JP6507512B2 - Estimation program, estimation method and estimation apparatus - Google Patents

Description

  The present invention relates to an estimation program and the like.

  In recent years, given a plurality of sample points, a regression function is identified by performing regression analysis on each sample point, and a new input value is given using the identified regression function. There is a prior art for estimating an output value corresponding to an input value.

  The regression analysis used in the prior art includes kernel regression analysis and polynomial regression analysis. In kernel regression analysis, an optimum value of a parameter called a bandwidth matrix is searched, and the searched parameter is used to specify a regression function. In polynomial regression analysis, the degree of polynomial and the optimum value of the coefficient of each term are searched to specify a regression function.

Nadaraya, E. A. (1964). "On Estimating Regression". Theory of Probability and its Applications 9 (1): 141-2. Watson, G. S. (1964). "Smooth Regression Analysis". Sankhya: The Indian Journal of Statistics, Series A 26 (4): 359-372. Wand, P. and Jones, C. (1994). "Kernel Smoothing". Chapman & Hall / CRC Monographs on Statistics & Applied Probability, Taylor & Francis.

  However, the above-described prior art has a problem that the amount of calculation for estimating the output value corresponding to the input value is large.

  When the input value is high-dimensional, or when each sample point has a complicated distribution shape, the process of adjusting the bandwidth of kernel regression and the time of adjusting the order and coefficients of polynomial regression become longer.

  In one aspect, it is an object of the present invention to provide an estimation program, estimation method, and estimation apparatus capable of estimating an output value corresponding to an input value while preventing an increase in calculation amount.

  In the first proposal, the estimation program causes the computer to execute the following processing. The computer is made to execute processing for specifying neighboring points present in the vicinity of each point for a plurality of points included in the sample set. The computer is caused to execute a process of selecting a point included in the sample set and a neighboring point corresponding to the point. Each point included in the sample set is a process of using a computer to average the vector values of the selected points based on the selected points and neighboring points, and specify a kernel function whose shape is to be determined based on the neighboring points Execute the process to be executed for. A process of estimating an output value corresponding to an input value by weighted averaging of an output value corresponding to each point included in a sample set with a value of a kernel function of the input value when the computer acquires an input value Run

  According to one embodiment of the present invention, it is possible to prevent an increase in the amount of calculation and to estimate an output value corresponding to an input value.

FIG. 1 is a functional block diagram showing the configuration of the estimation apparatus according to the present embodiment. FIG. 2 is a diagram showing an example of a data structure of sample set information. FIG. 3 is a view showing an example of the data structure of the k-neighborhood information. FIG. 4 is a diagram illustrating an example of a data structure of a whitening sample set. FIG. 5 is a diagram for explaining the process of identifying nearby points. FIG. 6 is a diagram showing an example of the regression function according to the present embodiment. FIG. 7 is a diagram for explaining the process of the estimation unit. FIG. 8 is a flowchart showing the processing procedure of the estimation apparatus according to the present embodiment. FIG. 9 is a diagram illustrating an example of a computer that executes an estimation program.

  Hereinafter, embodiments of the estimation program, estimation method and estimation apparatus disclosed in the present application will be described in detail based on the drawings. The present invention is not limited by this embodiment.

  An example of a structure of the estimation apparatus which concerns on a present Example is demonstrated. FIG. 1 is a functional block diagram showing the configuration of the estimation apparatus according to the present embodiment. As shown in FIG. 1, the estimation apparatus 100 includes a communication unit 110, an input unit 120, a display unit 130, a storage unit 140, and a control unit 150.

  The communication unit 110 is a processing unit that executes data communication with another device via a network. For example, the communication unit 110 corresponds to a communication device or the like. For example, when the communication unit 110 receives proximity size information 141 and sample set information 142 described later from another device, the communication unit 110 outputs the received proximity size information 141 and sample set information 142 to the control unit 150.

  The input unit 120 is an input device that inputs various types of information to the estimation device 100. For example, the input unit 120 corresponds to a keyboard, a mouse, a touch panel, and the like. For example, the user may operate the input unit 120 to input the proximity size information 141 and the sample set information 142.

  The display unit 130 is a display device that displays various data output from the control unit 150. For example, the display unit 130 corresponds to a liquid crystal display, a touch panel, or the like.

  The storage unit 140 includes neighborhood size information 141, sample set information 142, k neighborhood information 143, and function parameter information 144. The storage unit 140 corresponds to, for example, a storage device such as a semiconductor memory element such as a random access memory (RAM), a read only memory (ROM), or a flash memory.

  The neighborhood size information 141 is a parameter for specifying the size when finding the k neighborhood of a certain point of the sample set information 142.

  The sample set information 142 includes information of a plurality of sample points. FIG. 2 is a diagram showing an example of a data structure of sample set information.

In the example shown in FIG. 2, the horizontal axis is an axis corresponding to an input variable such as “time zone”. The input variable is not limited to the time zone, but may be a multidimensional variable including a plurality of attributes such as the day of the week, the amount of precipitation, and the temperature. The vertical axis is an axis corresponding to an output variable such as "required time". For example, the required time is the time required to pass a certain section. The output variable is not limited to the required time but may be a multidimensional variable including a plurality of attributes such as tolls and fuel consumption. Also, sample set information 142, each point _X 1 _{to X 15} is disposed. Coordinates of each point are appropriately (x ₁ , y ₁ ),..., (X ₁₅ , y ₁₅ ). The sample set information 142 may have points other than the points X _{1 to} X ₁₅ .

  The k-neighborhood information 143 is information indicating the relationship between the point of the sample set information 142 and the near point existing near this point. FIG. 3 is a view showing an example of the data structure of the k-neighborhood information. As shown in FIG. 3, the k-neighborhood information 143 associates points included in the sample set information 142 with neighboring points. The number of neighboring points is the number specified by the neighborhood size k.

  The function parameter information 144 holds information of parameters of the cross kernel function of each point. The description of the function parameter information 144 will be described later.

  The control unit 150 includes an acquisition unit 151, a whitening processing unit 152, a specifying unit 153, an inverse whitening processing unit 154, an execution unit 155, and an estimation unit 156. The control unit 150 corresponds to, for example, an integrated device such as an application specific integrated circuit (ASIC) or a field programmable gate array (FPGA). Further, the control unit 150 corresponds to, for example, an electronic circuit such as a central processing unit (CPU) or a micro processing unit (MPU).

  The acquisition unit 151 acquires the proximity size information 141 and the sample set information 142 from the communication unit 110 or the input unit 120. The acquisition unit 151 registers the proximity size information 151 and the sample set information 142 in the storage unit 140.

The whitening processing unit 152 is a processing unit that executes whitening processing on the sample set information 142. The whitening processing unit 152 outputs the whitened sample set information 142 to the specifying unit 153. In the following description, the sample set information 142 subjected to the whitening processing is referred to as a whitened sample set. Incidentally, whitening unit 152, the _X 1 _{to X 15} of the sample set information _142, by performing the whitening process and adjust the values for x 1 _{~x 15,} the value of y 1 _{~y 15} as it is I assume.

Here, an example of the whitening process will be described. The whitening processing unit 152 sets the sample set information 142 so that the average vector of x ₁ to x ₁₅ contained in the sample set information 142 is 0 and the covariance matrix of x _{1 to} x ₁₅ is an identity matrix. Adjust the values of x ₁ to x ₁₅ at each point.

An example of processing in which the whitening processing unit 152 sets the average vector of x ₁ to x ₁₅ included in the sample set information 142 to 0 will be described. The whitening processing unit 152 calculates the average vector of each of x _{1 to} x ₁₅ of the sample set information 142, and subtracts the value of the average vector from the value of the vector of each point, thereby the points included in the sample set information 142 Set the mean vector of 0 to 0.

An example of processing in which the whitening processing unit 152 adjusts the covariance matrix of the points included in the sample set information 142 to be a unit matrix will be described. The whitening processing unit 153 updates the values of each of x _{1 to} x ₁₅ by performing processing such as enlarging or reducing the interval on each of x _{1 to} x ₁₅ of the sample set information 142, and after updating The covariance matrix is calculated by the values of x _{1 to} x ₁₅ of. The whitening processing unit 152 calculates an interval of x _{1 to} x _{15 in which} the covariance matrix is an identity matrix.

  The whitening processing unit 152 specifies the whitening matrix “T” that satisfies Equation (1) by executing the above-described process of setting the mean vector to 0 and the process of setting the covariance matrix as the unit matrix. In equation (1), in equation (1), “S” represents a sample matrix in which vector values of points included in sample set information 142 are arranged, and “E [TS]” is a whitened sample “Cov [TS]” indicates the average vector of the matrix TS, “Cov [TS]” indicates the covariance matrix of the whitening sample matrix TS, and “I” indicates the unit matrix.

  E [TS] = 0, Cov [TS] = I (1)

For example, the whitening processing unit 152 updates the values of each of x _{1 to} x ₁₅ by multiplying each of x _{1 to} x ₁₅ included in the sample set information 142 by the whitening matrix T, and sets the whitened sample Generate The whitening processing unit 152 outputs the information on the generated whitened sample set to the specifying unit 153. In addition, the whitening processing unit 152 outputs the information of the whitening matrix T to the inverse whitening processing unit 154.

FIG. 4 is a diagram illustrating an example of a data structure of a whitening sample set. In the example shown in FIG. 4, the horizontal axis is an axis corresponding to the first principal component A ₁ of whitening unit 152 is output. The vertical axis is an axis corresponding to the output variable of the required time. The whitening unit 152 by being processed whitened, the position of each point _X 1 _{to X 15} as shown in FIG. 3 has moved to the position of each point _X 1 _{to X 15} as shown in FIG.

  The specifying unit 153 is a processing unit that specifies, for each point of the whitened sample set, a nearby point present in the vicinity of the point. The identifying unit 153 registers, in the k-neighborhood information 143, the relationship between a point and a neighboring point corresponding to this point. The identifying unit 153 causes the number of neighboring points to correspond to the size of the neighboring point size information 141.

  An example of the process of the identifying unit 153 will be described. FIG. 5 is a diagram for explaining the process of identifying nearby points. Here, as an example, the size of the neighboring point size information 141 is k. The identifying unit 153 calculates, for each point, the Yougrid distance from another point. Each arrow in FIG. 5 indicates the Yougrid distance. The identifying unit 153 sets a point from the closest to the k-th point in the u grid distance to the reference point as a point near the reference point.

For example, it is assumed that the reference point is a point X ₁ , and the points whose u grid distance with the point X ₁ is close are X ₂ , X ₃ , X ₄ , X ₅ ,. Further, the neighborhood size k is set to “3”. In this case, the specifying unit 153 sets the vicinity points of the point X ₁ as “X ₂ , X ₃ , X ₄ ”, and registers the relationship between the points and the vicinity points in the k-neighborhood information 143. The identifying unit 153 identifies nearby points similarly for the other points X _{2 to} X ₁₅ and registers them in the k-neighborhood information 143.

It returns to the explanation of FIG. The inverse whitening processing unit 154 is a processing unit that executes inverse whitening processing by multiplying each of x _{1 to} x ₁₅ included in the whitening sample set by the inverse matrix “T ⁻¹ ” of the whitening matrix. is there. Information obtained by inverse whitening the whitened sample set is referred to as sample set information. The reverse whitening processing unit 154 outputs the sample set information to the execution unit 155.

The execution unit 155 selects a point included in the sample set information and a nearby point corresponding to this point, and based on the selected point and the nearby point, takes the vector value of the selected point as an average, and based on the nearby point A process of identifying a kernel function whose shape is to be determined is executed for each point included in the sample set. The execution unit 155 specifies the regression function corresponding to the sample set by performing weighted averaging of the output value corresponding to each point included in the sample set with the value of the kernel function in the input value. The execution unit 155 registers information on parameters corresponding to the regression function in the function parameter information 144.

An example of processing of the execution unit 155 will be described. For example, the execution unit 155 specifies a regression function based on Expression (2). K _i included in equation (2) is a kernel function. The kernel function K _i is defined by equation (3).

In Expression (3), a value corresponding to x of the point X selected by the execution unit 155 is set as x _i . For example, execution unit 155 selects the point _{X 1,} when obtaining the kernel function of the point _{X 1,} the value of _{x 1} is set to _{x i.}

In equation (3), H _i is a variable that determines the shape of the kernel function of point X, and is defined by equation (4). The d _ij included in Equation 3 is defined by Equation (5). "Formula (7)" included in Formula (5) is defined by Formula (6).

The variables x _ij included in the equations (5) and (6) respectively correspond to x in each of the neighboring points of the point X selected by the execution unit 155. The execution unit 155 specifies the relationship between the point X and the nearby point based on the k-neighborhood information 143. For example, if a point near the point _{X 1} and _{X 2, X 3, X 4} , variable _{x ij} corresponding to the point _{X 1} is _{"x 12, x 13, x 14".} Expression (6) is for finding the average value of x of the neighboring points. Expression (5) is for finding the norm of the average value of the neighboring points and x of a certain neighboring point.

For example, the execution unit 155 selects a point included in the sample set information and a neighboring point corresponding to this point, and based on the selected point and the neighboring point, and the equations (3) to (6), sample set information The kernel functions K _{i of the} points included in are calculated respectively. The execution unit 155 registers, in the function parameter information 144, the relationship between the point of the sample set information and the kernel function corresponding to this point.

  After calculating the kernel function of each point of the sample set information, the execution unit 155 specifies the regression function by Equation (2). FIG. 6 is a diagram showing an example of the regression function according to the present embodiment. In FIG. 6, the horizontal axis is an axis corresponding to an input variable such as “time zone” as in FIG. 2. The vertical axis is an axis corresponding to an output variable such as "required time" in the same manner as in FIG. The graph of the regression function 10 corresponds to the weighted average of the output value corresponding to each point of the sample set by the value of the kernel function at the input value.

  When the estimation unit 156 acquires an input value from the communication unit 110 or the input unit 120, an output value corresponding to the input value, an output value corresponding to each point of sample set information, a value of the kernel function in the input value Is a processing unit that estimates by weighted averaging.

  For example, the estimation unit 156 refers to the function parameter information 144 and specifies a kernel function corresponding to each point of sample set information. Further, the estimation unit 156 specifies a regression function in which the output value corresponding to each point of the sample set information is weighted and averaged by the value of the kernel function in the input value based on Expression (1). The estimation unit 156 may obtain information on the regression function from the execution unit 155.

  When the estimation unit 156 receives an input value, the estimation unit 156 compares the input value with the regression function to estimate an output value. FIG. 7 is a diagram for explaining the process of the estimation unit. In FIG. 7, the horizontal axis is an axis corresponding to an input variable such as “time zone” as in FIG. 2. The vertical axis is an axis corresponding to an output variable such as "required time" in the same manner as in FIG. The graph of the regression function 10 corresponds to the weighted average of the output value corresponding to each point of the sample set by the value of the kernel function at the input value.

Estimation unit 156 acquires the input value _{B 1,} based on the regression function 10 and the input values _{B 1,} to estimate the output value _{B 2.} The estimation unit 156 outputs information of the estimated output value to the display unit 130. Alternatively, the estimation unit 156 transmits information of the estimated output value to an external device via the network.

  Next, an example of the processing procedure of the estimation apparatus 100 according to the present embodiment will be described. FIG. 8 is a flowchart showing the processing procedure of the estimation apparatus according to the present embodiment. As shown in FIG. 8, the acquisition unit 151 of the estimation device 100 acquires the proximity size information 141 (step S101). The acquisition unit 151 acquires the sample set information 142 (step S102).

  The whitening processing unit 152 of the estimation apparatus 100 executes the whitening process on the sample set information 142 (step S103). The identifying unit 153 of the estimation device 100 calculates the k neighborhood of each point included in the whitening sample set (step S104).

  The reverse whitening processing unit 154 of the estimation apparatus 100 performs reverse whitening processing on the whitened sample set (step S105). The execution unit 155 of the estimation device 100 calculates the parameters of each kernel function corresponding to each point of the sample set information (step S106).

  The estimation unit 156 of the estimation device 100 acquires an input value (step S107) and calculates a regression function (step S108). The estimation unit 156 specifies an output value based on the input value and the regression function, and outputs the output value (step S109).

  Next, the effects of the estimation apparatus 100 according to the present embodiment will be described. The estimation apparatus 100 averages the points included in the sample set, and executes a process of specifying a kernel function whose shape is to be determined based on the neighboring points of each point included in the sample set. Then, when acquiring the input value, the estimation apparatus 100 performs weighted averaging of the output value corresponding to the input value by the output value corresponding to each point included in the sample set with the value of the kernel function of the input value. presume. For this reason, it is possible to prevent an increase in the amount of calculation and to estimate the output value corresponding to the input value.

  For example, the estimating apparatus 100 specifies a regression function by weighted averaging the output values corresponding to each point included in the sample set with the value of the kernel function in the input value, and compares the input value with the regression function. Estimate the output value. The regression function according to the present embodiment uses, for example, the equation (2). However, in the equation (2), since the calculation for finding the optimum value is not performed, the amount of calculation can be reduced.

  In addition, when using the kernel function, the estimation apparatus 100 prevents the kernel function from being deviated from the sample set point and the accuracy of the regression function is degraded because the average of the kernel function is adjusted to the sample set point. be able to.

  Since the estimation apparatus 100 uses the neighborhood points of the points of the sample set when obtaining the kernel function of each point, the shape of the kernel function of the points of the sample set can be accurately obtained without performing the bandwidth adjustment and the like. It can be estimated and an appropriate regression function can be obtained.

  Next, an example of a computer that executes an estimation program that realizes the same function as that of the estimation device 100 described in the above embodiment will be described. FIG. 9 is a diagram illustrating an example of a computer that executes an estimation program.

  As shown in FIG. 9, the computer 200 has a CPU 201 that executes various arithmetic processing, an input device 202 that receives input of data from a user, and a display 203. The computer 200 further includes a reading device 204 that reads a program or the like from a storage medium, and an interface device 205 that exchanges data with other computers via a network. The computer 200 also has a RAM 206 for temporarily storing various information, and a hard disk drive 207. The devices 201 to 207 are connected to the bus 208.

  The hard disk drive 207 has a specific program 207a, an execution program 207b, and an estimation program 207c. The CPU 201 reads out the specific program 207 a, the execution program 207 b, and the estimation program 207 c and develops the same in the RAM 206. The specific program 207a functions as a specific process 206a. The execution program 207b functions as an execution process 206b. The estimation program 207c functions as an estimation process 206c.

  The specific program 207a, the execution program 207b, and the estimation program 207c may not necessarily be stored in the hard disk drive 207 from the beginning. For example, each program is stored in a "portable physical medium" such as a flexible disk (FD), a CD-ROM, a DVD disk, a magneto-optical disk, an IC card or the like inserted into the computer 200. Then, the computer 200 may read and execute the specific program 207a, the execution program 207b, and the estimation program 207c.

  The following appendices will be further disclosed regarding the embodiment including the above-described respective examples.

(Supplementary Note 1)
For each of a plurality of points included in the sample set, identify the neighboring points existing in the vicinity of each point,
A point included in the sample set and a neighboring point corresponding to the point are selected, and based on the selected point and the neighboring point, vector values of the selected point are averaged and the shape is determined based on the neighboring point Executing a process of identifying a kernel function to be performed on each point included in the sample set;
A process of estimating an output value corresponding to the input value by weighted averaging the output value corresponding to each point included in the sample set with the value of the kernel function of the input value when acquiring the input value An estimation program characterized by having it run.

(Supplementary Note 2) The process of estimating specifies a regression function by weighted averaging the output values corresponding to each point included in the sample set with the value of the kernel function in the input value, and specifies the regression function. The estimation program according to appendix 1, wherein the output value is estimated by comparing with a regression function.

(Supplementary Note 3) The estimation program according to Supplementary note 1 or 2, wherein the specifying process identifies neighboring points present in the vicinity of the point based on the k neighborhood.

(Supplementary Note 4) In the process of identifying the kernel function, a kernel function is assigned to a selected point, the average of the kernel function is matched to the vector value of the selected point, and the moment of the kernel function is a point near the selected point The estimation program according to any one of Appendices 1, 2 or 3, characterized in that a kernel function corresponding to the point is specified by adjusting to a moment of.

(Supplementary Note 5) A computer implemented estimation method,
A computer implemented estimation method,
For each of a plurality of points included in the sample set, identify the neighboring points existing in the vicinity of each point,
A point included in the sample set and a neighboring point corresponding to the point are selected, and based on the selected point and the neighboring point, vector values of the selected point are averaged and the shape is determined based on the neighboring point Executing a process of identifying a kernel function to be performed on each point included in the sample set;
A process of estimating an output value corresponding to the input value by weighted averaging the output value corresponding to each point included in the sample set with the value of the kernel function of the input value when acquiring the input value An estimation method characterized by performing.

(Supplementary Note 6) The process of estimating specifies a regression function by weighted averaging the output values corresponding to each point included in the sample set with the value of the kernel function in the input value, and specifies the regression function. The estimation method according to appendix 5, wherein the output value is estimated by comparing with a regression function.

(Supplementary Note 7) The estimation method according to Supplementary note 5 or 6, wherein the identifying process identifies neighboring points present in the vicinity of the point based on the k neighborhood.

(Supplementary Note 8) In the process of identifying the kernel function, a kernel function is assigned to a selected point, the average of the kernel function is matched to the vector value of the selected point, and the moment of the kernel function is a point near the selected point The estimation method according to any one of Appendices 5, 6 or 7, wherein a kernel function corresponding to the point is specified by adjusting to a moment of.

(Supplementary note 9) A specification unit that specifies neighboring points present in the vicinity of each point for a plurality of points included in the sample set,
A point included in the sample set and a neighboring point corresponding to the point are selected, and based on the selected point and the neighboring point, vector values of the selected point are averaged and the shape is determined based on the neighboring point An execution unit that executes a process of specifying a kernel function to be executed on each point included in the sample set;
An estimation unit that estimates an output value corresponding to the input value by weighted averaging the output value corresponding to each point included in the sample set with the value of the kernel function of the input value when the input value is acquired An estimation apparatus characterized by having and.

(Supplementary Note 10) The estimation unit specifies a regression function by performing weighted averaging of the output value corresponding to each point included in the sample set with the value of the kernel function in the input value, and the input value and the regression are identified. The estimation apparatus according to appendix 9, wherein the output value is estimated by comparing with a function.

(Supplementary note 11) The estimation apparatus according to supplementary note 9 or 10, wherein the identification unit identifies neighboring points present in the vicinity of the point based on the neighborhood of k.

(Supplementary Note 12) The execution unit assigns a kernel function to a selected point, matches the average of the kernel function to the vector value of the selected point, and matches the moment of the kernel function to the moment of a point near the selected point The estimation apparatus according to any one of appendices 9, 10, or 11, characterized in that a kernel function corresponding to the point is specified.

100 estimation device 153 identification unit 155 execution unit 156 estimation unit

Claims (3)

On the computer
At a plurality of points included in n (n is an integer of 2 or more) sample sets which indicate the relationship between an input variable including at least one attribute and an output variable corresponding to the input variable by points on multidimensional coordinates Then, whitening sample set information is generated in which the average vector of the plurality of points is 0 and the covariance matrix of the sample set is a unit matrix, and the whitening sample set information is multiplied by the whitening matrix Generate the whitening matrix whose matrix is an identity matrix,
Generating a plurality of whitened points from the plurality of points using the whitening matrix;
For each of the plurality of whitened points, the neighboring points that are the points from the closest point to the predetermined position based on the multidimensional coordinates are specified,
For each of the plurality of whitened points, the distance between each whitened point and the corresponding neighboring point is calculated, and a matrix in which values obtained from each distance are arranged is generated, and the acquired input value x And the kernel function K _i (x) which is a real-valued function that determines the similarity of the input variable x _i of the i-th point (i is an integer of 1 or more and n or less) included in the sample set was said matrix to execute the given Ru processing as the parameter,
A value obtained by substituting the acquired input value x into a regression function E shown in the following equation (1) using a plurality of kernel functions K _i (x) corresponding to the plurality of whitened points respectively A program for estimating a program according to the present invention, characterized in that the output value corresponding to the input value x .

Here, y _i represents an output variable of the _ith point (x _i, y _i ) included in the sample set, and K _i (x) is a kernel corresponding to the input variable x _i of the _ith point Indicates a function.
A computer implemented estimation method,
At a plurality of points included in n (n is an integer of 2 or more) sample sets which indicate the relationship between an input variable including at least one attribute and an output variable corresponding to the input variable by points on multidimensional coordinates Then, whitening sample set information is generated in which the average vector of the plurality of points is 0 and the covariance matrix of the sample set is a unit matrix, and the whitening sample set information is multiplied by the whitening matrix Generate the whitening matrix whose matrix is an identity matrix,
Generating a plurality of whitened points from the plurality of points using the whitening matrix;
For each of the plurality of whitened points, the neighboring points that are the points from the closest point to the predetermined position based on the multidimensional coordinates are specified,
For each of the plurality of whitened points, the distance between each whitened point and the corresponding neighboring point is calculated, and a matrix in which values obtained from each distance are arranged is generated, and the acquired input value x And the kernel function K _i (x) which is a real-valued function that determines the similarity of the input variable x _i of the i-th point (i is an integer of 1 or more and n or less) included in the sample set was said matrix running give Ru processing as the parameter,
A value obtained by substituting the acquired input value x into a regression function E shown in the following equation (1) using a plurality of kernel functions K _i (x) corresponding to the plurality of whitened points respectively As an output value corresponding to the input value x .

Here, y _i represents an output variable of the _ith point (x _i, y _i ) included in the sample set, and K _i (x) is a kernel corresponding to the input variable x _i of the _ith point Indicates a function.
At a plurality of points included in n (n is an integer of 2 or more) sample sets which indicate the relationship between an input variable including at least one attribute and an output variable corresponding to the input variable by points on multidimensional coordinates Then, whitening sample set information is generated in which the average vector of the plurality of points is 0 and the covariance matrix of the sample set is a unit matrix, and the whitening sample set information is multiplied by the whitening matrix The whitening matrix is a unitary matrix, and the whitening matrix is used to generate a plurality of whitened points from the plurality of points, and for each of the plurality of whitened points, An identifying unit that identifies neighboring points that are points from the closest point to a predetermined number from the point based on the multidimensional coordinates ;
For each of the plurality of whitened points, the distance between each whitened point and the corresponding neighboring point is calculated, and a matrix in which values obtained from each distance are arranged is generated, and the acquired input value x And the kernel function K _i (x) which is a real-valued function that determines the similarity of the input variable x _i of the i-th point (i is an integer of 1 or more and n or less) included in the sample set was said matrix has an execution unit for executing given Ru processing as the parameter,
A value obtained by substituting the acquired input value x into a regression function E shown in the following equation (1) using a plurality of kernel functions K _i (x) corresponding to the plurality of whitened points respectively An estimation apparatus characterized in that the output value corresponding to the input value x .

Here, y _i represents an output variable of the _ith point (x _i, y _i ) included in the sample set, and K _i (x) is a kernel corresponding to the input variable x _i of the _ith point Indicates a function.

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