Information processor, information processing method and information processing program

Abstract

To provide a practical technology capable of quickly estimating a parameter posterior distribution for an object approximated to a linear gaussian state space model.SOLUTION: An information processor 100 includes: a generation part 121 which generates a Kalman filter for an object process for time series data input to a state space model, calculates a prediction distribution using the relevant Kalman filter, and generates a sample of a parameter of the state space model according to the relevant prediction distribution; and an update part 122 which calculates the prediction distribution using the Kalman filter generated by the generation part 121, and updates the weight of the sample of the parameter generated by the generation part 121 on the basis of the prediction distribution.SELECTED DRAWING: Figure 4

Description

[Technical field]

[0001] The present invention relates to an information processing device, an information processing method, and an information processing program. [Background technology]

[0002] Conventionally, there is a method called SMC2 (Sequential Monte Carlo2) as a parameter posterior distribution estimation method for state space models. [Prior art documents] [Non-patent literature]

[0003] [Non-Patent Document 1] N. Chopin and O. Papaspiliopoulos: An Introduction to Sequential Monte Carlo, Springer 2020. [Non-Patent Document 2] Naohisa Funaki, "Probability Theory: Mathematical Thinking Lecture 20," Asakura Publishing, 2004 Summary of the Invention [Problem to be solved by the invention]

[0004] In the conventional technology, there are cases where it is not possible to provide a practical technology for quickly estimating a parameter posterior distribution for an object approximated by a linear Gaussian state space model. For example, when considering industrial applications, high speed and real-time performance are required, but the conventional technology has a problem that it is not practical because of the large amount of calculations. The above-mentioned problem is an example of the problem that the present invention is intended to solve.

[0005] The present invention has been made in view of the above, and has an object to provide a practical technique for quickly estimating a parameter posterior distribution for an object approximated by a linear Gaussian state space model. [Means for solving the problem]

[0006] In order to solve the above-mentioned problems and achieve the object, the information processing device of the present invention is characterized by having a generation unit that generates a Kalman filter of a target process for time series data input to a state space model, calculates a predictive distribution using the Kalman filter, and generates samples of parameters of the state space model based on the predictive distribution, and an update unit that calculates a predictive distribution using the Kalman filter generated by the generation unit, and updates weights of the sample parameters generated by the generation unit based on the predictive distribution. Effect of the Invention

[0007] According to the present invention, it is possible to provide a practical technique for quickly estimating the posterior distribution of parameters for an object approximated by a linear Gaussian state space model. [Brief description of the drawings]

[0008] [Figure 1] FIG. 1 is a diagram for explaining the prior art. [Diagram 2] FIG. 2 is a diagram for explaining the conventional technology. [Diagram 3] FIG. 3 is a diagram for explaining the conventional technology. [Figure 4] FIG. 4 is a diagram illustrating an example of the configuration of the information processing device according to the embodiment. [Diagram 5] FIG. 5 is a flowchart showing an example of the overall process flow of parameter posterior distribution estimation. [Figure 6] FIG. 6 is a flowchart showing an example of the flow of particle generation by the system model. [Figure 7] FIG. 7 is a flowchart showing an example of the flow of weight correction based on an observation model. [Figure 8] FIG. 8 is a flowchart showing an example of a flow of predictive distribution calculation in the prior art. [Figure 9] FIG. 9 is a flowchart showing an example of a flow of predictive distribution calculation performed by the information processing device according to the embodiment. [Figure 10] FIG. 10 is a flowchart showing an example of a flow of predictive distribution calculation in the prior art. [Figure 11] FIG. 11 is a flowchart showing an example of a flow of predictive distribution calculation performed by the information processing device according to the embodiment. [Figure 12] FIG. 12 is a diagram for explaining an example of posterior distribution estimation by the information processing device according to the embodiment. [Figure 13] FIG. 13 is a diagram for explaining an example of posterior distribution estimation by the information processing device according to the embodiment. [Figure 14] FIG. 14 is a diagram for explaining an example of posterior distribution estimation by the information processing device according to the embodiment. [Figure 15] FIG. 15 is a diagram for explaining an example of posterior distribution estimation by the information processing device according to the embodiment. [Figure 16] FIG. 16 is a diagram for explaining an example of posterior distribution estimation by the information processing device according to the embodiment. [Figure 17] FIG. 17 is a diagram for explaining an example of posterior distribution estimation by the information processing device according to the embodiment. [Figure 18] FIG. 18 is a diagram for explaining an example of posterior distribution estimation by the information processing device according to the embodiment. [Figure 19] FIG. 19 is a diagram for explaining an example of posterior distribution estimation by the information processing device according to the embodiment. [Figure 20] FIG. 20 is a diagram illustrating an example of a computer that executes an information processing program. DETAILED DESCRIPTION OF THE PREFERRED EMBODIMENTS

[0009] Hereinafter, with reference to the drawings, an embodiment of an information processing device, an information processing method, and an information processing program according to the present application will be described in detail. Note that the present invention is not limited to this embodiment. In addition, in the description of the drawings, the same parts are denoted by the same reference numerals, and duplicated explanations will be omitted as appropriate. In addition, in the following, when the symbol A is written as "A^{~}", it is equivalent to "the symbol with "~" written directly above "A".

[0010] [Prior Art] Conventionally, the state variable x t There is a method called Sequential Monte Carlo (SMC) that approximates the distribution of x by a set of realizations. t are the hard-to-observe state variables, y t is the observable output variable. x t Since the sequence of is not observable, a set of candidate values ​​is prepared from the set prior distribution p(x0), and each candidate value is set as the initial value, and y obtained by solving the state equation is t The predicted values ​​of and y t The candidate values ​​are corrected by comparing the observed values ​​of x. In the sequential Monte Carlo method, the process of prediction, observation, and correction is repeated to obtain t The distribution of is approximated by a set of realizations.

[0011] Here, the idea behind parameter estimation using SMC is to estimate the state variables x t If we can formulate a state space model in which the parameter θ to be estimated is a value, then we can use SMC to sequentially estimate the posterior distribution p(θ|y) (Figure 2).

[0012] As shown in FIG. 3, SMC2 (Sequential Monte Carlo2), which is a parameter posterior distribution estimation method for a state space model, has the same structure as the state space model shown in FIG. t For each particle in , a particle filter (PF) of the state space model of the target process is run, and y tIn SMC2, we first consider a nonlinear non-Gaussian state space model of the process defined by the following Equations 1 and 2.

[0013]

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[0014]

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[0015] Next, the samples following the posterior distribution of the parameters θ of the state space model defined by the above Equations 1 and 2 are converted into the appropriately designed transition kernel m t We consider generating the vectors using Markov Chain Monte Carlo methods (MCMC) based on the above.

[0016]

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[0017] The Metropolis Hastings method, which is a type of MCMC, is used as a design method for the transition kernel. In the MH method, in order to appropriately design the transition kernel, the following formulas 4 and 5 are used to determine the arbitrary proposed kernel m t ^{~} is corrected by the MH ratio r.

[0018]

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[0019]

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[0020] There are two problems with the sample generation method described so far. The first is the likelihood p(y 0:t-1 The second problem is that the proposed kernel m t The problem is that tuning ^{~} is difficult.

[0021] Regarding the first problem, one solution is to calculate the likelihood numerically, as shown in Equation 6 below.

[0022]

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[0023] In the above formula 6, p k (y k |x k , θ) is the observation model of Equation 2 above, and p k (x k |y 0:k-1 ,θ)(k=1, ,t-1) is the predictive distribution. Also, the following formula 7 is the predictive distribution p k (x k |y 0:k-1 , θ) is a weighted sample for Monte Carlo approximation and can be obtained by PF.

[0024]

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[0025] Next, we will explain the solution to the second problem. First, as shown in the following formula 8, the proposed kernel m t ^{~}, the average θ t-1 ,Consider a Gaussian distribution with covariance matrix Σ.

[0026]

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[0027] At this time, the proposed kernel m tThe tuning parameters of ^{~} are the covariance matrix Σ of the Gaussian distribution and the number of transitions. Here, the sample (particle) θ generated at time t t and the output measurement value y 0:t-1 Using this, output y t The predictive distribution of can be calculated as follows:

[0028]

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[0029] However, the predictive distribution p t (x t |y 0:t-1 ,θ t ) can be calculated by PF. The above sample generation of the parameter θ by the MCMC kernel and the output prediction model based on the sample and past observation data can be combined to form a state space model as shown in Equation 3 above and Equation 10 below.

[0030]

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[0031] By applying PF to the state space model shown in Equation 3 and Equation 10 above, a distribution of the parameter θ that is corrected each time the output is observed can be obtained, and the tuning parameters, the covariance matrix Σ and the number of transitions, can be adaptively changed based on that distribution.

[0032] [Problems with the prior art] Next, the problems with the conventional technology will be described. In the conventional technology SMC2, a double PF was constructed to estimate the posterior distribution. In the first PF (hereinafter referred to as PF1), the PF is constructed by treating the sample generation of the parameter posterior distribution by MCMC and the output prediction based on that sample (formula 3 and formula 10 above) as a state space model. In the sample generation by this system model (formula 3 above), the likelihood p(y 0:t-1To calculate θ, we build a second PF (PF2) for the state space model of the target process (Equation 1 and Equation 2 above) based on each sample. The computational complexity of this algorithm is shown below.

[0033] First, we will show the amount of calculation when the state space model (Equation 3 and Equation 10) is advanced by one step. When the system model (Equation 3) is advanced by one step, the proposed kernel m t The sample generation by ^{~} and the evaluation of the sample based on the MH ratio r are performed for the transition number M t However, this transition count M t changes adaptively at each time. In the calculation of the MH ratio r, the likelihood p(y 0:t-1 |θ) is calculated based on PF2. The computational complexity of one step of PF2 is τ PF2 Then, the likelihood p(y 0:t-1 |θ) is a computational complexity of τ PF2 t. This calculation is done for the number of transitions M t Since this is done multiple times, the amount of calculation required to advance the system model (Equation 3 above) by one step is τ PF2 tM t It becomes.

[0034] Next, consider the calculation of the observation model (Equation 10 above). At this point, the only weighted samples of PF2 that have been obtained are those shown in Equation 11 below, so the predictive distribution p t (x t |y 0:t-1 ,θ t ) to generate weighted samples (Equation 12 below). Therefore, the total τ PF2 (tM t +1) computational complexity.

[0035]

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[0036]

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[0037] Next, we will show the computational complexity for the entire PF1. θ Then, in SMC2, N θ Therefore, the above "operation of advancing the state space model (the above formula 3 and formula 10) by one step" is carried out as N θ In other words, if the length of the target time series data is T, the total amount of calculations can be expressed by the following formula 13.

[0038]

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[0039] The number of particles in PF2 is N x Then, the computational complexity of one step of PF2 is O(N x ) Therefore, the number of transitions M t If we set the maximum value of M^{―}, the computational complexity is O(N x N θ The computational complexity is the number of particles in the parameter distribution, N θ , the number of particles in the state distribution of the target process, N x Since the computational complexity is proportional to the data length T, SMC2 is a computationally intensive method. Considering the industrial application of SMC2, it is required to be faster and perform in real time. However, when real time is required for estimating the parameter posterior distribution, the above computational complexity is not practical.

[0040] Therefore, the information processing device 100 according to the embodiment generates a Kalman filter of a target process for time series data input to a state space model, calculates a predictive distribution using the Kalman filter, generates samples of parameters of the state space model based on the predictive distribution, calculates a predictive distribution using the generated Kalman filter, and updates weights of the generated parameter samples based on the predictive distribution. This allows the information processing device 100 to provide a practical technique for quickly estimating a parameter posterior distribution for a target approximated to a linear Gaussian state space model.

[0041] [Configuration of information processing device] Next, the configuration of the information processing device 100 will be described with reference to Fig. 4. Fig. 4 is a diagram showing an example of the configuration of the information processing device 100. As shown in Fig. 4, the information processing device 100 has a communication unit 110, a control unit 120, and a storage unit 130. Note that each of these units may be held in a distributed manner in a plurality of devices. The processing of each unit will be described below.

[0042] The communication unit 110 is realized by a NIC (Network Interface Card) or the like, and enables communication between an external device and the control unit 120 via an electric communication line such as a LAN (Local Area Network) or the Internet. For example, the communication unit 110 enables communication between the external device and the control unit 120.

[0043] The storage unit 130 is realized by a semiconductor memory element such as a random access memory (RAM) or a flash memory, or a storage device such as a hard disk or an optical disk. Information stored in the storage unit 130 includes, for example, a state space model, time series data, various algorithms, samples, parameters, prior distributions, predictive distributions, likelihoods, and other information necessary for generating and updating samples. Note that the information stored in the storage unit 130 is not limited to the above examples.

[0044] The control unit 120 is realized using a CPU (Central Processing Unit), an NP (Network Processor), an FPGA (Field Programmable Gate Array), etc., and executes a processing program stored in a memory. As shown in Fig. 4, the control unit 120 has a generating unit 121 and an updating unit 122. Each unit of the control unit 120 will be described below.

[0045] The generation unit 121 generates a Kalman filter of a target process for time series data input to the state space model, calculates a predictive distribution using the Kalman filter, and generates a sample of parameters of the state space model based on the predictive distribution. For example, the generation unit 121 generates a Kalman filter of a target process for time series data input to the state space model, calculates a predictive distribution using the Kalman filter, and generates a sample of parameters of the state space model based on the predictive distribution by processing step S4 in the flowchart described below.

[0046] Furthermore, the generating unit 121 generates a Kalman filter of a target process for time series data input to the nonlinear non-Gaussian state space model, calculates a predictive distribution using the Kalman filter, and generates a sample of parameters of the nonlinear non-Gaussian state space model based on the predictive distribution. For example, the generating unit 121 generates a Kalman filter of a target process for time series data input to the nonlinear non-Gaussian state space model by processing in step S4 in the [flowchart] described later, calculates a predictive distribution using the Kalman filter, and generates a sample of parameters of the nonlinear non-Gaussian state space model based on the predictive distribution. Details of the processing performed by the generating unit 121 will be described in the [parameter posterior distribution estimation method using an information processing device] [flowchart] described later.

[0047] Furthermore, the generation unit 121 generates parameter samples of the state space model using an appropriately designed transition kernel. For example, the generation unit 121 generates parameter samples of the state space model using a transition kernel designed so that the parameter distribution of the state space model becomes a stationary distribution. Here, when the transition kernel satisfies ergodicity, the parameter distribution of the state space model becomes a stationary distribution. For details of ergodicity, see Non-Patent Document 2.

[0048] That is, the generation unit 121 generates samples of parameters of the state space model using a transition kernel designed to satisfy ergodicity. Here, the transition kernel that satisfies the detailed balance condition shown in the following formula 14 is a transition kernel designed so that the parameter distribution of the state space model becomes a stationary distribution. That is, the transition kernel that satisfies the detailed balance condition shown in formula 14 is a transition kernel set so that the parameters of the state space model become a stationary distribution.

[0049]

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[0050] More specifically, the generation unit 121 corrects the sample parameters of the state space model using the above-mentioned Equation 5, which is a correction coefficient for satisfying the detailed balance condition, and generates the sample parameters of the state space model using a transition kernel designed to satisfy the detailed balance condition shown in the above-mentioned Equation 14.

[0051] The update unit 122 calculates a predictive distribution using the Kalman filter generated by the generation unit 121, and updates (modifies) the weights of the parameter samples generated by the generation unit 121 based on the predictive distribution. For example, the update unit 122 calculates a predictive distribution using the Kalman filter generated by the generation unit 121 by the process of step S5 in the [flowchart] described later, and updates the weights of the parameter samples generated by the generation unit 121 based on the predictive distribution.

[0052] For example, the update unit 122 accumulates weights of the parameter samples generated by the generation unit 121 based on the predictive distribution, and updates the weights of the parameter samples generated by the generation unit 121. Details of the process performed by the update unit 122 will be described later in [Parameter posterior distribution estimation method using an information processing device] [Flowchart].

[0053] [Parameter posterior distribution estimation method using information processing device] A parameter posterior distribution estimation method using the information processing device 100 according to the embodiment will be described. First, consider a linear Gaussian state space model of a certain process defined by the following formulas 15, 16, and 17. Here, N(x|μ, Σ) in the following expressions represents a probability density function of a multivariate Gaussian distribution with mean μ and covariance matrix Σ.

[0054]

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[0055]

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[0056]

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[0057] The samples following the posterior distribution of the parameter θ of the state space model (Equation 1, Equation 2) are treated as the transition kernel m t In the MH method, in order to appropriately design the transition kernel, an arbitrary proposed kernel m t ^{~} is corrected by the MH ratio r.

[0058] Here, as in the conventional technology, the likelihood p(y 0:t-1|θ) cannot be analytically obtained, and the proposed kernel m t There is a problem in that tuning ^{~} is difficult.

[0059] Regarding the first issue, when approximating to a linear Gaussian state space model, as shown in Equation 18 below, the likelihood p(y 0:t-1 |θ) can be analytically determined.

[0060]

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[0061] In the above formula 18, p k (y k |x k , θ) is the observation model of Equation 2 above, and p k (x k |y 0:k-1 ,θ)(k=1,...,t-1) is the predictive distribution. In the case of the linear Gaussian state space model, the predictive distribution is a Gaussian distribution, and its mean value x k | k-1 and the covariance matrix v k | k-1 is calculated using a Kalman filter (KF). Therefore, the x obtained using the KF k | k-1 ,v k | k-1 Using (k=1, ,t-1), the following formula 19 is obtained, so the likelihood p(y 0:t-1 |θ) can be analytically determined.

[0062]

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[0063] Next, we will explain the solution to the second problem. First, as shown in the above formula 8, the proposed kernel m t ^{~}, the average θ t-1 Consider the Gaussian distribution of the covariance matrix Σ. Then, the proposed kernel mt The tuning parameters of ^{~} are the covariance matrix Σ of the Gaussian distribution and the number of transitions. Here, the sample m generated at time t t ^{~} and the measured output value y 0:t-1 Using this, output y t The predictive distribution of can be calculated as shown in Equation 9 above.

[0064] Here, the predictive distribution p t (x t |y 0:t-1 ,θ t ) can be calculated analytically using the KF:

[0065]

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[0066] The sample generation of the parameter θ by the above MCMC kernel and the output prediction model based on the sample and the observation data up to now can be combined to form a state space model such as the above Equation 3 and Equation 10.

[0067] By applying PF to the state space model shown in Equation 3 and Equation 10 above, a distribution of the parameter θ that is corrected each time the output is observed can be obtained, and the tuning parameters, the covariance matrix Σ and the number of transitions, can be adaptively changed based on that distribution.

[0068] Next, a description will be given of the principle of reducing the amount of calculation by the parameter posterior distribution estimation method using the information processing device 100 according to the embodiment. In the parameter posterior distribution estimation method using the information processing device 100 according to the embodiment, it is considered that the nonlinear non-Gaussian state space model of the target process (Formula 1, Formula 2) is approximated to a linear Gaussian state space model such as Formula 21, Formula 22, and Formula 23 below.

[0069]

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[0070]

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[0071]

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[0072] Here, the predictive distribution p k (x k |y 0:k-1 Consider using KF instead of PF as the sequential Bayes filter used in the calculation of (k=1, , t-1). If KF2 is used instead of PF2, the predictive distribution can be analytically obtained as a Gaussian distribution as shown in the following formulas 24, 25, and 26.

[0073]

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[0074]

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[0075]

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[0076] However, x t-1 | t-1 ,v t-1 | t-1 are the mean and variance-covariance matrix of the filter distribution at time, respectively. Therefore, the output y t The predictive distribution of can also be analytically obtained as a Gaussian distribution as shown in Equation 27 below.

[0077]

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[0078] Using KF eliminates the need for Monte Carlo approximation, which reduces the computation time. KF2 By replacing PF2 with KF2, the computational complexity shown in Equation 13 is reduced by τ PF2 All are τ KF2 The number of dimensions of KF2 is d x Then, the computational complexity of one step of KF2 is given by the following Equation 28.

[0079]

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[0080] Here, the number of particles in the state variables, which are configurable parameters of PF2, is N x In order to estimate with the same accuracy as KF2 using PF2, N x It was necessary to set it large and set it to the relationship shown in the following Formula 29. In other words, in the posterior distribution estimation method using the information processing device 100 according to the embodiment, which is an approximation method using KF2, it is possible to reduce the amount of calculation.

[0081]

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[0082] In this way, in the parameter posterior distribution estimation method using the information processing device 100 according to the embodiment, for SMC2, which is a parameter posterior distribution estimation method for a state space model, the amount of calculation can be reduced by limiting the target process to a linear Gaussian state space model and replacing the sequential Bayes filter used in the predictive distribution calculation from a particle filter to a Kalman filter.

[0083] [flowchart] The parameter posterior distribution estimation method using SMC2 of the related art and the information processing device 100 according to the embodiment has a common overall process flow. Here, a process flow common to the parameter posterior distribution estimation using SMC2 of the related art and the information processing device 100 according to the embodiment will be described with reference to Figs. 5 to 7. Fig. 5 is a flowchart showing an example of the overall process flow of parameter posterior distribution estimation. Note that the information processing device 100 according to the embodiment can perform the processes of steps S4 and S5 of the series of processes.

[0084] As shown in Figure 5, first, the time series data y o:T is input (step S1). Next, weighted samples are generated from the prior distribution as shown in the following formula 30 (step S2).

[0085]

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[0086] Next, resampling and weight initialization are performed according to the situation (step S3). For example, samples with low likelihood are deleted and samples with high likelihood are replicated. Next, particles are generated according to the MCMC system model shown in the above formula 3 (step S4). Step S4 is a process performed by the generation unit 121 of the information processing device 100.

[0087] Here, the process flow (step S4) in particle generation by the MCMC system model will be described with reference to Fig. 6. Fig. 6 is a diagram for explaining the process flow in particle generation by the system model. As shown in Fig. 6, first, data shown in the following formula 31 is input (step S4-1).

[0088]

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[0089] Next, an initial value is set as shown in the following formula 32 (step S4-2).

[0090]

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[0091] Next, a sample shown in the following formula 34 is generated using random numbers shown in the following formula 33 (step S4-3).

[0092]

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[0093]

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[0094] Next, the predictive distribution shown in the following formula 35 is calculated (step S4-4). The process in step S4-4 will be described later in detail.

[0095]

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[0096] Next, the sample shown in the above formula 34 is corrected by the following formula 36 based on the predictive distribution (step S4-5). Here, for example, the processes in steps S4-3 to S4-5 are processes for correcting the sample of the parameters of the state space model using the above formula 5.

[0097]

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[0098] Next, i=1, ,N θ (step S4-6), where i=1, . . . , N θ If it is determined that the difference is within the range (step S4-6 "Yes"), the process returns to step S4-3 and repeats the process up to step S4-5.

[0099] On the other hand, for i=1, ,N θ If it is determined that the sample (i=1 to N) shown in the above formula 34 is not within the range (step S4-6 “No”), θ ) satisfies the convergence condition (step S4-7). An example of the convergence condition is when the distribution does not change much from the previous calculation after repeated calculation. If it is determined that the convergence condition is not satisfied (step S4-7 "No"), the process shown in the following formula 37 is performed, and the process returns to step S4-3 and repeats the process up to step S4-6.

[0100]

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[0101] On the other hand, if it is determined that the convergence condition is satisfied (step S4-7 "Yes"), a weighted sample shown in the following formula 38 is output (step S4-8). In this manner, particle generation processing using the system model is performed.

[0102]

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[0103] Returning to Fig. 5 again, weight correction is then performed based on the MCMC observation model shown in the above-mentioned formula 10 (step S5). Step S5 is a process performed by the update unit 122 of the information processing device 100.

[0104] Here, the process flow (step S5) of weight correction based on an observation model will be described with reference to Fig. 7. Fig. 7 is a diagram for explaining the process flow of weight correction based on an observation model. As shown in Fig. 7, first, data shown in the following formula 39 is input (step S5-1).

[0105]

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[0106] Next, the predictive distribution shown in the following formula 40 is calculated (step S5-2). The process in step S5-2 will be described later in detail.

[0107]

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[0108] Next, the predictive distribution p t (x t |y 0:t-1 ,θ t ) to calculate the following formula 41 (step S5-3).

[0109]

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[0110] Next, the weights are accumulated as shown in the following formula 42 (step S5-4).

[0111]

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[0112] Then, the weighted sample shown in the following formula 43 is output (step S5-5). In this manner, the weight correction process based on the observation model is performed.

[0113]

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[0114] Returning to Fig. 5 again, the weights are then normalized as shown in the following formula 44 (step S6).

[0115]

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[0116] Next, the covariance matrix Σ of the proposed kernel t is corrected based on the weighted samples as shown in the following Equation 45 (step S7).

[0117]

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[0118] Next, it is determined whether or not it is within the range of t=1,...,T (step S8). If it is determined that it is within the range of t=1,...,T (step S8 "Yes"), the process returns to step S3 and repeats the processes up to step S7. On the other hand, if it is determined that it is not within the range of t=1,...,T (step S8 "No"), the particle sequence is output as shown in the following formula 46 (step S9). The above is the overall process flow common to the conventional SMC2 and the parameter posterior distribution estimation method using the information processing device 100 according to the embodiment.

[0119]

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[0120] So far, we have explained the process flow common to the parameter posterior distribution estimation method using the conventional technology SMC2 and the information processing device 100 according to the embodiment. From here, we will use Figures 8 to 11 to explain the differences in the methods of calculating the predictive distribution of state variables performed by the conventional technology SMC2 and the parameter posterior distribution estimation method using the information processing device 100 according to the embodiment (mainly steps S4-4 and S5-2).

[0121] First, the process of step S4-4 according to the conventional technology will be described with reference to FIG.

[0046] FIG. 1 is a diagram for explaining a process according to the conventional technology. First, data shown in the following formula 47 is input (step S4-4-1).

[0122]

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[0123] Next, a particle filter for the target process is generated as shown in the following formula 48 (step S4-4-2).

[0124]

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[0125] Next, the particle filter shown in the above Formula 48 is executed up to time k=1, . . . , t-1, and a Monte Carlo approximation of the predictive distribution (the above Formula 35) is calculated as shown in the following Formula 49 (step S4-4-3).

[0126]

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[0127] Then, the data shown in the following formula 50 is output (step S4-4-4).

[0128]

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[0129] Next, the process of step S4-4 by the information processing device 100 according to the embodiment will be described with reference to Fig. 9. Fig. 9 is a diagram showing an example of the process by the information processing device 100 according to the embodiment. First, data shown in the above formula 47 is input to the information processing device 100 (step S4-4-1*). Next, the information processing device 100 generates a Kalman filter for the target process as shown in the following formula 51 (step S4-4-2*).

[0130]

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[0131] Next, the information processing device 100 executes the Kalman filter shown in the above formula 51 up to time k=1,...,t-1, and calculates the predictive distribution (the above formula 35) as shown in the following formulas 52, 53, and 54 (step S4-4-3*).

[0132]

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[0133]

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[0135] Then, the information processing device 100 calculates the mean value x k | k-1 and the covariance matrix v k | k-1 (step S4-4-4*).

[0136] Next, the process of step S5-2 according to the conventional technology will be described with reference to Fig. 10. Fig. 10 is a diagram for explaining the process according to the conventional technology. First, data shown in the following formula 55 is input (step S5-2-1).

[0137]

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[0138] Next, using the results of the generated particle filter (Equation 48 above), the predictive distribution p t (x t |y 0:t-1 ,θ t ) is calculated (step S5-2-2).

[0139]

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[0140] Then, the data shown in the following formula 57 is output (step S5-2-3).

[0141]

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[0142] Next, the process of step S5-2 by the information processing device 100 according to the embodiment will be described with reference to Fig. 11. Fig. 11 is a diagram showing an example of the process by the information processing device 100 according to the embodiment. First, data represented by the above formula 55 is input to the information processing device 100 (step S5-2-1*).

[0143] Next, the information processing device 100 calculates the prediction distribution p using the execution result of the generated Kalman filter (the above-mentioned formula 51) as shown in the following formulas 58, 59, and 60. t (x t |y 0:t-1 ,θ t ) is calculated (step S5-2-2*).

[0144]

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[0145]

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[0147] Then, the information processing device 100 calculates the mean value x t | t-1 and the covariance matrix v t | t-1(Step S5-2-3*).

[0148] That is, the information processing device 100 according to the embodiment calculates the predictive distribution of the state variables using a Kalman filter instead of a particle filter, thereby providing a practical technique for quickly estimating the posterior distribution of parameters for an object approximated to a linear Gaussian state space model.

[0149] [Estimation results of parameter posterior distribution] Next, the estimation result of the parameter posterior distribution using the information processing device 100 according to the embodiment will be described with reference to Figs. 12 to 19. First, an example assuming a one-input, one-output system shown in the following formulas 61 to 64 will be described. Note that the true values ​​of the parameters are G1(-2), T1(15), alpha(0.01), and beta(0.1).

[0150]

number

[0151]

number

[0152]

number

[0153]

number

[0154] As shown in Figure 12, the input data u o:T and the output data y o:T With the above as input, an output is generated based on a state space model of the target process. Here, as a condition setting of the distribution estimation algorithm, the state space model is expressed by the above formulas 61 to 64. Also, the prior distribution of the parameters is expressed by the following formula 65.

[0155]

number

[0156] The number of particles is set to N θ =50,N x = 50. Note that N x is only used in SMC2.

[0157] As a result of estimating the parameter posterior distribution under the above-mentioned condition settings, the calculation time, the shape of the parameter posterior distribution, and the representative value of the parameter posterior distribution will be described. First, as shown in Fig. 13, the calculation time was 61.068 seconds in the conventional method SMC2, whereas the calculation time was 28.671 seconds in the parameter posterior distribution estimation using the information processing device 100 according to the embodiment. That is, the parameter posterior distribution estimation using the information processing device 100 according to the embodiment was able to reduce the calculation time to less than half compared to the conventional technology.

[0158] Next, using Figure 14, we consider the shape of the parameter posterior distribution, p(θ|u o:T ,y o:T 14 shows a representative example (G1) of a histogram of a sample (above formula 33) for approximating (G2). As shown in Fig. 14, the parameter posterior distribution estimated by the information processing device 100 according to the embodiment (Fig. 14(1)) and the parameter posterior distribution estimated by SMC2, which is a conventional technique (Fig. 14(2)), are both histograms with an average value around -2.1.

[0159] Next, a representative value of the parameter posterior distribution estimation using the information processing device 100 according to the embodiment and the parameter posterior distribution by SMC2, which is a conventional technique, will be described with reference to FIG. 15. As a representative example, G1 is described. As shown in FIG. 15, the parameter posterior distribution estimation using the information processing device 100 according to the embodiment has an average value of -2.110 and a variance of 0.00206, whereas the parameter posterior distribution estimation using SMC2, which is a conventional technique, has an average value of -2.105 and a variance of 0.00251. That is, the parameter posterior distribution estimation using the information processing device 100 according to the embodiment has a result in which the average value is approximately the same as that of the conventional technique and the variance is smaller than that of the conventional technique, and it is possible to estimate with the same or higher accuracy as the conventional technique. Note that only G1 has been described above, but similar results are also obtained for T1, alpha, and beta, as shown in FIG. 15.

[0160] Next, an example will be described assuming a 3-input, 1-output system shown in the above formula 61, formula 62, and the following formulas 66 to 68. The true values ​​of the parameters are G1(-2), G2(-0.5), G3(-1), T1(15), T2(15), T3(15), alpha(0.01), and beta(0.1).

[0161]

number

[0162]

number

[0163]

number

[0164] As shown in Figure 16, the input data u o:T and the output data y o:TWith the above as input, an output is generated based on a state space model of the target process. Here, as condition settings for the distribution estimation algorithm, the state space model is expressed by the above formulas 61, 62, and 66 to 67. In addition, the prior distribution of the parameters is expressed by the following formula 69.

[0165]

number

[0166] The number of particles is set to N θ =150,N x = 100. Note that N x is only used in SMC2.

[0167] As a result of estimating the parameter posterior distribution under the above-mentioned condition settings, the calculation time, the shape of the parameter posterior distribution, and the representative value of the parameter posterior distribution will be described. First, as shown in Fig. 17, the calculation time was 716.722 seconds in the conventional method SMC2, whereas the calculation time was 304.839 seconds in the parameter posterior distribution estimation using the information processing device 100 according to the embodiment. That is, the parameter posterior distribution estimation using the information processing device 100 according to the embodiment was able to reduce the calculation time to less than half compared to the conventional technology.

[0168] Next, using Figure 18, we consider the shape of the parameter posterior distribution, p(θ|u o:T ,y o:T 18 shows a representative example (G1) of a histogram of a sample (above formula 33) for approximating (G2). As shown in Fig. 18, the parameter posterior distribution estimated by the information processing device 100 according to the embodiment (Fig. 18(1)) and the parameter posterior distribution estimated by SMC2, which is a conventional technique (Fig. 18(2)), are both histograms with an average value around -2.07.

[0169] Next, a representative value of the parameter posterior distribution by the information processing device 100 according to the embodiment and SMC2, which is a conventional technique, will be described with reference to FIG. 19. As a representative example, G1 is described. As shown in FIG. 19, the information processing device 100 according to the embodiment has a mean value of -2.07445 and a variance of 0.008841, whereas the SMC2, which is a conventional technique, has a mean value of -2.07746 and a variance of 0.012498. That is, in the parameter posterior distribution estimation using the information processing device 100 according to the embodiment, the mean value is closer to the true value than the conventional technique, and the variance is smaller than the conventional technique, and it is possible to estimate with the same or higher accuracy as the conventional technique. Note that only G1 has been described above, but as shown in FIG. 15, similar tendencies are obtained for G2, G3, T1, T2, T3, alpha, and beta.

[0170] [effect] The information processing device 100 according to the embodiment includes a generation unit 121 that generates a Kalman filter of a target process for time series data input to a state space model, calculates a predictive distribution using the Kalman filter, and generates parameter samples of the state space model based on the predictive distribution, and an update unit 122 that calculates a predictive distribution using the Kalman filter generated by the generation unit 121, and updates weights of the parameter samples generated by the generation unit 121 based on the predictive distribution.

[0171] As a result, the information processing device 100 can provide a practical technique for quickly estimating the posterior distribution of parameters for an object approximated by a linear Gaussian state space model by using a Kalman filter to calculate the likelihood of the output measured value for the state space model.

[0172] Moreover, the information processing device 100 can reduce the calculation time required for estimating the parameter posterior distribution compared to the conventional technology. In the examples of Figures 12 to 15 and the examples of Figures 16 to 19, the calculation time can be reduced to less than half that of the conventional technology. Moreover, the information processing device 100 can estimate the parameter posterior distribution with accuracy equal to or higher than that of the conventional technology.

[0173] The generation unit 121 of the information processing device 100 according to the embodiment generates a Kalman filter of a target process for time series data input to a nonlinear non-Gaussian state space model, calculates a predictive distribution using the Kalman filter, and generates samples of parameters of the nonlinear non-Gaussian state space model based on the predictive distribution.

[0174] As a result, the information processing device 100 can provide a practical technique for quickly estimating parameter posterior distributions for an object approximated to a linear Gaussian state space model by using a Kalman filter to calculate the likelihood of output measured values ​​for a nonlinear non-Gaussian state space model.

[0175] The generation unit 121 of the information processing device 100 according to the embodiment generates a sample of parameters of a state space model using a transition kernel designed so that the parameter distribution becomes a stationary distribution. As a result, the information processing device 100 generates a sample of parameters of a state space model using a transition kernel designed so that the parameter distribution becomes a stationary distribution, performs likelihood calculation of an output measured value for the state space model using a Kalman filter, and can provide a practical technique for quickly estimating a parameter posterior distribution for an object approximated to a linear Gaussian state space model.

[0176] The generation unit 121 of the information processing device 100 according to the embodiment generates samples of parameters of a state space model by using a transition kernel designed to satisfy a detailed balance condition.

[0177] As a result, the information processing device 100 generates samples of parameters of a state space model using a transition kernel designed to satisfy a detailed balance condition, and performs likelihood calculations of output measured values ​​for the state space model using a Kalman filter, thereby providing a practical technique for quickly estimating parameter posterior distributions for an object approximated by a linear Gaussian state space model.

[0178] The update unit 122 of the information processing device 100 according to the embodiment accumulates weights of the parameter samples generated by the generation unit 121 based on the predictive distribution, and updates the weights of the parameter samples generated by the generation unit 121.

[0179] As a result, the information processing device 100 can provide a practical technique for quickly estimating a parameter posterior distribution for an object approximated by a linear Gaussian state space model by accumulating and updating the weights of parameter samples based on the predictive distribution calculated using a Kalman filter.

[0180] [Hardware configuration] The information processing device 100 according to the embodiment described above is realized, for example, by a computer 1000 having a configuration as shown in Fig. 20. Fig. 20 is a hardware configuration diagram showing an example of a computer that realizes the functions of the information processing device 100. The computer 1000 has a configuration in which a CPU 1100, a RAM 1200, a ROM 1300, an auxiliary storage device 1400, a communication I / F (interface) 1500, and an input / output I / F (interface) 1600 are connected by a bus 1800.

[0181] The CPU 1100 operates and controls each unit based on a program stored in the ROM 1300 or the auxiliary storage device 1400. The ROM 1300 stores a boot program executed by the CPU 1100 when the computer 1000 is started up, programs that depend on the hardware of the computer 1000, and the like.

[0182] The auxiliary storage device 1400 stores programs executed by the CPU 1100 and data used by the programs. The communication I / F 1500 receives data from other devices via a predetermined communication network and sends it to the CPU 1100, and transmits data generated by the CPU 1100 to other devices via the predetermined communication network. The CPU 1100 controls output devices such as a display and a printer, and an input / output device 1700 such as a keyboard and a mouse, via an input / output I / F 1600. The CPU 1100 acquires data from the input / output device 1700 via the input / output I / F 1600. The CPU 1100 also outputs the generated data to the input / output device 1700 via the input / output I / F 1600.

[0183] For example, when the computer 1000 functions as the information processing device 100 according to this embodiment, the CPU 1100 of the computer 1000 executes a program loaded onto the RAM 1200 to realize the functions of the control unit 120 .

[0184] [others] Of the processes described in the above embodiments and modifications, all or part of the processes described as being performed automatically can be performed manually, or all or part of the processes described as being performed manually can be performed automatically by a known method. In addition, the information including the processing procedures, specific names, various data and parameters shown in the above documents and drawings can be changed arbitrarily unless otherwise specified. For example, the various information shown in each drawing is not limited to the illustrated information.

[0185] In addition, each component of each device shown in the figure is a functional concept, and does not necessarily have to be physically configured as shown in the figure. In other words, the specific form of distribution and integration of each device is not limited to that shown in the figure, and all or part of them can be functionally or physically distributed and integrated in any unit according to various loads, usage conditions, etc.

[0186] The above-mentioned components include those that can be easily imagined by a person skilled in the art, those that are substantially the same, and those that are within the so-called equivalent range. Furthermore, the above-mentioned embodiments and modifications can be appropriately combined within a range that does not cause contradictions in the processing contents.

[0187] Moreover, the above-mentioned "section, module, unit" can be read as "means" or "circuit", etc. For example, a control section can be read as control means or a control circuit.

[0188] Although some of the embodiments of the present invention have been described in detail above with reference to the drawings, these are merely examples, and the present invention can be embodied in other forms that incorporate various modifications and improvements based on the knowledge of those skilled in the art, including the forms described in the Disclosure of the Invention section. [Explanation of symbols]

[0189] 100 Information processing device 110 Communications Department 120 Control section 121 Generation part 122 Update Department 130 Storage section

Claims

1. a generation unit that generates a Kalman filter of a target process for time series data input to a state space model, calculates a predictive distribution using the Kalman filter, and generates samples of parameters of the state space model based on the predictive distribution; an update unit that calculates a predictive distribution using the Kalman filter generated by the generation unit, and updates weights of samples of the parameters generated by the generation unit based on the predictive distribution; 13. An information processing device comprising:

2. The generation unit generates a Kalman filter of a target process for time series data input to a nonlinear non-Gaussian state space model, calculates a predictive distribution using the Kalman filter, and generates samples of parameters of the nonlinear non-Gaussian state space model based on the predictive distribution.

2. The information processing apparatus according to claim 1,

3. The information processing apparatus according to claim 1 , wherein the generation unit generates samples of the parameters of the state space model by using a transition kernel designed so that a parameter distribution of the state space model becomes a stationary distribution.

4. 4. The information processing apparatus according to claim 3, wherein the generation unit generates samples of the parameters of the state space model by using a transition kernel designed to satisfy a detailed balance condition.

5. The update unit accumulates weights of the samples of the parameters generated by the generation unit based on the predictive distribution, and updates the weights of the samples of the parameters generated by the generation unit.

2. The information processing apparatus according to claim 1,

6. An information processing method executed by an information processing device, a generation step of generating a Kalman filter of a target process for time series data input to a state space model, calculating a predictive distribution using the Kalman filter, and generating samples of parameters of the state space model based on the predictive distribution; an updating step of calculating a predictive distribution using the Kalman filter generated by the generating step, and updating weights of samples of the parameters generated by the generating step based on the predictive distribution; 13. An information processing method comprising:

7. a generation step of generating a Kalman filter of a target process for time series data input to a state space model, calculating a predictive distribution using the Kalman filter, and generating samples of parameters of the state space model based on the predictive distribution; an updating step of calculating a predictive distribution using the Kalman filter generated by the generating step, and updating weights of samples of the parameters generated by the generating step based on the predictive distribution; An information processing program characterized by causing a computer to execute the above.

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