SHOGUN
v3.0.0
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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. 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.
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].
BOOTSTRAPPING: 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 86 of file QuadraticTimeMMD.h.
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 19 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 24 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 37 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 49 of file QuadraticTimeMMD.cpp.
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virtual |
Definition at line 55 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 54 of file KernelTwoSampleTestStatistic.cpp.
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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 1196 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 1313 of file SGObject.cpp.
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protectedvirtual |
helper method to compute m*biased squared quadratic time MMD
Definition at line 116 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 177 of file QuadraticTimeMMD.cpp.
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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 CKernelTwoSampleTestStatistic.
Definition at line 154 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 CKernelTwoSampleTestStatistic.
Definition at line 215 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 256 of file QuadraticTimeMMD.cpp.
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protectedvirtual |
helper method to compute m*unbiased squared quadratic time MMD
Definition at line 76 of file QuadraticTimeMMD.cpp.
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virtualinherited |
A deep copy. All the instance variables will also be copied.
Definition at line 160 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) |
Definition at line 1217 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 379 of file QuadraticTimeMMD.cpp.
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inherited |
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inherited |
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virtualinherited |
Definition at line 80 of file KernelTwoSampleTestStatistic.h.
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inherited |
Definition at line 98 of file TwoDistributionsTestStatistic.h.
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inherited |
Definition at line 1100 of file SGObject.cpp.
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inherited |
Returns description of a given parameter string, if it exists. SG_ERROR otherwise
param_name | name of the parameter |
Definition at line 1124 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 1137 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 CKernelTwoSampleTestStatistic.
Definition at line 171 of file QuadraticTimeMMD.h.
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virtualinherited |
Getter for joint features, SG_REF's
Reimplemented in CLinearTimeMMD.
Definition at line 151 of file TwoDistributionsTestStatistic.cpp.
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returns the statistic type of this test statistic
Implements CTestStatistic.
Definition at line 177 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 268 of file SGObject.cpp.
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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::get_version_parameter() for this in most cases) |
file | file to load from |
prefix | prefix for members |
Definition at line 673 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 514 of file SGObject.cpp.
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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 |
param_version | (optional) a parameter version different to (this is mainly for testing, better do not use) |
Definition at line 345 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 CKernel, CWeightedDegreePositionStringKernel, CList, CAlphabet, CLinearHMM, CGaussianKernel, CInverseMultiQuadricKernel, CCircularKernel, and CExponentialKernel.
Definition at line 1029 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 >, CDynamicArray< uint64_t >, and CDynamicObjectArray.
Definition at line 1024 of file SGObject.cpp.
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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 711 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 918 of file SGObject.cpp.
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protectedvirtualinherited |
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 858 of file SGObject.cpp.
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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().
Reimplemented in CLinearTimeMMD.
Definition at line 50 of file TestStatistic.cpp.
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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.
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inherited |
prints all parameter registered for model selection and their type
Definition at line 1076 of file SGObject.cpp.
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virtualinherited |
prints registered parameters out
prefix | prefix for members |
Definition at line 280 of file SGObject.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 296 of file QuadraticTimeMMD.cpp.
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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 |
param_version | (optional) a parameter version different to (this is mainly for testing, better do not use) |
Definition at line 286 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 1039 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 >, CDynamicArray< uint64_t >, and CDynamicObjectArray.
Definition at line 1034 of file SGObject.cpp.
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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.
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inherited |
set generic type to T
Definition at line 41 of file SGObject.cpp.
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inherited |
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inherited |
set the parallel object
parallel | parallel object to use |
Definition at line 220 of file SGObject.cpp.
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set the version object
version | version object to use |
Definition at line 255 of file SGObject.cpp.
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Setter for the underlying kernel
kernel | new kernel to use |
Definition at line 71 of file KernelTwoSampleTestStatistic.h.
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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 463 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 457 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 CLinearTimeMMD.
Definition at line 143 of file TwoDistributionsTestStatistic.cpp.
void set_statistic_type | ( | EQuadraticMMDType | statistic_type | ) |
statistic_type | statistic type (biased/unbiased) to use |
Definition at line 469 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 151 of file SGObject.h.
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inherited |
unset generic type
this has to be called in classes specializing a template class
Definition at line 275 of file SGObject.cpp.
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Updates the hash of current parameter combination.
Definition at line 227 of file SGObject.cpp.
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io
Definition at line 514 of file SGObject.h.
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number of iterations for bootstrapping null-distributions
Definition at line 138 of file TestStatistic.h.
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parameters wrt which we can compute gradients
Definition at line 529 of file SGObject.h.
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Hash of parameter values
Definition at line 535 of file SGObject.h.
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underlying kernel
Definition at line 115 of file KernelTwoSampleTestStatistic.h.
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defines the first index of samples of q
Definition at line 110 of file TwoDistributionsTestStatistic.h.
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model selection parameters
Definition at line 526 of file SGObject.h.
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Defines how the the null distribution is approximated
Definition at line 141 of file TestStatistic.h.
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number of Eigenvalues for spectrum null-dstribution-approximation
Definition at line 265 of file QuadraticTimeMMD.h.
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number of samples for spectrum null-dstribution-approximation
Definition at line 262 of file QuadraticTimeMMD.h.
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concatenated samples of the two distributions (two blocks)
Definition at line 107 of file TwoDistributionsTestStatistic.h.
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map for different parameter versions
Definition at line 532 of file SGObject.h.
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parameters
Definition at line 523 of file SGObject.h.
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type of statistic (biased/unbiased)
Definition at line 268 of file QuadraticTimeMMD.h.
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parallel
Definition at line 517 of file SGObject.h.
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version
Definition at line 520 of file SGObject.h.