SHOGUN
3.2.1
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This class implements the quadratic time Maximum Mean Statistic as described in [1]. The MMD is the distance of two probability distributions \(p\) and \(q\) in a RKHS
\[ \text{MMD}[\mathcal{F},p,q]^2=\textbf{E}_{x,x'}\left[ k(x,x')\right]- 2\textbf{E}_{x,y}\left[ k(x,y)\right] +\textbf{E}_{y,y'}\left[ k(y,y')\right]=||\mu_p - \mu_q||^2_\mathcal{F} \]
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Given two sets of samples \(\{x_i\}_{i=1}^m\sim p\) and \(\{y_i\}_{i=1}^n\sim q\) the (unbiased) statistic is computed as
\[ \text{MMD}_u^2[\mathcal{F},X,Y]=\frac{1}{m(m-1)}\sum_{i=1}^m\sum_{j\neq i}^m k(x_i,x_j) + \frac{1}{n(n-1)}\sum_{i=1}^n\sum_{j\neq i}^nk(y_i,y_j) - \frac{2}{mn}\sum_{i=1}^m\sum_{j=1}^nk(x_i,y_j) \]
A biased version is
\[ \text{MMD}_b^2[\mathcal{F},X,Y]=\frac{1}{m^2}\sum_{i=1}^m\sum_{j=1}^m k(x_i,x_j) + \frac{1}{n^2}\sum_{i=1}^n\sum_{j=1}^nk(y_i,y_j) - \frac{2}{mn}\sum_{i=1}^m\sum_{j=1}^nk(x_i,y_j) \]
The type (biased/unbiased) can be selected via set_statistic_type(). Note that computing the statistic returns m*MMD; same holds for the null distribution samples.
Along with the statistic comes a method to compute a p-value based on different methods. Sampling from null is also possible. If unsure which one to use, sampling with 250 permutation iterations always is correct (but slow).
To choose, use set_null_approximation_method() and choose from.
If you do not know about your data, but want to use the MMD from a kernel matrix, just use the custom kernel constructor. Everything else will work as usual.
MMD2_SPECTRUM: for a fast, consistent test based on the spectrum of the kernel matrix, as described in [2]. Only supported if LAPACK is installed.
MMD2_GAMMA: for a very fast, but not consistent test based on moment matching of a Gamma distribution, as described in [2].
PERMUTATION: For permuting available samples to sample null-distribution
For kernel selection see CMMDKernelSelection.
[1]: Gretton, A., Borgwardt, K. M., Rasch, M. J., Schoelkopf, B., & Smola, A. (2012). A Kernel Two-Sample Test. Journal of Machine Learning Research, 13, 671-721.
[2]: Gretton, A., Fukumizu, K., & Harchaoui, Z. (2011). A fast, consistent kernel two-sample test.
Definition at line 107 of file QuadraticTimeMMD.h.
Public Member Functions | |
CQuadraticTimeMMD () | |
CQuadraticTimeMMD (CKernel *kernel, CFeatures *p_and_q, index_t m) | |
CQuadraticTimeMMD (CKernel *kernel, CFeatures *p, CFeatures *q) | |
CQuadraticTimeMMD (CCustomKernel *custom_kernel, index_t m) | |
virtual | ~CQuadraticTimeMMD () |
virtual float64_t | compute_statistic () |
virtual SGVector< float64_t > | compute_statistic (bool multiple_kernels) |
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 |
SGVector< float64_t > | sample_null_spectrum (index_t num_samples, index_t num_eigenvalues) |
void | set_num_samples_sepctrum (index_t num_samples_spectrum) |
void | set_num_eigenvalues_spectrum (index_t num_eigenvalues_spectrum) |
void | set_statistic_type (EQuadraticMMDType statistic_type) |
SGVector< float64_t > | fit_null_gamma () |
virtual void | set_kernel (CKernel *kernel) |
virtual CKernel * | get_kernel () |
virtual SGVector< float64_t > | sample_null () |
virtual void | set_p_and_q (CFeatures *p_and_q) |
virtual CFeatures * | get_p_and_q () |
index_t | get_m () |
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 () |
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void | set_generic () |
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void | set_generic () |
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void | set_generic () |
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void | set_generic () |
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void | set_generic () |
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void | set_generic () |
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void | set_generic () |
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void | set_generic () |
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void | set_generic () |
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void | set_generic () |
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void | set_generic () |
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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 | |
virtual float64_t | compute_unbiased_statistic () |
virtual float64_t | compute_biased_statistic () |
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) |
Definition at line 40 of file QuadraticTimeMMD.cpp.
CQuadraticTimeMMD | ( | CKernel * | kernel, |
CFeatures * | p_and_q, | ||
index_t | m | ||
) |
Constructor
p_and_q | feature data. Is assumed to contain samples from both p and q. First all samples from p, then from index m all samples from q |
kernel | kernel to use |
p_and_q | samples from p and q, appended |
m | index of first sample of q |
Definition at line 45 of file QuadraticTimeMMD.cpp.
CQuadraticTimeMMD | ( | CKernel * | kernel, |
CFeatures * | p, | ||
CFeatures * | q | ||
) |
Constructor. This is a convienience constructor which copies both features to one element and then calls the other constructor. Needs twice the memory for a short time
kernel | kernel for MMD |
p | samples from distribution p, will be copied and NOT SG_REF'ed |
q | samples from distribution q, will be copied and NOT SG_REF'ed |
Definition at line 58 of file QuadraticTimeMMD.cpp.
CQuadraticTimeMMD | ( | CCustomKernel * | custom_kernel, |
index_t | m | ||
) |
Constructor. This is a convienience constructor which copies allows to only specify a custom kernel. In this case, the features are completely ignored and all computations will be done on the custom kernel
custom_kernel | custom kernel for MMD, which is a kernel between the appended features p and q |
m | index of first sample of q |
Definition at line 70 of file QuadraticTimeMMD.cpp.
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Definition at line 76 of file QuadraticTimeMMD.cpp.
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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 1174 of file SGObject.cpp.
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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.
Definition at line 1291 of file SGObject.cpp.
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helper method to compute m*biased squared quadratic time MMD
Definition at line 137 of file QuadraticTimeMMD.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.
Not all methods for computing the p-value are compatible with all methods of computing the statistic (biased/unbiased).
statistic | statistic value to compute the p-value for |
Reimplemented from CTwoSampleTest.
Definition at line 197 of file QuadraticTimeMMD.cpp.
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Computes the squared quadratic time MMD for the current data. Note that the type (biased/unbiased) can be specified with set_statistic_type() method. Note that it returns m*MMD.
Implements CKernelTwoSampleTest.
Definition at line 175 of file QuadraticTimeMMD.cpp.
Same as compute_statistic(), but with the possibility to perform on multiple kernels at once
multiple_kernels | if true, and underlying kernel is K_COMBINED, method will be executed on all subkernels on the same data |
Implements CKernelTwoSampleTest.
Definition at line 234 of file QuadraticTimeMMD.cpp.
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.
Not all methods for computing the p-value are compatible with all methods of computing the statistic (biased/unbiased).
alpha | test level to reject null-hypothesis |
Reimplemented from CTwoSampleTest.
Definition at line 275 of file QuadraticTimeMMD.cpp.
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helper method to compute m*unbiased squared quadratic time MMD
Definition at line 97 of file QuadraticTimeMMD.cpp.
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A deep copy. All the instance variables will also be copied.
Definition at line 126 of file SGObject.h.
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 1195 of file SGObject.cpp.
Approximates the null-distribution by the two parameter gamma distribution. It works in O(m^2) where m is the number of samples from each distribution. Its very fast, but may be inaccurate. However, there are cases where it performs very well. Returns parameters of gamma distribution that is fitted.
Called by compute_p_value() if null approximation method is set to MMD2_GAMMA.
Note that when being used for constructing a test, the provided statistic HAS to be the biased version (see paper for details). Note that m*Null-distribution is fitted, which is fine since the statistic is also m*MMD.
See Gretton, A., Fukumizu, K., & Harchaoui, Z. (2011). A fast, consistent kernel two-sample test.
Definition at line 387 of file QuadraticTimeMMD.cpp.
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Definition at line 84 of file KernelTwoSampleTest.h.
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Definition at line 122 of file TwoSampleTest.h.
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Definition at line 1066 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 1090 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 1103 of file SGObject.cpp.
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Returns the name of the SGSerializable instance. It MUST BE the CLASS NAME without the prefixed `C'.
Implements CKernelTwoSampleTest.
Definition at line 192 of file QuadraticTimeMMD.h.
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Getter for joint features, SG_REF'ed
Reimplemented in CStreamingMMD.
Definition at line 161 of file TwoSampleTest.cpp.
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returns the statistic type of this test statistic
Implements CHypothesisTest.
Definition at line 198 of file QuadraticTimeMMD.h.
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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 |
Definition at line 234 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 639 of file SGObject.cpp.
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inherited |
loads some specified parameters from a file with a specified version The provided parameter info has a version which is recursively mapped until the file parameter version is reached. Note that there may be possibly multiple parameters in the mapping, therefore, a set of TParameter instances is returned
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 480 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 311 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 occurres. |
Reimplemented in CWeightedDegreePositionStringKernel, CKernel, CList, CAlphabet, CLinearHMM, CGaussianKernel, CInverseMultiQuadricKernel, CCircularKernel, and CExponentialKernel.
Definition at line 995 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 occurres. |
Reimplemented in CDynamicArray< T >, CDynamicArray< float64_t >, CDynamicArray< float32_t >, CDynamicArray< int32_t >, CDynamicArray< char >, CDynamicArray< bool >, and CDynamicObjectArray.
Definition at line 990 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 677 of file SGObject.cpp.
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protectedvirtualinherited |
creates a new TParameter instance, which contains migrated data from the version that is provided. The provided parameter data base is used for migration, this base is a collection of all parameter data of the previous version. Migration is done FROM the data in param_base TO the provided param info Migration is always one version step. Method has to be implemented in subclasses, if no match is found, base method has to be called.
If there is an element in the param_base which equals the target, a copy of the element is returned. This represents the case when nothing has changed and therefore, the migrate method is not overloaded in a subclass
param_base | set of TParameter instances to use for migration |
target | parameter info for the resulting TParameter |
Definition at line 884 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 824 of file SGObject.cpp.
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Definition at line 200 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 110 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 118 of file HypothesisTest.cpp.
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prints all parameter registered for model selection and their type
Definition at line 1042 of file SGObject.cpp.
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prints registered parameters out
prefix | prefix for members |
Definition at line 246 of file SGObject.cpp.
merges both sets of samples and computes the test statistic m_num_null_samples times. This version checks if a precomputed custom kernel is used, and, if so, just permutes it instead of re- computing it in every iteration.
Reimplemented from CTwoSampleTest.
Reimplemented in CStreamingMMD.
Definition at line 54 of file KernelTwoSampleTest.cpp.
Returns a set of samples of an estimate of the null distribution using the Eigen-spectrum of the centered kernel matrix of the merged samples of p and q. May be used to compute p_value (easy)
kernel matrix needs to be stored in memory
Note that the provided statistic HAS to be the biased version (see paper for details). Note that m*Null-distribution is returned, which is fine since the statistic is also m*MMD:
Works well if the kernel matrix is NOT diagonal dominant. See Gretton, A., Fukumizu, K., & Harchaoui, Z. (2011). A fast, consistent kernel two-sample test.
num_samples | number of samples to draw |
num_eigenvalues | number of eigenvalues to use to draw samples Maximum number of 2m-1 where m is the size of both sets of samples. It is usually safe to use a smaller number since they decay very fast, however, a conservative approach would be to use all (-1 does this). See paper for details. |
Definition at line 314 of file QuadraticTimeMMD.cpp.
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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 252 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 occurres. |
Reimplemented in CKernel.
Definition at line 1005 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 occurres. |
Reimplemented in CKernel, CDynamicArray< T >, CDynamicArray< float64_t >, CDynamicArray< float32_t >, CDynamicArray< int32_t >, CDynamicArray< char >, CDynamicArray< bool >, and CDynamicObjectArray.
Definition at line 1000 of file SGObject.cpp.
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Definition at line 41 of file SGObject.cpp.
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Definition at line 46 of file SGObject.cpp.
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Definition at line 51 of file SGObject.cpp.
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Definition at line 56 of file SGObject.cpp.
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Definition at line 61 of file SGObject.cpp.
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Definition at line 66 of file SGObject.cpp.
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Definition at line 71 of file SGObject.cpp.
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Definition at line 76 of file SGObject.cpp.
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Definition at line 81 of file SGObject.cpp.
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Definition at line 86 of file SGObject.cpp.
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Definition at line 91 of file SGObject.cpp.
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Definition at line 96 of file SGObject.cpp.
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Definition at line 101 of file SGObject.cpp.
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Definition at line 106 of file SGObject.cpp.
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Definition at line 111 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 180 of file SGObject.cpp.
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set the version object
version | version object to use |
Definition at line 221 of file SGObject.cpp.
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Setter for the underlying kernel
kernel | new kernel to use |
Definition at line 75 of file KernelTwoSampleTest.h.
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sets the method how to approximate the null-distribution
null_approximation_method | method to use |
Definition at line 59 of file HypothesisTest.cpp.
void set_num_eigenvalues_spectrum | ( | index_t | num_eigenvalues_spectrum | ) |
setter for number of eigenvalues to use in spectrum based p-value computation. Maximum is 2*m_m-1
num_eigenvalues_spectrum | number of eigenvalues to use to approximate null-distributrion |
Definition at line 466 of file QuadraticTimeMMD.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 65 of file HypothesisTest.cpp.
void set_num_samples_sepctrum | ( | index_t | num_samples_spectrum | ) |
setter for number of samples to use in spectrum based p-value computation.
num_samples_spectrum | number of samples to draw from approximate null-distributrion |
Definition at line 460 of file QuadraticTimeMMD.cpp.
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Setter for joint features
p_and_q | joint features from p and q to set |
Reimplemented in CStreamingMMD.
Definition at line 153 of file TwoSampleTest.cpp.
void set_statistic_type | ( | EQuadraticMMDType | statistic_type | ) |
statistic_type | statistic type (biased/unbiased) to use |
Definition at line 472 of file QuadraticTimeMMD.cpp.
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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 117 of file SGObject.h.
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unset generic type
this has to be called in classes specializing a template class
Definition at line 241 of file SGObject.cpp.
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Updates the hash of current parameter combination
Definition at line 187 of file SGObject.cpp.
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io
Definition at line 473 of file SGObject.h.
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parameters wrt which we can compute gradients
Definition at line 488 of file SGObject.h.
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Hash of parameter values
Definition at line 494 of file SGObject.h.
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underlying kernel
Definition at line 119 of file KernelTwoSampleTest.h.
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defines the first index of samples of q
Definition at line 134 of file TwoSampleTest.h.
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model selection parameters
Definition at line 485 of file SGObject.h.
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Defines how the the null distribution is approximated
Definition at line 167 of file HypothesisTest.h.
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number of Eigenvalues for spectrum null-dstribution-approximation
Definition at line 286 of file QuadraticTimeMMD.h.
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number of iterations for sampling from null-distributions
Definition at line 164 of file HypothesisTest.h.
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number of samples for spectrum null-dstribution-approximation
Definition at line 283 of file QuadraticTimeMMD.h.
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concatenated samples of the two distributions (two blocks)
Definition at line 131 of file TwoSampleTest.h.
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map for different parameter versions
Definition at line 491 of file SGObject.h.
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parameters
Definition at line 482 of file SGObject.h.
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type of statistic (biased/unbiased)
Definition at line 289 of file QuadraticTimeMMD.h.
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parallel
Definition at line 476 of file SGObject.h.
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version
Definition at line 479 of file SGObject.h.