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SHOGUN
4.1.0
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SHOGUN
4.1.0
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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 which we denote by
\[ \hat{\eta_k}=\text{MMD}[\mathcal{F},p,q]^2=\textbf{E}_{x,x'} \left[ k(x,x')\right]-2\textbf{E}_{x,y}\left[ k(x,y)\right] +\textbf{E}_{y,y'}\left[ k(y,y')\right]=||\mu_p - \mu_q||^2_\mathcal{F} \]
.
Given two sets of samples \(\{x_i\}_{i=1}^{n_x}\sim p\) and \(\{y_i\}_{i=1}^{n_y}\sim q\), \(n_x+n_y=n\), the unbiased estimate of the above statistic is computed as
\[ \hat{\eta}_{k,U}=\frac{1}{n_x(n_x-1)}\sum_{i=1}^{n_x}\sum_{j\neq i} k(x_i,x_j)+\frac{1}{n_y(n_y-1)}\sum_{i=1}^{n_y}\sum_{j\neq i}k(y_i,y_j) -\frac{2}{n_xn_y}\sum_{i=1}^{n_x}\sum_{j=1}^{n_y}k(x_i,y_j) \]
A biased version is
\[ \hat{\eta}_{k,V}=\frac{1}{n_x^2}\sum_{i=1}^{n_x}\sum_{j=1}^{n_x} k(x_i,x_j)+\frac{1}{n_y^2}\sum_{i=1}^{n_y}\sum_{j=1}^{n_y}k(y_i,y_j) -\frac{2}{n_xn_y}\sum_{i=1}^{n_x}\sum_{j=1}^{n_y}k(x_i,y_j) \]
When \(n_x=n_y=\frac{n}{2}\), an incomplete version can also be computed as the following
\[ \hat{\eta}_{k,U^-}=\frac{1}{\frac{n}{2}(\frac{n}{2}-1)}\sum_{i\neq j} h(z_i,z_j) \]
where for each pair \(z=(x,y)\), \(h(z,z')=k(x,x')+k(y,y')-k(x,y')- k(x',y)\).
The type (biased/unbiased/incomplete) can be selected via set_statistic_type(). Note that there are presently two setups for computing statistic. While using BIASED, UNBIASED or INCOMPLETE, the estimate returned by compute_statistic() is \(\frac{n_xn_y}{n_x+n_y}\hat{\eta}_k\). If DEPRECATED ones are used, then this returns \((n_x+n_y)\hat{\eta}_k\) in general and \((\frac{n}{2}) \hat{\eta}_k\) when \(n_x=n_y=\frac{n}{2}\). This holds for the null distribution samples as well.
Estimating variance of the asymptotic distribution of the statistic under null and alternative hypothesis can be done using compute_variance() method. This is internally done alongwise computing statistics to avoid recomputing the kernel.
Variance under null is computed as \(\sigma_{k,0}^2=2\hat{\kappa}_2=2(\kappa_2-2\kappa_1+\kappa_0)\) where \(\kappa_0=\left(\mathbb{E}_{X,X'}k(X,X')\right )^2\), \(\kappa_1=\mathbb{E}_X\left[(\mathbb{E}_{X'}k(X,X'))^2\right]\), and \(\kappa_2=\mathbb{E}_{X,X'}k^2(X,X')\) and variance under alternative is computed as
\[ \sigma_{k,A}^2=4\rho_y\left\{\mathbb{E}_X\left[\left(\mathbb{E}_{X'} k(X,X')-\mathbb{E}_Yk(X,Y)\right)^2 \right ] -\left(\mathbb{E}_{X,X'} k(X,X')-\mathbb{E}_{X,Y}k(X,Y) \right)^2\right \}+4\rho_x\left\{ \mathbb{E}_Y\left[\left(\mathbb{E}_{Y'}k(Y,Y')-\mathbb{E}_Xk(X,Y) \right)^2\right ] -\left(\mathbb{E}_{Y,Y'}k(Y,Y')-\mathbb{E}_{X,Y} k(X,Y) \right)^2\right \} \]
where \(\rho_x=\frac{n_x}{n}\) and \(\rho_y=\frac{n_y}{n}\).
Note that statistic and variance estimation can be done for multiple kernels at once as well.
Along with the statistic comes a method to compute a p-value based on different methods. Permutation test 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.
MMD2_SPECTRUM_DEPRECATED: For a fast, consistent test based on the spectrum of the kernel matrix, as described in [2]. Only supported if Eigen3 is installed.
MMD2_SPECTRUM: Similar to the deprecated version except it estimates the statistic under null as \(\frac{n_xn_y}{n_x+n_y}\hat{\eta}_{k,U}\rightarrow \sum_r\lambda_r(Z_r^2-1)\) instead (see method description for more details).
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
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.
For kernel selection see CMMDKernelSelection.
NOTE: \(n_x\) and \(n_y\) are represented by \(m\) and \(n\), respectively in the implementation.
[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 158 of file QuadraticTimeMMD.h.
Public Attributes | |
| SGIO * | io |
| Parallel * | parallel |
| Version * | version |
| Parameter * | m_parameters |
| Parameter * | m_model_selection_parameters |
| Parameter * | m_gradient_parameters |
| uint32_t | m_hash |
Protected Member Functions | |
| SGVector< float64_t > | compute_unbiased_statistic_variance (int m, int n) |
| SGVector< float64_t > | compute_biased_statistic_variance (int m, int n) |
| SGVector< float64_t > | compute_incomplete_statistic_variance (int n) |
| float64_t | compute_unbiased_statistic (int m, int n) |
| float64_t | compute_biased_statistic (int m, int n) |
| float64_t | compute_incomplete_statistic (int n) |
| 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) |
default constructor
Definition at line 48 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 m 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 53 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 60 of file QuadraticTimeMMD.cpp.
| CQuadraticTimeMMD | ( | CCustomKernel * | custom_kernel, |
| index_t | m | ||
| ) |
Constructor. This is a convienience constructor which 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 66 of file QuadraticTimeMMD.cpp.
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destructor
Definition at line 72 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 597 of file SGObject.cpp.
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Creates a clone of the current object. This is done via recursively traversing all parameters, which corresponds to a deep copy. Calling equals on the cloned object always returns true although none of the memory of both objects overlaps.
Definition at line 714 of file SGObject.cpp.
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Wrapper method for computing biased estimate of MMD^2
| m | number of samples from p |
| n | number of samples from q |
Definition at line 536 of file QuadraticTimeMMD.cpp.
Helper method to compute biased estimate of squared quadratic time MMD and variance estimate under null and alternative hypothesis
| m | number of samples from p |
| n | number of samples from q |
Definition at line 239 of file QuadraticTimeMMD.cpp.
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Wrapper method for computing incomplete estimate of MMD^2
| n | number of samples from p and q |
Definition at line 541 of file QuadraticTimeMMD.cpp.
Helper method to compute incomplete estimate of squared quadratic time MMD and variance estimate under null and alternative hypothesis
| n | number of samples from p and q |
Definition at line 385 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/incomplete).
| statistic | statistic value to compute the p-value for |
Reimplemented from CTwoSampleTest.
Definition at line 749 of file QuadraticTimeMMD.cpp.
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Computes the squared quadratic time MMD for the current data. Note that the type (biased/unbiased/incomplete) can be specified with set_statistic_type() method.
Implements CKernelTwoSampleTest.
Definition at line 546 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 663 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/incomplete).
| alpha | test level to reject null-hypothesis |
Reimplemented from CTwoSampleTest.
Definition at line 801 of file QuadraticTimeMMD.cpp.
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Wrapper method for computing unbiased estimate of MMD^2
| m | number of samples from p |
| n | number of samples from q |
Definition at line 531 of file QuadraticTimeMMD.cpp.
Helper method to compute unbiased estimate of squared quadratic time MMD and variance estimate under null and alternative hypothesis
| m | number of samples from p |
| n | number of samples from q |
Definition at line 92 of file QuadraticTimeMMD.cpp.
Wrapper for computing variance estimate of the asymptotic distribution of the statistic (unbisaed/biased/incomplete) under null and alternative hypothesis (see class description for details)
Definition at line 598 of file QuadraticTimeMMD.cpp.
Same as compute_variance(), 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 |
Definition at line 704 of file QuadraticTimeMMD.cpp.
| float64_t compute_variance_under_alternative | ( | ) |
Wrapper method for compute_variance()
Definition at line 658 of file QuadraticTimeMMD.cpp.
| float64_t compute_variance_under_null | ( | ) |
Wrapper method for compute_variance()
Definition at line 653 of file QuadraticTimeMMD.cpp.
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A deep copy. All the instance variables will also be copied.
Definition at line 198 of file SGObject.cpp.
Recursively compares the current SGObject to another one. Compares all registered numerical parameters, recursion upon complex (SGObject) parameters. Does not compare pointers!
May be overwritten but please do with care! Should not be necessary in most cases.
| other | object to compare with |
| accuracy | accuracy to use for comparison (optional) |
| tolerant | allows linient check on float equality (within accuracy) |
Definition at line 618 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). To use, set BIASED_DEPRECATED as statistic type. 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 1032 of file QuadraticTimeMMD.cpp.
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Definition at line 86 of file KernelTwoSampleTest.h.
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Definition at line 127 of file TwoSampleTest.h.
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Definition at line 498 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 522 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 535 of file SGObject.cpp.
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Implements CKernelTwoSampleTest.
Definition at line 280 of file QuadraticTimeMMD.h.
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Getter for joint features, SG_REF'ed
Reimplemented in CStreamingMMD.
Definition at line 171 of file TwoSampleTest.cpp.
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returns the statistic type of this test statistic
Implements CHypothesisTest.
Definition at line 286 of file QuadraticTimeMMD.h.
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If the SGSerializable is a class template then TRUE will be returned and GENERIC is set to the type of the generic.
| generic | set to the type of the generic if returning TRUE |
Definition at line 296 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 |
Definition at line 369 of file SGObject.cpp.
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Can (optionally) be overridden to post-initialize some member variables which are not PARAMETER::ADD'ed. Make sure that at first the overridden method BASE_CLASS::LOAD_SERIALIZABLE_POST is called.
| ShogunException | will be thrown if an error occurs. |
Reimplemented in CKernel, CWeightedDegreePositionStringKernel, CList, CAlphabet, CLinearHMM, CGaussianKernel, CInverseMultiQuadricKernel, CCircularKernel, and CExponentialKernel.
Definition at line 426 of file SGObject.cpp.
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Can (optionally) be overridden to pre-initialize some member variables which are not PARAMETER::ADD'ed. Make sure that at first the overridden method BASE_CLASS::LOAD_SERIALIZABLE_PRE is called.
| ShogunException | will be thrown if an error occurs. |
Reimplemented in CDynamicArray< T >, CDynamicArray< float64_t >, CDynamicArray< float32_t >, CDynamicArray< int32_t >, CDynamicArray< char >, CDynamicArray< bool >, and CDynamicObjectArray.
Definition at line 421 of file SGObject.cpp.
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Definition at line 262 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 113 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 121 of file HypothesisTest.cpp.
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prints all parameter registered for model selection and their type
Definition at line 474 of file SGObject.cpp.
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prints registered parameters out
| prefix | prefix for members |
Definition at line 308 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 55 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).
The estimate is computed as
\[ \frac{n_xn_y}{n_x+n_y}\hat{\eta}_{k,U}\rightarrow\sum_{l=1}^\infty \lambda_l\left(Z^2_l-1 \right) \]
where \({Z_l}\stackrel{i.i.d.}{\sim}\mathcal{N}(0,1)\) and \(\lambda_l\) are the eigenvalues of centered kernel matrix HKH.
kernel matrix needs to be stored in memory
Note that m*n/(m+n)*Null-distribution is returned, which is fine since the statistic is also m*n/(m+n)*MMD^2
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 m+n-1 where m and n are the sizes of samples from p and q respectively. |
Definition at line 854 of file QuadraticTimeMMD.cpp.
| SGVector< float64_t > sample_null_spectrum_DEPRECATED | ( | index_t | num_samples, |
| index_t | num_eigenvalues | ||
| ) |
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).
The unbiased version uses
\[ t\text{MMD}_u^2[\mathcal{F},X,Y]\rightarrow\sum_{l=1}^\infty \lambda_l\left((a_l\rho_x^{-\frac{1}{{2}}} -b_l\rho_y^{-\frac{1}{{2}}})^2-(\rho_x\rho_y)^{-1} \right) \]
where \(t=m+n\), \(\lim_{m,n\rightarrow\infty}m/t\rightarrow \rho_x\) and \(\rho_y\) likewise (equation 10 from [1]) and \(\lambda_l\) are estimated as \(\frac{\nu_l}{(m+n)}\), where \(\nu_l\) are the eigenvalues of centered kernel matrix HKH.
The biased version uses
\[ t\text{MMD}_b^2[\mathcal{F},X,Y]\rightarrow\sum_{l=1}^\infty \lambda_l\left((a_l\rho_x^{-\frac{1}{{2}}}- b_l\rho_y^{-\frac{1}{{2}}})^2\right) \]
kernel matrix needs to be stored in memory
Note that (m+n)*Null-distribution is returned, which is fine since the statistic is also (m+n)*MMD: except when m and n are equal, then m*MMD^2 is returned
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 m+n-1 where m and n are the sizes of samples from p and q respectively. 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 933 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 |
Definition at line 314 of file SGObject.cpp.
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Can (optionally) be overridden to post-initialize some member variables which are not PARAMETER::ADD'ed. Make sure that at first the overridden method BASE_CLASS::SAVE_SERIALIZABLE_POST is called.
| ShogunException | will be thrown if an error occurs. |
Reimplemented in CKernel.
Definition at line 436 of file SGObject.cpp.
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Can (optionally) be overridden to pre-initialize some member variables which are not PARAMETER::ADD'ed. Make sure that at first the overridden method BASE_CLASS::SAVE_SERIALIZABLE_PRE is called.
| ShogunException | will be thrown if an error occurs. |
Reimplemented in CKernel, CDynamicArray< T >, CDynamicArray< float64_t >, CDynamicArray< float32_t >, CDynamicArray< int32_t >, CDynamicArray< char >, CDynamicArray< bool >, and CDynamicObjectArray.
Definition at line 431 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 241 of file SGObject.cpp.
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set the version object
| version | version object to use |
Definition at line 283 of file SGObject.cpp.
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Setter for the underlying kernel
| kernel | new kernel to use |
Definition at line 77 of file KernelTwoSampleTest.h.
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| m | number of samples from first distribution p |
Definition at line 162 of file TwoSampleTest.cpp.
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sets the method how to approximate the null-distribution
| null_approximation_method | method to use |
Definition at line 61 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 m_m+m_n-1
| num_eigenvalues_spectrum | number of eigenvalues to use to approximate null-distributrion |
Definition at line 1124 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 67 of file HypothesisTest.cpp.
| void set_num_samples_spectrum | ( | 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 1118 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 154 of file TwoSampleTest.cpp.
| void set_statistic_type | ( | EQuadraticMMDType | statistic_type | ) |
| statistic_type | statistic type (biased/unbiased/incomplete) to use |
Definition at line 1130 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 192 of file SGObject.cpp.
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unset generic type
this has to be called in classes specializing a template class
Definition at line 303 of file SGObject.cpp.
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Updates the hash of current parameter combination
Definition at line 248 of file SGObject.cpp.
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io
Definition at line 369 of file SGObject.h.
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parameters wrt which we can compute gradients
Definition at line 384 of file SGObject.h.
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Hash of parameter values
Definition at line 387 of file SGObject.h.
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underlying kernel
Definition at line 121 of file KernelTwoSampleTest.h.
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defines the first index of samples of q
Definition at line 139 of file TwoSampleTest.h.
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model selection parameters
Definition at line 381 of file SGObject.h.
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Defines how the the null distribution is approximated
Definition at line 177 of file HypothesisTest.h.
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number of Eigenvalues for spectrum null-dstribution-approximation
Definition at line 479 of file QuadraticTimeMMD.h.
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number of iterations for sampling from null-distributions
Definition at line 174 of file HypothesisTest.h.
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number of samples for spectrum null-dstribution-approximation
Definition at line 476 of file QuadraticTimeMMD.h.
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concatenated samples of the two distributions (two blocks)
Definition at line 136 of file TwoSampleTest.h.
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parameters
Definition at line 378 of file SGObject.h.
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type of statistic (biased/unbiased/incomplete as well as deprecated versions of biased/unbiased)
Definition at line 484 of file QuadraticTimeMMD.h.
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parallel
Definition at line 372 of file SGObject.h.
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version
Definition at line 375 of file SGObject.h.