JP6397380B2 - Spatio-temporal variable prediction apparatus and program - Google Patents

Description

  The present invention relates to a spatiotemporal variable prediction apparatus and program.

As a conventional technique, there is a method of estimating or predicting an unobserved spatiotemporal variable from time series data of a spatiotemporal variable using a kriging model. Now, it is assumed that N observation values {t ₁ ,..., T _N } are obtained at N locations {x ₁ ,..., X _N } in a space having a geographical and temporal spread. In the kriging model, the predicted value t _* at the new observation point x _* is output as a probability distribution.

  The Kriging model is a model based on the assumption that the closer the input variables (position, time, etc.) are, the closer the values are. The definition of “similar” can be arbitrarily determined according to the problem setting, and is defined by functions called semivariogram and covariogram. Examples of typical semivariograms used for various modeling include a nugget effect model, a spherical model, an exponential model, and a Gaussian model. As preprocessing for prediction, selection of semivariogram and covariogram and parameter estimation are performed using all data. For model selection and parameter estimation, an “empirical variogram” is used, which is a linear sum of differences between spatiotemporal variable pairs of all data.

  A semivariogram model is fitted to an empirical variogram designed from data points according to the above definition, and model selection and parameter estimation are performed using a weighted least squares method. Simple kriging, universal kriging, block kriging and the like have been proposed as kriging models used for prediction. The model used for prediction can be arbitrarily determined according to the problem setting. All these models are based on the assumption that the random field of the spatiotemporal variable has intrinsic stationarity or stationarity similar to intrinsic stationarity.

Inherent stationarity means that (1) the mean value of spatiotemporal variables is constant regardless of input variables (time zone and region), and (2) the variance of any spatiotemporal variable pair is It is a property that depends only on similarity. Simple kriging is a model that assumes intrinsic stationarity in the random field of the spatiotemporal variable, and the predicted value t _* of the spatiotemporal variable in the new input variable {x _* } follows a Gaussian distribution having the following average and variance.

Here, γ is a covariogram, and C is an N × N covariance matrix having an element γ (x _n , x _m ) + β ⁻¹ δ _nm . Where δ _nm is the Kronecker delta and β is a constant. Simple kriging can be regarded as a kind of regression by Gaussian process. In the Gaussian process, kernel functions are used instead of covariograms. Stationarity condition (2) corresponds to the assumption that the definition of “similar” is uniquely determined.

As mentioned above, Kriging model (or regression by Gaussian process) predicts the value of spatiotemporal variables based on the assumption that “the definition of“ similar ”is uniquely determined”. The influence of spatiotemporal variables from “similar” input variables should differ depending on the characteristics of the input variables such as region and time zone. Therefore, a mixed Gaussian process was proposed that relaxed the stationary process of the Kriging model and introduced local unsteadiness. The mixed Gaussian process is a model in which a plurality of single Gaussian processes are mixed, and is expressed by superposition of a plurality of Gaussian processes. In the mixed Gaussian process, the predicted value t _* in the new input variable x _* follows a Gaussian distribution having the mean and variance of the following equation.

p (z _* = r | x _* ) is a probability that t _* is generated from the r-th element, and is called a burden rate. Data is classified into a plurality of clusters according to the characteristics (region, time zone, etc.) of each input variable, and a semivariogram that best describes the data is selected and parameters are estimated for each cluster. The cluster to which each data point belongs and the parameters of each cluster are estimated simultaneously using the EM algorithm (see Non-Patent Document 1 and Non-Patent Document 2).

S. De Iaco, D.E.Myers, and D. Posa. “Space-time analysis using a general product-sum model.”, Statistics & Probability Letters, 52 (1): p.21-28, 2001. Benedikt Gr¨aler, Lydia E. Gerharz, and Edzer J. Pebesma. “Spatio-temporal analysis and interpolation of PM10 measurements in Europe.”, Technical report, ETC / ACM, 2012.

  As described above, the mixed Gaussian process of the prior art classifies input data into clusters without taking into account the differences in dimensions of input variables and the properties of individual input variables. In practice, however, clusters of spatiotemporal variables should have temporal and spatial correlations. In the prior art, there has been a problem that the real-time spatiotemporal variable distribution with temporal and spatial correlation cannot be accurately predicted.

  The present invention has been made in view of the above points, and an object of the present invention is to provide a spatiotemporal variable prediction apparatus and program capable of accurately predicting values of spatiotemporal variables having temporal and spatial correlations. And

  In order to achieve the above object, the spatiotemporal variable prediction apparatus according to the present invention is based on an observation data set having observation values of spatiotemporal variables with respect to an input variable having position information and temporal information. And a spatio-temporal variable prediction apparatus for predicting a value of a spatio-temporal variable with respect to temporal information, wherein a plurality of Gaussian processes corresponding to a spatial spread and a temporal spread are based on the set of observation data. Similarity between the observation data for each of the plurality of Gaussian processes included in the model for predicting the value of the spatio-temporal variable with respect to the input variable modeled by the hierarchical mixed Gaussian process mixed in the hierarchy A parameter representing the contribution of each of the plurality of Gaussian processes to each of the observed data. It is configured to include a learning unit for learning the load ratio and a motor.

  Further, in the spatiotemporal variable prediction apparatus according to the present invention, the learning unit, for each of the observation data, based on the set of the observation data and each hyperparameter of the kernel function of the plurality of Gaussian processes, Estimating a unit burden rate, which is a parameter representing the contribution of each of a plurality of units consisting of a plurality of Gaussian processes, and a burden factor, a parameter representing the contribution of each of the plurality of Gaussian processes, to each of the observation data A burden rate estimating unit, a set of the observation data, and each of the observation data estimated by the burden rate estimation unit, for each of the unit burden rates of the plurality of units, and for each of the observation data, Based on the burden ratio of each of the plurality of Gaussian processes, the Gaussian process is determined for each of the plurality of Gaussian processes. A Gaussian process parameter estimation unit for estimating each hyperparameter of the channel function, and an iterative determination unit that repeats the estimation by the burden factor estimation unit and the estimation by the Gaussian process parameter estimation unit until a predetermined iteration end condition is satisfied. Can be included.

  In addition, the spatiotemporal variable prediction apparatus according to the present invention is a parameter that represents a contribution degree of each of the plurality of Gaussian processes to the input variable based on the input variable having unobserved position information and time information input thereto. Each of the hyperparameters of the kernel function for each of the plurality of Gaussian processes, and each of the plurality of Gaussian processes for the estimated input variable, learned by the learning unit. A spatiotemporal variable calculation unit that predicts a value of the spatiotemporal variable with respect to the input variable based on the burden rate may be further included.

  Moreover, the program of this invention is a program for functioning a computer as each part which comprises said space-time variable prediction apparatus.

  As described above, according to the spatiotemporal variable prediction apparatus and program of the present invention, a plurality of Gaussian processes based on a set of observation data having observation values of spatiotemporal variables with respect to input variables having position information and temporal information. Are included in a model for predicting the value of a spatio-temporal variable with respect to an input variable, modeled by a hierarchical mixed Gaussian process mixed at multiple hierarchies corresponding to spatial and temporal spread. For each Gaussian process, each hyperparameter of the kernel function, which is a function that defines the similarity between the observed data, and a burden ratio that is a parameter that represents the contribution of each of the multiple Gaussian processes to each of the observed data Can be used to accurately predict the values of spatio-temporal variables having temporal and spatial correlations. .

It is a block diagram of the spatiotemporal variable prediction apparatus in an embodiment of the present invention. It is a figure which shows an example of the collection of observation data. It is a figure which shows the structural example of an input part and an output part. It is a flowchart which shows the learning process routine of the spatiotemporal variable prediction apparatus in embodiment of this invention. It is a flowchart which shows the spatiotemporal variable calculation processing routine of the spatiotemporal variable prediction apparatus in embodiment of this invention.

  Hereinafter, embodiments of the present invention will be described in detail with reference to the drawings.

<Overview>

The embodiment of the present invention proposes a hierarchical mixed Gaussian process that is an extension of the mixed Gaussian process. The hierarchical mixed Gaussian process is a model in which Gaussian processes are mixed hierarchically. In a hierarchical mixed Gaussian process, the predicted value t _{* at} the new input variable x _* follows a Gaussian distribution with the mean of:

p (z _* ′ = r ′ | z _* = r, x _* ) is a probability that t _* is generated from the r-th element of the r′-th unit, and is called a burden rate. z and z ′ are latent variables corresponding to each cluster of input variables. The cluster to which each data point belongs and the parameters of each cluster are estimated simultaneously using the EM algorithm.

  The spatio-temporal variable prediction device according to the embodiment of the present invention uses time-series data of spatio-temporal variables (population distribution, speed / direction of human flow / traffic flow, reserves of mineral resources such as gold and diamond, and weather such as precipitation) Data, land prices, etc.) and can be applied flexibly according to observation data. Hereinafter, as an embodiment, a case will be described in which an unobserved spot or a future spatiotemporal variable distribution is estimated / predicted under the condition that time series data of population density distribution is given as observation data.

<Configuration of Spatiotemporal Variable Prediction Device According to Embodiment of the Present Invention>

  Next, the configuration of the spatiotemporal variable prediction apparatus according to the embodiment of the present invention will be described. As shown in FIG. 1, a spatiotemporal variable prediction apparatus 100 according to an embodiment of the present invention includes a CPU, a RAM, and a ROM that stores programs and various data for executing processing routines described later. It can be configured with a computer including. The spatiotemporal variable prediction apparatus 100 calculates the value of the spatiotemporal variable in the unobserved position information and the temporal information based on the set of observation data having the observed value of the spatiotemporal variable with respect to the input variable having the positional information and the temporal information. Predict. As shown in FIG. 1, the spatiotemporal variable prediction device 100 functionally includes an operation unit 10, a population density information storage unit 12, an input unit 13, a calculation unit 14, and a spatiotemporal variable calculation unit 26. And an output unit 28. The operation unit 10 and the calculation unit 14 are connected to the population density information storage unit 12.

  The operation unit 10 accepts various operations from the user for data stored in a population density information storage unit 12 described later. The various operations include operations for registering, correcting, and deleting information stored in the population density information storage unit 12. The input means of the operation unit 10 may be anything such as a keyboard, mouse, menu screen, or touch panel. The operation unit 10 can be realized by a device driver of an input unit such as a mouse or control software for a menu screen.

  The population density information storage unit 12 stores a set of observation data.

  The population density information storage unit 12 stores a set of observation data to be analyzed by the calculation unit 14 to be described later. In response to a request from the calculation unit 14, the observation data set is read and the information is transmitted to the calculation unit 14. To do.

The set of observation data stored in the population density information storage unit 12 is a set of combinations of the input variable x and the spatiotemporal variable t. In the embodiment of the present invention, the input variable x has a position and time, and the spatiotemporal variable t is population density. The population density at a certain position and at a certain time can be expressed as {x _i , t _i }. The population density information storage unit 12 is a Web server that holds Web pages, a database server that includes a database, or the like.

  FIG. 2 shows an example of a set of observation data. As shown in FIG. 2, for example, information representing a position ID and time is stored as an input variable x, and information representing a population density is stored as a spatiotemporal variable t.

  The input unit 13 accepts input variables having unobserved position information and time information.

  In the embodiment of the present invention, the population density, which is a spatio-temporal variable, is predicted based on each parameter obtained by the calculation unit 14 described later with respect to the position information and time information received by the input unit 13. Is called.

  The input means of the input unit 13 may be anything such as a keyboard, mouse, menu screen, or touch panel. The input unit 13 can be realized by a device driver of input means such as a mouse or control software for a menu screen.

  FIG. 3 shows a configuration example of the input unit 13 in the present embodiment. The configuration example of FIG. 3 shows a case where the input unit 13 and an output unit 28 described later are configured as one screen.

  As illustrated in FIG. 3, the input unit 13 receives an input variable including position information regarding a point or region to be predicted and time information for prediction. Further, the output unit 28 displays the value of the spatiotemporal variable for the input variable output by the spatiotemporal variable calculation unit 26 described later as a prediction result.

  The calculation unit 14 includes a learning unit 16, a burden rate parameter storage unit 22, and a Gaussian process parameter storage unit 24. The computing unit 14 calculates, as learning data, a hyper parameter of a Gaussian process and a parameter representing a burden rate from each Gaussian process.

  Here, the principle in the embodiment of the present invention will be described.

Assume that the population density information storage unit 12 stores a set of observation data D = {x _i , t _i } ^N _{i = 1} , which is spatiotemporal variable data. Here, x _i is a vector including a plurality of variables such as position and time, and t _i is a population density.

The problems to be solved here include (1) prediction of population density value t _{* in} a new input variable (unobserved position and time) x _* accepted by the input unit 13, and (2) unobserved points. A prediction of the value of population density in an area.

In the present embodiment, description will be given focusing on the problem (1). In order to obtain the prediction surface of the population density distribution, problem (1) should be solved repeatedly while moving x _* .

Assume that the model followed by the predicted value t _* of the spatiotemporal variable can be described by a mixed model of R Gaussian processes. Based on this assumption, the expected value of the value t _* corresponding to the new input variable x _* can be written down as in the following equation (1).

Here, X is a set of input variables x (X = {x _i } ^N _{i = 1} ), Θ = {θ _{rr ′} } ^{R, R ′} _{r, r ′ = 1} is the r-th unit of the r′-th unit. The hyperparameters z and z ′ of the Gaussian process are latent variables corresponding to classes classified according to the characteristics of the input variables (region, time zone, etc.). Assuming that x _* originates from the r-th Gaussian process, the predicted value t _* follows a Gaussian distribution with mean and variance of the following equation.

Here, k is a kernel function, k ^{rr ′} is a vector having an element k ^{rr ′} (x _* , X), and C ^{rr ′} is an element k ^{rr ′} (x _n , x _m ) + β ⁻ 1δ _nm + Ψ ^{rr ′} _nm It is an N × N covariance matrix. Also, t is a vector of the element and _{t i.} However, Ψ ^{rr ′} is a diagonal matrix defined by the following equation.

Here, σ _n is a constant. When the i-th data point does not receive the contribution of the r-th Gaussian process, that is, when the burden factor is p (z _i ′ = r ′, z _i = r | x _i ), C ^{rr ′} diverges infinitely. , (C ^{rr ′} ) ⁻¹ has (i, i) components of zero. That is, t _i does not affect the parameter estimation of the r th Gaussian process of the r ′ th unit.

In the conventional method, the kernel function k (x _n , x _m ) is uniquely determined at the preprocessing stage of prediction. On the other hand, in the embodiment of the present invention, kernel functions are freely designed, and parameters and weights of each kernel function are automatically estimated. In the embodiment of the present invention, two typical covariograms generally used in time-series geographic statistical analysis such as weather data and environmental data are given.

If the number of dimensions of the input variable _x is nx,

It is. The kernel function corresponding to the above covariogram can be written as:

  In the embodiment of the present invention, a new kernel function defined by the following equation defined by the linear sum of the above kernel functions is used.

  The hyperparameter of the Gaussian process to be estimated here is

It is. A set of observation data under the condition that a set of input variables X = {x _i } ^N _{i = 1} and a set of hyperparameters of Gaussian processes Θ = (θ ¹¹ ,..., Θ ^{RR ′} ) The likelihood function of D can be written as

Here, t _\ i represents a value obtained by removing the i-th element from t.

  In the embodiment of the present invention, an EM algorithm is used to estimate parameters Θ, z, z ′ that maximize the likelihood function of the above equation. In the EM algorithm, parameter estimation is performed by the following procedures [1] to [4]. In the embodiment of the present invention, the Softmax function shown in the following equations (4) to (5) is adopted as the prior distribution of the burden rate, and is used for prediction of the spatiotemporal variable by the spatiotemporal variable calculation unit 26 described later. The parameter of the Softmax function to be calculated is calculated.

[1] Select an initial value for each parameter.

[2] In the E step of the EM algorithm, for each observation data in the observation data set, the load ratio p (z of the r'th Gaussian process of the r'th unit for the input variable of the observation data according to the following equation: _i = r | z _i ′ = r ′, X, t, θ ^{rr ′} ), and the burden ratio p (z _i = r | X, t, θ ^{rr ′} ) of the r-th Gaussian process for the input variable of the observed data. ), The joint probability p (z _i = r, z _i ′ = r ′ | X, t, θ ^{rr ′} ) of the r th Gaussian process of the r ′ th unit is calculated for the input variable of the observed data.

Note that p (t _i | X, t, θ ^{rr ′} ) in the above equation (2) is expressed by a Gaussian distribution having mean μ _i ^{rr ′} and variance Σ _i ^{rr ′} shown in the following equation.

[3] The burden ratio p (z _i = r | z _i '= r of the r'th Gaussian process of the r'th unit calculated for each observation data in the observation data set in the M step of the EM algorithm. ', X, t, θ ^rr' ), the burden factor p (z _i = r | X, t, θ ^{rr '} ) of the r th Gaussian process, the joint probability p of the r th Gaussian process of the r ′ th unit Based on (z _i = r, z _i ′ = r ′ | X, t, θ ^{rr ′} ), the parameter Θ ^new of the Gaussian process that maximizes the likelihood function Q is calculated. Here, the likelihood function Q is preliminarily approximated by Q ^~ represented by the following equation.

Where π _i ^{rr ′} is

Defined by

Also, the derivative of the objective function Q ^to the ^{hyperparameter} θ ^{rr ′} of the Gaussian process can be written as

Here theta ^{rr _'j} is the j th hyper parameters r th Gaussian process. Descent gradient method, it is possible to obtain the parameters Θ of the Gaussian process that maximizes the objective function Q ^~ a the use of the quasi-Newton method or the like. The update formula can be written as:

Also, the update equation for the parameters v ^r and v ^{rr ′} of the Softmax function representing the burden rate can be written as the following equation.

Here, η is a constant.

[4] Check whether the convergence condition of the EM algorithm is satisfied.

To return to the above procedure [2]. If the convergence condition is satisfied, the process is terminated.

  Therefore, the learning unit 16 uses each of the hyperparameters of the kernel function, which is a function that defines the similarity between the observation data, based on the set of observation data stored in the population density information storage unit 12, and each of the observation data. And a burden factor, which is a parameter representing the degree of contribution of each of a plurality of Gaussian processes.

  The kernel function in the embodiment of the present invention is modeled by a hierarchical mixed Gaussian process in which a plurality of Gaussian processes are mixed in a plurality of hierarchies corresponding to a spatial spread and a temporal spread. In addition, the kernel function in the embodiment of the present invention is a function that defines the similarity between observation data for each of a plurality of Gaussian processes included in a model for predicting the value of a spatiotemporal variable with respect to an input variable. It is.

  The learning unit 16 includes a burden rate estimation unit 18, a Gaussian process parameter estimation unit 20, and an iterative determination unit 21.

  The burden rate estimator 18 includes a set of observation data stored in the population density information storage unit 12 and initial values of hyper parameters of kernel functions of a plurality of Gaussian processes, or previous estimated values by the Gaussian process parameter estimator 20. Based on the above, according to the above formulas (6) to (8), each of the unit data, which is a parameter representing the contribution of each of a plurality of units composed of a plurality of Gaussian processes, to each of the observation data, and each of the observation data A burden factor, which is a parameter representing each degree of contribution of a plurality of Gaussian processes, is estimated.

  The Gaussian process parameter estimator 20 includes a set of observation data stored in the population density information storage unit 12 and a unit burden rate of each of a plurality of units for each of the observation data estimated by the burden rate estimator 18. And a kernel function of the Gaussian process for each of the plurality of Gaussian processes according to the above formulas (9) to (10) and (13) based on the respective burden rates of the plurality of Gaussian processes for each of the observation data. Estimate each hyperparameter of.

In addition, the Gaussian process parameter estimation unit 20 calculates the parameters v ^r and v ^{rr ′} of the Softmax function according to the above formulas (11) to (12) based on the set of observation data stored in the population density information storage unit 12. presume.

The iteration determination unit 21 repeats the estimation by the burden rate estimation unit 18 and the estimation by the Gaussian process parameter estimation unit 20 until a predetermined iteration end condition is satisfied. The repetition determining unit 21, when a defined iteration termination condition is satisfied in advance, the parameters of the Softmax function estimated by the Gaussian process parameter estimator 20 v ^r and v ^{rr 'burden} ratio parameter storage section 22 And the hyperparameter estimated by the Gaussian process parameter estimation unit 20 is stored in the Gaussian process parameter storage unit 24.

The burden rate parameter storage unit 22 stores the parameters v ^r and v ^{rr ′ of} the Softmax function estimated by the Gaussian process parameter estimation unit 20.

  The Gaussian process parameter storage unit 24 stores each hyperparameter of the kernel function of the Gaussian process estimated by the Gaussian process parameter estimation unit 20.

  The spatiotemporal variable calculation unit 26 receives the input variable having the unobserved position information and time information received by the input unit 13 and the parameter of the Softmax function stored in the burden ratio parameter storage unit 22. A burden factor, which is a parameter representing the degree of contribution of each of a plurality of Gaussian processes, is estimated.

Specifically, the spatiotemporal variable calculation unit 26 uses the following equation (14) based on the input variable x _* and the parameters v ^r and v ^{rr ′ of} the Softmax function stored in the burden factor parameter storage unit 22. According to ˜ (15), a burden factor, which is a parameter representing each contribution degree of the plurality of Gaussian processes to the input variable x _* , is estimated.

The spatiotemporal variable calculation unit 26 then inputs the input variable according to the above equations (1) to (3) based on the burden rate for the estimated input variable x _* and the hyperparameter stored in the Gaussian process parameter storage unit 24. Predict the value of the spatiotemporal variable for.

  The output unit 28 outputs the value of the spatiotemporal variable for the input variable predicted by the spatiotemporal variable calculation unit 26 as a result.

  For example, as shown in FIG. 3, the output unit 28 calculates the population density that is the predicted value of the spatiotemporal variable at the point or region to be predicted and the time, and the congestion level that is information related to the population density. Output as a result.

<Operation of Spatiotemporal Variable Prediction Device According to Embodiment of the Present Invention>

  Next, the operation of the spatiotemporal variable prediction apparatus 100 according to the embodiment of the present invention will be described.

<Learning processing routine>
First, when the observation data set D is input from the operation unit 10, the spatiotemporal variable prediction apparatus 100 stores the observation data set D in the population density information storage unit 12. Then, the spatiotemporal variable prediction apparatus 100 executes a learning process routine shown in FIG.

  First, in step S100, the repetition variable j and each parameter are initialized.

Next, in step S102, the repetition determination unit 21 determines whether or not a predetermined repetition end condition is satisfied. As a predetermined condition, when the repetition variable j is less than a predetermined value N _iter , the process proceeds to step S110. On the other hand, if the repetition variable j is greater than or equal to N _iter , the process proceeds to step S104.

  In step S104, the burden ratio estimation unit 18 sets the initial values of the hyperparameters of the set of observation data stored in the population density information storage unit 12 and the kernel functions of the plurality of Gaussian processes set in step S100. Or, based on the hyperparameters of the kernel function of the Gaussian process previously estimated in step S106, which will be described later, according to the above formulas (6) to (8), the unit burden rate for each of the observation data and the observation data Estimate the burden rate for each.

  In step S106, the Gaussian process parameter estimation unit 20 sets the observation data stored in the population density information storage unit 12, the unit burden rate for each observation data estimated in step S104, and each of the observation data. On the basis of the burden rate for each, a hyperparameter of each kernel function of the Gaussian process is estimated for each of the plurality of Gaussian processes according to the above formulas (9) to (10) and (13). In addition, the Gaussian process parameter estimation unit 20 estimates the parameter of the Softmax function according to the above formulas (11) to (12) based on the set of observation data stored in the population density information storage unit 12.

  In step S108, the repetition variable j is incremented by 1, and the process returns to step S102.

  In step S110, the iterative determination unit 21 stores the parameter of the Softmax function estimated in step S106 in the burden factor parameter storage unit 22, and stores the estimated hyperparameter of the Gaussian process in the Gaussian process parameter storage unit 24. Then, the learning process routine ends.

  Next, the spatiotemporal variable calculation processing routine shown in FIG. 5 will be described.

The learning processing routine is executed, the parameter v ^{rr ′} of the Softmax function is stored in the burden factor parameter storage unit 22, the hyperparameter of the Gaussian process is stored in the Gaussian process parameter storage unit 24, and an unobserved target to be predicted is stored. When an input variable including position information and time information is input, the spatiotemporal variable prediction apparatus 100 executes a spatiotemporal variable calculation processing routine shown in FIG.

<Time-space variable calculation processing routine>
In step S200, the input unit 13 receives an input variable including unobserved position information and time information.

  In step S <b> 202, the spatiotemporal variable calculation unit 26 reads the parameters of the Softmax function stored in the burden factor parameter storage unit 22 and the Gaussian process hyperparameters stored in the Gaussian process parameter storage unit 24.

  In step S204, the spatiotemporal variable calculation unit 26 performs input according to the above formulas (14) to (15) based on the input variables received in step S200 and the parameters of the Softmax function read in step S202. A burden factor, which is a parameter representing each degree of contribution of a plurality of Gaussian processes to a variable, is estimated. The spatiotemporal variable calculation unit 26 then inputs the input variable according to the above formulas (1) to (3) based on the burden rate for the estimated input variable and the hyperparameter of the Gaussian process read in step S202. Predict the population density, which is the value of the spatiotemporal variable for.

  The output unit 28 outputs the population density, which is the value of the spatiotemporal variable with respect to the input variable, predicted by the spatiotemporal variable calculation unit 26, and ends the spatiotemporal variable calculation processing routine.

  As described above, according to the spatiotemporal variable prediction apparatus according to the embodiment of the present invention, based on a set of observation data having observation values of spatiotemporal variables with respect to an input variable having position information and temporal information, Included in models for predicting spatio-temporal variable values for input variables, modeled by hierarchical mixed Gaussian processes mixed at multiple levels corresponding to spatial and temporal spreads A hyperparameter of each of the kernel functions, which is a function that defines the similarity between the observation data for each of the plurality of Gaussian processes, and a parameter that represents the contribution of each of the plurality of Gaussian processes to each of the observation data By learning a certain burden ratio, the value of the spatiotemporal variable having temporal and spatial correlation can be predicted with high accuracy.

  In addition, the value of the spatiotemporal variable having temporal and spatial correlation can be accurately predicted.

  Note that the present invention is not limited to the above-described embodiment, and various modifications and applications are possible without departing from the gist of the present invention.

  For example, in the above embodiment, a case has been described in which the time series data of the population density distribution is estimated / predicted as an unobserved point or a future spatiotemporal variable distribution under the condition given as observation data. The present invention is intended for time series data of various other spatiotemporal variables (population distribution, speed / direction of human / traffic flow, reserves of mineral resources such as gold and diamond, meteorological data such as precipitation, land price, etc.) Can be applied.

  Further, in the embodiment of the present invention, a case where each parameter is estimated by the spatiotemporal variable prediction apparatus 100, and a population density that is a spatiotemporal variable corresponding to the input variable is predicted using each estimated parameter. Although described as an example, the present invention is not limited to this. For example, a process for estimating each parameter and a process for predicting a spatiotemporal variable using each estimated parameter may be configured as separate devices. In this case, an apparatus for estimating each parameter includes an operation unit 10, a population density information storage unit 12, and a calculation unit 14, and an apparatus for predicting a spatiotemporal variable using each estimated parameter includes an input unit 13, a spatiotemporal variable calculation unit 26, and an output unit 28.

  In the embodiment of the present invention, the Softmax function is adopted as the prior distribution of the burden ratio and the parameter of the Softmax function is estimated as an example. However, the present invention is not limited to this, and other functions are used. It may be used.

  The spatiotemporal variable prediction apparatus 100 described above has a computer system inside, but if the “computer system” uses a WWW system, a homepage providing environment (or display environment) is also provided. Shall be included.

  Further, in the present specification, the embodiment has been described in which the program is installed in advance. However, the program can be provided by being stored in a computer-readable recording medium or provided via a network. It is also possible to do.

DESCRIPTION OF SYMBOLS 10 Operation part 12 Population density information storage part 13 Input part 14 Calculation part 16 Learning part 18 Burden rate estimation part 20 Gaussian process parameter estimation part 21 Iteration determination part 22 Burden rate parameter storage part 24 Gaussian process parameter storage part 26 Spatio-temporal variable calculation 28 Output unit 100 Spatio-temporal variable prediction device

Claims (4)

A spatiotemporal variable prediction device that predicts spatiotemporal variable values for unobserved position information and temporal information based on a set of observation data having observation values of spatiotemporal variables for input variables having positional information and temporal information. And
A spatio-temporal variable for the input variable modeled by a hierarchical mixed Gaussian process in which a plurality of Gaussian processes are mixed in a plurality of hierarchies corresponding to a spatial spread and a temporal spread based on the set of observation data Each of the plurality of Gaussian processes included in the model for predicting the value of each of the hyperparameters of a kernel function, which is a function that defines the similarity between the observation data, and for each of the observation data A spatio-temporal variable prediction apparatus including a learning unit that learns a burden rate that is a parameter representing a contribution degree of each of the plurality of Gaussian processes. The learning unit
Based on the set of observation data and the hyperparameters of each of the kernel functions of the plurality of Gaussian processes, the parameter represents the contribution of each of a plurality of units composed of a plurality of Gaussian processes to each of the observation data. A unit burden factor, and a burden factor estimation unit that estimates a burden factor that is a parameter representing the degree of contribution of each of the plurality of Gaussian processes to each of the observation data;
Each of the plurality of Gaussian processes for each of the unit burden rates of each of the plurality of units and each of the observation data, for each of the observation data, estimated by the set of the observation data, and the burden rate estimation unit A Gaussian process parameter estimator for estimating a hyperparameter of each kernel function of the Gaussian process for each of the plurality of Gaussian processes based on
The spatiotemporal variable prediction apparatus according to claim 1, further comprising: an iterative determination unit that repeats the estimation by the burden rate estimation unit and the estimation by the Gaussian process parameter estimation unit until a predetermined iteration end condition is satisfied. Based on the input variable having unobserved position information and time information that is input, estimate a burden factor that is a parameter representing the contribution of each of the plurality of Gaussian processes to the input variable,
Based on the hyperparameters of each of the kernel functions for each of the plurality of Gaussian processes learned by the learning unit and the burden rates of the plurality of Gaussian processes for the estimated input variable, The spatiotemporal variable prediction apparatus according to claim 1, further comprising a spatiotemporal variable calculation unit that predicts a value of the spatiotemporal variable with respect to the input variable.   The program for functioning a computer as each part which comprises the spatiotemporal variable prediction apparatus of any one of Claims 1-3.

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