JP6343591B2 - Submatrix region extraction apparatus, method, and program - Google Patents

Description

  The present invention relates to a partial matrix region extraction device, method, and program, and more particularly, to a partial matrix region extraction device, method, and program for extracting a characteristic partial matrix.

  Many data handled in practice can be expressed in a two-dimensional table. These two-dimensional table data can be immediately expressed in matrix form. Therefore, many statistical machine learning methods for matrix data have been proposed. Here, in particular, an analysis task called submatrix extraction, which extracts only a part having some unique features from the entire matrix data, is targeted.

  A submatrix is a Cartesian product of a subset of rows and columns of given matrix data, that is, a small rectangular area in the matrix (including index permutation). The purpose of the submatrix extraction method is not to classify the entire data into patterns but to extract a small number of submatrixes having unique observation values. Submatrix extraction is more convenient for the task of extracting only specific patterns in the data, because it extracts only the observation area that seems to be of interest as a rectangle that is easy to recognize. A number of methods have been devised, including the Plaid method (see Non-Patent Document 1) (see Non-Patent Document 2 and Non-Patent Document 3). FIG. 7 shows an example of a conventional extraction method.

Lazzeroni and Owen, “Plaid Models for Gene Expression Data”, Statistica Sinica, vol. 12, pp. 61-86, 2002. Caldas and Kaski, “Bayesian Biclustering with the Plaid Model”, in Proceedings of the IEEE International Workshop on Machine Learning and Signal Processing (MLSP), 2008. Shabalin et al., “Finding Large Average Submatrices in High Dimensional Data”, The Annals of Applied Statistics, Vol. 3, issu 3, pp. 995-1012, 2009.

  However, the methods represented by Non-Patent Document 1 and Non-Patent Document 2 have the following drawbacks. That is, the number of “singular submatrix regions” that will exist in the matrix data must be determined prior to analysis. Even though the original purpose is to analyze by applying the submatrix region extraction method because the nature of the given matrix data is unknown, the number of submatrix regions must be specified prior to analysis. It has a contradictory problem of not becoming. In Non-Patent Document 3, the number of submatrix regions itself does not need to be specified, but has a similar problem in that a threshold parameter to be regarded as a submatrix must be set in advance.

  In addition, the number of submatrix regions greatly affects the accuracy of the region extraction task. Also, since it is generally difficult to evaluate the optimal number of submatrix regions afterwards, this problem has not been dealt with.

  The present invention has been made to solve the above problems, and an object of the present invention is to provide a submatrix area extraction apparatus, method, and program capable of extracting an optimal number of submatrix areas. .

  In order to achieve the above object, a submatrix region extraction apparatus according to a first aspect of the present invention is based on an observation matrix composed of observation values for a pair relationship between each object in the first domain and each object in the second domain. A partial matrix region extraction apparatus for extracting a partial matrix region having features, wherein the number of the partial matrix regions and the number of the partial matrix regions for each object in the first domain and each object in the second domain An initialization unit for initializing a submatrix region allocation estimation value indicating whether or not to be allocated to each of the submatrix regions existing in a number, and for each object of the first domain and each object of the second domain, For each of the submatrix regions present by the number of the submatrix regions, the observation matrix, the submatrix region allocation estimation value, and the object for each object When it is estimated that the object should belong to the submatrix region by estimating whether the object should belong to the submatrix region based on a submatrix region hyperparameter related to allocation of the submatrix region Assigns the submatrix region to the object and determines whether to generate a new submatrix region to represent the object based on the observation matrix and the submatrix region allocation estimate. Estimating and generating a new submatrix region when it is estimated that a new submatrix region should be generated, assigning the generated new submatrix region to the object, and The number of submatrix regions is updated, and the allocated number of objects in the first domain or the number of objects in the second domain is The partial matrix region allocation estimation unit that deletes the partial matrix regions that are less than or equal to the value and updates the number of the partial matrix regions, and the estimation and allocation by the partial matrix region allocation estimation unit until a predetermined repetition termination condition is satisfied And a repeat determination unit that repeats.

  In the partial matrix region extraction device according to the first aspect of the present invention, the partial matrix region allocation estimation unit exists for each object in the first domain and each object in the second domain by the number of the partial matrix regions. For each of the submatrix regions, a pre-fit degree indicating the degree to which the object is assigned or not assigned to the submatrix region, estimated based on the submatrix region allocation estimation value, and the observation matrix, Based on a partial matrix region allocation estimate and a likelihood of the object being assigned to the submatrix region, or a data fitness representing a likelihood of being not assigned, estimated based on the submatrix region hyperparameter, Calculate the possibility that the object belongs or does not belong to the submatrix region And by the object, it may be estimated whether or not to belong to the partial matrix region.

  In the partial matrix region extraction apparatus according to the first invention, the partial matrix region allocation estimation unit is estimated based on the partial matrix region allocation estimation value for each object of the first domain. A pre-matching degree related to the number of submatrix regions to be generated, a pre-matching degree indicating a degree to which an object of the second domain is allocated to a newly generated submatrix area, and an observation value in the new submatrix area The new submatrix region is determined based on the prior goodness of the observation parameter with respect to and the data suitability representing the degree to which the observation matrix can be well explained by increasing the number of the submatrix regions. By calculating the possibility of the added number of submatrix regions, a new submatrix region should be generated to represent the object. Whether or not, and for each object of the second domain, a pre-fit degree relating to the number of required sub-matrix regions estimated based on the sub-matrix region allocation estimate, and a newly generated Pre-adaptation degree indicating the degree to which the object of the first domain is assigned to the sub-matrix region, the pre-adaptation degree regarding the observation parameter for the observation value in the new sub-matrix region, and the number of the sub-matrix regions By calculating the possibility that it is the number of the submatrix regions to which the new submatrix region is added based on the data suitability representing the degree that the observation matrix can be well explained by the increase It may be estimated whether a new partial matrix region should be generated to represent the object.

  The submatrix region extraction apparatus according to the first invention further includes a submatrix region hyperparameter estimation unit that estimates the submatrix region hyperparameter based on the partial matrix region allocation estimation value, and the initialization unit Further initializes the partial matrix region hyperparameter, and the iterative determination unit sets a predetermined repetition end condition for estimation and allocation by the partial matrix region allocation estimation unit and estimation by the partial matrix region hyperparameter estimation unit. It may be repeated until it is satisfied.

  A program according to a second invention is a program for causing a computer to function as each part of the partial matrix region extraction device according to the first invention.

  According to the submatrix region extraction apparatus, method, and program of the present invention, for each submatrix region for each object in the first domain and each object in the second domain, an observation matrix, a submatrix region allocation estimation value, and Based on the submatrix area hyperparameter, it is estimated whether the object should belong to the submatrix area, and if it is estimated that the object should belong to the submatrix area, the submatrix area is set to the object. Assigning and estimating whether or not a new submatrix region should be generated to represent an object based on the observation matrix and the submatrix region allocation estimate, and generating a new submatrix region If it is estimated, a new submatrix region is generated, and the generated new submatrix region is assigned to the object. And deleting the submatrix region where the number of objects of the first domain assigned or the number of objects of the second domain is equal to or less than a predetermined value, and repeating the estimation and assignment until a predetermined repetition end condition is satisfied, An effect is obtained that a submatrix region that is an optimum number of features can be extracted.

It is the abstract figure explaining the example which optimizes the number of the optimal submatrix area | region automatically. It is a block diagram which shows the structure of the partial matrix area | region extraction apparatus which concerns on embodiment of this invention. It is a flowchart which shows the partial matrix area | region extraction processing routine in the partial matrix area | region extraction apparatus which concerns on embodiment of this invention. It is a flowchart which shows the presumed allocation process routine in the partial matrix area | region extraction apparatus which concerns on embodiment of this invention. It is a flowchart which shows the presumed allocation process routine in the partial matrix area | region extraction apparatus which concerns on embodiment of this invention. It is a figure which shows the example which visualized what was actually extracted. It is an abstract figure which shows the example of the conventional extraction method.

  Hereinafter, embodiments of the present invention will be described in detail with reference to the drawings.

<Outline according to Embodiment of the Present Invention>

  First, an outline of the embodiment of the present invention will be described. The submatrix region extracting apparatus according to the embodiment of the present invention is characterized in that a submatrix having characteristics is obtained from an observation matrix composed of observation values regarding a pair relationship between each object in the first domain and each object in the second domain. Extract regions. The partial matrix region extraction apparatus according to the present embodiment solves the above problem by using a partial matrix region automatic extraction method that can automatically optimize the number of partial matrix regions to be extracted. As shown in FIG. 1, the method proposed in the present embodiment automatically optimizes the number of sub-matrix regions that are optimal for capturing the properties of given matrix data. Therefore, it is possible to avoid problems such as accuracy degradation and selection criteria for the number of sub-matrix regions related to the presetting of the number of sub-matrix regions.

  In this embodiment, it is assumed that the algorithm is completely described by a probabilistic generation model. Taking advantage of this advantage, optimization is also realized in which other preset parameters (observed parameter estimated values, observed hyperparameter estimated values, etc.) are automatically fitted to the data. Therefore, it is possible to perform submatrix region extraction with high accuracy almost completely automatically without the effort of searching parameters for unknown data. Further, the algorithm used in the present embodiment can be applied to matrix data in various formats, for example, arbitrary matrix data such as continuous numerical values, discrete numerical values, symbol amounts, and the like of each part depending on the applied data format. The general composition can vary. In particular, the implementation of the observation parameter estimation value, the observation hyperparameter estimation value, the observation parameter estimation unit for estimating these estimation values, and the observation hyperparameter estimation unit varies greatly depending on each configuration. Therefore, the configuration of these parts is not an essential element of the present invention. For the estimated values, the configuration method of each part, the actual calculation algorithm, etc., the above-mentioned Non-Patent Documents 1 and 2, and Non-Patent Document 4 can be referred to.

[Non-Patent Document 4]: Bishop, “Pattern Recognition and Machine Learning”, Springer Japan, 2007.

<Configuration of Submatrix Region Extraction Device According to Embodiment of the Present Invention>

  Next, the configuration of the submatrix region extraction apparatus according to the embodiment of the present invention will be described. As shown in FIG. 2, the submatrix area extraction apparatus 100 according to the embodiment of the present invention includes a CPU, a RAM, and a ROM that stores a program and various data for executing a submatrix area extraction processing routine described later. And a computer including Functionally, the submatrix region extraction apparatus 100 includes an input unit 10, a calculation unit 20, and an output unit 50 as shown in FIG.

  The input unit 10 receives the observation matrix X. The observation matrix X is usually a matrix composed of N1 × N2 row and column elements, and generally assumes a real value. The row direction is called the first domain, and i = 1,..., N1 is indexed in the row direction. The column direction is called the second domain, and is indexed in the column direction as j = 1,..., N2. The configuration of data stored in the input data storage unit 28 depends on the purpose and task of the user. For this reason, the above data is provided at the minimum, but the present invention is not limited to this, and various data can be stored depending on the purpose and task.

  The calculation unit 20 includes an initialization unit 26, an input data storage unit 28, a variable estimation unit 30, and a variable storage unit 42.

  The initialization unit 26 represents the number of sub-matrix regions and the number of sub-matrix regions indicating the number of sub-matrix regions allocated to each object in the first domain and each object in the second domain. The allocation estimation value Z, the submatrix region hyperparameter α, and the observation hyperparameter β are initialized, and the initial value K (0) of the submatrix region number estimation value and the initial value Z (0) of the submatrix region allocation estimation value, respectively. ), The initial value α (0) of the partial matrix region hyperparameter, and the initial value β (0) of the observation hyperparameter are stored in the input data storage unit 28. The initial value K (0) of the number of submatrix regions is the number of submatrix regions assumed at the time of initialization. The number K of partial matrix regions is optimized by estimation by a partial matrix region allocation estimation unit 32 described later.

  Here, the partial matrix region allocation estimated value Z will be described in detail. In the present embodiment, matrix row and column indices each represent an object. For example, for the observation matrix X to be analyzed, consider the case where the customer ID is represented in the row direction and the purchased quantity data of the product is represented in the column direction. At this time, the index i in each row (first domain) corresponds to an individual customer, and the index j in each column (second domain) corresponds to an individual product. Each submatrix represents a subset of customers and a subset of products, and realizes simultaneous clustering of row objects and column objects such as “specific customers” who prefer to purchase “specific products”. The submatrix is indexed by k, and the total number of k is expressed by an estimated value K of the number of submatrix regions. The sub-matrix allocation estimated value Z represents this clustering result. The submatrix allocation estimated value Z includes an estimated value Z1 related to N1 first domain objects and an estimated value Z2 related to N2 second domain objects.

Z1 is

Z1 = {Z (1,1), ..., Z (1, N1)}

Z2 is expressed as

Z2 = {Z (2,1), ..., Z (2, N2)}

It is expressed. Each estimated value Z (1, i) and Z (2, j) is a binary variable that represents whether or not K partial matrix regions are included in the k-th partial matrix region, or 0. A real number between 1 and 1. That is,

Z (1, i) = {Z (1, i, 1), Z (1, i, 2), ..., Z (1, i, k), ..., Z (1, i, K)},

Z (2, j) = {Z (2, j, 1), Z (2, j, 2), ..., Z (2, j, k), ..., Z (2, j, K)},

And if Z (1, i, k) = 1, the i-th object of the first domain belongs to the k-th submatrix region, and if Z (1, i, k) = 0, it does not belong. It expresses. When using real values, the degree of affiliation is also expressed as an intermediate point. The same applies to Z (2, j, k). In the present embodiment, it is necessary to assume some mathematical model for the partial matrix allocation estimated value Z.

  Next, hyperparameters initialized by the initialization unit 26 will be described. The initial value α (0) of the submatrix region hyperparameter is an initial value of the parameter of the mathematical model used for estimating the submatrix region. The parameter set of the model employed in the present embodiment is a submatrix area hyperparameter α. The model is mainly intended to estimate Z and K when an observation matrix X is given. The initial value β (0) of the observation hyperparameter is an initial value of a parameter related to a mathematical model that expresses the feature of the observation value in the submatrix region. The representation of the observation hyperparameter β varies depending on the purpose and the nature of the data.

  The input data storage unit 28 stores an initial value K (0) of the number of submatrix regions initialized by the initialization unit 26 and an initial value α (0) of the submatrix region hyperparameter. The input data storage unit 28 stores an observation matrix X received from the input unit 10 and an end condition constant. The end condition constant is a constant used for determining the end condition of the repetition determination unit 40 described later. Usually, since the estimation calculation is performed by iterative calculation, a threshold value for the maximum number of repetitions or the fluctuation range of the evaluation value is set as the end condition constant.

  The variable storage unit 42 stores the number K of partial matrix regions estimated by the variable estimation unit 30 described later, the partial matrix allocation estimated value Z, the observed parameter estimated value θ, and the observed hyperparameter estimated value β.

  The variable estimation unit 30 includes a partial matrix region allocation estimation unit 32, a partial matrix region hyperparameter estimation unit 34, an observation parameter estimation unit 36, an observation hyperparameter estimation unit 38, and an iterative determination unit 40. ing.

  The submatrix region allocation estimation unit 32 performs first to third processes described below. In the first process, the existing submatrix region allocation merely estimates whether or not the object of interest belongs to each of the existing K submatrixes, which is an existing method. On the other hand, in the second process, “whether a new submatrix region is generated”, that is, a process of increasing the number of submatrixes is performed. The third process incorporates a process of “erasing unused submatrix regions”, that is, reducing the number of submatrixes. By appropriately implementing the second and third processes, it is possible to automatically determine the optimum number K of sub-matrix regions in accordance with the matrix data. This is a feature.

  In the first process, the submatrix area allocation estimation unit 32 performs an input data storage unit for each submatrix area that exists in the number of submatrix areas for each object in the first domain and each object in the second domain. 28 or a pre-fit degree indicating the degree to which the object is assigned to the partial matrix region, which is estimated based on the partial matrix region allocation estimated value Z, stored in the variable storage unit 42, and the input data storage unit 28. The object is estimated based on the stored observation matrix X, the partial matrix region allocation estimated value Z, and the partial matrix region hyperparameter α stored in the input data storage unit 28 or the variable storage unit 42. The object belongs to the submatrix area based on the data fit that represents the likelihood assigned to the submatrix area By calculating the likelihood that the object is to estimate whether or not to belong to the partial matrix region. Then, when it is estimated that it should belong to the submatrix region, the submatrix allocation estimated value Z updated so as to allocate the submatrix region to the object is stored in the variable storage unit 42.

  Details of the first process of the submatrix region allocation estimation unit 32 will be described below. In the first process, the sub-matrix region allocation is recalculated for all objects in the first domain and the second domain. In this recalculation, the update order of the objects is arbitrary. For convenience of explanation, it is assumed that the object i of the first domain is currently selected and estimated, and the submatrix area allocation estimated value Z is updated by allocation.

  In the first processing, whether or not an object is likely to belong to the submatrix region k is generally determined by “the affiliation status of other objects, that is, the object estimated from the submatrix region allocation estimated value Z is included in the submatrix region. "Degree of assignment or degree of assignment (affiliation degree)" and "How well the observation matrix X can be explained when it is decided that it belongs to or does not belong to the submatrix region k". The If the former is called the pre-matching degree and the latter is called the data matching degree, the possibility that an object belongs to the submatrix region k can be calculated, for example, as in the following formula (1) or (2). . Similarly, the possibility of not belonging can also be calculated.

(Possibility of belonging to submatrix region k)
= (Preliminary conformance) x (Data conformance when it is decided to belong) (1)

Or

(Possibility of belonging to submatrix region k)
= (Preliminary conformance) + (Data conformance when it is decided to belong) (2)

  In the first process of the submatrix region allocation estimation unit 32, the allocation Z of the object i to the submatrix region k in the submatrix region allocation estimation value Z based on the above equation (1) (or equation (2)). Recalculate and update (1, i, k). The calculation method is a method of assigning Z (1, i, k) as a binary value of 0 or 1, and assigning Z (1, i, k) from 0 to 1 As a real number, there is a method of apportioning by the possibility of belonging and the possibility of not belonging, and a method of selecting Z (1, i, k) as a binary value of 0 or 1 and probabilistically selecting based on the apportioning result. It should be noted that the method of calculating the pre-fit and the data fit is arbitrary, and the implementation that is considered to be statistically optimal in the first process is to calculate the possibility calculation formula based on multiplication based on Bayesian estimation. Specific examples in the case of formulation based on Bayesian estimation will be introduced in the embodiment, and either or both may be constants unless otherwise indicated, but mathematically If not calculated in the proper way Processing incompatible with the number of updates of the partial matrix region in a calculation method of calculation is due to the possibility of failure has to be matched.

  In the second process, the submatrix region allocation estimation unit 32 performs advance processing on the number K of required submatrix regions estimated based on the partial matrix region allocation estimation value Z for each object in the first domain. The degree of goodness, the degree of pre-fit that represents the degree to which the object of the second domain is assigned to the newly generated submatrix region, the degree of prefit for the observation parameter θ for the observation value in the new submatrix region, Possibility of the number K + L of the number of submatrix regions added with a new submatrix region, based on the data suitability indicating the degree to which the observation matrix X can be well explained by increasing the number of submatrix regions To calculate whether or not a new submatrix region should be generated to represent the object. Further, in the same way as in the first domain, the second domain is estimated based on the partial matrix region allocation estimated value Z for each object. The degree of prior fit indicating the degree to which objects of the first domain are assigned to the generated submatrix region, the degree of prior fit for the observation parameter θ for the observation value in the new submatrix region, and the number of submatrix regions The possibility that the number of submatrix regions is K + L with the addition of a new submatrix region is calculated on the basis of the data suitability representing the degree that the observation matrix X can be well explained by increasing Thus, it is estimated whether or not a new submatrix region should be generated to represent the object.

  In the second process of the submatrix region allocation estimation unit 32, as will be described below, in the first process, the affiliation allocation to the K submatrix regions for the object in the first domain or the second domain is performed. After the estimation is finished, calculate whether a new submatrix region is necessary for the object. This is because there is a possibility that the number of submatrix regions latent in the observation matrix X given the number K of submatrix regions is insufficient when the number K of submatrix regions is appropriately given by initialization or the like. Because there is. The calculation method is almost the same as the case where the possibility that an object belongs to the submatrix area of the first process is calculated.

  Specifically, in the second process of the partial matrix region allocation estimation unit 32, it is estimated whether or not L new partial matrix regions should be generated based on the following three elements. In the second process, as a first element to be considered, “the affiliation status of the partial matrix area of all objects in the first process, that is, estimated based on the partial matrix area allocation estimated value Z is required. Pre-fit degree regarding the number K of sub-matrix regions ”is estimated. Here, an evaluation value for evaluating whether to increase the number of submatrix regions by 0 or 2 is estimated. Here, when the number of submatrix regions is increased by L, it is assumed that the object i or object j that is focused on in the current process always belongs to L submatrix regions.

  Next, in the second process, as a second element, “the degree to which objects of the other domain are allocated to L newly generated sub-matrix regions” is calculated. For example, for L submatrix regions newly generated for an object in the first domain, an object in the second domain must belong to L submatrix regions (objects in the second domain). If it is not assigned to, it will not become a rectangle). Therefore, assuming that the object i in the first domain is focused on in the current processing, it is assumed that the N2 objects in the second domain belong to a new submatrix region with some index (eg, random). Calculate the degree of prior fit for the degree.

  In the second process, as a third element, an observation parameter θ for L newly generated sub-matrix regions is set based on the observation parameter θ stored in the variable storage unit 42, Obtain a prior goodness of the observation parameter β for the observation value in the new submatrix region.

  In the second process, when the above three elements are determined, “how well the observation matrix X can be explained by setting the number of submatrix regions to K + L” is calculated.

  By combining the above elements, whether or not to increase the number of submatrix regions is estimated based on the following equation (3).

(The number of submatrix regions may be K + L)

= (Pre-fitness for the required number of submatrix regions K estimated based on the submatrix region allocation estimate Z) × (L newly generated submatrix regions in the other domain Pre-adaptation degree indicating the degree to which an object is assigned) x (pre-adaptation degree regarding observation parameter β for observation value in new sub-matrix region) x (observation matrix X is explained well by increasing the number of sub-matrix regions Data suitability indicating the degree to which it is possible
... (3)

In the above equation (3), multiplication is used, but how to use addition, subtraction, multiplication, and division is arbitrary. In addition, the calculation of each fitness level depends on a mathematical model designed by the user. Note that the number of L is determined randomly. In the embodiments described later, an optimal implementation example is introduced in a probabilistic sense, but the implementation method is not limited to the implementation examples in the embodiment, and various mathematical models can be designed.

  Then, in the second process, when it is estimated that L partial matrix regions should be generated, L partial matrix regions are generated, and the generated L partial matrices are generated in the object. The submatrix allocation estimated value Z stored in the variable storage unit 42 is updated so as to allocate the area. Further, the number K of partial matrix regions in the variable storage unit 42 is updated with the number K of partial matrix regions set to K = K + L.

  In the third process, the submatrix region allocation estimation unit 32 determines whether the number of objects in the first domain or the number of objects in the second domain allocated for each submatrix region by the first process and the second process is a predetermined value. The following submatrix regions are deleted and the number of submatrix regions is updated.

  Specifically, in the third process of the submatrix region allocation estimation unit 32, as described below, the submatrix region allocation estimation value Z is checked for an unnecessary submatrix region and updated. Here, the size of each submatrix region is determined by “the number of objects belonging to the first domain” × “the number of objects belonging to the second domain”. However, since the “submatrix region” to be extracted is expressed by the direct product of the subsets of the first domain and the second domain, it is estimated that “the object belonging to either domain is below a certain value”. The submatrix area is deleted as an unnecessary submatrix area. For example, the total number of objects in the first domain belonging to the kth submatrix region can be calculated by summing Z (1, i, k) for all i. The submatrix area whose total number of objects is equal to or smaller than a certain value is deleted. Then, with the deletion of one submatrix region, the number K of submatrix regions is reduced to K = K-1. If necessary, the submatrix area allocation estimated value Z, the observation parameter θ, the observation hyperparameter β, and the like, and the index k of the submatrix area are appropriately designated again. The deletion procedure of the third process is executed after the second process in the present embodiment, but is not limited to this. For example, the estimation of each object of the first process or the second process is performed. Can be executed at an arbitrary timing, such as every time or after all objects have been estimated.

  The submatrix area hyperparameter estimation unit 34 is based on the submatrix area allocation estimated value Z stored in the variable storage unit 42 and is necessary for a mathematical model relating to the number of submatrix areas and the allocation of the submatrix areas. The hyper parameter α is estimated and stored in the variable storage unit 42.

  Based on the observation matrix X received by the input unit 10 and the partial matrix region allocation estimated value Z stored in the variable storage unit 42, the observation parameter estimation unit 36 performs mathematics used to represent the observation values in the partial matrix region. The model observation parameter θ is estimated and stored in the variable storage unit 42.

  The observation hyperparameter estimation unit 38 estimates the observation hyperparameter β necessary for the mathematical model of the observation parameter θ based on the observation parameter θ estimated by the observation parameter estimation unit 36 and stores it in the variable storage unit 42.

  Note that the initial values may be left as they are without performing the estimation by the partial matrix region hyperparameter estimation unit 34, the observation parameter estimation unit 36, and the observation hyperparameter estimation unit 38.

  The iterative determination unit 40 determines the estimation and allocation by the submatrix region allocation estimation unit 32 and the estimation by the submatrix region hyperparameter estimation unit 34, the observation parameter estimation unit 36, and the observation hyperparameter estimation unit 38 in advance. Repeat until Here, an end condition constant stored in the input data storage unit 28 may be used as the repeat end condition.

  Note that the configuration of each part of the variable estimation unit 30 is not limited to the above, and depends on the observation model assumed by the user, and therefore cannot be generally described. In an example described later, an implementation example according to a specifically designed model is introduced.

<Operation of Submatrix Region Extraction Device According to Embodiment of the Present Invention>

  Next, the operation of the submatrix region extraction apparatus 100 according to the embodiment of the present invention will be described. When the observation matrix X is received by the input unit 10, the observation matrix X is stored in the input data storage unit 28, and the partial matrix region extraction device 100 executes a partial matrix region extraction processing routine shown in FIG.

  First, in step S100, the number of submatrix regions, a submatrix region allocation estimation value indicating whether each submatrix region is allocated to each object in the first domain and each object in the second domain, and a submatrix region The hyper parameters and the observation hyper parameters are initialized and stored in the input data storage unit 28.

  Next, in step S102, the partial matrix allocation estimated value Z and the number K of partial matrix regions are estimated by the first to third processes of the partial matrix region allocation estimating unit 32.

  In step S104, the partial matrix region hyperparameter α is estimated based on the partial matrix allocation estimated value Z estimated in step S102.

  In step S106, the observation parameter θ is estimated based on the observation matrix X received in the input unit 10 and the partial matrix allocation estimated value Z estimated in step S102.

  In step S108, the observation hyperparameter β is estimated based on the observation parameter θ estimated in step S106.

  In step S110, it is determined whether a predetermined repetition end condition is satisfied. If not satisfied, the process proceeds to step S102, and the processes in steps S102 to S108 are repeated. If satisfied, the process proceeds to step S108.

  In step S112, the partial matrix allocation estimated value Z estimated in step S102 and the number K of partial matrix regions are output by the output unit 50, and the process ends.

  The process of step S102 is realized by the estimated allocation process routine shown in FIGS.

  In step S200, the object i of the first domain is initialized as i = 1.

  In step S204, the submatrix region k is selected.

  In step S206, with respect to the object i of the first domain, with respect to the submatrix region selected in step S204, according to the above equation (1), a pre-matching degree indicating the degree to which the object i is assigned to the submatrix region k; The possibility that the object i belongs to the submatrix region k is calculated based on the data suitability representing the likelihood that the object i is assigned to the submatrix region k.

  In step S208, based on the calculation result of step S206, it is determined whether or not it should belong to the submatrix region k, and the submatrix allocation estimated value Z updated according to the determination result is stored in the variable storage unit 42. To do.

  In step S210, it is determined whether all submatrix regions have been estimated for the object i. If there is a submatrix region that has not been estimated, the process returns to step S204 to select the submatrix region k. And if the process of step S206-S208 is repeated and all are estimated, it will transfer to step S212.

  In step S212, the possibility that the number of sub-matrix regions is K + L is calculated according to the above equation (3). At this time, the value of L may be determined at random.

  In step S214, it is determined whether or not L partial matrix regions should be generated based on the calculation result in step S212. If not, the process proceeds to step S218. If so, the process proceeds to step S216. Transition.

  In step S216, L submatrix regions are generated, each of the generated L submatrix regions is assigned to the object i, and the submatrix allocation estimation value Z stored in the variable storage unit 42 is set. Update. In addition, the number K of sub-matrix regions stored in the variable storage unit 42 is updated.

  In step S218, it is determined whether all objects i in the first domain have been estimated and assigned. If not, the process proceeds to step S220, i = i + 1 is set, and the processes in steps S204 to S216 are repeated to estimate all. If yes, the process proceeds to step S222.

  In step S222, the object j of the second domain is initialized as j = 1.

  Steps S224 to S236 are not described here because the same processing as that performed for the object i described in steps S204 to S216 may be performed for the object j in the second domain.

  In step S238, it is determined whether all objects j have been estimated and assigned. If not, the process proceeds to step S240, j = j + 1 is set, and the processes in steps S224 to S236 are repeated to estimate all. If so, the process proceeds to step S240.

  In step S242, based on the submatrix allocation estimated value Z, a submatrix region in which the allocated number of objects in the first domain or the number of objects in the second domain is equal to or smaller than a predetermined value is deleted and stored in the variable storage unit 42. The number of sub-matrix regions that have been updated is updated, and the estimated allocation processing routine ends.

<Example>

  An experimental result of the method according to the embodiment of the present invention will be described. Here, the design of a mathematical model that can be stochastically optimized when real matrix data is given, specific variables, and calculation formulas to be implemented are shown.

  In this embodiment, an example using a probabilistic model based on the Plaid model (see Non-Patent Document 1 and Non-Patent Document 2 above) will be described. The Plaid model is one of existing techniques for submatrix extraction. The Plaid model expresses and extracts continuous real observation values by superposing average value parameters of a plurality of submatrix regions. In this example, by applying and extending the configuration of the embodiment described above to this method, it is possible to automatically extract the number of necessary submatrix regions and automatically estimate various parameters. An experiment was conducted to realize a clear Plaid model.

  In accordance with a probabilistic mathematical model of a partial matrix region extraction model (extended Plaid model) to which the Plaid model technique is applied, shown in the following equations (4) to (8), a partial matrix region allocation estimated value Z, an observation parameter estimated value θ , And an observation matrix X.

Z (1, i, k) to BePBerP (α1) (4)
Z (2, j, k) to BePBerP (α2) (5)
θk to Normal-Wishart (βk) (6)
θ0-Normal-Wishart (β0) (7)
X (i, j) to Normal (m0 + Σ_ {k} mk, τ) (8)

  The observation matrix X, partial matrix region hyperparameter α, partial matrix region allocation estimated value Z, observation parameter θ, and observation hyperparameter β corresponding to each element of the variable storage unit 42 are shown below.

X = {X (i, j)}
α = (α1, α2)
Z = (Z1, Z2)
Z1 = {Z (1, i, k)}
Z2 = {Z (2, j, k)}
θ = (t, θ0, θ1, θ2,...)
θk = (mk, τk)
β = (β0, βk)

For details of the probability distribution in the above formulas (4) to (8), the above-mentioned Non-Patent Document 4 and Non-Patent Document 5 can be referred to.
[Non-Patent Document 5]: Griffiths and Ghahramani, “The Indian Buffet Process: An Introduction and Reivew”, Journal of Machine Learning Research, Vol. 12, pp. 1185-1224, 2011.
The implementation of the mathematical model in the embodiment of the present invention is completed by designing the functions of the pre-matching degree and the data fitting degree with respect to the probability models of the above formulas (4) to (8). The design method is arbitrary, but a stochastic and statistically optimal function design method is Bayesian estimation (see Non-Patent Document 4 above).

  According to the estimation process in the submatrix region allocation estimation unit 32, according to Bayesian estimation, the prior fitness is “prior distribution” when other parameters and variables are given, and the data fitness determines the value of the variable of interest. It is defined as “likelihood” of the case, and estimation of the parameter of interest can be implemented as “calculation of posterior distribution”.

  In the first process of the submatrix region allocation estimation unit 32 in the above embodiment, whether or not the object i should belong to the submatrix region k is estimated according to the following equation (9).

p (Z (1, i, k) | Z1 (−ik), Z2, θ)
∝p (X | Z (1, i, k), Z1 (−ik), Z2, θ) × p (Z (1, i, k) | Z1 (−ik)) (9)

  Here, Z1 (−ik) is obtained by removing the value of Z (1, i, k) from Z1. Further, p (X | Z (1, i, k), Z1 (-ik), Z2, [theta]) is the data suitability, and p (Z (1, i, k) | Z1 (-ik)) is Pre-fit degree.

  In the second process of the submatrix region allocation estimation unit 32 in the above embodiment, it is estimated whether or not new L submatrix regions should be generated to represent the object i according to the following equation (10). To do.

p (L, Z2 *, θ *) = min (1.0, p (X | Z1, Z2, Z1 *, Z2 *, L, θ, θ *) / p (X | Z1, Z2, θ)) ... (10)

  Here, Z1 * is assigned to L new submatrix areas and is a set of objects i of the first domain to which Z1 * belongs, and Z2 * is assigned to L new submatrix areas. , Belonging to the second domain object j, θ * is an observation parameter for L new submatrix regions.

  Next, the partial matrix region hyperparameter estimation unit 34 estimates the partial matrix region hyperparameters for the first domain and the second domain according to the following equations (11) and (12).

  The submatrix region hyperparameter α1 for the first domain is estimated according to the following equation (11).

p (α1 | Z1) ∝p (α1 | Z1) × p (α1) (11)

  The submatrix region hyperparameter α2 for the second domain is estimated according to the following equation (12).

p (α2 | Z2) ∝p (α2 | Z2) × p (α2) (12)

  Here, a gamma distribution is used for the prior distribution.

  The above formulas (9) to (12) are mathematical formulas when the Markov chain Monte Carlo method, which is one type of Bayesian estimation, is used. Non-patent literature 4 and non-patent literature 5 may be referred to for a specific calculation method of each expression.

  In addition, the observation parameter estimation unit 36 and the observation hyperparameter estimation unit 38 calculate the posterior distribution and the parameter estimation using the conjugate of the normal Wishart distribution and the normal distribution according to the above formulas (6) and (7). (Refer nonpatent literature 4).

  After the above mathematical model is implemented in each component, the iteration determination unit 40 uses the maximum number of iterations of the Markov chain Monte Carlo method as an end condition constant, and ends the estimation when the number of iterations of estimation reaches the constant. .

  Table 1 shows examples of experimental results.

In the experiment, a plurality of artificial data designed with 3 or 4 latent submatrix regions were prepared, and it was verified whether these submatrix regions could be extracted with high accuracy. For accuracy, Normalized Mutual Information (NMI) was used (see Non-Patent Document 6).

  The numbers in Table 1 are the calculated NMI. Larger values represent better results. The maximum value is 1.0, which means that the submatrix region can be completely extracted at this time. The comparison object is an existing method (Non-Patent Document 6) in which the number K of sub-matrix regions to be extracted is fixed and analyzed, and a proposed invention method in which this method is extended with the configuration of the above embodiment.

[Non-Patent Document 6]: Lancichinetti, Fortunato, and Kertesz, “Detecting the Overlapping and Hierarchical Community Structure of Complex Networks”, New Journal of Physics, Vol. 11 (3), 2009.
In both methods, the initial value K0 of K is changed to (1) the number of true submatrix regions, (2) K0 = 10, (3) K0 = 20, and the existing method is fixed at K = K0. The method automatically estimated K. As shown in Table 1, the proposed method obtains better values in all cases. In particular, in the artificial data 2 to 4, the NMI indicated by the proposed method maintains a high value regardless of the number of K. This is because it is possible to accurately extract the number of correct submatrix regions by extracting the submatrix regions while automatically estimating the number of potential submatrix regions due to the effect of the configuration of the proposed method. It is shown that.

  FIG. 6 is an example of visualization of what is actually extracted. This is an example in which the proposed method can extract submatrix regions neatly, but the existing method does not work.

  As described above, according to the partial matrix region extraction device according to the embodiment of the present invention, the observation matrix X and the partial matrix for each partial matrix region for each object in the first domain and each object in the second domain. When it is estimated that the object should belong to the submatrix area by estimating whether the object should belong to the submatrix area based on the matrix area allocation estimated value Z and the submatrix area hyperparameter α. Assigns a submatrix region to the object, and estimates whether a new submatrix region should be generated to represent the object based on the observation matrix X and the submatrix region allocation estimate Z. If it is estimated that a new submatrix region should be generated, a new submatrix region is generated, and the generated new submatrix region is assigned to the object. Rely, update the number K of submatrix regions, delete submatrix regions where the number of assigned objects in the first domain or the number of objects in the second domain is equal to or less than a predetermined value, and repeat estimation and assignment in advance. By repeating the process until the end condition is satisfied, it is possible to extract a submatrix region that is an optimal number of features.

  The present invention is not limited to the above-described embodiment, and various modifications and applications can be made without departing from the gist of the present invention.

  For example, in the above-described embodiment, an example in which the possibility that an object belongs to a submatrix area is calculated based on the prior fitness indicating the degree to which the object is assigned to the submatrix area and the data fitness is taken as an example. As described above, but not limited to this, the possibility that the object does not belong to the submatrix area is calculated based on the degree of prior fit indicating the degree to which the object is not assigned to the submatrix area and the data suitability. You may make it do. In this case, the partial matrix area allocation estimated value Z may be updated according to the possibility that the object does not belong to the partial matrix area.

DESCRIPTION OF SYMBOLS 10 Input part 20 Calculation part 26 Initialization part 28 Input data storage part 30 Variable estimation part 32 Submatrix area | region allocation estimation part 34 Partial matrix area | region hyperparameter estimation part 36 Observation parameter estimation part 38 Observation hyperparameter estimation part 40 Determination part 42 Variable Storage unit 50 Output unit 100 Submatrix region extraction device

Claims (5)

A partial matrix region extraction device that extracts a characteristic partial matrix region from an observation matrix composed of observation values for each pair of objects in the first domain and each object in the second domain,
The number of the submatrix regions and the submatrix region indicating whether each submatrix region is allocated to each object in the first domain and each object in the second domain by the number of the submatrix regions. An initialization unit for initializing the allocation estimated value;
For each object in the first domain and each object in the second domain, the observation matrix, the partial matrix region allocation estimate, and the object for each object When it is estimated that the object should belong to the submatrix area by estimating whether the object should belong to the submatrix area based on the submatrix area hyperparameter related to the allocation of the submatrix area Assigns the submatrix region to the object and determines whether to generate a new submatrix region to represent the object based on the observation matrix and the submatrix region allocation estimate. If it is estimated that a new partial matrix region should be generated, the new partial row Generates a region, to the object, it assigns the generated new said sub-matrix area, updating the number of the partial matrix region,
A submatrix region allocation estimation unit that deletes the submatrix region in which the number of assigned objects in the first domain or the number of objects in the second domain is a predetermined value or less, and updates the number of submatrix regions;
An iterative determination unit that repeats the estimation and allocation by the submatrix region allocation estimation unit until a predetermined repetition end condition is satisfied;
A submatrix region extraction apparatus including:   The submatrix region allocation estimation unit sets the submatrix region allocation estimation value for each submatrix region that exists as many as the number of submatrix regions for each object of the first domain and each object of the second domain. Estimated based on a pre-fit degree indicating the degree to which the object is assigned or not assigned to the sub-matrix region, the observation matrix, the sub-matrix region assignment estimate, and the sub-matrix region hyperparameter The object may or may not belong to the submatrix region based on the likelihood that the object is assigned to the submatrix region or the data fitness that represents the likelihood that the object is not assigned. By calculating the possibility, the object is located in the submatrix region. Submatrix region extraction apparatus according to claim 1 to estimate whether or not to not. The submatrix region allocation estimation unit is configured to estimate, for each object in the first domain, a pre-fit degree related to the number of required submatrix regions estimated based on the partial matrix region allocation estimation value, and A pre-fit degree indicating the degree to which the object of the second domain is assigned to the generated sub-matrix area; a pre-fit degree related to observation parameters for observation values in the new sub-matrix area; and Calculate the possibility of the number of submatrix regions to which the new submatrix region is added based on the data suitability representing the degree to which the observation matrix can be well explained by increasing the number To estimate whether to generate a new submatrix region to represent the object,
For each object in the second domain, a pre-fit degree related to the number of required submatrix regions estimated based on the submatrix region allocation estimate, and a newly generated submatrix region, The observation matrix is obtained by increasing the pre-adaptation degree indicating the degree to which the objects of the first domain are allocated, the pre-adaptation degree regarding the observation parameter for the observation value in the new sub-matrix region, and the number of the sub-matrix regions. To express the object by calculating the possibility of the number of the submatrix regions to which the new submatrix region is added based on the data suitability representing the degree to which can be well explained The partial matrix region extraction apparatus according to claim 2, wherein whether or not a new partial matrix region is to be generated is estimated. A partial matrix region hyperparameter estimation unit for estimating the partial matrix region hyperparameter based on the partial matrix region allocation estimation value;
The initialization unit further initializes the partial matrix region hyperparameter,
The said repetition determination part repeats the estimation and allocation by the said partial matrix area | region allocation estimation part, and the estimation by the said partial matrix area | region hyper parameter estimation part until it satisfies predetermined repetition termination conditions. Submatrix region extraction device.   The program for functioning a computer as each part of the partial matrix area | region extraction apparatus of any one of Claims 1-4.