JP2017126028A - Disturbance data reconstruction error estimation device, disturbance data reconstruction error estimation method and program - Google Patents

Abstract

PROBLEM TO BE SOLVED: To provide a disturbance data reconstruction error estimation device for estimating an error Vbetween a probability density function representing generation distribution of source data and a histogram equivalent with a probability density function representing generation distribution of reconstruction source data without estimating the histogram equivalent with the probability density function representing the generation distribution of the reconstruction source data.SOLUTION: The disturbance data reconstruction error estimation device comprises: a transition probability matrix calculation part 110 for calculating a transition probability matrix Pwhile using a conditional probability Pwhich is defined in accordance with whether an attribute A(1≤j≤M) is a category attribute or a numerical attribute; and an error calculation part 120 for calculating variance (1≤m,n≤|A|, m=n) of xor covariance (1≤m,n≤|A| m≠n) of xand xas an error Vwhile using a number N of sample data items, a frequency Hx (x) (1≤i≤|A|) of xthat appears in the sample data, and the transition probability matrix P(1≤j≤M).SELECTED DRAWING: Figure 3

Description

  The present invention relates to a technique for estimating a statistical value of individual data from data obtained by concealing individual data in a database by a probabilistic technique, and more particularly to estimating an error between individual data and individual data reconstructed after disturbance.

  Concealing the original data in the database using a probabilistic method is called disturbance. Estimating a statistical value of original data (hereinafter referred to as reconstructed data) from concealed data (hereinafter referred to as disturbance data) is referred to as reconstruction. That is, disturbance means creating a secret database composed of disturbance data, and reconstruction means performing statistical analysis on the secret database to obtain reconstructed data.

  As such disturbance / reconstruction techniques, there are maintenance replacement disturbance (Non-Patent Document 1, Non-Patent Document 2, Non-Patent Document 4), and bounded Laplace noise addition (Non-Patent Document 3). In the reconstruction processing in these techniques, when estimating the reconstruction data from the disturbance data, a probability density function representing the generation distribution of the original data is estimated in order to enable various statistical analyses. This is because knowing the probability density function is equivalent to knowing the data generation rules, and any statistical analysis is possible.

  In the technique of Non-Patent Document 4, the probability density function is expressed as a histogram, and the generation distribution of the original data is estimated. Hereinafter, “estimation of probability density function” and “estimation of histogram” are treated as equivalent.

  A conventionally proposed disturbance method will be described according to Non-Patent Document 2 and Non-Patent Document 3. For this purpose, first, the attributes, category attributes, and numerical attributes will be described with examples.

  Let M be the number of attributes of the data to be disturbed, that is, the original data. Further, it is assumed that the attributes include a mixture of category attributes and numerical attributes. Examples of category attributes include gender, and the set of attribute values is {male, female}. An example of a numerical attribute is height, and the set of attribute values is {t | t is 0 cm to 200 cm}.

j th attribute (i.e., a set of attribute values) to represent and _{A j} (1 ≦ j ≦ M), the set of all attributes and _{A = A 1 × ... × A} M. In addition, when A _j is a category attribute, the density of the set A _j , and when A _j is a numeric attribute, the number (quantization number) K _j that divides the possible range (value range of the numeric attribute) of the numeric attribute value _Is represented by | A _j |. Furthermore, | A | = | A ₁ | ×... × | A _M | In the example of the height above, the range is {t | t is 0 cm to 200 cm}, and is divided (quantized) into partial sections [0, 20], [20, 40], ... [180, 200]. , K _j = 10. At this time, a numerical attribute having a range of {[0, 20], [20, 40],... [180, 200]} is referred to as a quantized numerical attribute.

FIG. 1 shows an example of sample data of A = A ₁ × A ₂ when A ₁ and A ₂ are category attributes representing gender and occupation, respectively. At this time, the number of sample data N = 10, M = 2, A ₁ = {male, female}, A ₂ = {researcher, developer, doctor}, A = {(male, researcher), (male, developer) , (Male, doctor), (female, researcher), (female, developer), (female, doctor)}, | A ₁ | = 2, | A ₂ | = 3, | A | = 6.

The elements of A _j are numbered {1, 2,..., | A _j |}. Researcher is an element of A _2, the developer for the physician, when assigning a 1,2,3, the example which has numbering the elements of A _2. Similarly, the elements of A are also numbered by {1, 2,..., | A |}. For example, x ₁ = (male, researcher), x ₂ = (male, developer), x ₃ = (male, doctor), x ₄ = (female, researcher), x ₅ = (female, developer), x ₆ = This is an example in which (female, doctor) numbers the elements of A. Also referred to x _i and the combination of the i-th attribute values.

Next, some symbols will be described. P _X (x) (where x is an element of A) is a probability density function of the original data, and P _Y (y) (where y is an element of A) is the probability density function of the disturbance data. Each of P _X (x) and P _Y (y) is a function whose domain is A = A ₁ ×... × A _M and whose range is [0, 1]. Variables x and y are treated as M-dimensional column vectors. Also, the conditional probability P _{Y | X} (y | x) represents the probability that the original data x is randomly disturbed to become disturbed data y.

The sample data of the original data is sampled N times from a probability distribution having a probability density P _X (x). Further, for the i-th attribute value combination x _i (1 ≦ i ≦ | A |), P _X (x _i ) is obtained by dividing the frequency of the i-th attribute value combination x _{i by} the number of sample data, H _X (x _i ) represents the frequency of the i-th attribute value combination x _i . Therefore, P _X (x _i ) = H _X (x _i ) / N. Considering the example of FIG. 1, P _x (x ₃ ) = 2/10 and H _x (x ₃ ) = 2 for x ₃ = (male, doctor).

Similar symbols are used for the disturbance data. That is, for the i-th attribute value combination y _i (1 ≦ i ≦ | A |), P _Y (y _i ) is obtained by dividing the frequency of the i-th attribute value combination y _{i by} the number of sample data, H _Y (y _i ) represents the frequency of the i-th attribute value combination y _i .

  Finally, the disturbance method will be explained. When generating the disturbance data, the maintenance replacement disturbance described in Non-Patent Document 2 is used for the category attribute, and the bounded Laplace noise addition described in Non-Patent Document 3 is used for the numerical attribute.

と表される（ｊは１≦ｊ≦Ｍを満たす整数）。先述の通り、｜Ａ_ｊ｜は集合Ａ_ｊの濃度を示す。 (Categorical attribute disturbance method)
For the category attribute, a process of concealing data is performed by maintaining the attribute value with a maintenance probability ρ and randomly changing the attribute value with a probability of 1−ρ. That is, the conditional probability P _{Y | X} ^Aj (v ^′ | v) that the attribute value v of a certain category attribute A _j changes to the attribute value v ^′ is obtained using the maintenance probability ρ _j of the attribute A _j.

(J is an integer satisfying 1 ≦ j ≦ M). As described above, | A _j | indicates the concentration of the set A _j .

In the disturbance for the category attribute, random processing according to the conditional probability is performed. Further, the maintenance probability ρ _j of the attribute A _j is disclosed.

The conditional probability P _{Y | X} ^Aj (v ^′ | v) can be expressed by a matrix P _{j of} | A _j | × | A _j | (hereinafter, P _{j is referred} to as a transition probability matrix). The transition probability matrix P _j is a matrix that represents the probability that the attribute value v changes to the attribute value v ^′ , and is expressed as in Expression (2). In defining the transition probability matrix P _j , numbering of elements of A _j may be used.

となる（ｊは１≦ｊ≦Ｍを満たす整数）。ここでγ_ｊ（ｖ）はラプラス分布を有界にしたことによって生じる有界ラプラス分布を調整するための関数、[ａ_ｊ，ｂ_ｊ]は属性Ａ_ｊの値域（ただし、ａ_ｊ，ｂ_ｊはａ_ｊ≦ｂ_ｊを満たす実数）である。 (Numerical attribute disturbance method)
For numerical attributes, disturbance is performed by adding noise according to the bounded Laplace distribution to the attribute value (that is, adding bounded Laplace noise). The bounded Laplace distribution is a Laplace distribution in which the upper and lower limits of the platform are fixed, and the original data is concealed by applying noise according to the bounded Laplace distribution. The conditional probability density P _{Y | X} ^Aj (v ^′ | v) for changing the attribute value v of a numerical attribute A _{j to} the attribute value v ^′ is obtained by using the parameter φ _j of the bounded Laplace distribution of the attribute A _j.

(J is an integer satisfying 1 ≦ j ≦ M). Here, γ _j (v) is a function for adjusting the bounded Laplace distribution generated by making the Laplace distribution bounded, and [a _j , b _j ] is the range of the attribute A _j (where a _j , b _j Is a real number satisfying a _j ≦ b _j ).

In the disturbance for the numerical attribute, random processing according to this conditional probability density is performed. The parameter phi _j bounded Laplace distribution attribute A _j shall be published.

ただし、｜Ｉ_ｋ｜は区間の長さ、Δは部分区間Ｉ_ｋに含まれる属性値ｖを部分区間Ｉ_ｋ’に含まれる属性値にランダムに変えるランダム化アルゴリズムを表す。 As shown in Non-Patent Document 3, the section [a _j , b _j ] is divided into an appropriate number K _j partial sections I ₁ ,..., I _Kj , and the attribute value included in the partial section I _k is the partial section I. _{k 'probability} changing the attribute value included in the ^{_{P Y | X Aj (I k}} ' | I k) ( hereinafter, the subinterval _{I k} is called the conditional probability turn into subintervals I _{k ')} be quantized using Thus, the transition probability matrix P _j can also be defined for the bounded Laplace noise addition.

However, | I _k | represents the length of the section, and Δ represents a randomization algorithm that randomly changes the attribute value v included in the partial section I _k to the attribute value included in the partial section I _{k ′} .

That is, the transition probability matrix _{P j} is subinterval _{I k} is subinterval I _{k 'to} change the conditional probability ^{_{P Y | X Aj (I k}} ' | K j × a _{I k)} as an element of the k-th row k 'column This is a matrix of K _j (= | A _j | × | A _j |).

When the section [a _j , b _j ] is _equally divided into K _j partial sections I ₁ ,..., I _Kj , the transition probability matrix P _j uses φ _j , a _j , b _j , K _j . Can be calculated. In general, the interval [a _j , b _j ] is divided into I ₁ = [t ₀ , t ₁ ], I ₂ = [t ₁ , t ₂ ],..., I _Kj = [t _Kj−1 , t _Kj ] ( If t ₀ (= a _j ) <t ₁ <... <t _Kj (= b _j )), the transition probability matrix P _j is φ _j , t ₀ (= a _j ), t ₁ ,..., t _Kj ( = B _j ). Hereinafter, t ₀ , t ₁ ,..., T _Kj are referred to as dividing points of the section [a _j , b _j ].

Therefore, it is possible to define the transition probability matrix by quantizing the numerical attribute and defining the conditional probability P _{Y | X} ^Aj (I _{k ′} | I _k ). It can be handled in a common framework.

Therefore, the numerical attribute is quantized using an appropriate partial section I ₁ ,..., I _Kj and the conditional probability P _{Y | X} ^Aj (I _{k ′} | I _k ) can be defined. And Further, as described above, the conditional probability P _{Y | X} ^Aj (I _{k ′} | I _k ) is defined using the divisions I ₁ , I ₂ ,..., I _Kj of the interval [a _j , b _j ]. Quantized bounded Laplace noise addition is called quantized bounded Laplace noise addition.

ｘ_ｎ ^（ｊ）、ｙ_ｍ ^（ｊ）はそれぞれＭ次元ベクトルｘ_ｎ、ｙ_ｍのｊ番目の要素（ｊ番目の属性Ａ_ｊの属性値）を表す。属性がカテゴリ属性である場合は式（１）、数値属性である場合は式（３）を用いて式（４）を計算することができる。 (Multi-attribute disturbance method)
The conditional probabilities P _{Y | X} (y _m | x _n ) of all attributes A = A ₁ ×... × A _M are the products of the conditional probabilities for each attribute (where x _n and y _m are elements) A combination of the nth attribute value of the data and a combination of the mth attribute value of the disturbance data (representing 1 ≦ n, m ≦ | A |)).

x _{n ^(j),} representing a ^{y m (j)} the j-th element of each of the M-dimensional vector _x n, _{y m} (attribute value of the j-th attribute _{A j).} Formula (4) can be calculated using Formula (1) when the attribute is a category attribute, and Formula (3) when it is a numerical attribute.

This conditional probability P _{Y | X} (y _m | x _n ) can also be expressed using a matrix. When the P _j and the transition probability matrix for the j-th attribute _{A j,} the conditional probability _P Y | _X | transition probability matrix P matrix is a representation of _(y m _{x n)} is the _{P j} by the equation (5) Expressed as Kronecker product.

Igarashi Univ., Koji Senda, Katsumi Takahashi, “Efficient Privacy Protection Cross Tabulation Applicable to Multi-valued Attributes”, Computer Security Symposium 2008 Proceedings, October 2008, Vol. 2008, pp.497-502 Igarashi Univ., Koji Senda, Katsumi Takahashi, “Extension to k-anonymity probabilistic index and its application example”, Proceedings of Computer Security Symposium 2009, October 2009, Vol. 2009, pp.1-6 Igarashi Univ., Satoshi Hasegawa, Tatsuya Naya, Ryo Kikuchi, Koji Chida, “Applicable to numeric attributes, privacy protection cross tabulation that guarantees k-anonymity by randomization”, Computer Security Symposium 2012 Proceedings, October 2012 , Pp.639-646 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

Probability density function representing the generated distribution of the reconstructed original data is the probability density function P _{X (x)} and a result of reconstructing the probability density function representing the generated distribution of the original data representing the generated distribution of the original data P ^ X _(x It was necessary to obtain P ^ _X (x) once as to how much the error generated during For example, in Non-Patent Document 4, P ^ _X (x) is obtained by maximizing a likelihood function L shown below (hereinafter referred to as a maximum likelihood estimation method). Specifically, P ^ _X (x) is obtained as a histogram using an Expectation Maximization algorithm.

In the following, variables x and y are omitted, and P ^ _X (x), _HY (y), and _{PY | X} (y | x) are simply expressed as P ^ _X , _HY , _{PY | X.} There is also.

In the maximum likelihood estimation method such as the expected value maximization algorithm, it is necessary to execute two processes, a disturbance process and an estimation process of P ^ _X (equivalent histogram). For this reason, it is necessary to repeatedly perform numerical experiments for each database in order to estimate the error between P _X and P ^ _X (the error between histograms equivalent to each probability density function), and the evaluation of the error is very costly. It was.

Therefore, in the present invention, without performing the estimation of the probability density function P ^ _X equivalent histogram representing the generated distribution of the reconstructed original data, generation of the probability density function P _X and reconstruction based on data representing the generated distribution of the original data An object of the present invention is to provide a disturbance data reconstruction error estimation device that estimates an error between histograms equivalent to a probability density function P ^ _X representing a distribution.

In one embodiment of the present invention, M is the number of attributes of original data that is data to be disturbed, A _j is a j-th attribute (where j is an integer satisfying 1 ≦ j ≦ M), and disturbance for the attribute A _{j is performed} . Indicates that the attribute A _j is a category attribute, the maintenance replacement disturbance with a maintenance probability ρ _j , the attribute A _j is a numerical attribute (the value range is [a _j , b _j ] (where a _j and b _j are a _j ≦ b _j )), the bounded Laplace noise addition with the parameter of the bounded Laplace distribution as φ _j is divided into the range [a _j , b _j ] I ₁ = [t ₀ , t ₁ ], I ₂ = [t ₁ , t ₂ ],..., I _Kj = [t _Kj−1 , t _Kj ] (where t ₀ (= a _j ) <t ₁ <... <t _Kj (= b _j), the _{following, t 0, t 1, ...} , quantization bounded Laplace noise quantized using the called equinox) _{t Kj} And a calculation, | A | a _{| A | = | A 1 |} × ... × | A M | ( However, if the attribute _{A j} is a number attribute _| A _{j |} = _K j) comprising an _{integer, x} i ( 1 ≦ i ≦ | A |) is a combination of the i-th attribute values of the original data, N is the number of sample data of the original data, and H _X (x _i ) is the frequency of x _i appearing in the sample data, and the sample A disturbance data reconstruction error estimation device for estimating an error between the original data and the reconstructed original data from the number of data N and the frequency H _X (x _i ), and the maintenance probability ρ when the attribute A _j is a category attribute _j and the conditional probability P _{Y | X} ^Aj (v ^′ | v) that the attribute value v calculated using | A _j | changes to the attribute value v ^′, and if the attribute A _j is a numerical attribute, the parameter Using φ _j , the range [a _j , b _j ] and the _dividing points t ₁ ,..., t _Kj−1 Transition using the conditional probability P _{Y | X} ^Aj (I _{k ′} | I _k ) (1 ≦ k, k ′ ≦ | K _j |) where the calculated sub-section I _k is changed to the sub-section I _{k ′} A transition probability matrix calculator for calculating a probability matrix P _j , the number of sample data N, the frequency H _X (x _i ) (1 ≦ i ≦ | A |), and the transition probability matrix P _j (1 ≦ j ≦ M) ) using the variance of _{x m (1 ≦ m, n} ≦ | a |, m = n) or the covariance of _{x m} and _{x n (1 ≦ m, n} ≦ | a |, m ≠ n) the And an error calculation unit for calculating as an error.

According to the present invention, by defining the error between P ^ _X equivalent histogram obtained in P _X and maximum likelihood estimation method as a dispersion, it is possible to determine the error analytically. As a result, an error can be estimated without estimating a histogram equivalent to P ^ _X.

The figure which shows an example of sample data. FIG. 3 is a diagram illustrating a variance-covariance calculation algorithm according to the first embodiment. 1 is a block diagram illustrating a configuration of a disturbance data reconstruction error estimation device 100 according to a first embodiment. 5 is a flowchart illustrating the operation of the disturbance data reconstruction error estimation device 100 according to the first embodiment. FIG. 3 is a block diagram illustrating a configuration of an error calculation unit 120 according to the first embodiment. 5 is a flowchart illustrating an operation of an error calculation unit 120 according to the first embodiment. FIG. 10 is a diagram illustrating a variance-covariance calculation algorithm according to the second embodiment. The block diagram which shows the structure of the disturbance data reconstruction error estimation apparatus 200 of Example 2. FIG. FIG. 6 is a block diagram illustrating a configuration of an error calculation unit 220 according to the second embodiment. 9 is a flowchart illustrating the operation of an error calculation unit 220 according to the second embodiment. The figure which shows the variance covariance calculation algorithm of Example 3 (modified example of Example 1). The figure which shows the variance covariance calculation algorithm of Example 3 (modified example of Example 2). The block diagram which shows the structure of the error calculation part 320 of Example 3 (modified example of Example 1). The block diagram which shows the structure of the error calculation part 325 of Example 3 (modified example of Example 2).

  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.

The numerical attribute A _j is quantized using an appropriate partial interval I ₁ ,..., I _Kj , quantized bounded Laplace noise addition, conditional probability P _{Y | X} ^Aj (I _{k ′} | I _k ), and a transition probability matrix P _j is defined. As described above, each probability density function such as P _X is treated as an equivalent histogram (identified), and the number of sample data generated according to P _X is N. Also, x _i and y _k represent the combination of the i-th attribute value of the original data and the combination of the k-th attribute value of the disturbance data (1 ≦ i, k ≦ | A |).

The P ^ _X Equation (6), an error (hereinafter, referred to as original data and the reconstruction of the original data errors) between P _X and P ^ _X equivalent histogram when determined by the maximum likelihood estimation method (7) It is defined as the covariance of P _X and P _{^ X.} Here, P _X and the P ^ _X of covariance, the combination x _i of the i-th attribute values of the original data was generated is regarded as a random variable | A | dimensional column vector (x 1 _... x | _{A |} ) Is defined as a covariance matrix of | A | × | A |, where (i, j) elements for ^T are covariances of x _i and x _j (the variance of x _i when i = j). Is.

が成立する。 By defining in this way, an error between histograms equivalent to P _X and P ^ _X can be analytically obtained (reference non-patent document 5).
(Reference Non-Patent Document 5) Xumeng Cao and James C. Spall, “Relative Performance of Expected and Observed Fisher Information in Covariance Estimation for Maximum Likelihood Estimates”, In American Control Conference (ACC), 2012, IEEE, June 2012, pp. 1871-1876.
Specifically, the frequency P _x (x _i ) obtained by dividing the frequency H _x (x _i ) of the i-th attribute value combination x _i (1 ≦ i ≦ | A |) by the number N of sample data is used. by treating _X, covariance of P _X and P ^ _X can be calculated using the Fisher information matrix. _If the variance-covariance matrix of P _X and P ^ _X is V and the Fisher information matrix is I, V and I are | A | × | A |

Is established.

By obtaining a variance covariance matrix V as an inverse matrix of the Fisher information matrix I and accessing each element V _ij of the variance covariance matrix V (V _ij represents an element of i rows and j columns of V), Dispersion can be obtained. V _ij represents the variance of x _i (when i = j) or the covariance of x _i and x _j (when i ≠ j).

Hereinafter, a calculation method of the Fisher information matrix I will be described. The Fisher information matrix I is defined by the expected value (Expression (9)) of the Hessian matrix H of Expressions (6) and (7).

ここで、Ｐ_Ｙ（ｙ_ｋ）は撹乱データの確率密度関数であり、

である。Ｐ_Ｘ（ｘ_ｉ）はｉ番目の属性値の組み合わせｘ_ｉの度数Ｈ_Ｘ（ｘ_ｉ）をサンプルデータ数Ｎで割ったものを用いる。 Each value H _ij of Hesse matrix H (element of i rows and j columns of matrix H) can be calculated by equation (10). Note that L is a likelihood function of Expression (7).

Where P _Y (y _k ) is the probability density function of the disturbance data,

It is. P _{X (x i)} is used after divided frequency _H X combination _{x i} of i-th attribute values _{(x i)} in the sample data number N.

である。 Therefore, the element I _ij of i row j column of the Fisher information matrix I is

It is.

In summary, the variance-covariance calculation algorithm shown in FIG. 2 is obtained. Each parameter characterizing the disturbance (conditional probability P _{Y | X} ^Aj of each attribute A _j and each frequency H _X (x _i ) of the sample data generated according to the parameter of the parameter A _Y and the probability density function P _X of the original data N) Is the input. That is, if j is an integer satisfying 1 ≦ j ≦ M, ρ _j , | A _j | when A _j is a category attribute, and φ _j , a _j , b _j , t ₁ when A _j is a numerical attribute, ..., t _Kj-1 , i is an integer satisfying 1 ≦ i ≦ | A |, and the number N of sample data and the frequency H _X (x _i ) of x _i appearing in the sample data are input. M and n are integers of 1 ≦ m and n ≦ | A | for designating an error (variance or covariance) to be generated.

In S110, a transition probability matrix _Pj of attribute _Aj is calculated. In S121, the probability density function P _Y (y _k ) of the disturbance data is calculated using Expression (11). In S122, each element _Iij of the Fisher information matrix I is calculated using Expression (12). In S123, the variance-covariance matrix V is calculated as an inverse matrix of the Fisher information matrix I. In S124, the error _{V mn} to be obtained, i.e., | A | outputs the covariance of the m elements _{x m} and the n element _{x n} of the ^T (dispersion) dimensional column vector _{(x 1 ... x | | A} ).

  Hereinafter, the disturbance data reconstruction error estimation apparatus 100 according to the first embodiment will be described with reference to FIGS. FIG. 3 is a block diagram illustrating the configuration of the disturbance data reconstruction error estimation apparatus 100 according to the first embodiment. FIG. 4 is a flowchart illustrating the operation of the disturbance data reconstruction error estimation device 100 according to the first embodiment. As shown in FIG. 3, the disturbance data reconstruction error estimation device 100 includes a transition probability matrix calculation unit 110 and an error calculation unit 120.

When the attribute A _j is a category attribute, the transition probability matrix calculation unit 110 performs the maintenance probability ρ _j and the concentration | A _j |, and when the attribute A _j is a numerical attribute, the bounded Laplace distribution parameter φ _j , the range [ a _{j, b j],} equinox _t 1, _..., using a _{t Kj-1,} calculates the transition probability matrix _{P j} for each attribute _{a j} (S110). In the case of a category attribute, it can be calculated using equation (1). In the case of a numerical attribute, calculation can be performed using Equation (3) (the specific method is described in Non-Patent Document 3). Error calculator 120 uses the frequency _H X _(x i), the sample data number N of combinations _{x i} of the i-th attribute value of the transition probability matrix _{P j,} the original data sample data for each attribute _{A j,} the error V _mn is calculated (S120).

  Hereinafter, the error calculation unit 120 according to the first embodiment will be described with reference to FIGS. FIG. 5 is a block diagram illustrating a configuration of the error calculation unit 120 according to the first embodiment. FIG. 6 is a flowchart illustrating the operation of the error calculation unit 120 according to the first embodiment. As shown in FIG. 5, the error calculation unit 120 includes a disturbance data probability density function calculation unit 121, a Fisher information matrix calculation unit 122, a variance covariance matrix calculation unit 123, and an output result generation unit 124.

The disturbance data probability density function calculation unit 121 includes the transition probability matrix P _j (1 ≦ j ≦ M) calculated by the transition probability matrix calculation unit 110 and the number N of sample data that is input to the disturbance data reconstruction error estimation device 100. Using the frequency H _X (x _i ) (1 ≦ i ≦ | A |), the probability density function P _Y (y _k ) (1 ≦ k ≦ | A |) is calculated by equation (11) (S121). The Fisher information matrix calculation unit 122 includes a transition probability matrix P _j (1 ≦ j ≦ M) calculated by the transition probability matrix calculation unit 110 and a probability density function P _Y (y _k ) calculated by the disturbance data probability density function calculation unit 121. Each element I _ij (1 ≦ i, j ≦ | A |) of the Fischer information matrix I is calculated by Equation (12) using (1 ≦ k ≦ | A |) and the number of sample data N (S122). The variance-covariance matrix calculation unit 123 calculates an inverse matrix of the Fisher information matrix I calculated by the Fisher information matrix calculation unit 122 as a variance-covariance matrix V (S123). The inverse matrix may be obtained using a Gaussian elimination method or the like. The output result generation unit 124 is an element V of m rows and n columns of the variance-covariance matrix V with respect to m and n (1 ≦ m, n ≦ | A |) that are inputs to the disturbance data reconstruction error estimation device 100. _mn is extracted and output (S124). V _mn is the covariance of _{x m} and _{x n} (dispersion).

In the invention of Example 1, by defining the error between P ^ _X equivalent histogram obtained in P _X and maximum likelihood estimation method as covariance, calculating the variance-covariance matrix, the P ^ _X The error can be estimated without estimation. Thus, conventionally it has been difficult, to fit in the specified range errors caused during the probability density function P ^ _X representing the generated distribution of the probability density function P _X and reconstruction based on data representing the generated distribution of the original data It is possible to perform a disturbance reconstruction process.

In Example 1, the variance / covariance was calculated using the Fisher information matrix I. In this method, in order to obtain one variance or covariance value, it is necessary to calculate an inverse matrix of I, the spatial complexity is O (| A | ² ), and the time complexity is O (| A | ³ ) Required. Since | A | increases exponentially as the number M of attributes increases, the calculation becomes difficult.

Therefore, in the second embodiment, in order to efficiently calculate both the space calculation amount and the time calculation amount, the distribution / cooperation is based on the expression of the transition probability matrix P using the Kronecker product of the transition probability matrix P _j of each attribute A _j. A method for calculating the variance will be described.

ここで、１はすべての要素が１である｜Ａ｜次元列ベクトル、ｐ_ｙはＰ_Ｙ（ｙ_ｋ）を第ｋ要素としてもつ｜Ａ｜次元列ベクトル、．／は要素ごとの除算（element-wise division）、ｄｉａｇ（λ）はλ＝（λ_１，…，λ_Ｌ）^Ｔを対角成分とする対角行列を表す。 Using equation (12) and equation (5) to transform equation (8),

Where 1 all elements are 1 | A | dimensional column vector, _{p y} has a k-th element _{P Y (y k) | A} | dimensional column vector. / Represents an element-wise division, and diag (λ) represents a diagonal matrix having λ = (λ ₁ ,..., Λ _L ) ^T as a diagonal component.

Equation (13) shows that it is not necessary to calculate the inverse matrix of the Fisher information matrix I of | A | × | A |, and the inverse matrix of the matrix P _j of | A _j | × | A _j | It shows that it only has to be calculated. Therefore, the amount of space calculation is O (| A |) or max _{1 ≦ j ≦} MO (| A _j | ² ).

ここで、ｉｄｘ（ｍ,ｊ）、ｉｄｘ（ｎ,ｊ）は、ｍ番目の属性値の組み合わせｘ_ｍの属性Ａ_ｊの要素に対応する遷移確率行列Ｐ_ｊの列番号、ｎ番目の属性値の組み合わせｘ_ｎの属性Ａ_ｊの要素に対応する遷移確率行列Ｐ_ｊの列番号を表し、Ｐ_ｊ ^−１［：，ｉｄｘ（ｍ,ｊ）］、Ｐ_ｊ ^−１［：，ｉｄｘ（ｎ,ｊ）］は、行列Ｐ_ｊ ^−１の第ｉｄｘ（ｍ,ｊ）列のベクトル、行列Ｐ_ｊ ^−１の第ｉｄｘ（ｎ,ｊ）列のベクトルを表す。＊は、ベクトル同士の要素積を表す。なお、Ｑ_１、Ｑ_２はいずれも｜Ａ｜×１の行列（つまり、｜Ａ｜次元列ベクトル）となっている。 For example, when V _mn is obtained, the following may be performed.

Here, idx (m, j), idx (n, j) is, m-th attribute value combination _{x m} attribute _{A j} transition probability matrix _{P j} column numbers that correspond to elements of, n th attribute values _Represents the column number of the transition probability matrix P _j corresponding to the element of the attribute A _j of the combination x _n of P _j ⁻¹ [:, idx (m, j)], P _j ⁻¹ [:, idx (n, j)] represents the idx (m matrices _P ^{j -1,} j) column vector, the vector of the idx (n, j) column of the matrix _P ^{j -1.} * Represents an element product between vectors. Note that Q ₁ and Q ₂ are both | A | × 1 matrices (that is, | A | dimensional column vectors).

Considering the example of FIG. 1, idx (2, 2) is a transition probability matrix corresponding to “developer” which is an element of attribute A ₂ of the second attribute value combination x ₂ = (male, developer). indicate the column number of the P _2. Here, researcher of the numbering of the _{A 2} element, the developer, to the doctor, and the 1, 2, 3, and idx (2,2) = 2.

In this case, the time calculation amount may be O (| A |) or max _{1 ≦ j ≦} MO (| A _j | ³ ).

  A variance-covariance calculation algorithm based on Expression (14) is shown in FIG. The input is the same as the variance-covariance calculation algorithm of FIG.

In S110 and S121, the transition probability matrix P _j and the probability density function P _Y (y _k ) are calculated as in the first embodiment. In S222, an inverse matrix P _j ⁻¹ of the transition probability matrix P _j is calculated. In S223, the error V _mn to be obtained is calculated using equation (14).

  Hereinafter, the disturbance data reconstruction error estimation apparatus 200 according to the second embodiment will be described with reference to FIG. FIG. 8 is a block diagram illustrating a configuration of the disturbance data reconstruction error estimation apparatus 200 according to the second embodiment. As shown in FIG. 8, the disturbance data reconstruction error estimation device 200 includes a transition probability matrix calculation unit 110 and an error calculation unit 220.

The transition probability matrix calculation unit 110 is the same as that in the first embodiment. Error calculator 220 uses the frequency _H X _(x i), the sample data number N of combinations _{x i} of the i-th attribute value of the transition probability matrix _{P j,} the original data sample data for each attribute _{A j,} the error V _mn is calculated. The calculation procedure of the error V _mn is different from the error calculation unit 110 of the first embodiment.

  Hereinafter, the error calculation unit 220 according to the second embodiment will be described with reference to FIGS. 9 to 10. FIG. 9 is a block diagram illustrating a configuration of the error calculation unit 220 according to the second embodiment. FIG. 10 is a flowchart illustrating the operation of the error calculation unit 220 according to the second embodiment. As shown in FIG. 9, the error calculation unit 220 includes a disturbance data probability density function calculation unit 121, a transition probability matrix inverse matrix calculation unit 222, and a variance covariance calculation unit 223.

The disturbance data probability density function calculation unit 121 calculates the probability density function P _Y (y _k ) (1 ≦ k ≦ | A |) according to the equation (11) as in the first embodiment (S121). The transition probability matrix inverse matrix calculation unit 222 calculates the inverse matrix P _j ⁻¹ for the transition probability matrix P _j (1 ≦ j ≦ M) calculated by the transition probability matrix calculation unit 110 (S223). The inverse matrix may be obtained using a Gaussian elimination method or the like. The variance covariance calculation unit 223 includes the inverse matrix P _j ⁻¹ (1 ≦ j ≦ M) of the transition probability matrix calculated by the transition probability matrix inverse matrix calculation unit 222 and the probability density calculated by the disturbance data probability density function calculation unit 121. Using the function P _Y (y _k ) (1 ≦ k ≦ | A |), the error V _mn is calculated by the equation (14) (S223). V _mn is the covariance of _{x m} and _{x n} (dispersion).

In the invention of the second embodiment, instead of calculating the inverse matrix of the Fisher information matrix I of | A | × | A |, the transition probability matrix P _j (1 ≦ j ≦ M) of | A _j | × | A _j | The variance / covariance is calculated by calculating the inverse matrix of. That is, the inverse matrix necessary for the calculation of variance / covariance is reduced to a smaller matrix. Thereby, it is possible to suppress both the space calculation amount and the time calculation amount as compared with the first embodiment. Specifically, the space calculation amount is O (| A |) or max _{1 ≦ j ≦ M} O (| A _j | ² ), and the time calculation amount is O (| A |) or max _{1 ≦ j ≦ M} O. (| A _j | ³ ).

In Examples 1 and 2, Equation (11) was used to calculate the probability density function P _Y (y _k ) of the disturbance data. The calculation of Expression (11) requires H _X (x _i ).

In Example 3, a method will be described which does not require the frequency H _X combination x _i of the i-th attribute value of sample data of the original data _{(x i).}

とすることでＰ_Ｙ（ｙ_ｋ）を近似的に求めることができる。 P _Y (y _k ) is _obtained using actually disturbed data. That is, the frequency H _Y (y _k ) of the disturbance data y _k observed after disturbance of N sample data is divided by N. That is,

By doing so, P _Y (y _k ) can be obtained approximately.

FIG. 11 and FIG. 12 show modified examples of the variance-covariance calculation algorithm of Example 1 and Example 2 that calculate P _Y (y _k ) using Expression (15). Further, an error calculation unit 320 and an error calculation unit 325 of Example 3 which are modifications of the error calculation unit 120 of Example 1 and the error calculation unit 220 of Example 2 are shown in FIGS. 13 and 14, respectively. The difference between the error calculation unit 120 and the error calculation unit 320 and the difference between the error calculation unit 220 and the error calculation unit 325 use a disturbance data probability density function calculation unit 321 instead of the disturbance data probability density function calculation unit 121. Is a point. The disturbance data probability density function calculation unit 321 calculates the probability density function P _Y (y _k ) (1 ≦ k ≦ | A |) based on the equation (15) with H _Y (y _k ) and N as inputs.

In the invention of Embodiment 3, P _Y (y _k ) is approximately calculated using H _Y (y _k ) instead of H _X (x _i ). Thereby, it is possible to easily calculate the variance / covariance without using the equation (11).
<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 (8)

M is the number of attributes of the original data that is the disturbance target data, A _j is the jth attribute (where j is an integer 1 ≦ j ≦ M),
Disturbance for the attribute _{A j,} if the attribute _{A j} is the category attribute maintained substituted disturbance to maintain probability and [rho _j, attribute _{A j} is a numeric attribute (the value range _{[a j, b} j] (but, _{a j} , B _j is a real number satisfying a _j ≦ b _j )), the bounded Laplace noise addition with the parameter of the bounded Laplace distribution as φ _j is the division I _{1 of the} range [a _j , b _j ] = [T ₀ , t ₁ ], I ₂ = [t ₁ , t ₂ ],..., I _Kj = [t _Kj−1 , t _Kj ] (where t ₀ (= a _j ) <t ₁ <... < Quantized bounded Laplace noise addition quantized using t _Kj (= b _j ), hereinafter referred to as t ₀ , t _{1 ,.}
| A | is an integer of | A | = | A ₁ | ×... || A _M | (where, if attribute A _j is a numerical attribute, | A _j | = K _j ), x _i (1 ≦ i ≦ | A |) is the combination of the i-th attribute value of the original data, N is the number of sample data of the original data, and H _X (x _i ) is the frequency of x _i appearing in the sample data,
A disturbance data reconstruction error estimation device for estimating an error between the original data and the reconstructed original data from the sample data number N and the frequency H _X (x _i ),
When the attribute A _j is a category attribute, the conditional probability P _{Y | X} ^Aj (v ^′ | v where the attribute value v calculated using the maintenance probability ρ _j and the | A _j | is changed to the attribute value v ^′. ), When the attribute A _j is a numerical attribute, the partial interval I _k calculated using the parameter φ _j , the range [a _j , b _j ], and the _dividing points t ₁ ,..., T _Kj−1 is Transition probability for calculating the transition probability matrix P _j using the conditional probability P _{Y | X} ^Aj (I _{k ′} | I _k ) (1 ≦ k, k ′ ≦ | K _j |) that changes to the partial interval I _{k ′} A matrix calculator,
Using the sample data number N, the frequency H _X (x _i ) (1 ≦ i ≦ | A |), and the transition probability matrix P _j (1 ≦ j ≦ M), the variance of x _m (1 ≦ m, n ≦ | a |, m = n) or _{x m} and _{x n} covariance (1 ≦ m, n ≦ | a |, disturbance data reconstruction including the error calculator for calculating a m ≠ n) as the error Error estimation device. The disturbance data reconstruction error estimation device according to claim 1,
Let y _k (1 ≦ k ≦ | A |) be the combination of the k th attribute value of the disturbance data,
The error calculator is
Using the sample data number N, the frequency H _X (x _i ) (1 ≦ i ≦ | A |), and the transition probability matrix P _j (1 ≦ j ≦ M), the probability density function P _Y ( y _k ) (1 ≦ k ≦ | A |), a disturbance data probability density function calculation unit;
Fisher calculating a Fisher information matrix I using the transition probability matrix P _j (1 ≦ j ≦ M), the probability density function P _Y (y _k ) (1 ≦ k ≦ | A |), and the number N of sample data. An information matrix calculator,
A variance-covariance matrix calculator that calculates an inverse matrix of the Fisher information matrix I as a variance-covariance matrix V;
Element _{V mn (1 ≦ m, n} ≦ | A |) of m rows and n columns of the covariance matrix V of the _{x m} of the dispersion (1 ≦ m, n ≦ | A |, m = n) or the x A disturbance data reconstruction error estimation device including an output result generation unit that outputs _m and _xn as covariance (1 ≦ m, n ≦ | A |, m ≠ n). The disturbance data reconstruction error estimation device according to claim 1,
Let y _k (1 ≦ k ≦ | A |) be the combination of the k th attribute value of the disturbance data,
The error calculator is
Using the sample data number N, the frequency H _X (x _i ) (1 ≦ i ≦ | A |), and the transition probability matrix P _j (1 ≦ j ≦ M), the probability density function P _Y ( y _k ) (1 ≦ k ≦ | A |), a disturbance data probability density function calculation unit;
A transition probability matrix inverse matrix calculator that calculates an inverse matrix P _j ⁻¹ of the transition probability matrix P _j (1 ≦ j ≦ M);
Using the probability density function P _Y (y _k ) (1 ≦ k ≦ | A |) and the inverse matrix P _j ⁻¹ (1 ≦ j ≦ M), the variance of x _m (1 ≦ m, n ≦ | a |, m = n) or the _{x m} and _x covariance _{n (1 ≦ m, n ≦} | a |, m ≠ n) disturbance data reconstruction error estimation and a variance-covariance calculation unit for calculating the apparatus. M is the number of attributes of the original data that is the disturbance target data, A _j is the jth attribute (where j is an integer 1 ≦ j ≦ M),
Disturbance for the attribute _{A j,} if the attribute _{A j} is the category attribute maintained substituted disturbance to maintain probability and [rho _j, attribute _{A j} is a numeric attribute (the value range _{[a j, b} j] (but, _{a j} , B _j is a real number satisfying a _j ≦ b _j )), the bounded Laplace noise addition with the parameter of the bounded Laplace distribution as φ _j is the division I _{1 of the} range [a _j , b _j ] = [T ₀ , t ₁ ], I ₂ = [t ₁ , t ₂ ],..., I _Kj = [t _Kj−1 , t _Kj ] (where t ₀ (= a _j ) <t ₁ <... < Quantized bounded Laplace noise addition quantized using t _Kj (= b _j ), hereinafter referred to as t ₀ , t _{1 ,.}
| A | is an integer of | A | = | A ₁ | ×... || A _M | (where, if attribute A _j is a numerical attribute, | A _j | = K _j ), y _k (1 ≦ k ≦ | A |) is a combination of the k-th attribute values of the disturbance data, N is the number of sample data of the original data, and H _Y (y _k ) is the frequency of y _k appearing in the disturbance data obtained by disturbing the sample data,
A disturbance data reconstruction error estimation device for estimating an error between original data and reconstructed original data from the number of sample data N and the frequency H _Y (y _k ),
When the attribute A _j is a category attribute, the conditional probability P _{Y | X} ^Aj (v ^′ | v where the attribute value v calculated using the maintenance probability ρ _j and the | A _j | is changed to the attribute value v ^′. ), When the attribute A _j is a numerical attribute, the partial interval I _k calculated using the parameter φ _j , the range [a _j , b _j ], and the _dividing points t ₁ ,..., T _Kj−1 is Transition probability for calculating the transition probability matrix P _j using the conditional probability P _{Y | X} ^Aj (I _{k ′} | I _k ) (1 ≦ k, k ′ ≦ | K _j |) that changes to the partial interval I _{k ′} A matrix calculator,
Using the sample data number N, the frequency H _Y (y _k ) (1 ≦ k ≦ | A |), and the transition probability matrix P _j (1 ≦ j ≦ M), the variance of x _m (1 ≦ m, n ≦ | a |, m = n) or _{x m} and _{x n} covariance (1 ≦ m, n ≦ | a |, disturbance data reconstruction including the error calculator for calculating a m ≠ n) as the error Error estimation device. The disturbance data reconstruction error estimation device according to claim 4,
The error calculator is
The probability density function P _Y (y _k ) (1 ≦ k ≦ | A |) of the disturbance data is calculated using the sample data number N and the frequency H _Y (y _k ) (1 ≦ k ≦ | A |). A disturbance data probability density function calculation unit,
Fisher calculating a Fisher information matrix I using the transition probability matrix P _j (1 ≦ j ≦ M), the probability density function P _Y (y _k ) (1 ≦ k ≦ | A |), and the number N of sample data. An information matrix calculator,
A variance-covariance matrix calculator that calculates an inverse matrix of the Fisher information matrix I as a variance-covariance matrix V;
Element _{V mn (1 ≦ m, n} ≦ | A |) of m rows and n columns of the covariance matrix V of the _{x m} of the dispersion (1 ≦ m, n ≦ | A |, m = n) or the x A disturbance data reconstruction error estimation device including an output result generation unit that outputs _m and _xn as covariance (1 ≦ m, n ≦ | A |, m ≠ n). The disturbance data reconstruction error estimation device according to claim 4,
The error calculator is
The probability density function P _Y (y _k ) (1 ≦ k ≦ | A |) of the disturbance data is calculated using the sample data number N and the frequency H _Y (y _k ) (1 ≦ k ≦ | A |). A disturbance data probability density function calculation unit,
A transition probability matrix inverse matrix calculator that calculates an inverse matrix P _j ⁻¹ of the transition probability matrix P _j (1 ≦ j ≦ M);
Using the probability density function P _Y (y _k ) (1 ≦ k ≦ | A |) and the inverse matrix P _j ⁻¹ (1 ≦ j ≦ M), the variance of x _m (1 ≦ m, n ≦ | a |, m = n) or the _{x m} and _x covariance _{n (1 ≦ m, n ≦} | a |, m ≠ n) disturbance data reconstruction error estimation and a variance-covariance calculation unit for calculating the apparatus. M is the number of attributes of the original data that is the disturbance target data, A _j is the jth attribute (where j is an integer 1 ≦ j ≦ M),
Disturbance for the attribute _{A j,} if the attribute _{A j} is the category attribute maintained substituted disturbance to maintain probability and [rho _j, attribute _{A j} is a numeric attribute (the value range _{[a j, b} j] (but, _{a j} , B _j is a real number satisfying a _j ≦ b _j )), the bounded Laplace noise addition with the parameter of the bounded Laplace distribution as φ _j is the division I _{1 of the} range [a _j , b _j ] = [T ₀ , t ₁ ], I ₂ = [t ₁ , t ₂ ],..., I _Kj = [t _Kj−1 , t _Kj ] (where t ₀ (= a _j ) <t ₁ <... < Quantized bounded Laplace noise addition quantized using t _Kj (= b _j ), hereinafter referred to as t ₀ , t _{1 ,.}
| A | is an integer of | A | = | A ₁ | ×... || A _M | (where, if attribute A _j is a numerical attribute, | A _j | = K _j ), x _i (1 ≦ i ≦ | A |) is the combination of the i-th attribute value of the original data, N is the number of sample data of the original data, and H _X (x _i ) is the frequency of x _i appearing in the sample data,
A disturbance data reconstruction error estimation method for estimating an error between the original data and the reconstructed original data from the sample data number N and the frequency H _X (x _i ),
When the attribute A _j is a category attribute, the conditional probability P _{Y | X} ^Aj (v ^′ | v where the attribute value v calculated using the maintenance probability ρ _j and the | A _j | is changed to the attribute value v ^′. ), When the attribute A _j is a numerical attribute, the partial interval I _k calculated using the parameter φ _j , the range [a _j , b _j ], and the _dividing points t ₁ ,..., T _Kj−1 is Transition probability for calculating the transition probability matrix P _j using the conditional probability P _{Y | X} ^Aj (I _{k ′} | I _k ) (1 ≦ k, k ′ ≦ | K _j |) that changes to the partial interval I _{k ′} Matrix calculation step;
Using the sample data number N, the frequency H _X (x _i ) (1 ≦ i ≦ | A |), and the transition probability matrix P _j (1 ≦ j ≦ M), the variance of x _m (1 ≦ m, n ≦ | a |, m = n) or _x covariance of _m and _{x n (1 ≦ m, n} ≦ | a |, re disturbance data to perform the error calculation step of calculating the m ≠ n) as the error Construction error estimation method.   The program for functioning a computer as a disturbance data reconstruction error estimation apparatus of any one of Claim 1 thru | or 6.

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