JP6316773B2 - Statistical data reconstruction device, statistical data reconstruction method, program - Google Patents

Description

  The present invention relates to a statistical data reconstruction device, a statistical data reconstruction method, and a program for reconstructing statistical data from disturbance data generated by performing disturbance processing on original data.

  Conventionally, for example, Non-Patent Documents 1, 2, and 3 have been disclosed as techniques for reconstructing only the cross tabulation result while concealing individual data in a database by a probabilistic method.

University of Igarashi, Koji Senda, Katsumi Takahashi, “Randomization method that satisfies k-anonymity in numerical attributes”, Proceedings of the Computer Security Symposium 2011, October 2011, Vol. 2011, pp.450-455 University of Igarashi, Koji Senda, Katsumi Takahashi, “Extension to k-anonymity and its application example”, 2009 Proceedings of Computer Security Symposium, October 2009, Vol. 1, pp.1-6 Rakesh Agrawal, Ramakrishnan Srikant, and Dilys Thomas, "Privacy preserving OLAP," In Proceedings of the 2005 ACM SIGMOD international conference on Management of data, 2005, pp. 251-262

  In the prior art, no assumptions are made on the original data, and the cross tabulation table is estimated (hereinafter referred to as reconstruction processing) from concealed data (hereinafter referred to as disturbance data). In order to construct it, it was a problem that a huge amount of data was required.

  Therefore, an object of the present invention is to provide a statistical data reconstruction apparatus that can reconstruct statistical data with a smaller number of data than before.

  The present invention is a statistical data reconstruction device for reconstructing statistical data from disturbance data generated by performing disturbance processing on original data, a parameter initialization unit, an E step calculation unit, an M step calculation unit, Includes a statistical data reconstruction unit.

  The parameter initialization unit assumes that the original data before the disturbance processing can be described by a linear sum of a finite number of Gaussian distributions with non-diagonal elements 0 of the covariance matrix, and calculates the weight, average, and variance of each Gaussian distribution. initialize. The E step calculation unit calculates the probability density of the predetermined disturbance data in the denominator for all combinations of the predetermined number of disturbance data generated by adding the predetermined noise to the original data and the Gaussian distribution to which the predetermined number of noises are added. In addition, a process of calculating a burden rate including the probability density of the predetermined number of disturbance data in the predetermined number of Gaussian distribution in which noise is added to the numerator is repeatedly executed. The M step calculation unit updates the weight of each Gaussian distribution based on the average value of the burden rate, sets the unknown function as a logarithm value of the likelihood density of the disturbance data, and sets the variable as the average of the Gaussian distribution. Use differential to update the average of each Gaussian distribution in the direction that maximizes the likelihood, use the unknown function as the logarithm value of the likelihood density of the disturbance data, and the variable as the variance of the Gaussian distribution Is used to repeatedly execute the process of updating the variance of each Gaussian distribution in the direction in which the likelihood is maximized. The statistical data restructuring unit reconstructs statistical data using the average, variance, and weight converged by repeatedly executing the processing for calculating the burden rate and the processing for updating the weight, average, and variance.

  According to the statistical data reconstruction apparatus of the present invention, statistical data can be accurately reconstructed with a smaller number of data than before.

1 is a block diagram illustrating a configuration of a statistical data reconstruction apparatus according to a first embodiment. 5 is a flowchart illustrating the operation of the statistical data reconstruction apparatus according to the first embodiment. FIG. 6 is a block diagram illustrating a configuration of a statistical data reconstruction apparatus according to a second embodiment. 10 is a flowchart illustrating the operation of the statistical data reconstruction apparatus according to the second embodiment. FIG. 9 is a block diagram illustrating a configuration of a statistical data reconstruction apparatus according to a third embodiment. 10 is a flowchart illustrating the operation of the statistical data reconstruction apparatus according to the third embodiment. FIG. 9 is a block diagram illustrating a configuration of an E step calculation unit according to the third embodiment. 10 is a flowchart illustrating the operation of an E step calculation unit according to the third embodiment. FIG. 10 is a diagram for explaining data rotation executed by a principal component analysis unit according to the third embodiment. FIG. 10 is a block diagram illustrating a configuration of an M step calculation unit according to the third embodiment. 10 is a flowchart illustrating the operation of an M step calculation unit according to the third embodiment.

  Hereinafter, embodiments of the present invention will be described in detail. In addition, the same number is attached | subjected to the structure part which has the same function, and duplication description is abbreviate | omitted.

とする。また、元データを実施例１において１次元のベクトルx、実施例２、３においてd次元ベクトル

で表すこととする。 <Notation>
In the following description, a vector is described in bold type, and a scalar value is described in italic type (or typeface other than bold type). In addition, the j-th element of the vector is described as a (j). The number of observed disturbance data is N, and the i-th (1 ≦ i ≦ N) disturbance data is a one-dimensional vector y _i in Example 1, d-dimensional vector in Examples 2 and 3 (d is 2 or more) integer)

And The original data is a one-dimensional vector x in the first embodiment, and the d-dimensional vector in the second and third embodiments.

It shall be expressed as

<Preparation>
The statistical data reconstruction apparatus of the present invention is an apparatus for reconstructing statistical data from disturbance data generated by performing disturbance processing on original data. The statistical data disturbance and reconstruction process is roughly divided into the following two steps.
Disturbance: Disturbance processing is performed on the original data to create disturbance data.
Reconstruction: Statistical analysis is performed on disturbance data, and statistical results (statistical data) are obtained.

Examples of the disturbance processing include noise addition in Non-Patent Document 1 and maintenance replacement disturbance in Non-Patent Document 2. Statistical analysis includes a method of estimating cross tabulation in Reference Non-Patent Document 1 and a method of performing a t-test. The present invention belongs to the latter reconstruction (statistical analysis).
(Reference Non-Patent Document 1: University of Igarashi, Koji Senda, Katsumi Takahashi, “Efficient Privacy Protection Cross Tabulation Applicable to Multi-valued Attributes”, Proceedings of Computer Security Symposium 2008, October 2008, Vol. 2008, pp .497-502)

に従うノイズ(以後、ラプラスノイズと呼ぶ)を付加したものとし、パラメータφは公開されているものとする。なお実施例２、３においては、元データに対し、平均0、パラメータφのd次元ラプラス分布

に従うノイズ(以後、ラプラスノイズと呼ぶ)を付加したものとする。本発明の統計データ再構築装置は、以上の操作が行われた撹乱データy_i,(i=1,...,N)を用いて、再構築処理を行う。 In the following embodiments, the technique of Non-Patent Document 1 is used as the disturbance process. That is, in the first embodiment, the original data is a one-dimensional Laplace distribution with an average of 0 and a parameter φ.

It is assumed that noise (hereinafter referred to as Laplace noise) is added and the parameter φ is disclosed. In Examples 2 and 3, d-dimensional Laplace distribution with an average of 0 and parameter φ with respect to the original data

Is added (hereinafter referred to as Laplace noise). The statistical data reconstruction apparatus of the present invention performs the reconstruction process using the disturbance data y _i (i = 1,..., N) on which the above operations have been performed.

  Hereinafter, the configuration and operation of the statistical data reconstruction apparatus according to the first embodiment will be described with reference to FIGS. 1 and 2. FIG. 1 is a block diagram illustrating a configuration of a statistical data reconstruction apparatus 1 according to the present embodiment. FIG. 2 is a flowchart showing the operation of the statistical data reconstruction apparatus 1 of this embodiment.

  As shown in FIG. 1, the statistical data reconstruction device 1 according to the present exemplary embodiment includes a parameter initialization unit 11, a likelihood maximization unit 12, and a statistical data reconstruction unit 13. The likelihood maximizing unit 12 includes an E step calculating unit 121 and an M step calculating unit 122.

で記述できるものと仮定し、k番目の各ガウス分布の重みρ_k、平均μ_k、および分散σ_k ²をランダムに初期化する（Ｓ１１）。（２）式において

は、平均μ、分散σ²に従う共分散行列の非対角要素0のガウス分布を表すものとする。詳細には、zを任意の変数として、

と表される。なお、（２）式で述べたように、全ての（∀）kに対して、重みρ_kが０以上１以下、重みρ_kの総和が１となる制約条件の下でρ_kをランダムに初期化するものとする。 The parameter initialization unit 11 is a Gaussian distribution in which the original data before the disturbance processing is a finite number of non-diagonal elements 0 of the covariance matrix (K, K is a natural number, and k is a natural number satisfying 1 ≦ k ≦ K). A linear sum of

The weight ρ _k , mean μ _k , and variance σ _k ² of each kth Gaussian distribution are initialized at random (S11). In equation (2)

Denote the Gaussian distribution of off-diagonal element 0 of the covariance matrix with mean μ and variance σ ² . Specifically, let z be an arbitrary variable

It is expressed. Note that, as described in the equation (2), for all (∀) k, ρ _k is randomly set under the constraint that the weight ρ _k is 0 or more and 1 or less and the sum of the weights ρ _k is 1. It shall be initialized.

In the following steps S121 and S122, the likelihood maximization unit 12 solves the following likelihood maximization problem.

と表される。ここで、erfc(x)は、相補誤差関数と呼ばれているものであり、詳細には、

と表される。式を簡略化するために関数ｆを

と定義すれば、i番目の撹乱データの確率分布を表す関数ｇは

と表される。 Note that the probability distribution of the i-th disturbance data is represented by the sum of the two random variables (Equations (1) and (3)) described above. That is, the function g representing the probability distribution of the i-th disturbance data is

It is expressed. Here, erfc (x) is called a complementary error function.

It is expressed. To simplify the equation, the function f is

The function g representing the probability distribution of the i-th disturbance data is

It is expressed.

In this embodiment, a generalized EM algorithm as used in Reference Non-Patent Document 2 is applied as a method for solving the above likelihood maximization problem.
(Reference Non-Patent Document 2: Jeff A Bilmes, et al., "A Gentle Tutorial of the EM Algorithm and its Application to Parameter Estimation for Gaussian Mixture and Hidden Markov Models," International Computer Science Institute, Vol. 4, No. 510 , 1998, pp. 1-13)

を含み、分子にノイズを付加した所定番目(l番目、lを1≦l≦Kを充たす自然数とする)のガウス分布における所定番目(i番目)の撹乱データの確率密度、すなわち

を含む負担率γ(i,l)を計算する処理を繰り返し実行する（Ｓ１２１）。負担率γ(i,l)は、

と表される。負担率γ(i,l)は、所定番目のガウス分布への負担度合いを表すことから、負担率と呼ばれる。 The E step calculation unit 121 adds a predetermined number (N) of disturbance data generated by adding a predetermined noise (Laplace noise) to the original data x and all of the Gaussian distributions with the predetermined number (K) of noise added. For the combination (i = 1, ..., N, l = 1, ... K), the probability density of the predetermined (i-th) disturbance data in the denominator, ie

And the probability density of the predetermined order (i-th) disturbance data in a predetermined Gaussian distribution with noise added to the numerator (l, where l is a natural number satisfying 1 ≦ l ≦ K), that is,

The process of calculating the burden rate γ (i, l) including is repeatedly executed (S121). The burden factor γ (i, l) is

It is expressed. The burden rate γ (i, l) is called a burden rate because it represents the degree of burden on the predetermined Gaussian distribution.

と計算する（Ｓ１２２）。ここでN_lは、

であるため、更新後の重みρ_l ^newは、l番目のガウス分布における負担率の平均値ということができる。Ｍステップ計算部１２２は、未知関数を撹乱データの確率密度の尤度を対数化した値L、すなわち

とし、変数をl番目のガウス分布の平均μ_lとした偏微分、すなわち

を用いて、尤度が最大化する方向にガウス分布それぞれの平均を更新する（Ｓ１２２）。具体的には、Ｍステップ計算部１２２は、

により、l番目のガウス分布の平均μ_lを更新する（Ｓ１２２）。αは学習率と呼ばれ、ステップＳ１２２実行前に、ユーザが適切な値を決めておくものとする。同様に、Ｍステップ計算部１２２は、未知関数を撹乱データの確率密度の尤度を対数化した値（（８）式）とし、変数をl番目のガウス分布の分散σ_lとした偏微分を用いて、尤度が最大化する方向にガウス分布それぞれの分散を更新する（Ｓ１２２）すなわち、Ｍステップ計算部１２２は、

により、l番目のガウス分布の分散σ_lを更新する（Ｓ１２２）。Ｍステップ計算部１２２は、上述の重みρ_l、平均μ_l、分散σ_lを更新する処理を繰り返し実行する（Ｓ１２２）。なお、

は、偏微分であり微分可能であるため、解が必ず求まる。ステップＳ１２１とステップＳ１２２は、収束条件を充たすまで交互に、繰り返し実行される。収束条件については、μ^new、σ^new、ρ^newから計算されるL^new、μ^old、σ^old、ρ^oldから計算されるL^oldに違いがあまりない状態(L^newとL^oldの差分が所定の閾値以下となる状態)であれば、収束とする。 Next, the M step calculation unit 122 updates the weight of each Gaussian distribution based on the average value of the burden rate. That is, the M step calculation unit 122 sets the updated weight ρ _l ^new to

Is calculated (S122). Where N _l is

Therefore, the updated weight ρ _l ^new can be said to be the average value of the burden rate in the l-th Gaussian distribution. The M step calculation unit 122 is a value L obtained by logarithmizing the likelihood of the probability density of the disturbance data from the unknown function, that is,

Partial differential, i.e. where the to the variable and the mean mu _l of l-th Gaussian distribution

The average of each Gaussian distribution is updated in the direction in which the likelihood is maximized (S122). Specifically, the M step calculation unit 122

Thus, the average μ _l of the l-th Gaussian distribution is updated (S122). α is called a learning rate, and it is assumed that the user determines an appropriate value before executing step S122. Similarly, the M step calculation unit 122 performs partial differentiation with the unknown function as a logarithmic value of the likelihood density probability probability of the disturbance data (equation (8)) and the variable as the variance σ _l of the l-th Gaussian distribution. The variance of each Gaussian distribution is updated in the direction in which the likelihood is maximized (S122), that is, the M step calculation unit 122

Thus, the variance σ _l of the l-th Gaussian distribution is updated (S122). The M step calculation unit 122 repeatedly executes the process of updating the above weight ρ _l , average μ _l , and variance σ _l (S122). In addition,

Is a partial differentiation and is differentiable, so a solution is always obtained. Step S121 and step S122 are repeatedly executed alternately until the convergence condition is satisfied. The convergence ^{condition, μ new, σ new, ρ} L new calculated from ^{new, μ old, σ old,} the difference between the [rho state there is not much difference in L ^old calculated from ^old (L ^{new new} and L ^old is given If the threshold value is less than or equal to the threshold value, convergence is assumed.

  Next, the statistical data reconstruction unit 13 reconstructs the statistical data by using the average, variance, and weight converged by repeatedly executing the processing for calculating the burden rate and the processing for updating the weight, average, and variance alternately. (S13).

  Thus, according to the statistical data reconstruction apparatus 1 of the present embodiment, statistical data can be reconstructed with high accuracy with a smaller number of data than before.

  Hereinafter, the statistical data reconstruction apparatus according to the second embodiment in which the dimension of the original data is expanded to the d dimension (d is an integer of 2 or more) will be described with reference to FIGS. FIG. 3 is a block diagram illustrating the configuration of the statistical data reconstruction apparatus 2 according to the present embodiment. FIG. 4 is a flowchart showing the operation of the statistical data reconstruction apparatus 2 of the present embodiment.

As shown in FIG. 3, the statistical data reconstruction device 2 of this embodiment includes a parameter initialization unit 11, a likelihood maximization unit 22, and a statistical data reconstruction unit 13. The likelihood maximizing unit 22 includes an E step calculating unit 221 and an M step calculating unit 222. The configuration requirements other than the likelihood maximizing unit 22 are the same as those in the first embodiment. In this embodiment, the likelihood maximization unit 22 solves the following likelihood maximization problem.

は、

である。具体的には、Ｅステップ計算部２２１は、各次元において負担率を計算し（Ｓ２２１）、Ｍステップ計算部２２２は、各次元において重みと平均と分散を更新する（Ｓ２２２）。 here,

Is

It is. Specifically, the E step calculation unit 221 calculates the burden rate in each dimension (S221), and the M step calculation unit 222 updates the weight, average, and variance in each dimension (S222).

  As described above, the statistical data reconstruction apparatus 2 according to the present embodiment performs partial differentiation for each dimension, and applies the generalized EM algorithm similar to that of the first embodiment for each dimension. Although the constraint of non-diagonal elements (0) of the covariance matrix is imposed, it can be applied even when the original data has multiple dimensions (d dimensions), so it supports a wider range of data formats than the first embodiment. it can.

  Hereinafter, when the original data is described by a linear sum of a general multi-dimensional mixed Gaussian distribution by removing the condition of non-diagonal element 0 of the covariance matrix, which was a constraint condition in the above-described embodiment. A statistical data reconstruction apparatus according to the third embodiment that can be used will be described. When the off-diagonal elements of the covariance matrix are not 0, that is, when a general multidimensional mixed Gaussian distribution is considered, it is difficult to analytically calculate the integral required for calculating g. Therefore, by partially changing the configuration of the statistical data reconstruction apparatus 2 of the second embodiment, it is possible to handle the original data when the covariance matrix is general.

  Hereinafter, the configuration and operation of the statistical data reconstruction apparatus of this embodiment will be described with reference to FIGS. FIG. 5 is a block diagram illustrating the configuration of the statistical data reconstruction device 3 according to the present embodiment. FIG. 6 is a flowchart showing the operation of the statistical data reconstruction device 3 of this embodiment.

  As shown in FIG. 5, the statistical data reconstruction device 3 according to the present exemplary embodiment includes a parameter initialization unit 31, a likelihood maximization unit 32, and a statistical data reconstruction unit 13. The likelihood maximization unit 32 includes an E step calculation unit 321 and an M step calculation unit 322, and the statistical data reconstruction unit 13 is common to the first and second embodiments.

  The parameter initialization unit 31 assumes that the original data before the disturbance processing can be described by a linear sum of a finite number of multidimensional Gaussian distributions, and initializes the weight, average, and variance of each Gaussian distribution (S31). . Step S31 and step S11 have different assumptions about the original data. However, since the dispersion initialization needs to be executed in a state where the covariance is 0, step S31 is executed by the same method as step S11 described above. The E step calculation unit 321 executes the E step (described later) in this embodiment (S321). The M step calculation unit 322 executes the M step (described later) in this embodiment (S322). Step S13 is the same as described above.

Hereinafter, the configuration and operation of the E step calculation unit 321 according to the present embodiment will be described with reference to FIGS. FIG. 7 is a block diagram illustrating a configuration of the E step calculation unit 321 of the present embodiment. FIG. 8 is a flowchart showing the operation of the E step calculation unit 321 of this embodiment. As illustrated in FIG. 7, the E step calculation unit 321 according to the present exemplary embodiment includes a burden rate calculation unit 3211, a data set generation unit 3212, and a principal component analysis unit 3213. The burden factor calculation unit 3211 calculates the burden factor in the same manner as step S221 in the second embodiment (S3211). Specifically, the burden factor calculation unit 3211 calculates the burden factor γ (i, l) as follows (S3211).

The data set generation unit 3212 has a Gaussian distribution corresponding to the highest burden ratio among the burden ratios γ (i, k), k = 1,... K corresponding to the predetermined (i-th) disturbance data. A predetermined number (K) of data sets corresponding to all the disturbance data so that the predetermined (i-th) disturbance data is included in the data set (for example, D _k ) corresponding to the number (for example, k-th). (D ₁ ,..., D _K ) are generated (S3212).

  The principal component analysis unit 3213 performs principal component analysis on each of the K data sets, rotates the data in each data set to be uncorrelated, and generates uncorrelated K data sets. (S3213).

  The operation of the principal component analysis unit 3213 will be supplemented with reference to FIG. FIG. 9 is a diagram illustrating data rotation executed by the principal component analysis unit 3213 of the present embodiment. 9A, 9B and 9C are graphs in which the distribution of data is expressed by contour lines when the x-axis is dimension: height data, the y-axis is dimension: weight data, and the frequency is expressed on the z-axis. When the original data consists of two correlated dimensions, such as height and weight shown in FIG. 9, the non-diagonal elements of the covariance matrix are non-zero, as shown in FIG. Analysis by 322 becomes difficult. Therefore, the principal component analysis unit 3213 according to the present embodiment performs principal component analysis, rotates the data set as illustrated in FIG. 9B or FIG. 9C, for example, and correlates the data in each data set (covariance matrix). Of non-diagonal elements 0).

  Hereinafter, with reference to FIGS. 10 and 11, the configuration and operation of the M-step calculation unit 322 of the present embodiment will be described. FIG. 10 is a block diagram illustrating a configuration of the M step calculation unit 322 of the present embodiment. FIG. 11 is a flowchart showing the operation of the M step calculation unit 322 of this embodiment. As illustrated in FIG. 10, the M step calculation unit 322 of this embodiment includes a parameter update unit 3221, a reverse rotation unit 3222, and a convergence determination unit 3223.

The parameter update unit 3221 updates the weight, average, and variance in the same manner as in step S222 in the second embodiment (S3211). Specifically, the parameter update unit 3221 updates the weight, average, and variance according to the following formula (S3211).

For _N1 , Equation (7) is used as it is. The inverse rotation unit 3222 applies a principal component to the diagonal matrix Σ ^new in which the elements of the updated variance σ _l ^new are arranged as diagonal elements, the updated average μ _l ^new , and the data set generated in step S3212. The rotation opposite to the rotation in the analysis is executed (S3222). Convergence determination unit 3223 determines convergence when the change in likelihood L ^old before update and likelihood L ^new after update is equal to or less than a predetermined threshold (S3223).

  According to the statistical data reconstruction apparatus 3 of the present embodiment, even if the original data is described by a linear sum of general multidimensional mixed Gaussian distributions, statistical data can be accurately obtained with a smaller number of data than before. Can be rebuilt.

  The ingenuity of the statistical data reconstruction device 3 of the present embodiment is that it corresponds to the case where there is a correlation in the original data. When there is no correlation in the original data, it is possible to analytically solve the necessary integral calculation as disclosed in the first embodiment, but when the original data has a correlation, the integral calculation is performed analytically. Becomes difficult. Therefore, as shown in step S3213, by rotating the data to make it uncorrelated, it is possible to deal with a case where the original data has a correlation.

  In Non-Patent Documents 1, 2, and 3, the frequency cross tabulation table is handled as a statistical value that can be reconstructed. However, in the present invention, reconstruction that obtains the probability distribution of the original data as a statistic is realized. If the probability distribution of the original data can be obtained, a percentage cross tabulation table can be obtained, and if the number of data is obtained therefrom, a frequency cross tabulation table can also be obtained. Therefore, in the present invention, the probability distribution is reconstructed, but is essentially the same as reconstructing the frequency cross tabulation table.

  In common with the statistical data reconstruction apparatuses of the first, second, and third embodiments described above, what is the value of K is a problem. K may be determined using the Akaike information criterion or Bayesian information criterion.

<Supplementary note>
The apparatus of the present invention includes, for example, a single hardware entity as an input unit to which a keyboard or the like can be connected, an output unit to which a liquid crystal display or the like can be connected, and a communication device (for example, a communication cable) capable of communicating outside the hardware entity. Can be connected to a communication unit, a CPU (Central Processing Unit, may include a cache memory or a register), a RAM or ROM that is a memory, an external storage device that is a hard disk, and an input unit, an output unit, or a communication unit thereof , A CPU, a RAM, a ROM, and a bus connected so that data can be exchanged between the external storage devices. If necessary, the hardware entity may be provided with a device (drive) that can read and write a recording medium such as a CD-ROM. A physical entity having such hardware resources includes a general-purpose computer.

  The external storage device of the hardware entity stores a program necessary for realizing the above functions and data necessary for processing the program (not limited to the external storage device, for example, reading a program) It may be stored in a ROM that is a dedicated storage device). Data obtained by the processing of these programs is appropriately stored in a RAM or an external storage device.

  In the hardware entity, each program stored in an external storage device (or ROM or the like) and data necessary for processing each program are read into a memory as necessary, and are interpreted and executed by a CPU as appropriate. . As a result, the CPU realizes a predetermined function (respective component requirements expressed as the above-described unit, unit, etc.).

  The present invention is not limited to the above-described embodiment, and can be appropriately changed without departing from the spirit of the present invention. In addition, the processing described in the above embodiment may be executed not only in time series according to the order of description but also in parallel or individually as required by the processing capability of the apparatus that executes the processing. .

  As described above, when the processing functions in the hardware entity (the apparatus of the present invention) described in the above embodiments are realized by a computer, the processing contents of the functions that the hardware entity should have are described by a program. Then, by executing this program on a computer, the processing functions in the hardware entity are realized on the computer.

  The program describing the processing contents can be recorded on a computer-readable recording medium. As the computer-readable recording medium, for example, any recording medium such as a magnetic recording device, an optical disk, a magneto-optical recording medium, and a semiconductor memory may be used. Specifically, for example, as a magnetic recording device, a hard disk device, a flexible disk, a magnetic tape or the like, and as an optical disk, a DVD (Digital Versatile Disc), a DVD-RAM (Random Access Memory), a CD-ROM (Compact Disc Read Only). Memory), CD-R (Recordable) / RW (ReWritable), etc., magneto-optical recording medium, MO (Magneto-Optical disc), etc., semiconductor memory, EEP-ROM (Electronically Erasable and Programmable-Read Only Memory), etc. Can be used.

  The program is distributed by selling, transferring, or lending a portable recording medium such as a DVD or CD-ROM in which the program is recorded. Furthermore, the program may be distributed by storing the program in a storage device of the server computer and transferring the program from the server computer to another computer via a network.

  A computer that executes such a program first stores, for example, a program recorded on a portable recording medium or a program transferred from a server computer in its own storage device. When executing the process, the computer reads a program stored in its own recording medium and executes a process according to the read program. As another execution form of the program, the computer may directly read the program from a portable recording medium and execute processing according to the program, and the program is transferred from the server computer to the computer. Each time, the processing according to the received program may be executed sequentially. Also, the program is not transferred from the server computer to the computer, and the above-described processing is executed by a so-called ASP (Application Service Provider) type service that realizes the processing function only by the execution instruction and result acquisition. It is good. Note that the program in this embodiment includes information that is used for processing by an electronic computer and that conforms to the program (data that is not a direct command to the computer but has a property that defines the processing of the computer).

  In this embodiment, a hardware entity is configured by executing a predetermined program on a computer. However, at least a part of these processing contents may be realized by hardware.

Claims (7)

A statistical data reconstruction device that reconstructs statistical data from disturbance data generated by performing disturbance processing on original data,
Assuming that the original data before the disturbance processing can be described by a linear sum of a finite number of Gaussian distributions with non-diagonal elements 0 of the covariance matrix, the initial parameter values for initializing the weight, mean, and variance of each Gaussian distribution And
For all combinations of a predetermined number of disturbance data generated by adding predetermined noise to the original data and a Gaussian distribution to which a predetermined number of noises are added, the denominator includes the probability density of the predetermined number of the disturbance data, and the numerator An E step calculation unit that repeatedly executes a process of calculating a burden rate including the probability density of the predetermined number of disturbance data in the predetermined number of Gaussian distributions to which the noise is added;
The weight of each of the Gaussian distributions is updated based on the average value of the burden ratio, the unknown function is a value obtained by logarithmizing the likelihood density of the disturbance data, and the partial differentiation with the variable being the average of the Gaussian distribution. And updating the average of each of the Gaussian distributions in the direction in which the likelihood is maximized, the unknown function is a logarithm value of the likelihood density of the disturbance data, and the variable is the variance of the Gaussian distribution An M-step calculation unit that repeatedly executes a process of updating the variance of each of the Gaussian distributions in a direction in which the likelihood is maximized using partial differentiation;
A statistical data restructuring unit that reconstructs the statistical data using the average, variance, and weight converged by repeatedly executing the process of calculating the burden rate, the process of updating the weight, the average, and the variance;
Statistical data reconstruction device including The statistical data reconstruction device according to claim 1,
The original data is data having a plurality of dimensions,
The E step calculation unit
Calculate the burden rate in each dimension,
The M step calculator is
A statistical data reconstruction device that updates the weight, the average, and the variance in each dimension. A statistical data reconstruction device that reconstructs statistical data from disturbance data generated by performing disturbance processing on original data,
The original data is data having a plurality of dimensions,
Assuming that the original data before the disturbance processing can be described by a linear sum of a finite number of multidimensional Gaussian distributions, a parameter initialization unit that initializes the weight, average, and variance of each Gaussian distribution;
For all combinations of a predetermined number of disturbance data generated by adding predetermined noise to the original data and a Gaussian distribution to which a predetermined number of noises are added, the denominator includes the probability density of the predetermined number of the disturbance data, and the numerator A burden factor calculation unit that repeatedly executes a burden factor including a probability density of predetermined disturbance data in the predetermined Gaussian distribution with the noise added to each dimension, and
Corresponding to all disturbance data so that the predetermined disturbance data is included in the data set corresponding to the Gaussian distribution number corresponding to the highest burden ratio among the burden ratios corresponding to the predetermined disturbance data A data set generation unit that generates a predetermined number of data sets;
A principal component analysis unit that performs principal component analysis on each of the data sets and rotates the data in each data set to be uncorrelated;
The weight of each of the Gaussian distributions is updated for each dimension based on the average value of the burden ratio, the unknown function is a value obtained by logarithmizing the likelihood density of the disturbance data, and the variable is the average of the Gaussian distribution Using partial differentiation, the average of each of the Gaussian distributions is updated for each dimension in the direction in which the likelihood is maximized, the unknown function is a logarithm value of the likelihood density of the disturbance data, and the variable is A parameter updating unit that repeatedly performs a process of updating the variance of each of the Gaussian distributions for each dimension in a direction in which the likelihood is maximized, using a partial derivative that is a variance of a Gaussian distribution;
A diagonal matrix in which the elements of the updated variance are arranged as diagonal elements, an average after the update, and a reverse rotation unit that performs a rotation opposite to the rotation in the principal component analysis on the generated data set;
A convergence determination unit that determines convergence when the likelihood before update and the change in likelihood after update are equal to or less than a predetermined threshold;
A statistical data reconstruction unit that reconstructs the statistical data using the converged mean, variance and weight;
Statistical data reconstruction device including A statistical data reconstruction method executed by a statistical data reconstruction device that reconstructs statistical data from disturbance data generated by performing disturbance processing on original data,
Assuming that the original data before the disturbance processing can be described by a linear sum of a finite number of Gaussian distributions with non-diagonal elements 0 of the covariance matrix, the initial parameter values for initializing the weight, mean, and variance of each Gaussian distribution Step,
For all combinations of a predetermined number of disturbance data generated by adding predetermined noise to the original data and a Gaussian distribution to which a predetermined number of noises are added, the denominator includes the probability density of the predetermined number of the disturbance data, and the numerator An E step calculation step of repeatedly executing a process of calculating a burden rate including the probability density of the predetermined number of disturbance data in the predetermined number of Gaussian distributions to which the noise is added;
The weight of each of the Gaussian distributions is updated based on the average value of the burden ratio, the unknown function is a value obtained by logarithmizing the likelihood density of the disturbance data, and the partial differentiation with the variable being the average of the Gaussian distribution. And updating the average of each of the Gaussian distributions in the direction in which the likelihood is maximized, the unknown function is a logarithm value of the likelihood density of the disturbance data, and the variable is the variance of the Gaussian distribution An M-step calculation step of repeatedly executing a process of updating the variance of each of the Gaussian distributions in a direction in which the likelihood is maximized using partial differentiation;
A statistical data restructuring step of reconstructing the statistical data using the average, variance, and weight converged by repeatedly executing the process of calculating the burden rate, the process of updating the weight, the average, and the variance;
Statistical data reconstruction method including The statistical data reconstruction method according to claim 4,
The original data is data having a plurality of dimensions,
The E step calculation step includes:
Calculate the burden rate in each dimension,
The M step calculation step includes:
A statistical data reconstruction method for updating the weight, the average, and the variance in each dimension. A statistical data reconstruction method executed by a statistical data reconstruction device that reconstructs statistical data from disturbance data generated by performing disturbance processing on original data,
The original data is data having a plurality of dimensions,
Assuming that the original data before the disturbance processing can be described by a linear sum of a finite number of multi-dimensional Gaussian distributions, a parameter initialization step for initializing the weight, mean, and variance of each Gaussian distribution;
For all combinations of a predetermined number of disturbance data generated by adding predetermined noise to the original data and a Gaussian distribution to which a predetermined number of noises are added, the denominator includes the probability density of the predetermined number of the disturbance data, and the numerator A burden factor calculating step of repeatedly executing, for each dimension, a burden factor including the probability density of the predetermined disturbance data in the predetermined Gaussian distribution with the noise added thereto;
Corresponding to all disturbance data so that the predetermined disturbance data is included in the data set corresponding to the Gaussian distribution number corresponding to the highest burden ratio among the burden ratios corresponding to the predetermined disturbance data A data set generation step for generating a predetermined number of data sets;
Performing principal component analysis on each of the data sets, and rotating the principal component analysis steps so that the data in each data set is uncorrelated;
The weight of each of the Gaussian distributions is updated for each dimension based on the average value of the burden ratio, the unknown function is a value obtained by logarithmizing the likelihood density of the disturbance data, and the variable is the average of the Gaussian distribution Using partial differentiation, the average of each of the Gaussian distributions is updated for each dimension in the direction in which the likelihood is maximized, the unknown function is a logarithm value of the likelihood density of the disturbance data, and the variable is A parameter updating step that repeatedly executes a process of updating the variance of each of the Gaussian distributions for each dimension in a direction in which the likelihood is maximized, using partial differentiation as a variance of the Gaussian distribution,
A reverse rotation step of performing a rotation opposite to the rotation in the principal component analysis on the diagonal matrix in which the elements of the updated variance are arranged as diagonal elements, the average after the update, and the generated data set;
A convergence determination step for determining convergence when the likelihood before update and the change in likelihood after update are equal to or less than a predetermined threshold;
A statistical data reconstruction step of reconstructing the statistical data using the converged mean, variance and weight;
Statistical data reconstruction method including   A program for causing a computer to function as the statistical data reconstruction device according to any one of claims 1 to 3.

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