Information processing device, information processing method, and program

Abstract

An information processing device according to one aspect of this invention comprises a processor and a storage unit. The storage unit comprises a first storage area and a first storage area. The first storage area stores event occurrence data about a location of occurrence of an event to be analyzed. A second storage area stores covariate data observed in an observation area of the event. The processor has a kernel function designation unit, a calculation method designation unit, and an intensity function estimation unit. The kernel function designation unit accepts designation of a kernel function in a Gaussian process. The calculation method designation unit accepts designation of a calculation method of an equivalent kernel function. The intensity function estimation unit calculates an equivalent kernel function on the basis of the specified kernel function and calculation method, and estimates an intensity function with respect to the covariate using the calculated equivalent kernel function.

Description

[Technical Field]

[0001] One aspect of the present invention relates to an information processing device, an information processing method, and a program for estimating the occurrence probability (intensity function) of an event with respect to a covariate based on data relating to the event occurrence position and the covariate. [Background technology]

[0002] Consider a situation in which a point event (hereinafter referred to as an event) occurs probabilistically in a space where covariates are defined at an arbitrary point. This situation can be expressed, for example, as (space, covariates, observed data) = (latitude / longitude, crowd density, location of accident event occurrence). A technique using kernel density estimation is known as a technique for estimating the probability of an event occurring with respect to covariates (also called an intensity function) (see, for example, Non-Patent Document 1).

[0003] Recently, methods that can achieve higher accuracy than kernel density estimation have been reported. For example, Bayesian estimation, which uses a Gaussian process as a prior distribution, is known to achieve higher accuracy than kernel density estimation (see, for example, Non-Patent Documents 2 and 3). [Prior art documents] [Non-patent literature]

[0004] [Non-Patent Document 1] Baddeley et al., "Nonparametric estimation of the dependence of a spatial point process on spatial covariates", Statistics and Its Interface, 5, pp.221-236, 2012. [Non-patent document 2] Lloyd et al., "Variational inference for Gaussian process modulated Poisson processes", ICML2015, 2015. [Non-patent document 3] Donner et al., "Efficient Bayesian inference for a Gaussian process density model", UAI, 1, pp.53-62, 2018. Summary of the Invention [Problem to be solved by the invention]

[0005] Although there are known techniques that use kernel density estimation to estimate the intensity function for covariates, there are no known techniques that can use Bayesian estimation with a Gaussian process prior, which is expected to be more accurate than kernel density estimation.

[0006] This invention has been made in light of the above circumstances, and its purpose is to provide a technique that makes it possible to estimate an intensity function for a covariate using a Bayesian estimation method that uses a Gaussian process as a prior distribution. [Means for solving the problem]

[0007] An information processing device according to one aspect of the present invention includes a processor and a storage unit. The storage unit includes a first storage area and a second storage area. The first storage area stores event occurrence data related to the occurrence position of an event to be analyzed. The second storage area stores covariate data observed within the observation area of ​​the event. The processor includes a kernel function designation unit, a calculation method designation unit, and an intensity function estimation unit. The kernel function designation unit accepts designation of a kernel function in a Gaussian process. The calculation method designation unit accepts designation of a calculation method for an equivalent kernel function. The intensity function estimation unit calculates an equivalent kernel function based on the designated kernel function and calculation method, and estimates an intensity function for a covariate using the calculated equivalent kernel function. [Effects of the Invention]

[0008] According to one aspect of the present invention, it is possible to provide a technique that can estimate an intensity function for a covariate based on a Bayesian estimation method that uses a Gaussian process as a prior distribution. [Brief explanation of the drawings]

[0009] [Figure 1] FIG. 1 is a functional block diagram illustrating an example of an information processing apparatus according to an embodiment. [Figure 2] FIG. 2 is a functional block diagram showing an example of the information processing device 1 shown in FIG. [Figure 3] FIG. 3 is a flowchart showing an example of a processing procedure of the information processing device 1 shown in FIG. DETAILED DESCRIPTION OF THE INVENTION

[0010] Hereinafter, an embodiment of the present invention will be described with reference to the drawings. <Configuration> FIG. 1 is a functional block diagram illustrating an example of an information processing apparatus according to an embodiment. The information processing device 1 is a computer including a processor and a memory. The information processing device 1 includes a processor 11, an input / output interface 12, and a storage unit 13. The processor 11, the input / output interface 12, and the storage unit 13 are connected to each other via a bus so as to be able to communicate with each other. The processor 11 controls the information processing device 1. The processor 11 is an arithmetic processing device such as a CPU (Central Processing Unit) or an MPU (Micro Processing Unit).

[0011] The input / output interface 12 is an interface that enables transmission and reception of information between the input device 2 and the output device 3. The input / output interface 12 may include a wired or wireless communication interface. That is, the information processing device 1, the input device 2, and the output device 3 may transmit and receive information via a network such as a LAN or the Internet.

[0012] The storage unit 13 is a storage medium. The storage unit 13 is configured by combining a nonvolatile memory that can be written to and read from at any time, such as a hard disk drive (HDD) or a solid state drive (SSD), a nonvolatile memory such as a read-only memory (ROM), and a volatile memory such as a random access memory (RAM). The storage unit 13 has a storage area including a program storage area and a data storage area. The program storage area stores an operating system (OS), middleware, and application programs required to execute various processes.

[0013] The input device 2 includes, for example, a keyboard, a pointing device, etc., which are used by the owner of the information processing device 1 (for example, an assignor, an administrator, or a supervisor) to input instructions to the information processing device 1. The input device 2 may also include a reader for reading data to be stored in the storage unit 13 from a memory medium such as a USB memory, or a disk device for reading such data from a disk medium. Furthermore, the input device 2 may include an image scanner.

[0014] The output device 3 includes a display that displays output data to be presented to the owner from the information processing device 1, a printer that prints the output data, etc. The output device 3 may also include a writer that writes data to be input to another information processing device 1 such as a PC or a smartphone onto a memory medium such as a USB memory, and a disk device that writes such data onto a disk medium.

[0015] Fig. 2 is a functional block diagram showing an example of the information processing device 1 shown in Fig. 1. In Fig. 2, a storage unit 13 stores a program 10 that causes a processor 11 to function as the information processing device 1. Furthermore, the storage unit 13 includes a first storage area 131, a second storage area 132, and a third storage area 133. The first storage area 131 stores event occurrence data 100. The event occurrence data 100 is data relating to the occurrence locations of events to be analyzed, and includes at least the number of observed events, a sequence of event locations, and an observation area. The second storage area 132 stores the covariate data 101 observed within the observation area of ​​the event to be analyzed. The third storage area 133 stores the intensity function distribution 105 calculated by the processor 11.

[0016] The processor 11 includes, as processing functions according to the embodiment, a kernel function designation unit 102, a calculation method designation unit 103, an intensity function estimation unit 112, and an output control unit 114. The kernel function designation unit 102, the calculation method designation unit 103, the intensity function estimation unit 112, and the output control unit 114 are functional processes realized by the arithmetic processing of the processor 11 based on the program 10.

[0017] The kernel function designation unit 102 receives a designation of a kernel function in a Gaussian process. The kernel function is designated by the user by operating the input device 2, for example. The calculation method designation unit 103 accepts designation of a calculation method for the equivalent Kernel function. The calculation method may also be designated by the user operating the input device 2, for example.

[0018] The intensity function estimation unit 112 calculates an equivalent Kernel function based on the specified Kernel function and calculation method. Furthermore, the intensity function estimation unit 112 estimates an intensity function for the covariate using the calculated equivalent Kernel function. The intensity function distribution 105 is stored in the third storage area 133.

[0019] The output control unit 114 outputs the intensity function distribution 105 to the output device 3 via the input / output interface 12. The output device 3 visualizes and displays the calculated intensity function distribution 105 on a display, for example.

[0020] Next, the operation of the above configuration will be described. <effect> (overview) First, an outline of the operation will be described. In the embodiment, the processor 11 mainly executes the processes (1) to (4) to realize estimation of the intensity function for the covariate based on the Bayesian estimation method with a Gaussian process as the prior distribution.

[0021] (1) For variables that follow a Gaussian process defined in covariate space, the square of the variable is defined as the intensity function. In this way, the estimate of the square root of the intensity function that maximizes the posterior probability (maximum a posteriori probability estimate or MAP estimate) is given as the solution to a system of equations with N unknowns, where N is the number of observed data. This can be expressed as the representative theorem holding. This fact makes it easy to numerically solve the estimate of the square root of the intensity function.

[0022] (2) The estimation error of the square root of the intensity function is calculated using Laplace approximation. That is, the Hessian matrix of the MAP estimate of the logarithmic posterior probability distribution to which the square root of the intensity function follows is calculated. The inverse matrix of this Hessian matrix is ​​then multiplied by -1 to obtain the covariance matrix of the estimate of the square root of the intensity function.

[0023] (3) Under the Laplace approximation in (2), we obtain a gamma distribution that the estimated value of the intensity function follows. Obtaining a probability distribution for this estimated value is the final goal of intensity function estimation. (4) The hyperparameters required for estimating the intensity function are estimated from observed data using the empirical Bayes method. The empirical Bayes method is a method that estimates the hyperparameters that maximize the marginal likelihood. A typical example of a hyperparameter is the parameter of the kernel function in a Gaussian process.

[0024] Fig. 3 is a flowchart showing an example of a processing procedure of the information processing device 1 shown in Fig. 1. In Fig. 3, the processor 11 accepts a user's specification of a kernel function in a Gaussian process (step SST21). Next, the processor 11 accepts a user's specification of a calculation method for an equivalent kernel function (step SST22).

[0025] Next, the processor 11 calculates an equivalent Kernel function based on the specified Kernel function and calculation method (step ST23). Furthermore, the processor 11 estimates an intensity function for the covariate using the calculated equivalent Kernel function (step S24).

[0026] (detail) Next, the details of the operation will be explained with reference to mathematical formulas.

[0027] [About event occurrence data] Data on the occurrence location of the event to be analyzed is given as input. The event occurrence data includes the following (A), (B), and (C).

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[0028] [About covariate data] Covariate data observed within the observation domain T (equation (C)) of an event are given as a function (equation (E)) that takes any point (equation (D)) within the observation domain (equation (C)) as input and outputs the covariate. In many applications, information about the covariate is only available at a finite number of points within the observation domain (equation (C)). In such cases, it is assumed that the function (equation (E)) is constructed using a regression model or an interpolation technique such as kriging.

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[0029] Regarding [Specifying the kernel function in Gaussian processes] To use a Gaussian process, you specify a kernel function, which determines the smoothness of the function you are modeling. You also specify the values ​​of the parameters (hyperparameters) included in the function. In the embodiment, the function to be modeled is an intensity function for the covariates. The kernel function for any two points (equation (F)) in the covariate space is expressed as (equation (G)).

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[0030] An example of the kernel function is the Gaussian kernel given by equation (1).

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[0031] Alternatively, an example of the kernel function is the kernel of equation (2) which is expressed as an inner product of a finite-dimensional feature mapping vector (equation (H)).

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[0032] About [Specifying the calculation method for the equivalent kernel function] A method for calculating the equivalent kernel function is given as input, which includes the type of calculation method and the number of points for Monte Carlo integration (Equation (I)).

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[0033] The options for the type of calculation method are type 1 and type 2, and type 2 can be selected only when the kernel function is given by the inner product of finite-dimensional feature mapping vectors.

[0034] About [Estimation of Intensity Function] Based on the information given above, the processor 11 calculates the equivalent kernel function (equation (J)).

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[0035] Then, the processor 11 uses the equivalent kernel function (J) to estimate the intensity function (equation (K)) for the covariates.

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[0036] First, the equivalent kernel function (equation (J)) is defined as the solution of the integral equation of equation (3).

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[0037] In preparation for numerically solving the above integral equation, the integral part is approximated by Monte Carlo integration to obtain equation (4).

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[0038] where (equation (L)) is the covariate on the m-th sample point.

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[0039] When type 1 is specified as the type of calculation method, the equivalent kernel function (equation (N)) is obtained as shown in equation (5) by solving equation (4) as a matrix equation for the column vector function (equation (M)).

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[0040] When type 2 is specified as the type of calculation method, the equivalent kernel function (equation (O)) is obtained from equation (4) in the form of equation (6), assuming that the kernel function is given by the inner product of finite-dimensional feature mapping vectors as in equation (2).

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[0041] Next, the equivalent kernel function (Equation (J)) is used to calculate the MAP estimate of the square root of the intensity function using Equation (7).

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[0042] However, (Equation (P)) can be obtained by solving the following simultaneous equations (8).

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[0043] Next, under the Laplace approximation, we assume that the square root of the intensity function follows a normal distribution with the MAP estimated value as the mean, and calculate its covariance matrix (equation (Q)) using equation (9).

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[0044] However, the number (10) holds.

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[0045] Finally, the probability distribution of the intensity function estimate at each covariate value (equation (R)) is calculated as a gamma distribution whose scale parameter (equation (S)) and shape parameter (equation (T)) are given by equation (11).

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[0046] In the process of [Estimating the Intensity Function], the validity of the hyperparameters specified in [Specifying the Kernel Function in the Gaussian Process] can also be evaluated based on the marginal likelihood function. When optimizing the hyperparameters, the hyperparameters that maximize the marginal likelihood function are searched for, and equation (11) is recalculated using the values.

[0047] Regarding the estimated intensity function distribution The probability distribution of the intensity function calculated in [Estimation of Intensity Function] is output. The output is a function that outputs a gamma distribution value with the scale and shape parameters given by equation (11) for any covariate value (equation (R)).

[0048] <Effects> As described above, according to the embodiment, it is possible to estimate an intensity function for a covariate using a Bayesian estimation method that uses a Gaussian process as a prior distribution. It should be noted that the present invention is not limited to the above-described embodiment. For example, the selection of the kernel function is not limited to equation (1) or equation (2). Furthermore, in the implementation stage, this invention can be embodied by modifying the components within the scope of the embodiments. Furthermore, various inventions can be formed by appropriately combining multiple components disclosed in the above embodiments. For example, some components may be deleted from all the components shown in the embodiments. Furthermore, components across different embodiments may be appropriately combined. [Explanation of symbols]

[0049] 1...Information processing device 2...Input device 3...Output device 10. Program 11...Processor 12...Input / output interface 13...Storage section 131...1st storage area 132…Second storage area 133…Third storage area 100...Event occurrence data 101...Covariate data 102...Kernel function specification section 103...Calculation method specification section 105...Intensity function distribution 112...Intensity function estimation unit 114...Output control section.

Claims

1. In an information processing device including a processor and a storage unit, The storage unit a first storage area for storing event occurrence data relating to the occurrence position of an event to be analyzed; a second storage area for storing covariate data observed within the observation domain of the event; The processor: a kernel function specification unit that accepts specification of a kernel function in a Gaussian process; a calculation method designation unit that accepts designation of a calculation method for the equivalent Kernel function; an intensity function estimation unit that calculates the equivalent Kernel function based on the specified Kernel function and calculation method, and estimates an intensity function for a covariate using the calculated equivalent Kernel function.

2. The information processing apparatus according to claim 1 , wherein the event occurrence data includes at least the number of observed events, a sequence of event positions, and an observation area.

3. The information processing device according to claim 1 , wherein the kernel function specifying unit further accepts specification of hyperparameter values ​​of the kernel function.

4. The information processing apparatus according to claim 1 , wherein the calculation method designation unit accepts designation of at least a type of calculation method for the equivalent Kernel function and a designation of the number of points for Monte Carlo integration.

5. 1. An information processing method for an information processing device including a memory unit that stores event occurrence data relating to an occurrence position of an event to be analyzed and covariate data observed within an observation area of ​​the event, and a processor, a step of receiving, by the processor, a specification of a kernel function in a Gaussian process; a step of receiving, by the processor, a designation of a method for calculating an equivalent kernel function; the processor calculating the equivalent kernel function based on the specified kernel function and calculation method; and a step in which the processor estimates an intensity function for a covariate using the calculated equivalent kernel function.

6. A program that causes a computer to function as each of the units of the information processing device according to claim 1 .

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