approximation_method (ENullApproximationMethod null_approximation_method) |
| virtual CSGObject * | shallow_copy () const |
| virtual CSGObject * | deep_copy () const |
| virtual bool | is_generic (EPrimitiveType *generic) const |
| template<class T > | |
| void | set_generic () |
| template<> | |
| void | set_generic () |
| template<> | |
| void | set_generic () |
| template<> | |
| void | set_generic () |
| template<> | |
| void | set_generic () |
| template<> | |
| void | set_generic () |
| template<> | |
| void | set_generic () |
| template<> | |
| void | set_generic () |
| template<> | |
| void | set_generic () |
| template<> | |
| void | set_generic () |
| template<> | |
| void | set_generic () |
| template<> | |
| void | set_generic () |
| template<> | |
| void | set_generic () |
| template<> | |
| void | set_generic () |
| template<> | |
| void | set_generic () |
| template<> | |
| void | set_generic () |
| void | unset_generic () |
| virtual void | print_serializable (const char *prefix="") |
| virtual bool | save_serializable (CSerializableFile *file, const char *prefix="") |
| virtual bool | load_serializable (CSerializableFile *file, const char *prefix="") |
| void | set_global_io (SGIO *io) |
| SGIO * | get_global_io () |
| void | set_global_parallel (Parallel *parallel) |
| Parallel * | get_global_parallel () |
| void | set_global_version (Version *version) |
| Version * | get_global_version () |
| SGStringList< char > | get_modelsel_names () |
| void | print_modsel_params () |
| char * | get_modsel_param_descr (const char *param_name) |
| index_t | get_modsel_param_index (const char *param_name) |
| void | build_gradient_parameter_dictionary (CMap< TParameter *, CSGObject * > *dict) |
| virtual void | update_parameter_hash () |
| virtual bool | parameter_hash_changed () |
| virtual bool | equals (CSGObject *other, float64_t accuracy=0.0, bool tolerant=false) |
| virtual CSGObject * | clone () |
Public 属性 | |
| SGIO * | io |
| Parallel * | parallel |
| Version * | version |
| Parameter * | m_parameters |
| Parameter * | m_model_selection_parameters |
| Parameter * | m_gradient_parameters |
| uint32_t | m_hash |
Protected 成员函数 | |
| SGMatrix< float64_t > | compute_helper (SGMatrix< float64_t > m) |
| SGMatrix< float64_t > | get_kernel_matrix_K () |
| SGMatrix< float64_t > | get_kernel_matrix_L () |
| virtual void | load_serializable_pre () throw (ShogunException) |
| virtual void | load_serializable_post () throw (ShogunException) |
| virtual void | save_serializable_pre () throw (ShogunException) |
| virtual void | save_serializable_post () throw (ShogunException) |
Protected 属性 | |
| 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 |
|
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.
| dict | dictionary of parameters to be built. |
在文件 SGObject.cpp 第 597 行定义.
|
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.
在文件 SGObject.cpp 第 714 行定义.
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 |
重载 CHypothesisTest .
|
virtual |
Computes the NOCCO statistic (see class description) for underlying kernels and data.
Note that since kernel matrices have to be stored, it has quadratic space costs.
实现了 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 |
重载 CHypothesisTest .
|
virtualinherited |
A deep copy. All the instance variables will also be copied.
在文件 SGObject.cpp 第 198 行定义.
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) |
在文件 SGObject.cpp 第 618 行定义.
|
inherited |
|
inherited |
|
inherited |
在文件 KernelIndependenceTest.cpp 第 158 行定义.
在文件 KernelIndependenceTest.cpp 第 184 行定义.
|
virtualinherited |
Getter for kernel for features from p, SG_REF'ed
在文件 KernelIndependenceTest.cpp 第 146 行定义.
|
virtualinherited |
Getter for kernel for features from q, SG_REF'ed
在文件 KernelIndependenceTest.cpp 第 152 行定义.
|
inherited |
在文件 SGObject.cpp 第 498 行定义.
|
inherited |
Returns description of a given parameter string, if it exists. SG_ERROR otherwise
| param_name | name of the parameter |
在文件 SGObject.cpp 第 522 行定义.
|
inherited |
Returns index of model selection parameter with provided index
| param_name | name of model selection parameter |
在文件 SGObject.cpp 第 535 行定义.
|
virtual |
|
virtualinherited |
|
virtualinherited |
|
virtual |
|
virtualinherited |
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 |
在文件 SGObject.cpp 第 296 行定义.
|
virtualinherited |
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 |
在文件 SGObject.cpp 第 369 行定义.
|
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. |
被 CKernel, CWeightedDegreePositionStringKernel, CList, CAlphabet, CLinearHMM, CGaussianKernel, CInverseMultiQuadricKernel, CCircularKernel , 以及 CExponentialKernel 重载.
在文件 SGObject.cpp 第 426 行定义.
|
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. |
被 CDynamicArray< T >, CDynamicArray< float64_t >, CDynamicArray< float32_t >, CDynamicArray< int32_t >, CDynamicArray< char >, CDynamicArray< bool > , 以及 CDynamicObjectArray 重载.
在文件 SGObject.cpp 第 421 行定义.
|
virtualinherited |
在文件 SGObject.cpp 第 262 行定义.
|
virtualinherited |
Performs the complete two-sample test on current data and returns a p-value.
This is a wrapper that calls compute_statistic first and then calls compute_p_value using the obtained statistic. In some statistic classes, it might be possible to compute statistic and p-value in one single run which is more efficient. Therefore, this method might be overwritten in subclasses.
The method for computing the p-value can be set via set_null_approximation_method().
被 CStreamingMMD 重载.
在文件 HypothesisTest.cpp 第 113 行定义.
|
inherited |
Performs the complete two-sample test on current data and returns a binary answer wheter null hypothesis is rejected or not.
This is just a wrapper for the above perform_test() method that returns a p-value. If this p-value lies below the test level alpha, the null hypothesis is rejected.
Should not be overwritten in subclasses. (Therefore not virtual)
| alpha | test level alpha. |
在文件 HypothesisTest.cpp 第 121 行定义.
|
inherited |
prints all parameter registered for model selection and their type
在文件 SGObject.cpp 第 474 行定义.
|
virtualinherited |
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.
|
virtualinherited |
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 |
在文件 SGObject.cpp 第 314 行定义.
|
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. |
被 CKernel 重载.
在文件 SGObject.cpp 第 436 行定义.
|
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. |
被 CKernel, CDynamicArray< T >, CDynamicArray< float64_t >, CDynamicArray< float32_t >, CDynamicArray< int32_t >, CDynamicArray< char >, CDynamicArray< bool > , 以及 CDynamicObjectArray 重载.
在文件 SGObject.cpp 第 431 行定义.
| void set_epsilon | ( | float64_t | epsilon | ) |
|
inherited |
在文件 SGObject.cpp 第 41 行定义.
|
inherited |
在文件 SGObject.cpp 第 46 行定义.
|
inherited |
在文件 SGObject.cpp 第 51 行定义.
|
inherited |
在文件 SGObject.cpp 第 56 行定义.
|
inherited |
在文件 SGObject.cpp 第 61 行定义.
|
inherited |
在文件 SGObject.cpp 第 66 行定义.
|
inherited |
在文件 SGObject.cpp 第 71 行定义.
|
inherited |
在文件 SGObject.cpp 第 76 行定义.
|
inherited |
在文件 SGObject.cpp 第 81 行定义.
|
inherited |
在文件 SGObject.cpp 第 86 行定义.
|
inherited |
在文件 SGObject.cpp 第 91 行定义.
|
inherited |
在文件 SGObject.cpp 第 96 行定义.
|
inherited |
在文件 SGObject.cpp 第 101 行定义.
|
inherited |
在文件 SGObject.cpp 第 106 行定义.
|
inherited |
在文件 SGObject.cpp 第 111 行定义.
|
inherited |
set generic type to T
|
inherited |
|
inherited |
|
inherited |
|
virtualinherited |
Setter for kernel for features from distribution p, SG_REFs it
| kernel_p | kernel for features from p |
在文件 KernelIndependenceTest.cpp 第 130 行定义.
|
virtualinherited |
Setter for kernel for features from distribution q, SG_REFs it
| kernel_q | kernel for features from q |
在文件 KernelIndependenceTest.cpp 第 138 行定义.
|
virtualinherited |
sets the method how to approximate the null-distribution
| null_approximation_method | method to use |
在文件 HypothesisTest.cpp 第 61 行定义.
|
virtualinherited |
sets the number of permutation iterations for sample_null()
| num_null_samples | how often permutation shall be done |
在文件 HypothesisTest.cpp 第 67 行定义.
|
virtual |
|
virtual |
|
virtualinherited |
A shallow copy. All the SGObject instance variables will be simply assigned and SG_REF-ed.
被 CGaussianKernel 重载.
在文件 SGObject.cpp 第 192 行定义.
|
inherited |
unset generic type
this has to be called in classes specializing a template class
在文件 SGObject.cpp 第 303 行定义.
|
virtualinherited |
Updates the hash of current parameter combination
在文件 SGObject.cpp 第 248 行定义.
|
inherited |
io
在文件 SGObject.h 第 369 行定义.
|
inherited |
parameters wrt which we can compute gradients
在文件 SGObject.h 第 384 行定义.
|
inherited |
Hash of parameter values
在文件 SGObject.h 第 387 行定义.
|
protectedinherited |
underlying kernel for p
在文件 KernelIndependenceTest.h 第 137 行定义.
|
protectedinherited |
underlying kernel for q
在文件 KernelIndependenceTest.h 第 140 行定义.
|
inherited |
model selection parameters
在文件 SGObject.h 第 381 行定义.
|
protectedinherited |
Defines how the the null distribution is approximated
在文件 HypothesisTest.h 第 177 行定义.
|
protectedinherited |
number of iterations for sampling from null-distributions
在文件 HypothesisTest.h 第 174 行定义.
|
protectedinherited |
samples of the distribution p
在文件 IndependenceTest.h 第 116 行定义.
|
inherited |
parameters
在文件 SGObject.h 第 378 行定义.
|
protectedinherited |
samples of the distribution q
在文件 IndependenceTest.h 第 119 行定义.
|
inherited |
parallel
在文件 SGObject.h 第 372 行定义.
|
inherited |
version
在文件 SGObject.h 第 375 行定义.
