JPH0765168A - Device and method for function approximation - Google Patents

Abstract

PURPOSE:To provide the function approximation device and method which can find a precise function approximation expression from data for learning and further derive reliability to an obtained output. CONSTITUTION:The function approximation device has a learning part 100, a recording part 110, and an execution part 120. The learning part 100 has a cluster generation part 102 for classifying input data, an approximation expression generation part 104 for deriving an approximation expression for calculating outputs from inputs, cluster by cluster, and an approximation expression evaluation part 106 which evaluates the precision of the approximation expression so that the different approximation expression can be applied to input data having different features. The execution part 120 has an affiliation cluster decision part 122 for selecting which cluster the input data belong to and an approximation value calculation part 124 which derives the output by using the approximation expression of the selected cluster.

Description

Detailed Description of the Invention

[0001]

BACKGROUND OF THE INVENTION 1. Field of the Invention The present invention relates to a function approximating apparatus and method having a learning function applied to control problems and prediction problems.

[0002]

2. Description of the Related Art As an example of a conventional representative function approximation method, multiple regression analysis (reference: multivariate analysis method, pp. 25-1
58, Tadashi Okuno et al., Nikkan Giren, and hierarchical neural networks (reference: Neurocomputer Engineering, Kazuo Kuma et al., Industrial Research Committee). Multiple regression analysis finds the polynomial that best approximates the output variable from the input variables. There are first-order (linear) and multi-order (non-linear) equations. Further, the approximation by the hierarchical neural network is to learn the relation between the input and output variables by adjusting the connection weight between the elements.

[0003]

The linear multiple regression analysis is
There is a problem that the approximation error is large for non-linear objects. Further, the nonlinear multiple regression analysis has a problem that it is difficult to select an order or an approximate expression for accurately approximating the entire range.

Further, in the approximation by the hierarchical neural network, the output value cannot be trusted for the input of the pattern that has not been learned at all, so that it is difficult to use it for problems such as control.

In addition, when the learning data group is added, learning must be performed from the beginning using all the learning data groups and the additional learning data group, and there is a problem that the learning takes time. It was

[0006]

In order to solve the above problems, according to the present invention, when a function approximation formula is created, (1) a learning data group is similar to a data group having similar characteristics before performing the function approximation. It has a first means for clustering for each cluster, and (2) a second means for obtaining an approximate expression for each created cluster.

Further, in order to improve the approximation accuracy,
(3) Third means for evaluating the created approximate expression, and (4)
If the evaluation is good for all the clusters, the learning is terminated, and if there is a bad evaluation, the fourth means for changing the clustering, and (5) the fifth means for changing the approximate expression
It has the means of.

Further, when performing the function approximation, (6)
A sixth means for determining the cluster to which the input data group belongs, and (7) seventh means for obtaining output data or an output data group from the input data group using the approximate expression of the determined cluster. There is.

Further, in order to quickly determine the cluster to which the input data group belongs, (8) a data group representative of the characteristics of the data group in the cluster is provided, and this data group is compared with the input data group. , And the eighth means for determining the closest one as the belonging cluster.

Further, in order to obtain the reliability for the output data group, (9) the output obtained by calculating the evaluation amount for judging the reliability for the input data group belonging to the cluster It has a ninth means for deriving the reliability of the data or the output data group.

Further, when adding the learning data group,
Since high-speed learning can be performed, (10) high-speed learning is possible by re-learning only the cluster to which the additional learning data group has been added and the clusters around it.
It has 0 means.

[0012]

By the above (1), (2), (6) and (7),
Since the approximation is performed using the function approximation formula derived from the input data groups having similar tendencies, there is an effect that the approximation can be performed with high accuracy.

Further, according to (3), (4) and (5),
Even when the classification according to (1) is insufficient, there is an effect that the data group in the cluster is reclassified so that accurate approximation can be performed.

Further, (8) has the effect that the user can be notified of the high reliability of the output result.

Further, (9) has the effect of speeding up the determination of the cluster to which the input data group belongs.

Further, according to (10), the learning can be performed at high speed even if the learning data group is added.

[0017]

EXAMPLES Examples of the present invention will be described below. Figure 2
It is a whole block diagram of this invention. The configuration and operation of the entire device will be described. The devices are a control device (CPU) 200 and a CRT2.
02, keyboard 204, main memory 206, and magnetic disk 208. A program and database for performing function approximation are stored in the magnetic disk 208 and are stored in the main memory 206 as needed. Keyboard 20
4 is used for the instruction of the program. CRT20
2 is used for displaying the result.

Next, the system configuration and operation of this function approximating apparatus will be described with reference to FIG. FIG. 1 is a block diagram of this function approximating apparatus. The function approximating apparatus is composed of a learning unit 100, a recording unit 102, and an executing unit 104.

In the present embodiment, the learning data group given as a sample is clustered into those having similar characteristics and an approximate expression is obtained for each cluster, so that it is more accurate than approximation by one approximate expression. With characteristics. Then, by changing the clustering state by changing the clustering state such as dividing a cluster having an approximation formula with poor accuracy into smaller clusters, the approximation formula can be obtained from the learning data group with less variation. Is characterized by improving.

The learning unit 100 will be described. Main learning section 100
Is an approximate expression for calculating an output data group from an input data group using a plurality of learning data groups given as samples. The learning unit 100 does not obtain only one approximate expression corresponding to all learning data groups, but obtains a plurality of approximate expressions. At runtime, the most appropriate of these equations is selected according to the characteristics of the input data group. If the accuracy of the approximation formula created once is low, the approximation formula is changed.

The operation of the learning unit 100 and the recording unit 110 will be described. The cluster creating unit 102 collectively clusters the learning data groups recorded in the learning data group recording unit 112 into data groups having similar characteristics. Information on the created cluster is stored in the cluster information recording unit 11
Recorded in 4. The approximate expression creating unit 104 finds an approximate expression using the learning data group for each cluster created in 102. The approximate expression created in 104 is recorded in the approximate expression recording unit 116. The approximate expression evaluation unit 106 evaluates the accuracy of the approximate expression created by the approximate expression creating unit 104.
If the evaluation result is bad, a command for changing the state of clustering or changing the approximate expression is issued.

The execution unit 120 will be described. The execution unit 120 selects the most appropriate approximate expression for the given input data group and obtains the output data group.

The operations of the execution unit 120 and the recording unit 110 will be described. The belonging cluster determination unit 122 determines the cluster to which the input data group belongs from the clusters recorded in the cluster information recording unit 114 for the input data group to be approximated. The approximate value calculation unit 124 calls from the approximate expression recording unit 116 an approximate expression for a cluster determined to belong to the input data group, and uses the expression to obtain an output data group from the input data group. This value is an approximate value.

In this embodiment, the cluster creating unit 102,
An approximate expression creating unit 104 that obtains an approximate expression for each cluster,
Approximate expression evaluation unit 10 for evaluating the approximate expression created in 104
6, a learning data group recording unit 112 that records a learning data group, a cluster information recording unit 114 that records information about the cluster created in 102, and an approximation created in 104. An input data group is classified by providing a recording unit 110 having three approximate expression recording units 116 for recording expressions and an execution unit 120 having two belonging cluster determination units 122 and approximate value calculation units 124. An output data group that is an approximate value can be obtained using the most appropriate approximation formula for the input data group.
The feature is that the state of the cluster and the approximation formula can be changed in order to improve the approximation accuracy.

Next, the operation of the learning section 100 will be described in detail. The cluster creation unit 102 calculates the degree of similarity for a plurality of learning data groups, and clusters into a suitable number of clusters with high degree of similarity. To realize clustering, cluster analysis in multivariate analysis (reference: multivariate analysis method, pp. 391-412, Tadashi Okuno et al., Nikka Giren) and neural network (reference: clustering type using AIC information criterion) Conventional methods such as neuro optimization, Yutaka Sako, 1992 Spring Conference of the Institute of Electronics, Information and Communication Engineers, 6-35), etc. can be used.

In the present embodiment, the sake odor clustering method is used for explanation, so this method will be explained.
In general, clustering has a problem of how many clusters the data is classified into. The sake odor clustering method is a method that can determine the optimal number of clusters. This method is an LVQ (Learning Vector Qua), which is a method of a neural network.
(learning vector quantization) is applied to clustering. However, since it is not possible to determine the optimum number of clusters with a normal LVQ, the number of clusters is determined using the information amount reference AIC. At this time, the AIC is used as an index indicating the goodness of clustering. Specifically, the AIC is calculated for each of the results classified by different cluster numbers, and AIC is calculated.
The number of clusters that minimizes the IC value is the optimum number of clusters. On the other hand, at the time of execution, when new data is input, the cluster to which the input data belongs is selected.

The approximate expression creating unit 104 creates an approximate expression for obtaining the output data group from the input data group for each cluster created by the cluster creating unit 102. here,
To create an approximate expression, multiple regression analysis in multivariate analysis (reference: multivariate analysis method, pp. 25-158, Tadashi Okuno, Nikka Giren) and neural network (reference: neurocomputer engineering, Kazuo Kuma) Others, Industrial Research Committee)
Conventional methods such as, etc. can be used.

Next, the operation of the recording section 110 will be described in detail. The learning data group recording unit 112 stores a learning data group. The learning data group is used as sample data for clustering and approximation formula derivation.

The cluster information recording unit 114 records the information of the cluster created by the cluster creating unit 102 for each cluster. At the time of execution, belonging cluster determination unit 1
22 provides a cluster which is determined to belong to the input data group sent from S.22.

In the approximate expression recording unit 116, the approximate expression creating unit 1
The approximate expression created in 04 is recorded. At run time,
An approximate expression corresponding to the cluster sent from the approximate value calculation unit 124 is provided.

Next, an example of changing the clustering state in the learning section 100 will be described. In the present embodiment, the accuracy is improved by classifying a cluster with poor evaluation into smaller clusters. Note that classifying the inside of the cluster into smaller clusters will be referred to as hierarchical clustering.

An example of hierarchical clustering in the learning unit 100 will be described with reference to FIGS. 4 to 6.
In the figure, the two-dimensional data is used for display in order to make it easier to visually understand the state of clustering, but in reality, any number of dimensions may be used. FIG. 4 is a result of clustering targeting all the learning data groups. C1 to C8 are the names of each cluster. FIG. 5 shows C1 and C
It is an example in which the regions are hierarchically clustered because the accuracy is evaluated to be low in 2 and C4. A cluster of C1 is a cluster of C11 to C14. The same applies to C2 and C4. Furthermore, FIG.
Is an example in which the accuracy is evaluated to be low in C12 and C21, and thus the region is hierarchically clustered.
What clustered in C12 is C121 to C
There are 123 clusters. The same applies to C21. FIG. 7 shows a hierarchical relationship of clusters created as a result of performing the processes of FIGS. 4 to 6.

Next, an embodiment in which the execution unit 120 determines the cluster to which the input data group belongs from the hierarchically structured clusters will be described with reference to FIG. First, it is determined which cluster, among C1 to C8 in the highest hierarchy, belongs to. If it is determined to be a cluster such as C3 or C5 to C8, there is no cluster in the lower hierarchy, and the belonging cluster is determined. On the other hand, if it is determined that the cluster has a cluster in a lower hierarchy like C1, C2, and C4, the determination is further continued. This process is continued until a cluster having no lower cluster is found.

The selection process will be specifically described below by taking the case of selecting C212 as an example. First, it is determined which one of C1 to C8 the input data group belongs to. Here, when C2 is selected, C21 to C
It determines which one of 24 it belongs to. Where C2
When 1 is selected, C211 to C213
Which of which it belongs to is determined. When C212 is selected, there are no clusters in the hierarchy below it, so the cluster selection processing ends, and it is finally determined that it belongs to C212.

Next, an embodiment of a method for algorithmically realizing FIG. 1 will be described with reference to FIGS. 22 and 23. The clustering changing method in this embodiment uses hierarchical clustering.

FIG. 22 realizes a part of the function of the learning unit 100 and the function of the recording unit 110 in FIG. First, in 2204, the learning data group is clustered. Next, in 2206, an approximate expression is obtained for each cluster created in 2204. Then 2208
In, the accuracy of the approximate expression of each cluster obtained in 2206 is evaluated. At 2210, it is determined whether there are large error clusters. If it does not exist, learning ends. If present, the entire process 2200 of FIG.
Is repeated recursively. This process is performed for all clusters determined to have a large error (2214).

FIG. 23 realizes a part of the functions of the execution unit 120 and the recording unit 110 in FIG. When the input data group is given, the cluster to which the input data group belongs is determined in 2302. Next, in 2304, it is determined whether or not there is a lower-level cluster within the determined cluster area. If there is a cluster in the lower layer, the process returns to 2302. If there is no cluster in the lower layer, in 2306, an output data group that is an approximate value is obtained from the input data group using the approximation formula of the cluster selected in 2302.

Next, an example of determining the belonging cluster of the input data group by using the codebook provided in the cluster will be described with reference to FIGS. 8 and 9. FIG. 8 shows the distribution of the learning data group. FIG. 9 shows the result of classifying the same data into eight clusters. C1 to C8
Represents a cluster. The points CB1 to CB8 represent the codebooks C1 to C8, respectively. As one embodiment of the method of determining the cluster to which the input data group 900 belongs by using the codebook, the distance between the input data group and each codebook is obtained, and the cluster to which the codebook closest to the input data group belongs is selected. There is a way to do it. Figure 9
In this example, the codebook closest to the input data group 900 is CB1 (902), so it is determined that the codebook belongs to the cluster C1.

The present embodiment has a feature that the belonging cluster of the input data group can be determined at high speed by providing the codebook. When a codebook is used, the number of distance calculations for determination is limited to the number of codebooks. On the other hand, when the codebook is not used, since the cluster (C1) to which the data group (904) closest to the input data group (900) belongs is selected, the distance calculation must be performed by the number of all data groups. I won't. Therefore, by using the codebook, the number of distance calculations can be reduced, and the speed can be increased.

An example of how to obtain the codebook will be shown. As a first example, the average of each data of all the data groups in the cluster is used. In the second example, the centroid of the cluster area is used as the codebook. Other than this, any method may be selected as long as it is representative of the data in the cluster.

Next, one embodiment of the processing of the reliability deriving unit 300 shown in FIG. 3 will be described with reference to FIGS. 10 to 12. This embodiment can be realized by clustering a data group. In this embodiment, it is assumed that the cluster has a codebook.

The reliability of the output data group in the present invention is
It can be judged by the following factors. (1) Distance between input data group and codebook (2) Number of learning data group in cluster First, an example of obtaining reliability using the distance between the input data group and codebook in the above (1) will be described. To do. For the derivation of the approximate expression, consider a case where all the learning data groups in the cluster are created by using least square approximation. At this time, since the input data group close to the codebook is generally more accurate,
The criterion for deriving the reliability can be the distance between the input data group and the codebook. However, since the distribution state of the data within the cluster and the size of the cluster differ from one cluster to another, it is not possible to compare them simply by distance.

An embodiment of a method for comparing the distance reliability between clusters having different distribution states will be described. First, the distribution state of the learning data group in the cluster is
Assume a normal distribution centered on the codebook.
As the reliability standard, the value of the distribution density of the normal distribution at the coordinates of the input data group can be taken. As described above, when the value of the distribution density is used as the reliability standard, the input data group close to the codebook has high reliability, and the distant input data group has low reliability. Further, even if the distance between the codebook and the input data group is the same, the higher the density of the learning data group in the cluster near the center, the higher the value of the normal distribution.

A specific example of reliability derivation using a normal distribution will be described with reference to FIGS. 10 to 12. 10 and 11 show the distribution state of the data group in the cluster. Figure 11
10, the same reference numerals as in FIG. 10 indicate the same things. In the figure, 1000 is a codebook, and 1002 and 10
Reference numeral 04 is an input data group. FIG. 12 shows a distribution density function when the distributions of the data groups of FIGS. 10 and 11 are assumed to be normal distributions. Reference numeral 1200 represents the distribution state of FIG. 10, and 1202 represents the distribution state of FIG. 11. Reference numeral 1204 denotes a distance 1006 between the data group 1002 and the codebook 1000 in FIGS. 10 and 11, and 1206 denotes the data groups 1004 and 100.
A distance 1008 from 0 is shown. According to FIG. 12, the input data group 1002 has higher reliability than 1004.

On the other hand, for the input data group 1002,
Distance from codebook 100 in FIGS. 10 and 11.
8 is the same, but the distribution density function is different.
The distribution of clusters of is more reliable. Further, the input data group 1004 has higher reliability in the distribution state of FIG. 11.

Next, an example of obtaining the reliability by using the number of learning data groups in the cluster in the above (2) will be described. When the approximation formula is derived so as to minimize the error of all data groups in the cluster, it is generally more accurate to derive it using a large amount of data. Therefore, the number of data groups in the cluster can be used as a criterion for deriving the reliability.

FIG. 13 shows the distribution state of the data group in the cluster. 13, the same reference numerals as those in FIG. 10 indicate the same things. In FIG. 13, the codebook (1000), the input data group (1002, 1004),
Distance between codebook and data group (1006, 100
8), the distribution density is the same as in FIG. 10, but the number of data groups used to derive the approximate expression is small. Therefore, the distribution of FIG. 10 is considered to have higher reliability than that of FIG.

The methods (1) and (2) described here can be used independently or in combination. Furthermore, it can be used in combination with other methods. As a concrete example, it can be realized by multiplying the reliability obtained by each method or by adding weights.

Next, a method of discarding an output with low reliability will be described with reference to FIG. 1400 is a distribution density function when the distribution of the data group in the cluster is assumed to be a normal distribution. 1402 is a standard for discarding. 1404
And 1406 are the distances between the codebook and the input data group. For the 1404 input data group whose distribution density value exceeds 1402, the obtained output data group is adopted.
Regarding the input data group of 1406 which does not exceed 02, the obtained output data group is discarded.

The present embodiment is characterized in that the distance between the input data group and the codebook can be evaluated and the output with low reliability can be discarded. Therefore, when applied to a control problem or the like, unpredictable output will not occur for unknown inputs.

Next, an example in which the present invention is applied to a simple function value prediction by using the functions shown in Expressions 1 and 2 will be described.

[0052]

[Equation 1]

[0053]

[Equation 2]

In Equation 1, f (0) = 0 and the value of k is 0.
To 19 and 20 pieces of data from f (0) to f (19) are treated as one set. Number 2 is number 1
Is a random number in the range of −0.5 to 0.5. In this embodiment, a method of setting an input data group, an output data group, and a learning data group, a learning method, an execution method, a result display method, and a method of creating an approximate expression from a cluster having a small number of data will be specifically described. explain.

First, the method of setting the input data group, the output data group, and the learning data group in this embodiment will be described with reference to FIG. Input data group is continuous 5
It is individual data (1600 to 1608). The output data group is the data next to the last data of the input data group (16
10). The learning data group is 6 continuous data. Of these, the five data from the front correspond to the input data group, and the last data correspond to the output data group. At the time of learning, clustering is performed using the data corresponding to the input data group 1600 to 1608. When creating the approximate expression, a calculation expression for deriving data 1610 corresponding to the output data group from 1600 to 1608 is obtained.

In this embodiment, the number of data in one set is 20.
Since it is a single piece, a learning data group composed of 6 consecutive data pieces can be set in 15 types per set. The input data group and the output data group are also set in 15 types per set according to the learning data group.

In the present embodiment, one set of the function shown in Formula 1 and 9 sets of the function shown in Formula 2 are prepared for a total of 10 sets. Of these, the learning data group has a total of 8 sets, that is, one set based on Formula 1 and 7 sets based on Formula 2, that is, 1
5 × 8 120 data groups are used. Two sets not used for learning are used to evaluate the learning results.

FIG. 15 shows all 10 sets. The 1st to 8th sets of data are used for learning, and
The 10th set of data is test data. In FIG. 15, in the description of ab on the horizontal axis, the value on that axis is the first data of the learning data group of data number a and the last data of the learning data group of data number b. It means that there is. For example, the learning data group of data number 4 is composed of six data from the value on the axis of (4- *) to the value on the axis of (9-4). In the example of the data group of FIG. 16, it is described as (P1-P6).

Next, the learning method of this embodiment will be described. In the present embodiment, it is assumed that the above-mentioned method based on liquor is used for clustering and linear multiple regression analysis is used for deriving an approximate expression. For clustering, only the five data from the front of the learning data group are used. After completion of the clustering, an approximate expression for obtaining one data corresponding to the output data group is derived from the five data by using all the data in the cluster. Equation 3 shows an example of the approximation formula. Where a1
To a5 represent data values of the input data group, and b represent data values of the output data group.

[0060]

[Equation 3]

Next, the execution method of this embodiment will be described. In the present embodiment, first, the codebook is used to determine the cluster to which the input data group belongs. The codebook is the average value of the values of five data of the learning data group in the cluster. The cluster having the shortest distance between the input data group and the codebook is selected as the cluster to which the input data group belongs. After the cluster is determined, the value of each data in the input data group is substituted into the approximate expression of the corresponding cluster to obtain the output data group.

Next, the hierarchical clustering in this embodiment will be described. In the present embodiment, the approximation formula is verified using all the learning data groups in the cluster, and when the maximum error exceeds 0.5, the area of the corresponding cluster is subdivided. As a result, in each cluster almost ±
Prediction is possible with an error of 0.5 or less.

Next, the display device for displaying the results of this embodiment will be described with reference to FIG. FIG. 17 shows an example of the clustering result in this embodiment. In this embodiment, clusters are displayed using a tree structure as a method of displaying the relationship between clusters. Reference numerals 1700 to 1744 represent clusters. The number in the parentheses is the head data number of the learning data group existing in the cluster, and the number below is the number of the learning data group in the cluster. The leading data number represents the type of data group. For example, in cluster 1724, all eight data groups starting with 2 are grouped into one cluster. The display device shown in FIG.
The whole or part of the diagram showing the relationship between the clusters having the hierarchical structure as shown in FIG. 7 is displayed to the user as the execution result. From this display, it is possible to recognize how the learning data group has been classified.

A method of highlighting the result in this display will be described with reference to FIG. 18, the same reference numerals as those in FIG. 17 refer to the same parts. If the cluster determined to belong to the input data group is blinked (180
0), it becomes easy to recognize the selected cluster. With this display, the positioning of the entire input data group can be easily understood. In addition to the blinking display,
Examples such as black and white reverse display are also available.

A method of displaying the reliability as additional information in this display will be described with reference to FIG. 19, reference numerals that are the same as those in FIG. 17 refer to the same things. In addition to displaying the clusters from 1700 to 1744, displaying the reliability (1900) provides a criterion for how much the user should trust the value.

A method of displaying an approximate expression as additional information in this display will be described with reference to FIG. 20, the same reference numerals as those in FIG. 17 refer to the same parts. In addition to displaying the clusters from 1700 to 1744, by displaying an approximate expression (2000), it is possible to show the user how much the input value affects the output.

Next, an embodiment of a method of changing the number and types of input data and output data of an approximate expression for each cluster in this embodiment will be described. As a method of changing the number and type, there is a method of selecting data with large dispersion. This has a feature that when the number of data groups in the cluster is smaller than the number of input data of the approximate expression, the approximate expression can be created by reducing the number of input data of the approximate expression. In FIG. 17, the number of data groups in the cluster 1738 is four. Since the number of input data of the approximate expression is 5, 4 out of 5 are selected to create the approximate expression.

Next, with reference to FIG. 21, an example in which the present apparatus is used for detecting abnormality in a plant or the like will be described. The configuration of the plant abnormality detection device is shown in FIG. In this embodiment, the function approximating device functions to obtain the predicted value of the next output from the past input / output data group of the plant. In this embodiment, the output of the plant 2100 that is the target of abnormality diagnosis and the predicted value by the function approximation device 2110 are used as the abnormality detection device 21.
Compare in 12. As a result, if the error is large,
It is determined that an unexpected behavior is occurring, and the operator is warned.

The operation of this apparatus will be described below.
The plant 2100 receives the input 2114 and outputs the output 211.
6 is output. 2102 to 2106 are time delay elements. This element stores the output 2116 for a certain period of time and sends the output 2118 before the temporary point to the preprocessing device 2108. The preprocessing device 2108 receives the plant input 2114 and the output 2118 before the temporary point, accumulates data for several time points, and generates a learning data group and an input data group 2120 of the function approximating device 2110. The function approximation device 2110 uses the input data group 212 created by the preprocessing unit.
Upon receiving 0, the predicted value 2122 of the output is generated. The abnormality detection device 2112 uses the output data group 2116 and the predicted value 21.
22 is compared, and when the difference is large, the user is warned that the operation of the plant is abnormal.

Next, an example in which the present invention is applied to image restoration and used to predict a spatial pattern will be described. At the time of learning, image data is stored as a learning data group, and the cluster creating unit 106 clusters them. At the time of execution, an image with a part missing is input and it is determined to which cluster the image belongs. Furthermore, the missing part of the input image is predicted from the images in the cluster. A simple concrete example is shown. Images of humans, monkeys, dogs, and other animals' faces are stored in the database. At the time of execution, an image with a part of the face missing is input. First, it is determined which of a human, a monkey and a dog is an animal. When the clustering is performed hierarchically, similar ones are clustered for each animal, and therefore a cluster of faces similar to the input image is selected. Next, the missing parts from the image in the cluster and the input image are repaired.

Next, an additional learning method in the function approximating apparatus according to the present invention will be described. This embodiment is applied when hierarchical clustering is performed in the present invention. In the present embodiment, when a learning data group is newly added after the learning has been completed once, the learning results up to that point are partially improved and dealt with. Therefore, the learning is faster than the case where the learning is performed again from the beginning using all the learning data groups.
Further, even if the characteristics of the approximation target change, the function approximating apparatus can follow this characteristic change only by additionally learning the changed data group.

The following four application methods can be considered as the additional learning method. (1) It is applied every time an additional learning data group is added. (2) The additional learning data group is evaluated by the conventional approximation formula, and is applied only when the error exceeds the standard. (3) It is applied only when the number of additional learning data groups reaches a preset number. (4) The additional learning data group is evaluated by the conventional approximation formula, and is applied only when the number of times the error exceeds the reference reaches a preset number of times.

First, referring to FIG. 24, an additional learning method 1 which is an embodiment of the additional learning will be described. In this additional learning method, a group of data for additional learning belongs, and
Learning is performed only within the region of the cluster Xm of the hierarchy. The value of this m can be determined arbitrarily.

The processing procedure of this embodiment will be described below. First, in 2402, the cluster Xm to which the additional learning data group belongs is selected from the m-th layer clusters. If the cluster to which the additional learning data group belongs does not exist in the m-th layer, a cluster in a layer higher than the m-th layer is selected. Next, in 2404, only in the area of this cluster Xm, 2 in FIG.
The additional learning is ended by performing the processing of 200 and re-calculating the cluster and the approximate expression. Note that, thereafter, performing the processing of 2200 in the area of a certain cluster is called re-learning.

In the present embodiment, re-learning is performed only in the area of the cluster of the m-th layer to which the additional learning data group belongs, and re-learning is not performed for the other portions, so that it is faster than new learning. There is a feature that you can learn.

This embodiment will be specifically described with reference to FIGS. 27 to 29. 27 to 29, C represents a cluster. In this embodiment, the number m of layers to be re-learned is 2. The processing procedure of this embodiment will be described below. First, the cluster to which the additional learning data group belongs is determined. Consider a case where it is determined that the cluster to which the additional learning data group belongs is C2 in the first hierarchy, C21 in the second hierarchy, and C212 in the third hierarchy. Since the value of m is 2, C21 is selected as the cluster to be relearned. Therefore, as shown in FIG. 28, the ends C211 and C212 are discarded, and re-learning is performed using all the learning data groups and the additional learning data group in the area C21. Figure 2
2900 of 9 is a cluster newly created by additional learning. In this example, since the approximation accuracy of the cluster C214 is poor, the clusters C2141 and C214 of the fourth layer are
2 has been created.

Next, with reference to FIG. 25, an additional learning method 2 which is another embodiment of the additional learning will be described. 25, the same reference numerals as those in FIG. 24 indicate the same things.
In this embodiment, re-learning is performed on the following two types of clusters.

(1) The cluster for the additional learning data belongs and is adjacent to the cluster of the mth layer from the top (2) The cluster of the nth layer from the top. Predetermine the value arbitrarily. In addition,
When there is no cluster to which the additional learning data group belongs in the m-th layer, a cluster in a layer higher than the m-th layer is selected. The same applies to n.

In this embodiment, learning is performed only in the area of the m-th layer cluster to which the additional learning data group belongs and the clusters around it, and the other portions are not learned. There is a feature that you can learn faster than you learn. Further, since the clusters that are affected by the re-learning of the cluster to which the additional learning data group belongs are re-learned, the additional learning method is characterized by higher accuracy.

The processing procedure of this embodiment will be described below. First, in 2402, the cluster Xm of the m-th layer to which the additional learning data group belongs and is selected. Next, in 2404, the process of 2200 in FIG. 22 is performed only for the region of the cluster Xm to relearn. Next, at 2502, the cluster Yn of the nth hierarchy is selected from the clusters adjacent to the cluster Xm. Next, in 2504, an evaluation amount for determining whether or not to re-learn the area of the cluster Yn is obtained,
At 2506, it is determined whether it is necessary to re-learn in the area of the cluster Yn. If there is no need for re-learning, the process proceeds to 2510. If re-learning is necessary, the process 2250 in FIG.
I do. In this processing, the clustering state of the cluster Yn is not changed, an approximate expression is obtained using all the learning data groups in the area of Yn, and if the accuracy of the approximate expression is poor, the processing 2200 in FIG. 22 is performed. Is. Finally,
In 2510, it is determined whether the process 2506 has been performed for all clusters in the nth layer adjacent to the cluster X, and the additional learning ends when all clusters have been processed. In the processing of 2508, 225
It is also possible to perform the processing of 2200 instead of 0. This is to change the approximation formula after changing the clustering state of the cluster Yn.

This embodiment will be specifically described with reference to FIGS. 30 to 32. First, m and n are 1, that is,
An example is shown in which both the cluster to which the additional learning data group belongs and the adjacent clusters are relearned with respect to the cluster of the first layer. 30 to 33, CB represents a codebook. The codebook is assumed to be the average of all training data sets in the cluster. Further, the boundary between the clusters is a vertical bisector (line) between the codebooks representing each cluster. In the following description, a cluster having CBn (n is an integer) as a codebook will be referred to as a “CBn cluster”.

FIG. 30 shows the state of the cluster before the additional learning.

First, an embodiment of re-learning of the cluster to which the additional learning data group belongs will be described with reference to FIG. FIG. 31 is an example in which the additional learning data group (3100) is added to the area of the cluster of CB1 (3102). Since the codebook is the average of all learning data groups, the position of the codebook is CB due to the influence of the additional learning data group.
Move to 1 '(3104). As CB1 moves, C
The boundary between the clusters, which is the vertical bisector between B, also moves. After CB1 moves to CB1 ', CB1'
Retrain in the region of the cluster of.

Next, an embodiment of re-learning of a cluster adjacent to the cluster to which the additional learning data group belongs will be described. 3108 is CB4 and CB5 along with the movement of CB1.
This is an area that is added to the area of the cluster of CB1 from the cluster of. Here, it is not necessary to re-learn the clusters of CB4 and CB5. The area of these clusters decreases, but
This is because all the data groups remaining in the area can be accurately approximated by the conventional approximation formulas of CB4 and CB5. However, these clusters can also be relearned.

Further, an embodiment of re-learning of a cluster adjacent to the cluster to which the additional learning data group belongs will be described. Reference numeral 3106 denotes an area which is deviated from the area of the cluster of CB1 due to the movement of the codebook and added to the clusters of CB2, CB3, and CB6. Where CB2 and CB
3. The cluster of CB6 needs to be relearned. Before the additional learning, the region 3100 was the region CB1, so the data group in the region 3106 was approximated by the formula CB1. Therefore, there is no guarantee that the data group in the region 3106 can be approximated with high accuracy by the approximation formula of the clusters CB2, CB3, and CB6.

When re-learning these clusters, there are two possible ways to handle the codebook. There are a case where only the approximate expression is changed without moving the codebook, and a case where the approximate expression is changed after moving the codebook. The former corresponds to 2508 in FIG. 29, and the latter is 2508.
Corresponds to performing the process 2200 instead of the process 2250. How does this distinction deal with the problem of re-learning effects propagating one after another when moving the codebooks of adjacent clusters and further changing the boundaries of those clusters with adjacent clusters? Due to that. If the codebook is not moved, the boundaries do not move, so there is no need to relearn other clusters. If you want to move the codebook,
It is necessary to continue to re-learn clusters whose boundaries change until the effect of moving the boundaries disappears.

Next, an embodiment of a method for evaluating a method for determining a cluster to be relearned will be described. This is 25 in FIG.
Equivalent to 04. When the learning data group does not exist in the region 3106, there is no change in the learning data group for deriving the approximate expression in the clusters CB2, CB3, and CB6, and thus it is not necessary to re-learn. Therefore, 3
It is determined whether or not the learning data exists in the region of 106, and finally it is decided whether or not to re-learn.

Next, with reference to FIG. 32, an example will be shown in which n is 2, that is, relearning is performed for the cluster of the second layer of the adjacent cluster. In the second layer, 3200 is an area adjacent to the cluster (CB1 ′) to which the additional learning data group belongs and that satisfies the condition described with reference to FIG. Therefore, this area is relearned.

Next, with reference to FIG. 26, an additional learning method 3, which is another embodiment of the additional learning, will be described. In the present embodiment, the additional learning data group belongs and learning is performed only in the area of the cluster Xk of the kth layer. However, the value of k can be determined appropriately according to each additional learning data group.

The present embodiment is characterized in that learning is performed only in the area around the additional learning data group, and learning is not performed in the other portions, so that learning can be performed faster than new learning. Further, since the most appropriate re-learning hierarchy k can be obtained according to each additional learning data group, it is fast without re-learning an area larger than necessary.

The processing procedure of this embodiment will be described below. First, in 2602, the evaluation amount of the cluster to which the additional learning data group belongs in each layer is calculated. Next, in 2604, the cluster Xk of the nth hierarchy from the bottom is selected from the clusters whose evaluation amount is equal to or less than the reference.
Next, in 2606, the processing of 2200 in FIG. 22 is performed only within the region of the cluster Xk, and the additional learning is ended by changing the clustering state and the approximate expression.

Here, an example of the evaluation amount in 2602 and 2604 will be described. This evaluation amount determines the hierarchy of the cluster to be re-learned according to each additional learning data group. The cluster to be relearned is determined as follows. (1) When the additional learning data group exists near the boundary between the clusters, re-learning is performed in a layer above that layer (the region for re-learning is expanded). (2) When the additional learning data group exists near the codebook, relearning is performed within the area of the cluster.

The grounds for this are as follows. (A) When the additional learning data group exists near the boundary between the clusters, the amount of movement of the codebook also increases, and the boundary between the clusters and the approximate expression may change significantly. (B) When the additional learning data group exists near the boundary between the clusters, the distance from the center of the cluster is large, and thus the accuracy of the approximation formula may be poor. Therefore, it is possible to determine the criterion as “the lowest cluster among the clusters in which the distance between the codebook and the additional learning data group is less than or equal to a predetermined criterion”. However, it does not have to be the bottom layer,
Arbitrary layers such as the second from the bottom and the third from the bottom are possible.

The distance criterion for judgment can be changed depending on each cluster. For example, the average value or standard deviation of the distances from the codebook for all learning data groups in a cluster is multiplied by a predetermined constant α, or the distance between adjacent clusters is predetermined. The value multiplied by the constant α that has been used can be considered.

Furthermore, the reference can be changed for each layer by making α different for each layer. For example, if the reference value is set to be higher in the higher clusters, re-learning can be performed in a lower hierarchy as compared with the case where the same reference value is used. This is characterized by the fact that the cluster for the lower hierarchy has a smaller area for re-learning, so that the time for additional learning becomes shorter.

This embodiment will be specifically described with reference to FIGS. 33 and 34. FIG. 33 shows a state of clustering and a data group for additional learning. CB1 to CB6 are codebooks of clusters in the first layer. CB11 to CB13 are codebooks of clusters in the second layer. 3300, 3302, and 3304 are additional learning data groups. Reference numerals 3311 to 3316 denote areas of CB1 to CB6 in which the distance between the codebook and the additional learning data group is within the evaluation criterion. 3321 to 3323 are C
Regarding B11 to CB13, the distance between the codebook and the additional learning data group is within the evaluation standard. When the additional learning data group is added to the area within the evaluation standard, re-learning is performed using all the data groups in the area of the corresponding cluster.

In this embodiment, the value of k in FIG. 26 is set to 1
And That is, among the clusters in which the distance between the codebook and the additional learning data group is equal to or less than the reference value, the cluster in the lowest hierarchy is relearned.

The determination of the re-learning hierarchy of the three additional learning data groups 3300, 3302 and 3304 will be described below. First, the case where the additional learning data group 3300 is added will be described. Since this data group is within the area within the evaluation criteria of CB11, re-learning is performed within the area of the cluster of CB11.

Next, the case where the additional learning data group 3302 is added will be described. Since this data group is outside the area within the evaluation criteria of CB12, relearning is not performed within the area of the cluster of CB12. Therefore, the cluster of CB1 which is a cluster in a layer above CB12 is evaluated. In CB1, since it is in the area within the evaluation criteria, CB
Re-learning is performed within the area of one cluster.

Finally, the case where the additional learning data group 3302 is added will be described. Since this data group is outside the area within the evaluation criteria for CB13 and CB1, re-learning is performed using all data groups.

FIG. 34 shows the clusters of FIG. 33 hierarchically. 3400 is a data group 33 for additional learning
This is a region to be relearned by 00. All data groups in the area of CB11 are used for re-learning. An area 3402 is an area to be relearned by the additional learning data group 3302. The clusters CB11 to CB13 are discarded and C
All data groups in the area of B1 are used for re-learning. 34
Reference numeral 04 is an area to be relearned by the additional learning data group 3304. All clusters are discarded and all data sets are used for retraining.

It is also possible to use the above three types of additional learning methods together. For example, it is usually possible to use the additional learning method 3 shown in FIG. 29 and use the additional learning method 2 or 1 only when it is determined to be in the upper hierarchy. The additional learning method 3 alone has a drawback that the number of data groups required for the additional learning increases when it is determined to re-learn in the upper hierarchy. There is a feature that this defect can be prevented by using a plurality of methods together.

Next, one embodiment of a method of discarding the recorded learning data group in the above additional learning method will be described. As a method of discarding, there are a method of discarding an old data group and a method of discarding a data group that exceeds or does not meet a predetermined evaluation standard. The discarded data group is not used for clustering or creation of an approximate expression in the subsequent additional learning. The present embodiment is characterized in that it is possible to prevent the storage capacity of the function approximating apparatus from becoming insufficient and the processing speed from becoming slow due to the increase of the learning data group.

[0104]

According to the present invention, since the clustering of data is performed before performing the function approximation, the approximation can be executed by using the approximation formula most suitable for the input data group. It will be possible. Furthermore, by clustering the data group and performing statistical processing, the reliability of the approximate value can be obtained. further,
By having the additional learning function, the learning can be performed at high speed even when the learning data group is added. Furthermore, by having an additional learning function, even for approximation targets whose characteristics change,
Only by additionally learning the changed data group, the present function approximating apparatus can follow the change in characteristics.

[Brief description of drawings]

FIG. 1 is a block diagram showing processing of a function approximating apparatus showing an embodiment of the present invention.

FIG. 2 is an overall configuration diagram of a function approximating apparatus showing an embodiment of the present invention.

FIG. 3 is a block diagram showing a process of a function approximating device for deriving the reliability of an output data group.

FIG. 4 is an example of clustering.

5 is an example of hierarchical clustering of some of the clusters in FIG.

FIG. 6 is an example in which some clusters in FIG. 5 are further hierarchically clustered.

FIG. 7 is a hierarchical structure of clusters obtained by the clustering of FIGS. 4 to 6;

FIG. 8 is an example of a distribution of a learning data group.

9 is an example in which the learning data group of FIG. 8 is clustered, a codebook is arranged, and an input data group is added.

FIG. 10 is an example of a cluster in which a learning data group is densely distributed near the center of the cluster.

11 is an example of a cluster having the same number of data as FIG. 10 but a different distribution state.

FIG. 12 is a density function showing the distribution state of FIGS. 10 and 11.

FIG. 13 is an example of a cluster having the same distribution density function as that of FIG. 10, but having a different number of data.

14 is a density function showing the distribution state of FIG.

FIG. 15 is a function used in an embodiment of the present invention.

FIG. 16 is an example of a method for setting an input data group and an output data group according to the present invention.

17 is a hierarchical clustering result learned for the function of FIG.

FIG. 18 shows an example of blinking display in FIG.

FIG. 19 is an example in which the reliability of output is displayed.

FIG. 20 is an example in which an approximate expression used for approximation is displayed.

FIG. 21 is a block diagram of an abnormality detection device showing an embodiment of the present invention.

FIG. 22 is a flowchart showing an embodiment of the learning method of the present invention.

FIG. 23 is a flowchart showing an embodiment of an execution method of the present invention.

FIG. 24 is a flowchart showing an example of additional learning.

FIG. 25 is a flowchart showing an example of additional learning.

FIG. 26 is a flowchart showing an example of additional learning.

FIG. 27 is a hierarchical structure of clusters before additional learning.

FIG. 28 is a hierarchical structure of clusters when a cluster to be relearned by additional learning is determined.

FIG. 29 is a hierarchical structure of clusters after additional learning.

FIG. 30 is an example of a state of a cluster before additional learning.

FIG. 31 is an example of a state in which the codebook is moved based on the additional learning data group.

FIG. 32 is an area in which there is a possibility of re-learning when a cluster in the second hierarchy is selected as an adjacent cluster.

FIG. 33 shows a state of a cluster before additional learning and an evaluation criterion for determining a hierarchy of the cluster for additional learning.

34 is a hierarchical structure of the cluster shown in FIG.

[Explanation of symbols]

100 ... Learning unit, 102 ... Cluster creation unit, 104 ... Approximation formula creation unit, 106 ... Approximation formula evaluation unit, 110 ... Recording unit,
112 ... Learning data group recording unit, 114 ... Cluster information recording unit, 116 ... Approximate expression recording unit, 120 ... Execution unit, 12
2 ... Affiliated cluster determination unit, 124 ... Approximate value calculation unit.

 ─────────────────────────────────────────────────── ─── Continuation of the front page (72) Inventor Hisao Ogata 1-280, Higashi Koikekubo, Kokubunji, Tokyo Metropolitan Research Center, Hitachi, Ltd.

Claims (19)

[Claims] 1. From an input data group which is a set of input data,
In a function approximation device that obtains output data or an output data group that is a set of output data, (1) calculation formula for function approximation for obtaining output data or output data group from input data group (hereinafter referred to as approximation formula) (A) means for clustering a learning data group, which is an input data group used for learning, into data groups having similar characteristics, and (b) an approximate expression for each created cluster. , (C) a means for evaluating the created approximate expression, (d) learning is ended when (c) is good for all clusters, and clustering is performed when there is a bad evaluation. A function approximation apparatus comprising: a function approximation expression learning means having a means for changing the approximation expression and a means for changing the approximation expression according to the evaluations of (e) and (c). 2. An input data group, which is a set of input data,
In a function approximation device that obtains output data or an output data group that is a set of output data, (1) calculation formula for function approximation for obtaining output data or output data group from input data group (hereinafter referred to as approximation formula) (A) means for clustering a learning data group, which is an input data group used for learning, into data groups having similar characteristics, and (b) an approximate expression for each created cluster. , (C) a means for evaluating the created approximate expression, (d) learning is ended when (c) is good for all clusters, and clustering is performed when there is a bad evaluation. And (e) and (c) according to the evaluation of the approximation formula changing means learning means, and (2) (a) (1) (a) clustering means To And means for recording learning data group can use this in, (b)
(1) In the clustering means of (a), means for recording information on the created cluster, and (c)
(1) In the means for obtaining the approximate expression in (b), a recording means having a means for recording the created approximate expression, and (3) When obtaining and executing output data or output data group from the input data group, , (A) means for determining the cluster to which the input data group belongs, and (b) means for obtaining output data or output data group from the input data group using the approximate expression of the determined cluster And a means for approximating a function. 3. The function approximating apparatus according to claim 2, wherein for each created cluster, the input data is set by setting a codebook that is a data group representing the characteristics of the data group existing in the cluster. A function approximating apparatus characterized by calculating a distance between an input data group and a codebook when selecting a cluster to which the group belongs and selecting a cluster to which the codebook closest to the input data group belongs. 4. The function approximating device according to claim 2, wherein (1) when changing the approximation formula of a cluster with a poor evaluation by means for changing the clustering of (d), (f) the corresponding cluster Means for performing hierarchical clustering by clustering the region of into smaller clusters,
(G) means for creating an approximate expression for each cluster created in (f), and a function approximating apparatus. 5. The function approximating apparatus according to claim 2, wherein a cluster to which the input data group is determined to belong to
A function approximating apparatus comprising means for deriving the reliability of the obtained output data or output data group by calculating the evaluation amount of the input data group. 6. The function approximating apparatus according to claim 5, wherein for the clusters to which the input data group is determined to belong,
A function approximating apparatus having means for discarding the output data or output data group when the reliability does not exceed a reference. 7. The function approximating apparatus according to claim 2, further comprising means for displaying a classification relation between the created clusters and displaying a relation between a cluster to which the input data group belongs and another cluster. Characteristic function approximation device. 8. The function approximating apparatus according to claim 2, further comprising means for displaying the reliability of the derived output data group. 9. The function approximating apparatus according to claim 2, further comprising means for displaying a function approximating expression derived for each cluster. 10. The function approximating apparatus according to claim 2, wherein when the time series data is handled, one data or a set of a plurality of data is output from the time series data obtained from a plurality of times. A means for obtaining a calculation formula for deriving an output data group from the input data group for each cluster, using a data group and a data set other than the output data group as the input data group.
The function approximating device having the function approximating formula learning means and deriving an approximate value of the output data group using the calculated formula. 11. The function approximating apparatus according to claim 2, wherein the time series data (P
1, P2 ,. ． ． , Pt), (P1, P
2 ,. ． ． , Pt-1) as the input data group and Pt as the output data group, and when the error between the output data group and the actual measurement value Rt at time t exceeds the reference value, it is determined that the behavior is not predicted. Then, the function approximating apparatus is provided with means for detecting that the target system is behaving abnormally, and means for displaying the detected abnormal behavior. 12. The function approximating apparatus according to claim 2, wherein the function approximating means (1) is used to obtain a calculation formula for deriving an output data group related to another location from an input data group measured at a plurality of locations. A function approximating device, characterized in that the function approximating means has means for deriving a predicted value or an interpolated value of an output data group. 13. The function approximating apparatus according to claim 2, wherein when an approximate expression is created, data of an input data group and an output data group of the approximate expression (that is, an input variable and an output variable of the approximate expression) for each cluster. A function approximating apparatus characterized by having a means for changing the number and type) of. 14. The function approximating apparatus according to claim 13, wherein when an approximate expression cannot be created because the number of data in the cluster is small, an input variable of the approximate expression is applied to the corresponding cluster. And a function approximation device characterized by reducing the number of output variables. 15. The function approximating apparatus according to claim 4, wherein when hierarchical clustering is performed, an additional learning data group, which is a new learning data group, is added in addition to the existing learning data group. When learning, the value of m is arbitrarily determined in advance, and among the clusters created by the learning performed in advance, the cluster Xm to which the additional learning data group belongs and which is in the mth hierarchy from the top is selected. A function approximating apparatus characterized by having means for performing re-learning by selecting and using the learning data group and additional learning data group in the cluster Xm. 16. The function approximating apparatus according to claim 4, wherein when hierarchical clustering is performed, an additional learning data group, which is a new learning data group, is added in addition to the existing learning data group. When learning, the values of m and n are arbitrarily determined in advance, and among the clusters created by the learning performed in advance, the cluster for the additional learning data group and in the mth hierarchy from the top Xm is selected, learning is performed using the learning data group and the additional learning data group in the cluster Xm, the cluster Yn adjacent to the cluster Xm and in the nth hierarchy from the top is selected. A function approximating apparatus having means for determining whether or not to perform re-learning, and having means for re-learning using a learning data group in the determined cluster, and performing additional learning. 17. The function approximating apparatus according to claim 4, wherein when hierarchical clustering is performed, an additional learning data group, which is a new learning data group, is added in addition to the existing learning data group. At the time of learning, the value of k is arbitrarily determined in advance, the evaluation amount is calculated for the cluster to which the additional learning data group belongs in each layer, and among the clusters whose evaluation amount is equal to or less than the reference, A function approximating apparatus having means for selecting a cluster Xk and performing learning on a learning data group and additional learning data group in the cluster Xk, and performing additional learning. 18. The function approximating apparatus according to claim 15, 16, or 17, wherein when the number or capacity of learning data groups exceeds a reference due to the addition of learning data groups, the old A function approximating apparatus characterized in that a learning data group is deleted from a learning data group recording unit, and the deleted learning data group is not used in subsequent additional learning. 19. In a function approximation method for obtaining output data or an output data group from an input data group, (1) (a) a learning data group is clustered for each data group having similar characteristics, and (b) is created. An approximate expression is obtained for each cluster, (c) the created approximate expression is evaluated, and (d)
If the evaluation of (c) is good for all clusters, the learning is terminated, if there is a cluster with a bad evaluation, the clustering is changed, and the approximation formula is changed according to the evaluations of (e) and (c). (2) (a) Determine the cluster to which the input data group belongs,
(B) A function approximation method characterized by performing output by obtaining output data or an output data group from an input data group using an approximate expression of the determined cluster.