This class implements the quadratic time Maximum Mean Statistic as described in [1]. The MMD is the distance of two probability distributions 
and 
in a RKHS
![\[ \text{MMD}}[\mathcal{F},p,q]^2=\textbf{E}_{x,x'}\left[ k(x,x')\right]- 2\textbf{E}_{x,y}\left[ k(x,y)\right] +\textbf{E}_{y,y'}\left[ k(y,y')\right]=||\mu_p - \mu_q||^2_\mathcal{F} \]](form_313.png)
.
Given two sets of samples 
and 
the (unbiased) statistic is computed as
![\[ \text{MMD}_u^2[\mathcal{F},X,Y]=\frac{1}{m(m-1)}\sum_{i=1}^m\sum_{j\neq i}^m k(x_i,x_j) + \frac{1}{n(n-1)}\sum_{i=1}^n\sum_{j\neq i}^nk(y_i,y_j) - \frac{2}{mn}\sum_{i=1}^m\sum_{j=1}^nk(x_i,y_j) \]](form_320.png)
A biased version is
![\[ \text{MMD}_b^2[\mathcal{F},X,Y]=\frac{1}{m^2}\sum_{i=1}^m\sum_{j=1}^m k(x_i,x_j) + \frac{1}{n^2}\sum_{i=1}^n\sum_{j=1}^nk(y_i,y_j) - \frac{2}{mn}\sum_{i=1}^m\sum_{j=1}^nk(x_i,y_j) \]](form_321.png)
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. Bootstrapping, is also possible. If unsure which one to use, bootstrapping with 250 iterations always is correct (but slow).
To choose, use set_null_approximation_method() and choose from
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].
BOOTSTRAPPING: For permuting available samples to sample null-distribution
A very basic method for kernel selection when using CGaussianKernel is to use the median distance of the underlying data. See examples how to do that. More advanced methods will follow in the near future. However, the median heuristic works in quite some cases. See [1].
[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 84 of file QuadraticTimeMMD.h.
Definition at line 17 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 22 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 35 of file QuadraticTimeMMD.cpp.
| ~CQuadraticTimeMMD | ( | ) | [virtual] |
Definition at line 47 of file QuadraticTimeMMD.cpp.
merges both sets of samples and computes the test statistic m_bootstrap_iteration 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 CTwoDistributionsTestStatistic.
Reimplemented in CLinearTimeMMD.
Definition at line 57 of file KernelTwoSampleTestStatistic.cpp.
| void build_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.
| dict | dictionary of parameters to be built. |
Definition at line 1201 of file SGObject.cpp.
| float64_t compute_biased_statistic | ( | ) | [protected, virtual] |
helper method to compute m*biased squared quadratic time MMD
Definition at line 108 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 CTwoDistributionsTestStatistic.
Definition at line 169 of file QuadraticTimeMMD.cpp.
| float64_t compute_statistic | ( | ) | [virtual] |
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 CTestStatistic.
Definition at line 146 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 CTwoDistributionsTestStatistic.
Definition at line 207 of file QuadraticTimeMMD.cpp.
| float64_t compute_unbiased_statistic | ( | ) | [protected, virtual] |
helper method to compute m*unbiased squared quadratic time MMD
Definition at line 68 of file QuadraticTimeMMD.cpp.
| virtual CSGObject* deep_copy | ( | ) | const [virtual, inherited] |
A deep copy. All the instance variables will also be copied.
Definition at line 131 of file SGObject.h.
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 318 of file QuadraticTimeMMD.cpp.
| SGIO * get_global_io | ( | ) | [inherited] |
| Parallel * get_global_parallel | ( | ) | [inherited] |
| Version * get_global_version | ( | ) | [inherited] |
| SGStringList< char > get_modelsel_names | ( | ) | [inherited] |
Definition at line 1108 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
| param_name | name of the parameter |
Definition at line 1132 of file SGObject.cpp.
| index_t get_modsel_param_index | ( | const char * | param_name | ) | [inherited] |
Returns index of model selection parameter with provided index
| param_name | name of model selection parameter |
Definition at line 1145 of file SGObject.cpp.
| virtual const char* get_name | ( | ) | const [virtual] |
Returns the name of the SGSerializable instance. It MUST BE the CLASS NAME without the prefixed `C'.
Implements CKernelTwoSampleTestStatistic.
Definition at line 149 of file QuadraticTimeMMD.h.
| bool is_generic | ( | EPrimitiveType * | generic | ) | const [virtual, inherited] |
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 278 of file SGObject.cpp.
| DynArray< TParameter * > * load_all_file_parameters | ( | int32_t | file_version, | |
| int32_t | current_version, | |||
| CSerializableFile * | file, | |||
| const char * | prefix = "" | |||
| ) | [inherited] |
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_PARAMETER for this in most cases) | |
| file | file to load from | |
| prefix | prefix for members |
Definition at line 679 of file SGObject.cpp.
| DynArray< TParameter * > * load_file_parameters | ( | const SGParamInfo * | param_info, | |
| int32_t | file_version, | |||
| CSerializableFile * | file, | |||
| const char * | prefix = "" | |||
| ) | [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 523 of file SGObject.cpp.
| bool load_serializable | ( | CSerializableFile * | file, | |
| const char * | prefix = "", |
|||
| int32_t | param_version = VERSION_PARAMETER | |||
| ) | [virtual, inherited] |
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) |
Reimplemented in CModelSelectionParameters.
Definition at line 354 of file SGObject.cpp.
| void load_serializable_post | ( | ) | throw (ShogunException) [protected, virtual, inherited] |
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 CLinearHMM, CAlphabet, CANOVAKernel, CCircularKernel, CExponentialKernel, CGaussianKernel, CInverseMultiQuadricKernel, CKernel, CWeightedDegreePositionStringKernel, and CList.
Definition at line 1033 of file SGObject.cpp.
| void load_serializable_pre | ( | ) | throw (ShogunException) [protected, virtual, inherited] |
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. |
Definition at line 1028 of file SGObject.cpp.
| void map_parameters | ( | DynArray< TParameter * > * | param_base, | |
| int32_t & | base_version, | |||
| DynArray< const SGParamInfo * > * | target_param_infos | |||
| ) | [inherited] |
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 717 of file SGObject.cpp.
| TParameter * migrate | ( | DynArray< TParameter * > * | param_base, | |
| const SGParamInfo * | target | |||
| ) | [protected, virtual, inherited] |
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 923 of file SGObject.cpp.
| void one_to_one_migration_prepare | ( | DynArray< TParameter * > * | param_base, | |
| const SGParamInfo * | target, | |||
| TParameter *& | replacement, | |||
| TParameter *& | to_migrate, | |||
| char * | old_name = NULL | |||
| ) | [protected, virtual, inherited] |
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 864 of file SGObject.cpp.
| float64_t perform_test | ( | ) | [virtual, inherited] |
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 CLinearTimeMMD.
Definition at line 50 of file TestStatistic.cpp.
| bool perform_test | ( | float64_t | alpha | ) | [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. |
Definition at line 58 of file TestStatistic.cpp.
| void print_modsel_params | ( | ) | [inherited] |
prints all parameter registered for model selection and their type
Definition at line 1084 of file SGObject.cpp.
| void print_serializable | ( | const char * | prefix = "" |
) | [virtual, inherited] |
prints registered parameters out
| prefix | prefix for members |
Definition at line 290 of file SGObject.cpp.
Definition at line 247 of file QuadraticTimeMMD.cpp.
| bool save_serializable | ( | CSerializableFile * | file, | |
| const char * | prefix = "", |
|||
| int32_t | param_version = VERSION_PARAMETER | |||
| ) | [virtual, inherited] |
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) |
Reimplemented in CModelSelectionParameters.
Definition at line 296 of file SGObject.cpp.
| void save_serializable_post | ( | ) | throw (ShogunException) [protected, virtual, inherited] |
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 1043 of file SGObject.cpp.
| void save_serializable_pre | ( | ) | throw (ShogunException) [protected, virtual, inherited] |
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.
Definition at line 1038 of file SGObject.cpp.
| void set_bootstrap_iterations | ( | index_t | bootstrap_iterations | ) | [virtual, inherited] |
sets the number of bootstrap iterations for bootstrap_null()
| bootstrap_iterations | how often bootstrapping shall be done |
Definition at line 44 of file TestStatistic.cpp.
| void set_generic< floatmax_t > | ( | ) | [inherited] |
set generic type to T
| void set_global_io | ( | SGIO * | io | ) | [inherited] |
| void set_global_parallel | ( | Parallel * | parallel | ) | [inherited] |
set the parallel object
| parallel | parallel object to use |
Definition at line 230 of file SGObject.cpp.
| void set_global_version | ( | Version * | version | ) | [inherited] |
set the version object
| version | version object to use |
Definition at line 265 of file SGObject.cpp.
| void set_null_approximation_method | ( | ENullApproximationMethod | null_approximation_method | ) | [virtual, inherited] |
sets the method how to approximate the null-distribution
| null_approximation_method | method to use |
Definition at line 38 of file TestStatistic.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 394 of file QuadraticTimeMMD.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 388 of file QuadraticTimeMMD.cpp.
| void set_p_and_q | ( | CFeatures * | p_and_q | ) | [virtual, inherited] |
Setter for joint features
| p_and_q | joint features from p and q to set |
Definition at line 144 of file TwoDistributionsTestStatistic.cpp.
| void set_statistic_type | ( | EQuadraticMMDType | statistic_type | ) |
| statistic_type | statistic type (biased/unbiased) to use |
Definition at line 400 of file QuadraticTimeMMD.cpp.
| virtual CSGObject* shallow_copy | ( | ) | const [virtual, inherited] |
A shallow copy. All the SGObject instance variables will be simply assigned and SG_REF-ed.
Reimplemented in CGaussianKernel.
Definition at line 122 of file SGObject.h.
| void unset_generic | ( | ) | [inherited] |
unset generic type
this has to be called in classes specializing a template class
Definition at line 285 of file SGObject.cpp.
| bool update_parameter_hash | ( | ) | [protected, virtual, inherited] |
Updates the hash of current parameter combination.
Definition at line 237 of file SGObject.cpp.
io
Definition at line 462 of file SGObject.h.
index_t m_bootstrap_iterations [protected, inherited] |
number of iterations for bootstrapping null-distributions
Definition at line 129 of file TestStatistic.h.
uint32_t m_hash [inherited] |
Hash of parameter values
Definition at line 480 of file SGObject.h.
underlying kernel
Definition at line 83 of file KernelTwoSampleTestStatistic.h.
defines the first index of samples of q
Definition at line 102 of file TwoDistributionsTestStatistic.h.
Parameter* m_model_selection_parameters [inherited] |
model selection parameters
Definition at line 474 of file SGObject.h.
ENullApproximationMethod m_null_approximation_method [protected, inherited] |
Defines how the the null distribution is approximated
Definition at line 132 of file TestStatistic.h.
index_t m_num_eigenvalues_spectrum [protected] |
number of Eigenvalues for spectrum null-dstribution-approximation
Definition at line 237 of file QuadraticTimeMMD.h.
index_t m_num_samples_spectrum [protected] |
number of samples for spectrum null-dstribution-approximation
Definition at line 234 of file QuadraticTimeMMD.h.
concatenated samples of the two distributions (two blocks)
Definition at line 99 of file TwoDistributionsTestStatistic.h.
ParameterMap* m_parameter_map [inherited] |
map for different parameter versions
Definition at line 477 of file SGObject.h.
Parameter* m_parameters [inherited] |
parameters
Definition at line 471 of file SGObject.h.
EQuadraticMMDType m_statistic_type [protected] |
type of statistic (biased/unbiased)
Definition at line 240 of file QuadraticTimeMMD.h.
parallel
Definition at line 465 of file SGObject.h.
version
Definition at line 468 of file SGObject.h.
