JP2010218521A - Relationship data processing method - Google Patents

Abstract

<P>PROBLEM TO BE SOLVED: To represent non-similarity data having non-symmetry while utilizing the advantages of the classic MDS (multi-dimensional scaling) in order to handle the non-similarity data having non-symmetry, although the advantage of the classic MDS that a contribution rate can be directly evaluated by eigenvalue decomposition is not utilized because data were handled by methods different from the classic MDS in the prior arts for expanding MDS having the problem of not utilizing the advantage of the classic MDS. <P>SOLUTION: According to this method, the coordinates of 2n-number of points in a space composed of (n-1)-number of coordinate axes or less are calculated corresponding to the number n of objects represented by non-similarity data to a given non-similarity data. <P>COPYRIGHT: (C)2010,JPO&INPIT

Description

  The present invention relates to a technique for giving spatial coordinates to each object when data representing the relationship between the objects is given.

  Multidimensional Scaling (MDS) is a data analysis method that visually expresses the relationship by spatial arrangement of objects when given data representing the relationship between objects. Marketing and psychology It is applied in such fields.

  Various methods of MDS have been devised. In particular, when the relevance is expressed as dissimilarity, the dissimilarity is regarded as the distance between the points representing the object, and A method based on a method called principal coordinate analysis (PCO) that restores the coordinates of each point by eigenvalue decomposition of an inner product matrix between position vectors is well known as classical MDS.

  Classical MDS is characterized by associating dissimilarity with distance, but there are many dissimilarity data that are actually handled, such as having asymmetry, that cannot be directly associated with distance. Examples of dissimilarity data with asymmetry include the mutual trade balance between multiple countries and the brand switching matrix in marketing. Various ideas for extending MDS to handle data having such asymmetry have been made so far.

Chino: Asymmetric multidimensional scale construction method, Modern Mathematics, 1997. Cox and Cox, Multidimensional Scaling, Chapman & Hall / CRC, 2001. Borg and Groenen, Modern Multidimensional Scaling, Springer, 2005. Saito and Yadohisa, Data Analysis of Asymmetric Structures, Marcel Dekker, 2005.

  The classic MDS described above has an advantage that the importance of each coordinate axis can be evaluated in the form of a contribution rate by calculating the eigenvalue of the inner product matrix consisting of the inner products of the position vectors of the points representing the object. Thereby, it is possible to easily evaluate the accuracy of approximation when the coordinate axis having a low contribution rate is omitted and the object is approximately expressed in a lower dimensional space. In particular, the target is often expressed visually by plotting it in a two-dimensional or three-dimensional space. For this purpose, select two or three coordinate axes in descending order of contribution, and omit the other coordinate axes. It will be good.

  On the other hand, all of the conventional techniques for extending MDS so that dissimilarity dissimilarity data can be handled are to process data and express objects by a method different from that of classical MDS. For this reason, there is a problem that the advantage of the classical MDS as described above that the contribution rate can be directly evaluated by eigenvalue decomposition cannot be utilized.

  The present invention solves the problem of expressing dissimilarity data having asymmetry while taking advantage of the classic MDS described above.

  In order to solve the above problems, in the present invention, the given dissimilarity data includes (n-1) or less coordinate axes according to the number n of objects represented by the dissimilarity data. Calculate the coordinates of 2n points in space. Each of the objects is associated with a pair of points consisting of one point belonging to the first group and one point belonging to the second group. A distance from a point belonging to the first group associated with the first object constituting the combination to a point belonging to the second group associated with the second object constituting the combination is the input In the second group associated with the first object constituting the combination, which is represented as a monotonic map to the dissimilarity of the second object with respect to the first object in the dissimilarity data obtained The distance from the point belonging to the point belonging to the first group associated with the second target constituting the combination is the first target with respect to the second target in the input dissimilarity data As expressed as monotonic mapping to dissimilarity, it calculates the coordinates of the 2n number of points.

最適化問題の解として得られるものとする。

ここでλ_ｉ（ｉ＝１，…，ｎ）は任意に与えることのできる正の定数である。この最適化問題は、

る内積行列が求まるので、古典的ＭＤＳと同様の考え方で点Ｐ_ｉおよびＱ_ｉの座標が求まる。Specifically, the calculation is performed as follows. Consider a situation where there are n objects i (i = 1,..., n), and a dissimilarity d _ij between different objects i and j is given. In the present invention, an object i is a point P _i (i = 1,..., N).

It shall be obtained as a solution to the optimization problem.

Here, λ _i (i = 1,..., N) is a positive constant that can be arbitrarily given. This optimization problem is

Therefore, the coordinates of the points P _i and Q _i can be obtained in the same way as the classic MDS.

すると以下が得られる。

点は（ｎ−１）次元ユークリッド空間に配置することが可能であることが分かる。Now, the number of dimensions of the space necessary for arranging 2n points represented by P _i and Q _i will be considered below. By introducing a Lagrange undetermined multiplier λ _ij for dealing with the above-mentioned optimization problem (i ≠ j), consider the following Lagrange function L.

Then you get:

It can be seen that the points can be arranged in the (n−1) -dimensional Euclidean space.

  Compared to the conventional technique of extending MDS to handle asymmetric relationship data, the present invention obtains the coordinates of the points representing each object based on the eigenvalue decomposition of the inner product matrix in the same manner as in classical MDS. There is an effect that the accuracy of approximation when an object is represented in an arbitrary dimension can be evaluated through a scale of cumulative contribution rate. In the present invention, all objects are represented by 2n points, but the required number of dimensions is n-1 at the maximum as in the case of classical MDS for n points. In particular, when the dissimilarity satisfies the distance axiom, the coordinates of two points representing each object are the same, and if this is regarded as a single point, it is exactly the same as the conventional classical MDS. Thus, it can be said that the present invention includes classical MDS as a special case.

1 is a configuration diagram illustrating a system in Embodiment 1. FIG. It is the block diagram which showed the data and program which a memory stores. It is the flowchart which showed the operation | movement of the data processing program. It is a figure which shows the example of the square of dissimilarity data. It is the figure which showed the example of an output of the plot of the point showing an object. It is the figure which showed the example of an output of an eigenvalue, a contribution rate, and a cumulative contribution rate. It is the block diagram which showed the system in Example 2. FIG.

  The mode for carrying out the present invention will be described below with reference to examples.

  FIG. 1 is a system configuration diagram in the first embodiment of the present invention. This system includes an input device 101, a computer 102, and an output device 107. The input device 101 is a device for inputting data to the computer 102 such as a keyboard and a mouse. The computer 102 is a device such as a PC or a server for processing input data and outputting a result, and includes an input interface 103, a CPU 104, a memory 105, an output interface 106, and the like. The input interface 103 is an interface for connecting the input device 101 to the computer 102. The CPU 104 is a device that performs calculations by referring to data and programs stored in the memory 105. The memory 105 is a device that stores data such as input / output data and programs. The output interface 106 is an interface for connecting the output device 107 to the computer 102. The output device 107 is a device such as a display that displays a screen for prompting data input or displays a result processed by the computer 102.

  FIG. 2 is a configuration diagram showing data and programs stored in the memory 105. The memory 105 stores input data 201, a data processing program 202, and output data 203. The input data 201 is data input by the input device 101, and a specific example thereof will be described later with reference to FIG. The data processing program 202 is a program that allows the CPU 104 to process the input data 201 and output the result to the output device 107. Output data 203 is output data generated by the CPU 104 executing the data processing program 202.

値行列であることを意味する。ｄ_ｉｊ ^２は非類似度データを２乗したものとして入力される入力データ２０１である。ステップ３０２では、Ｘ^＊を固有値分解し、点Ｐ_ｉ（ｉ＝１，…，ｎ）およびＱ_ｉ（ｉ＝１，…，ｎ）の座標と固有値、またそれらから生成される値を出力データ２０３としてメモリ１０５に格納する。ステップ３０３では、出力データ２０３として格納されたデータを出力インタフェース１０６を通して出力装置１０７へ出力する。FIG. 3 is a flowchart showing the operation of the data processing program 202. In step 301, after specifying the target number n from the input data, the following semi-definite programming problem using the (i, j) component x _ij of the 2n × 2n matrix X as a variable is solved, and the matrix X ^* Ask for.

Means a value matrix. d _ij ² is input data 201 that is input as the square of dissimilarity data. In step 302, X ^* is decomposed into eigenvalues, and the coordinates and eigenvalues of points P _i (i = 1,..., N) and Q _i (i = 1,..., N), and values generated therefrom are output data. 203 is stored in the memory 105. In step 303, the data stored as the output data 203 is output to the output device 107 through the output interface 106.

Hereinafter, an example in which output data 203 is obtained from specific input data 201 input by the input device 101 and output to the output device 107 will be described. Here, positive constant lambda _i given optionally _{(i = 1, ..., n} ) value of all the 1.

れる入力データ２０１は、ｄ_１２ ^２＝９，ｄ_１３ ^２＝４，ｄ_１４ ^２＝５，ｄ_２１ ^２＝１２，ｄ_２３ ^２＝１６，ｄ_２４ ^２＝５，ｄ_３１ ^２＝４，ｄ_３２ ^２＝２０，ｄ_３４ ^２＝２５，ｄ_４１ ^２＝６，ｄ_４２ ^２＝６，ｄ_４３ ^２＝１５という等式などとして入力される。ここでは、対象Ａ，Ｂ，Ｃ，Ｄにそれぞれ１，２，３，４という添字を割り当てて入力データ２０１を表している。ステップ３０１では、上記の入力データ２０１から対象の数を４であると特定し、以下の半正定値計画問題を解く。

ｓ．ｔ．ｘ_１１＋ｘ_６６−２ｘ_１６＝９，ｘ_１１＋ｘ_７７−２ｘ_１７＝４，ｘ_１１＋ｘ_８８−２ｘ_１８＝５，
ｘ_２２＋ｘ_５５−２ｘ_２５＝１２，ｘ_２２＋ｘ_７７−２ｘ_２７＝１６，ｘ_２２＋ｘ_８８−２ｘ_２８＝５，
ｘ_３３＋ｘ_５５−２ｘ_３５＝４，ｘ_３３＋ｘ_６６−２ｘ_３６＝２０，ｘ_３３＋ｘ_８８−２ｘ_３８＝２５，
ｘ_４４＋ｘ_５５−２ｘ_４５＝６，ｘ_４４＋ｘ_６６−２ｘ_４６＝６，ｘ_４４＋ｘ_７７−２ｘ_４７＝１５，

一般的に半正定値計画問題には計算機を用いて数値的に解くための効率的なアルゴリズムがいくつか知られているので、上記問題は容易に解くことができ、解として以下の行列を得る。

FIG. 4 is an example in which the dissimilarity data that is the source of the input data 201 is expressed by squaring. In this example, the mutual dissimilarities of the target A, the target B, the target C, and the target D are expressed by squaring. For example, the dissimilarity of the target A with respect to the target B is 3, and the target of the target B Vs A

The input data 201 is: d ₁₂ ² = 9, d ₁₃ ² = 4, d ₁₄ ² = 5, d ₂₁ ² = 12, d ₂₃ ² = 16, d ₂₄ ² = 5, d ₃₁ ² = 4, d ₃₂ ² = 20, d ₃₄ ² = 25, d ₄₁ ² = 6, d ₄₂ ² = 6, d ₄₃ ² = 15, etc. Here, the input data 201 is represented by assigning subscripts 1, 2, 3, and 4 to the objects A, B, C, and D, respectively. In step 301, the number of objects is specified as 4 from the above input data 201, and the following semi-definite value programming problem is solved.

s. t. x ₁₁ + x ₆₆ -2x ₁₆ = 9, x ₁₁ + x ₇₇ -2x ₁₇ = 4, x ₁₁ + x ₈₈ -2x ₁₈ = 5
x ₂₂ + x ₅₅ -2x ₂₅ = 12, x ₂₂ + x ₇₇ -2x ₂₇ = 16, x ₂₂ + x ₈₈ -2x ₂₈ = 5
x ₃₃ + x ₅₅ -2x ₃₅ = 4, x ₃₃ + x ₆₆ -2x ₃₆ = 20, x ₃₃ + x ₈₈ -2x ₃₈ = 25,
_{x 44 + x 55 -2x 45 =} 6, x 44 + x 66 -2x 46 = 6, x 44 + x 77 -2x 47 = 15,

In general, there are some efficient algorithms for solving semi-definite programming problems numerically using a computer, so the above problem can be easily solved, and the following matrix is obtained as a solution .

これにより、各対象を表す点Ｐ_ｉ，Ｑ_ｉ（ｉ＝１，２，３，４）の位置ベクトルが以下のように求まる。

に２４．４，６．６３，２．１０，０，０，０，０，０と計算される。Next, in step 302, the matrix X ^* is decomposed into X ^* = AA ', as in the well-known classic MDS. Here, A ′ means a transposed matrix of A. Such A is obtained through eigenvalue decomposition of X ^* as follows.

Thereby, the position vectors of the points P _i and Q _i (i = 1, 2, 3, 4) representing each object are obtained as follows.

24.4, 6.63, 2.10, 0, 0, 0, 0, 0.

Computer 102, output data 203, calculated points _P i, as described _above, the coordinates and the Q i (i = 1,2,3,4), the matrix A obtained in the process of calculation, intermediate, such as ^{X *} It is generated as data or in the form shown in FIGS. 5 and 6 and output to the output device 107.

FIG. 5 shows an example in which a plot of the coordinates of the points P _i and Q _i (i = 1, 2, 3, 4) obtained as described above is output to the output device 107 in step 303. Points plotted with white circles and black circles represent points P _i and Q _i , respectively. Here, an example is shown in which the third component of each coordinate is omitted and two-dimensional plotting is performed, but three-dimensional plotting may be performed including the third coordinate.

FIG. 6 shows an example in which the eigenvalue of X ^* , the contribution rate calculated thereby, and the cumulative contribution rate are output to the output device 107 in step 303. In this example, the cumulative contribution rate up to the second coordinate axis is 93.7%, and the contribution rate by the third coordinate axis, which is omitted when the two-dimensional plot as shown in FIG. 5 is performed, is 6.3%. It is expressed that.

  FIG. 7 is a system configuration diagram in the second embodiment of the present invention. This system includes a client 701, a server 702, and a network 706. The client 701 is a computer such as a PC. The server 702 is a computer that provides a service in response to a request from the client 701, and includes a network interface 703, a CPU 704, a memory 705, and the like. The network interface 703 is an interface such as a network card for connecting the server 702 to the network 706. The CPU 704 is a device that performs calculations by referring to data and programs stored in the memory 705. The memory 705 is a device that stores data such as input / output data and programs. The network 706 is a LAN, the Internet, or the like, and includes relay devices such as switches and routers, cables, and the like.

  The data and program stored in the memory 705 are the same as those described in the first embodiment with reference to FIG. 2, and the operation of the data processing program 202 is also the same as that described in the first embodiment with reference to FIG. is there.

In the first embodiment, the computer 102 processes the input data 201 input by the input device 101 and outputs the result as output data 203 to the output device 107, whereas in this embodiment, the server 702 has a client The input data 201 received from 701 is processed, and the result is transmitted as output data 203 to the client 701. The method in which the CPU 704 obtains the output data 203 by executing the data processing program based on the input data 201 received from the client 701 is the same as that described in the first embodiment. In step 303, the CPU 704 outputs the output data 203 as intermediate data such as coordinates of points P _i and Q _i (i = 1, 2, 3, 4) and matrices A and X ^* obtained in the course of calculation. Alternatively, it is generated in the form shown in FIGS. 5 and 6 and transmitted to the client 701.

Claims (2)

A computer that outputs the coordinates of 2n points in a space composed of (n-1) or less coordinate axes for the inputted dissimilarity data in accordance with the number n of objects represented by the dissimilarity data. Each of the objects is associated with a pair of points consisting of one point belonging to the first group and one point belonging to the second group, and a combination of two different objects. The distance from the point belonging to the first group associated with the first object constituting the combination to the point belonging to the second group associated with the second object constituting the combination Is represented as a monotonic map to the dissimilarity of the second object with respect to the first object in the input dissimilarity data, and is associated with the first object constituting the combination. Belongs to 2 groups A distance from a point to a point belonging to the first group associated with the second object constituting the combination is a distance between the first object and the second object in the input dissimilarity data. A computer that outputs coordinates of the 2n points so as to be expressed as a monotone map to dissimilarity. A method of calculating coordinates of 2n points in a space composed of (n−1) or less coordinate axes for given dissimilarity data in accordance with the number n of objects represented by the dissimilarity data. Each of the objects is associated with a pair of points consisting of one point belonging to the first group and one point belonging to the second group, and a combination of two different objects. The distance from the point belonging to the first group associated with the first object constituting the combination to the point belonging to the second group associated with the second object constituting the combination Is represented as a monotonic map to the dissimilarity of the second object with respect to the first object in the input dissimilarity data, and is associated with the first object constituting the combination. Belong to group 2 To the point belonging to the first group associated with the second object constituting the combination is a non-first object non-similarity with respect to the second object in the input dissimilarity data. A calculation method comprising calculating coordinates of the 2n points so as to be expressed as a monotone map to similarity.

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