Information processing apparatus, information processing method, and computer-readable storage medium

Abstract

Provided are an information processing device, an information processing method, and a computer-readable storage medium that estimate a posterior distribution of parameters at a high speed for an object that approximates a linear Gaussian state space model. An information processing device (100) according to an embodiment is provided with: a generation unit (121) that generates a Kalman filter for a subject process of time-series data input to a state-space model, calculates a predicted distribution using the Kalman filter, and generates a specimen of parameters of the state-space model on the basis of the predicted distribution; and an update unit (122) that calculates a predicted distribution using the Kalman filter generated by the generation unit (121), and updates the weights of the parameter specimens generated by the generation unit (121) on the basis of the predicted distribution.

Description

Technical Field

[0001] The present invention relates to an information processing device, an information processing method and a computer-readable storage medium. Background Art

[0002] Conventionally, there is a method called Sequential Monte Carlo 2 (SMC2) as a method for estimating the posterior distribution of parameters for a state-space model.

[0003] [Prior art literature]

[0004] [Non-patent literature]

[0005] [Non-patent document 1] N. Chopin and O. Papaspiliopoulos: An Introduction to Sequential Monte Carlo, Springer 2020.

[0006] [Non-patent document 2] Naohisa Funaki, “Probability Theory Lectures: Mathematical Thoughts 20”, Asakura Publishing, 2004 Summary of the invention

[0007] [Problems to be solved by the invention]

[0008] In the prior art, it is sometimes impossible to provide a practical technology for quickly estimating the posterior distribution of parameters for an object that approximates a linear Gaussian state space model. For example, in consideration of industrial applications, high speed and real-time performance are required, but the prior art has the problem of being impractical due to the large amount of calculation. As a problem to be solved by the present invention, the above problem can be cited as an example.

[0009] The present invention is made in view of the above situation, and its object is to provide a practical technology for estimating the posterior distribution of parameters at high speed for an object that approximates a linear Gaussian state-space model.

[0010] [Technical means to solve the problem]

[0011] In order to solve the above-mentioned problems and achieve the purpose, the information processing device of the present invention is characterized in that it has: a generating unit, which generates a Kalman filter for an object process of time series data input into a state-space model, and uses the Kalman filter to calculate a predicted distribution, and generates samples of parameters of the state-space model based on the predicted distribution; and an updating unit, which uses the Kalman filter generated by the generating unit to calculate a predicted distribution, and updates the weights of the samples of the parameters generated by the generating unit based on the predicted distribution.

[0012] [Effects of the Invention]

[0013] According to the present invention, a practical technique for quickly estimating the posterior distribution of parameters for an object approximated by a linear Gaussian state-space model can be provided. BRIEF DESCRIPTION OF THE DRAWINGS

[0014] Figure 1 It is a diagram for explaining the prior art.

[0015] Figure 2 It is a diagram for explaining the prior art.

[0016] Figure 3 It is a diagram for explaining the prior art.

[0017] Figure 4 It is a diagram showing an example of the structure of an information processing device according to an embodiment.

[0018] Figure 5 This is a flowchart showing an example of the overall processing flow of parameter posterior distribution estimation.

[0019] Figure 6 This is a flowchart showing an example of the flow of particle generation based on the system model.

[0020] Figure 7 This is a flowchart showing an example of the flow of weight correction based on the observation model.

[0021] Figure 8 This is a flowchart showing an example of a flow of prediction distribution calculation in the prior art.

[0022] Fig. 9 This is a flowchart showing an example of a flow of predicted distribution calculation performed by the information processing device according to the embodiment.

[0023] Fig.10 This is a flowchart showing an example of a flow of prediction distribution calculation in the prior art.

[0024] Fig.11 This is a flowchart showing an example of a flow of predicted distribution calculation performed by the information processing device according to the embodiment.

[0025] Fig.12 This is a diagram for explaining an example of posterior distribution estimation by the information processing device according to the embodiment.

[0026] Fig.13 This is a diagram for explaining an example of posterior distribution estimation by the information processing device according to the embodiment.

[0027] Fig.14 (1) Fig.14 (2) is a diagram for explaining an example of posterior distribution estimation by an information processing device according to an embodiment.

[0028] Fig.15 This is a diagram for explaining an example of posterior distribution estimation by the information processing device according to the embodiment.

[0029] Fig.16 This is a diagram for explaining an example of posterior distribution estimation by the information processing device according to the embodiment.

[0030] Fig.17 This is a diagram for explaining an example of posterior distribution estimation by the information processing device according to the embodiment.

[0031] Fig.18 (1) Fig.18 (2) is a diagram for explaining an example of posterior distribution estimation by an information processing device according to an embodiment.

[0032] Fig.19 This is a diagram for explaining an example of posterior distribution estimation by the information processing device according to the embodiment.

[0033] Fig. 20 It is a diagram showing an example of a computer that executes an information processing program.

[0034] [Explanation of Symbols]

[0035] 100: Information processing device

[0036] 110: Ministry of Communications

[0037] 120: Control Department

[0038] 121: Generation Department

[0039] 122: Update Department

[0040] 130: Storage DETAILED DESCRIPTION

[0041] Hereinafter, the implementation modes of the information processing device, information processing method and information processing program of the present application will be described in detail with reference to the accompanying drawings. In addition, the present invention is not limited to the described implementation modes. In addition, in the description of the accompanying drawings, the same parts are marked with the same symbols to represent them, and repeated descriptions are appropriately omitted. In addition, when the symbol A is recorded as "A^{~}" below, it is the same as "the symbol "~" is recorded directly above "A".

[0042] [Prior art]

[0043] In the past, there was a method called Sequential Monte Carlo (SMC), which approximated the state variable x at each moment using a set of realization values. t The distribution of Figure 1 As shown, x t is a state variable that is difficult to observe, y t is an observable output variable. t The sequence cannot be observed, so a candidate value group is prepared from the set prior distribution p(x0), and each candidate value is used as the initial value to solve the state equation y t The predicted value group and y t In the successive Monte Carlo method, by repeating the above-mentioned prediction, observation, and correction steps, the set of realized values ​​is used to approximate x. t distribution.

[0044] Here, the idea of ​​using SMC for parameter estimation is that if the state variable x t If the state space model with the estimated parameter θ as the value can be formulated, then SMC can be used in parameter estimation to estimate the posterior distribution p(θ|y) successively ( Figure 2 ).

[0045] And, if Figure 3 As shown in the figure, Sequential Monte Carlo 2 (SMC2), a method for estimating the posterior distribution of parameters for state-space models, uses Figure 1 The state space model has the same structure as shown. t The particle filter (PF: Particle Filter) of the state space model of the particle execution object process is repeated t In the SMC2, the nonlinear non-Gaussian state space model of the process defined by the following equations 1 and 2 is first considered.

[0046] [Number 1]

[0047] x t ~p t (x t |x t-1 ,θ)

[0048] [Number 2]

[0049] y t ~p t (y t |x t ,θ)

[0050] Next, consider the transition kernel m represented by the following equation 3 based on appropriate design. t The Markov Chain Monte Carlo method (MCMC) is used to generate samples of the posterior distribution of the parameter θ of the state space model defined by the above equations 1 and 2.

[0051] [Number 3]

[0052] θ t ~m t (θ t |θ t-1 )

[0053] As a transition kernel design method, the Metropolis-Hastings method (MH method) which is a type of MCMC is used. In the MH method, in order to appropriately design a transition kernel, the following equations 4 and 5 are used to correct an arbitrary proposed kernel m by the MH ratio r. t ^{~}.

[0054] [Number 4]

[0055]

[0056] [Number 5]

[0057]

[0058] The sample generation method described so far has two problems. The first problem is that it is impossible to analytically determine the likelihood p(y 0:t-1 |θ). The second point is to propose kernel m t ^{~} is difficult to tune.

[0059] Regarding the first problem, there is a solution to obtain the likelihood in numerical form as shown in the following equation 6.

[0060] [Number 6]

[0061]

[0062] p in the above equation 6 k (y k |x k , θ) is the observation model of formula 2, p k (x k |y 0:k-1 ,θ)(k=1,…,t-1) is the predicted distribution. In addition, the following formula 7 is used to calculate the predicted distribution pk (x k |y 0:k-1 ,θ) weighted samples for Monte Carlo approximation can be obtained through PF.

[0063] [Number 7]

[0064]

[0065] Next, the solution to the second problem is described. First, as shown in the following equation 8, the proposed core m t ^{~}, considering the average θ t-1 , a Gaussian distribution with covariance matrix Σ.

[0066] [Number 8]

[0067]

[0068] At this time, the so-called proposal kernel m t The tuning parameters of ^{~} are the covariance matrix Σ and the number of transitions of the Gaussian distribution. Here, the sample (particle) θ generated at time t is used. t And the output measured value y observed before 0:t-1 , output y t The predicted distribution of can be calculated as shown in the following formula 9.

[0069] [Number 9]

[0070] p t (y t |y 0:t-1 ,θ t )=∫p t (y t |x t ,θ t ) t (x t |y 0:t-1 ,θ t )dx t Among them, the prediction distribution p t (x t |y 0:t-1 ,θ t ) can be calculated by PF. The above sample generation of the parameter θ using the MCMC kernel, the sample and the output prediction model of the observed data to date are combined, which can be understood as a state space model such as the above formula 3 and the following formula 10.

[0071] [Number 10]

[0072] y t ~p t (y t|y 0:t-1 ,θ t )

[0073] By applying PF to the state space model shown in Equations 3 and 10, the distribution of the parameter θ corrected when the output is observed at each moment can be obtained. Therefore, based on the distribution, the covariance matrix Σ and the number of transitions as tuning parameters can be adaptively changed.

[0074] [Problems of the prior art]

[0075] Next, the problems of the prior art are explained. In SMC2, which is the prior art, PF is constructed doubly and posterior distribution estimation is performed. In the first PF (hereinafter referred to as PF1), the sample generation based on the parameter posterior distribution using MCMC and the output prediction of the sample (Formula 3, Formula 10) are regarded as the state space model to construct the PF. In the sample generation using the system model (Formula 3), in the calculation of the MH ratio r (Formula 4, Formula 5), ​​in order to obtain the likelihood p(y 0:t-1 , θ), and constructs a second PF (hereinafter referred to as PF2) related to the state space model of the target process specified by each sample (Formula 1, Formula 2). The following shows the amount of calculation in the algorithm.

[0076] First, the amount of calculation when the state space model (Formula 3 and Formula 10) is advanced by one step is shown. When the system model (Formula 3) is advanced by one step, the number of transitions M is repeated. t The use of the proposed kernel m t ^{~} sample generation and evaluation of the sample based on the MH ratio r. Wherein, the number of transfers M t In the calculation of the MH ratio r, the likelihood p(y 0:t-1 |θ) is calculated based on PF2. If the amount of calculation for one step of PF2 is set to τ PF2 , then the likelihood p(y 0:t-1 |θ) The amount of calculation becomes τ PF2 t. The calculation is performed for the number of transfers M t times, so the amount of calculation when the system model (Formula 3) is advanced one step becomes τ PF2 T t .

[0077] Next, let us consider the calculation of the observation model (Formula 10). At this point in time, the weighted samples of PF2 are only those shown in Formula 11 below. Therefore, in order to generate the approximate prediction distribution p t (x t |y 0:t-1 ,θ t) weighted samples (Formula 12 below), it is necessary to make PF2 one step forward. Therefore, the total τ PF2 (tM t +1) computational effort.

[0078] [Number 11]

[0079]

[0080] [Number 12]

[0081]

[0082] Next, the calculation amount of PF1 as a whole is shown. If the number of particles of PF1 is set to N θ , then construct N in SMC2 θ PF2. Therefore, in N θ The above-mentioned “operation of advancing the state space model (Formula 3 and Formula 10) by one step” is performed at each time point. That is, if the length of the target time series data is set to T, the total amount of calculation can be expressed by the following Formula 13.

[0083] [Number 13]

[0084]

[0085] If the number of particles of PF2 is set to N x , then the computational complexity of one step of PF2 is O(N x ). Therefore, if the number of transfers M t The maximum value of is set to M^{-}, then the amount of calculation is O(N x N θ TM^{-}). The amount of calculation and the number of particles N of parameter distribution θ , the number of particles N of the state distribution of the object process x , data length T, so SMC2 is a method with a large amount of calculation. If SMC2 is considered for industrial application, higher speed and real-time performance are required. However, when real-time performance is required for the estimation of the posterior distribution of parameters, there is a problem that the amount of calculation is not practical.

[0086] Therefore, the information processing device 100 of the embodiment generates a Kalman filter for the object process of the time series data input to the state space model, and uses the Kalman filter to calculate the prediction distribution, and generates samples of the parameters of the state space model based on the prediction distribution, uses the generated Kalman filter to calculate the prediction distribution, and updates the weights of the generated parameter samples based on the prediction distribution. Thus, the information processing device 100 can provide a practical technology for estimating the posterior distribution of parameters at high speed for an object that approximates a linear Gaussian state space model.

[0087] [Structure of information processing device]

[0088] Next, use Figure 4 The configuration of the information processing device 100 will be described. Figure 4 1 is a diagram showing an example of the structure of the information processing device 100. Figure 4 As shown in FIG. 1 , the information processing device 100 includes a communication unit 110, a control unit 120, and a storage unit 130. In addition, each of these units may be held in a distributed manner by a plurality of devices. The processing of each unit will be described below.

[0089] The communication unit 110 is implemented by a network interface card (NIC) or the like, and can communicate with the control unit 120 via an electric communication line such as a local area network (LAN) or the Internet. For example, the communication unit 110 can communicate with the control unit 120.

[0090] The storage unit 130 is implemented by a semiconductor memory element such as a random access memory (RAM), a flash memory (FlashMemory), or a storage device such as a hard disk or an optical disk. The information stored in the storage unit 130 includes, for example, state space models, time series data, various algorithms, samples, parameters, prior distributions, predicted distributions, likelihoods, and other information required for generating and updating samples. In addition, the information stored in the storage unit 130 is not limited to the above examples.

[0091] The control unit 120 is implemented using a central processing unit (CPU), a network processor (NP), or a field programmable gate array (FPGA), etc., and executes a processing program stored in a memory. Figure 4 As shown in FIG. 1 , the control unit 120 includes a generating unit 121 and an updating unit 122. Hereinafter, each unit included in the control unit 120 will be described.

[0092] The generator 121 generates a Kalman filter for the target process of the time series data input to the state-space model, calculates a prediction distribution using the Kalman filter, and generates a sample of the parameter of the state-space model based on the prediction distribution. For example, the generator 121 generates a Kalman filter for the target process of the time series data input to the state-space model, calculates a prediction distribution using the Kalman filter, and generates a sample of the parameter of the state-space model based on the prediction distribution through the processing of step S4 in the [flow chart] described later.

[0093] In addition, the generation unit 121 generates a Kalman filter for the object process of the time series data input to the nonlinear non-Gaussian state space model, and uses the Kalman filter to calculate the predicted distribution, and generates a sample of the parameter of the nonlinear non-Gaussian state space model based on the predicted distribution. For example, the generation unit 121 generates a Kalman filter for the object process of the time series data input to the nonlinear non-Gaussian state space model, and uses the Kalman filter to calculate the predicted distribution, and generates a sample of the parameter of the nonlinear non-Gaussian state space model based on the predicted distribution through the processing of step S4 in the [flowchart] described later. The details of the processing performed by the generation unit 121 are described in the [parameter posterior distribution estimation method using an information processing device] [flowchart] described later.

[0094] In addition, the generator 121 generates samples of the parameters of the state-space model using a properly designed transition kernel. For example, the generator 121 generates samples of the parameters of the state-space model using a transition kernel designed in such a way that the distribution of the parameters of the state-space model becomes a stable distribution. Here, when the transition kernel satisfies the ergodicity, the distribution of the parameters of the state-space model becomes a stable distribution. For details on the ergodicity, please refer to non-patent document 2.

[0095] That is, the generator 121 generates samples of the parameters of the state-space model using the transition kernel designed in a manner that satisfies the ergodicity. Here, the transition kernel that satisfies the detailed balance condition shown in the following formula 14 is a transition kernel designed in a manner that the parameter distribution of the state-space model becomes a stable distribution. That is, the transition kernel that satisfies the detailed balance condition shown in formula 14 is a transition kernel set in a manner that the parameter distribution of the state-space model becomes a stable distribution.

[0096] [Number 14]

[0097] q(θ1):q(θ0)=m(θ1|θ0):m(θ0|θ1)

[0098] More specifically, the generator 121 corrects the sample of the state-space model parameters using the equation 5 as the correction coefficient for satisfying the detailed balance condition, and generates the sample of the state-space model parameters using the transition kernel designed to satisfy the detailed balance condition shown in the equation 14.

[0099] The updating unit 122 calculates the prediction distribution using the Kalman filter generated by the generating unit 121, and updates (corrects) the weight of the sample of the parameter generated by the generating unit 121 based on the prediction distribution. For example, the updating unit 122 calculates the prediction distribution using the Kalman filter generated by the generating unit 121 through the processing of step S5 in the [flow chart] described later, and updates the weight of the sample of the parameter generated by the generating unit 121 based on the prediction distribution.

[0100] For example, the updating unit 122 accumulates the weights of the samples of the parameters generated by the generating unit 121 based on the predicted distribution, and updates the weights of the samples of the parameters generated by the generating unit 121. Details of the processing performed by the updating unit 122 will be described in [Parameter posterior distribution estimation method using information processing device] [Flowchart] described later.

[0101] [Parameter posterior distribution estimation method using information processing device]

[0102] The method for estimating the posterior distribution of parameters of the information processing device 100 using the embodiment is described. First, consider the linear Gaussian state space model of a certain process defined by the following equations 15, 16, and 17. In which, N(x|μ, Σ) represents the probability density function of the multivariate Gaussian distribution with mean μ and covariance matrix Σ.

[0103] [Number 15]

[0104] x t =A(θ)x t-1 +B(θ)u t +H(θ)v t

[0105] [Number 16]

[0106] y t =C(θ)x t +w t

[0107] [Number 17]

[0108] p(v t )=N(v t |0,Q t (θ)),p(w t )=N(wt |0,R t (θ)

[0109] Consider the transfer kernel m shown in the above formula 3 based on appropriate design t MCMC generates samples of the posterior distribution of the parameter θ that follows the state-space model (Formula 1, Formula 2). As in the prior art, the MH method is used as a design method for the transition kernel. In the MH method, in order to appropriately design the transition kernel, the above-mentioned Formula 4 and Formula 5 are used to correct any proposed kernel m by the MH ratio r. t ^{~}.

[0110] Therefore, as in the prior art, there is a problem that the likelihood p(y 0:t-1 |θ) and the proposed core m t ^{~} is a difficult issue to tune.

[0111] For the first problem, as shown in the following equation 18, the likelihood p(y 0:t-1 |θ).

[0112] [Number 18]

[0113]

[0114] p in the above equation 18 k (y k |x k , θ) is the observation model of formula 2, p k (x k |y 0:k-1 ,θ)(k=1,…,t-1) is the prediction distribution. In the case of a linear Gaussian state space model, the prediction distribution becomes a Gaussian distribution with a mean value x k | k-1 And the covariance matrix v k | k-1 It is obtained by Kalman filter (KF). Therefore, we use the x obtained by KF. k | k-1 、v k | k-1 (k=1,…,t-1), we can obtain the following equation 19, and thus we can analytically obtain the likelihood p(y 0:t-1 |θ).

[0115] [Number 19]

[0116] N(y k |C(θ)x k|k-1 ,C(θ)Vk|k-1 C(θ) T +R k (θ))(k=1,...,t-1)

[0117] Next, the solution to the second problem is described. First, as shown in the above formula 8, as the proposed core m t ^{~}, considering the average θ t-1 , the Gaussian distribution of the covariance matrix Σ. At this time, the so-called proposed kernel m t The tuning parameters of ^{~} are the covariance matrix Σ and the number of transitions of the Gaussian distribution. Here, we use the sample m generated at time t. t ^{~} and the output measured value y observed before 0:t-1 , output y t The predicted distribution of can be calculated as described in Formula 9.

[0118] Here, the predicted distribution p t (x t |y 0:t-1 ,θ t ) can be calculated by KF, so it can be calculated analytically as shown in the following formula 20.

[0119] [Number 20]

[0120] =N(C(θ)x t|t-1 ,C(θ)V t|t-1 C(θ) T +R t (θ)

[0121] The above sample generation of the parameter θ using the MCMC kernel, the sample generation, and the output prediction model based on the observed data to date are combined, which can be understood as a state space model such as the above-mentioned equations 3 and 10.

[0122] By applying PF to the state space model shown in Equations 3 and 10, the distribution of the parameter θ corrected when the output is observed at each moment can be obtained. Therefore, based on the distribution, the covariance matrix Σ and the number of transitions as tuning parameters can be adaptively changed.

[0123] Next, the principle of compressing the amount of calculation by the parameter posterior distribution estimation method using the information processing device 100 of the embodiment is described. In the parameter posterior distribution estimation method using the information processing device 100 of the embodiment, it is considered to approximate the nonlinear non-Gaussian state space model (Formula 1, Formula 2) of the target process to a linear Gaussian state space model such as the following Formula 21, Formula 22, and Formula 23.

[0124] [Number 21]

[0125] x t =Ax t-1 +Bu t +Hv t

[0126] [Number 22]

[0127] y t =Cx t +w t

[0128] [Number 23]

[0129] v t ~N(0,Q t ),w t ~N(0,R t )

[0130] Here, consider the prediction distribution p k (x k |y 0:k-1 The successive Bayesian filter used in the calculation of (k=1, ..., t-1) is set to KF instead of PF. If the KF used instead of PF2 is set to KF2, the predicted distribution can be analytically obtained as a Gaussian distribution such as the following formula 24, formula 25, and formula 26.

[0131] [Number 24]

[0132] p k (x k |y 0:k-1 ,θ)=N(x t|t-1 ,V t|t-1 )

[0133] [Number 25]

[0134] x t|t-1 =Ax t-1|t-1 +Bu t

[0135] [Number 26]

[0136] V t|t-1 =AV t-1|t-1 A T +HQ t H T

[0137] Among them, x t-1 | t-1 、v t-1 | t-1 are the mean and variance covariance matrix of the filter distribution at each moment. Therefore, the output y tThe predicted distribution of is also analytically obtained as a Gaussian distribution as shown in the following equation 27.

[0138] [Number 27]

[0139] p t (y t |y 0:t-1 ,θ t )=N(Cx t|t-1 ,CV t|t-1 C T +R t )

[0140] If KF is used, Monte Carlo approximation is not required, so the calculation time can be shortened. Let the calculation amount of one step of KF2 be τ KF2 By setting PF2 to KF2, the τ in the calculation amount shown in Formula 13 is PF2 Replace all with τ KF2 If the dimension of KF2 is set to d x , then the amount of calculation for one step of KF2 is expressed by the following formula 28.

[0141] [Number 28]

[0142]

[0143] Here, the number of particles of the state variable which is a parameter that can be set for PF2 is N. x In order to use PF2 to estimate with the same accuracy as KF2, N x By increasing and setting, the relationship shown in the following equation 29 is formed. In other words, in the posterior distribution estimation method using the information processing device 100 of the embodiment as the approximation method using KF2, the amount of calculation can be compressed.

[0144] [Number 29]

[0145]

[0146] Thus, in the parameter posterior distribution estimation method of the information processing device 100 using the implementation method, for SMC2, which is a parameter posterior distribution estimation method of a state space model, the object process is limited to a linear Gaussian state space model, and the successive Bayesian filter used for predictive distribution calculation is replaced from a particle filter to a Kalman filter, thereby compressing the amount of calculation.

[0147] [flow chart]

[0148] In the SMC2 of the prior art and the parameter posterior distribution estimation method using the information processing device 100 of the embodiment, the overall processing flow is common. Figure 5 to Figure 7A description will be given of a processing flow common to the SMC2 of the related art and the estimation of the posterior distribution of parameters using the information processing device 100 of the embodiment. Figure 5 : is a flowchart showing an example of the overall processing flow of parameter posterior distribution estimation. In addition, the information processing device 100 of the embodiment can perform the processing of step S4 and step S5 in the series of processing.

[0149] like Figure 5 As shown, first input the time series data y o:T (Step S1) Next, as shown in the following equation 30, weighted samples are generated based on the prior distribution (Step S2).

[0150] [Number 30]

[0151]

[0152] Then, resampling and weight initialization are performed according to the situation (step S3). For example, samples with low likelihood are deleted, or samples with high likelihood are copied. Then, particles are generated based on the MCMC system model shown in the above formula 3 (step S4). Step S4 is processed by the generation unit 121 of the information processing device 100.

[0153] Here, use Figure 6 The process flow (step S4 ) of particle generation in the system model based on MCMC will be described. Figure 6 This is a diagram for explaining the process flow of particle generation based on the system model. Figure 6 As shown, first, data represented by the following equation 31 is input (step S4-1).

[0154] [Number 31]

[0155]

[0156] Next, an initial value is set as shown in the following equation 32 (step S4-2).

[0157] [Number 32]

[0158]

[0159] Next, based on the random number shown in the following equation 33, a sample represented by the following equation 34 is generated (step S4-3).

[0160] [Number 33]

[0161]

[0162] [Number 34]

[0163]

[0164] Next, the predicted distribution represented by the following equation 35 is calculated (step S4-4). The details of the processing in step S4-4 will be described later.

[0165] [Number 35]

[0166]

[0167] Then, the samples expressed by the above equation 34 are corrected by the following equation 36 based on the predicted distribution (step S4-5). Here, for example, the processing of steps S4-3 to S4-5 is a processing for correcting the samples of the parameters of the state space model using the above equation 5.

[0168] [Number 36]

[0169]

[0170] Then, whether i=1, ..., N θ Here, if it is determined that i=1, ..., N θ If it is within the range (step S4-6 "Yes"), return to step S4-3 and repeat the processing up to step S4-5.

[0171] On the other hand, when it is determined that i=1, ..., N θ In the case where the sample (i=1 to N) represented by the above formula 34 is within the range (step S4-6 "No"), θ ) satisfies the convergence condition to determine (step S4-7). In addition, as a convergence condition, for example, the situation where the distribution after repeated calculations does not change much from the previous calculation can be listed. Here, when it is determined that the convergence condition is not satisfied (step S4-7 "No"), the processing shown in the following formula 37 is performed, and the process returns to step S4-3 again, and the processing up to step S4-6 is repeated.

[0172] [Number 37]

[0173]

[0174] On the other hand, when it is determined that the convergence condition is satisfied (step S4-7: Yes), the weighted samples expressed by the following equation 38 are output (step S4-9). In this way, the particle generation process based on the system model is performed.

[0175] [Number 38]

[0176]

[0177] Return to Figure 5 Then, the weight of the observation model is corrected based on the MCMC represented by the above-mentioned equation 10 (step S5). Step S5 is a process performed by the updating unit 122 of the information processing device 100.

[0178] Here, use Figure 7 The processing flow (step S5) in weight correction based on the observation model will be described. Figure 7 This is a diagram for explaining the processing flow in weight correction based on the observation model. Figure 7 As shown, first, data represented by the following equation 39 is input (step S5-1).

[0179] [Number 39]

[0180]

[0181] Next, the prediction distribution represented by the following equation 40 is calculated (step S5-2). The details of the processing in step S5-2 will be described later.

[0182] [Number 40]

[0183]

[0184] Then, according to the predicted distribution p t (x t |y 0:t-1 ,θ t ) Calculate the following formula 41 (step S5-3).

[0185] [Number 41]

[0186]

[0187] Then, the weights are accumulated as shown in the following equation 42 (step S5-4).

[0188] [Number 42]

[0189]

[0190] Then, the weighted sample represented by the following equation 43 is output (step S5-5). In this way, the weight correction process based on the observation model is performed.

[0191] [Number 43]

[0192]

[0193] Return to Figure 5 Then, the weight is normalized as shown in the following equation 44 (step S6).

[0194] [Number 44]

[0195]

[0196] Then, the covariance matrix Σ of the proposed kernel is calculated based on the weighted standard shown in the following formula 45: t Correction is performed (step S7).

[0197] [Number 45]

[0198]

[0199] Then, it is determined whether it is within the range of t=1, ..., T (step S8). Here, when it is determined that it is within the range of t=1, ..., T (step S8 "Yes"), return to step S3 and repeat the processing up to step S7. On the other hand, when 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 processing flow common to the parameter posterior distribution estimation method of SMC2 of the prior art and the information processing device 100 using the implementation method.

[0200] [Number 46]

[0201]

[0202] So far, the common processing flow in the method for estimating the posterior distribution of parameters in the SMC2 of the prior art and the information processing device 100 using the embodiment has been described. However, from now on, Figure 8 to Figure 11 , the differences (mainly steps S4-4 and S5-2) in the method of calculating the predicted distribution of state variables using the prior art SMC2 and the parameter posterior distribution estimation method using the information processing device 100 of the implementation method are explained.

[0203] First, use Figure 8 The processing of step S4-4 according to the conventional technology will be described. Figure 8 This is a diagram for explaining the processing based on the prior art. First, the data represented by the following equation 47 is input (step S4-4-1).

[0204] [Number 47]

[0205]

[0206] Next, as shown in the following equation 48, a particle filter of the target process is generated (step S4-4-2).

[0207] [Number 48]

[0208]

[0209] Next, the particle filter shown in the above formula 48 is executed until the time k=1, ..., t-1, and the Monte Carlo approximation of the predicted distribution (the above formula 35) is calculated as shown in the following formula 49 (step S4-4-3).

[0210] [Number 49]

[0211]

[0212] Then, the data represented by the following equation 50 is output (step S4-4-4).

[0213] [Number 50]

[0214]

[0215] Next, use Fig. 9 The process of step S4 - 4 by the information processing device 100 according to the embodiment will be described. Fig. 9 4-1-1*). Then, the information processing device 100 generates a Kalman filter of the target process as shown in the following equation 51 (step S4-4-2*).

[0216] [Number 51]

[0217]

[0218] Next, the information processing device 100 executes the Kalman filter shown in the above-mentioned formula 51 until the time k=1, ..., t-1, and calculates the predicted distribution (the above-mentioned formula 35) as shown in the following formulas 52, 53, and 54 (step S4-4-3*).

[0219] [Number 52]

[0220]

[0221] [Number 53]

[0222]

[0223] [Number 54]

[0224]

[0225] Then, the information processing device 100 outputs the mean value x of the predicted distribution. k| k-1 , and the covariance matrix v k | k-1 , (step S4-4-4*).

[0226] Next, use Fig.10 The processing of step S5-2 according to the conventional technology will be described. Fig.10 This is a diagram for explaining the processing based on the prior art. First, the data represented by the following equation 55 is input (step S5-2-1).

[0227] [Number 55]

[0228]

[0229] Then, using the execution result of the generated particle filter (Formula 48 above), the prediction distribution p is calculated as shown in the following Formula 56. t (x t |y 0:t-1 ,θ t ) (step S5-2-2).

[0230] [Number 56]

[0231]

[0232] Then, the data represented by the following equation 57 is output (step S5-2-3).

[0233] [Number 57]

[0234]

[0235] Next, use Fig.11 The process of step S5 - 2 by the information processing device 100 according to the embodiment will be described. Fig.11 55 is a diagram showing an example of processing by the information processing device 100 according to the embodiment. First, the data represented by the above-mentioned equation 55 is input to the information processing device 100 (step S5-2-1*).

[0236] Then, the information processing device 100 uses the execution result of the generated Kalman filter (Formula 51) to calculate the prediction distribution p as shown in the following Formulas 58, 59, and 60. t (x t |y 0:t-1 ,θ t )(Step S5-2-2*).

[0237] [Number 58]

[0238]

[0239] [Number 59]

[0240]

[0241] [Number 60]

[0242]

[0243] Then, the information processing device 100 outputs the mean value x of the predicted distribution. t | t-1 , and the covariance matrix v t | t-1 (Step S5-2-3*).

[0244] That is, the information processing device 100 of the embodiment uses a Kalman filter instead of a particle filter to calculate the predicted distribution of the state variable. This provides a practical technique for quickly estimating the posterior distribution of parameters for an object that approximates a linear Gaussian state space model.

[0245] [Estimation results of parameter posterior distribution]

[0246] Next, use Figures 12 to 19 The estimation result of the posterior distribution of the parameters of the information processing device 100 using the embodiment is described. First, the example of the system assuming 1 input and 1 output shown in the following equations 61 to 64 is described. In addition, the true value of the parameter is G1(-2), T1(15), alpha (alpha) (0.01), beta (beta) (0.1).

[0247] [Number 61]

[0248] x k =Ax k-1 +Bu k +v k

[0249] [Number 62]

[0250] y k =Cx k +w k

[0251] [Number 63]

[0252]

[0253] [Number 64]

[0254] x0~N(0,1),v k ~N(0,α 2 ),w k ~N(0,β2 )

[0255] like Fig.12 As shown, the input data u of the object process o:T And the output data y o:T As input, an output is generated based on the state space model of the target process. Here, as a condition setting of the distribution estimation algorithm, the state space model is represented by the above-mentioned equations 61 to 64. In addition, the prior distribution of the parameter is represented by the following equation 65.

[0256] [Number 65]

[0257] G1~N(2,10 2 ),T1~U(5,20),α,β~Ga(1,2)

[0258] In addition, the number of particles is set to N θ =50, N x =50. In addition, N x Only used in SMC2.

[0259] As a result of estimating the posterior distribution of the parameters by setting the above conditions, the calculation time, the shape of the posterior distribution of the parameters, and the representative value of the posterior distribution of the parameters are described respectively. First, Fig.13 As shown, in SMC2 as the conventional method, the calculation time is 61.068 seconds, whereas in the estimation of the posterior distribution of parameters using the information processing device 100 of the embodiment, the calculation time is 28.671 seconds. That is, in the estimation of the posterior distribution of parameters using the information processing device 100 of the embodiment, the calculation time can be shortened to less than half compared with the conventional technology.

[0260] Next, use Fig.14 (1) Fig.14 (2), for the shape of the posterior distribution of the parameter, represents the approximate posterior distribution p(θ|u o:T ,y o:T ) is a representative example (G1) of the histogram of the sample (Formula 33). Fig.14 (1) Fig.14 As shown in (2), the parameter posterior distribution ( Fig.14 (1)), and the parameter posterior distribution estimated by SMC2 as the prior art ( Fig.14 (2)) are all histograms with the average value near -2.1.

[0261] Next, use Fig.15The following describes the parameter posterior distribution estimation based on the information processing device 100 using the embodiment and the representative value of the parameter posterior distribution of SMC2 as the prior art. When G1 is described as a representative example, Fig.15 As shown, in the estimation of the posterior distribution of parameters using the information processing device 100 of the embodiment, the mean value is -2.110 and the variance is 0.00206, while in SMC2 as the prior art, the mean value is -2.105 and the variance is 0.00251. That is, in the estimation of the posterior distribution of parameters using the information processing device 100 of the embodiment, the results tend to be that the mean value is the same as that of the prior art and the variance is smaller than that of the prior art, and the estimation can be performed with the same or higher accuracy than that of the prior art. In addition, only G1 is described in the above content, but as shown in Fig.15 As shown, the same results were obtained in T1, alpha, and beta.

[0262] Next, an example of a system assuming 3 inputs and 1 output, represented by the above-mentioned equations 61 and 62 and the following equations 66 to 68, is described. In addition, 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).

[0263] [Number 66]

[0264]

[0265] [Number 67]

[0266]

[0267] [Number 68]

[0268] x0~N(0,I n ),v k ~N(0,α 2 I n ),w k ~N(0,β 2 I n )

[0269] like Fig.16 As shown, the input data u of the object process o:T And the output data y o:T As input, an output is generated based on the state space model of the target process. Here, as a condition setting of the distribution estimation algorithm, the state space model is represented by the above-mentioned equations 61, 62, 66 and 67. In addition, the prior distribution of the parameter is represented by the following equation 69.

[0270] [Number 69]

[0271] G1~N(1,10 2 ),G2~N(1,10 2 ),G3~N(1,10 2 ),T1,T2,T3~N(5,20),α,β~Ga(1,2)

[0272] In addition, the number of particles is set to N θ =150, N x =100. In addition, N x Only used in SMC2.

[0273] As a result of estimating the posterior distribution of the parameters by setting the above conditions, the calculation time, the shape of the posterior distribution of the parameters, and the representative value of the posterior distribution of the parameters are described respectively. First, Fig.17 As shown, in SMC2 as the conventional method, the calculation time is 716.722s, whereas in the estimation of the posterior distribution of parameters using the information processing device 100 of the embodiment, the calculation time is 304.839s. That is, in the estimation of the posterior distribution of parameters using the information processing device 100 of the embodiment, the calculation time can be shortened to less than half compared with the conventional technology.

[0274] Next, use Fig.18 (1) Fig.18 (2), for the shape of the posterior distribution of the parameter, represents the approximate posterior distribution p(θ|u o:T ,y o:T ) is a representative example (G1) of the histogram of the sample (Formula 33). Fig.18 (1) Fig.18 As shown in (2), the parameter posterior distribution ( Fig.18 (1)), and the parameter posterior distribution estimated by SMC2 as the prior art ( Fig.18 (2)) are all histograms with the average value near -2.07.

[0275] Next, use Fig.19 The representative values ​​of the parameter posterior distribution of the information processing device 100 according to the embodiment and the SMC2 as the prior art are described. When G1 is described as a representative example, Fig.19As shown, in the information processing device 100 of the embodiment, the mean value is -2.07445 and the variance is 0.008841, while in SMC2 as the prior art, the mean value is -2.07746 and the variance is 0.012498. That is, in the estimation of the posterior distribution of the parameters using the information processing device 100 of the embodiment, the results obtained are that the mean value is closer to the true value than the prior art, the variance is smaller than the prior art, and the estimation can be performed with the same degree or higher accuracy than the prior art. In addition, only G1 is described in the above content, but as Fig.19 As shown, the same trend of results was obtained in G2, G3, T1, T2, alpha, and beta.

[0276] [Effect]

[0277] The information processing device 100 of the implementation mode comprises: a generating unit 121, which generates a Kalman filter for an object process of time series data input into a state-space model, calculates a prediction distribution using the Kalman filter, and generates samples of parameters of the state-space model based on the prediction distribution; and an updating unit 122, which calculates a prediction distribution using the Kalman filter generated by the generating unit 121, and updates the weights of the samples of the parameters generated by the generating unit 121 based on the prediction distribution.

[0278] Thus, the information processing device 100 can provide a practical technology for using a Kalman filter to perform likelihood calculations for output measured values ​​of a state-space model, and for estimating parameter posterior distributions at high speed for an object approximating a linear Gaussian state-space model.

[0279] In addition, the information processing device 100 can shorten the calculation time required for estimating the posterior distribution of parameters compared to the prior art. Figure 12 to Figure 15 Examples and Figure 16 to Figure 19 In the example of , the calculation time can be shortened to less than half of that of the prior art. In addition, the information processing device 100 can estimate the posterior distribution of the parameter with the same degree of accuracy or higher than that of the prior art.

[0280] The generation unit 121 of the information processing device 100 of the embodiment generates a Kalman filter for the target process of time series data input to the nonlinear non-Gaussian state-space model, calculates a prediction distribution using the Kalman filter, and generates a sample of the parameters of the nonlinear non-Gaussian state-space model based on the prediction distribution.

[0281] Therefore, the information processing device 100 can provide a practical technology for using a Kalman filter to calculate the likelihood of the output measured value of a nonlinear non-Gaussian state space model, and for an object that approximates a linear Gaussian state space model, to quickly estimate the posterior distribution of parameters.

[0282] The generation unit 121 of the information processing device 100 of the embodiment generates samples of the parameters of the state space model using a transition kernel designed so that the parameter distribution becomes a stable distribution. Thus, the information processing device 100 can provide the following practical technology: using the transition kernel designed so that the parameter distribution becomes a stable distribution to generate samples of the parameters of the state space model, using the Kalman filter to perform likelihood calculation for the output measured value of the state space model, and for an object that approximates a linear Gaussian state space model, a practical technology for quickly estimating the posterior distribution of parameters.

[0283] The generation unit 121 of the information processing device 100 according to the embodiment generates samples of parameters of the state-space model using transition kernels designed so as to satisfy the detailed balance condition.

[0284] Therefore, the information processing device 100 can provide the following practical technology: using a transition kernel designed in a manner to satisfy a detailed balance condition to generate samples of the parameters of the state-space model, using a Kalman filter to calculate the likelihood of the output measured values ​​of the state-space model, and for an object that approximates a linear Gaussian state-space model, a practical technology for quickly estimating the posterior distribution of parameters.

[0285] The updating unit 122 of the information processing device 100 according to the embodiment accumulates the weights of the samples of the parameters generated by the generating unit 121 based on the predicted distribution, and updates the weights of the samples of the parameters generated by the generating unit 121 .

[0286] Thus, the information processing device 100 can provide the following practical technology: by accumulating and updating the weights of parameter samples based on the prediction distribution calculated using the Kalman filter, a practical technology for quickly estimating the posterior distribution of parameters for an object that approximates a linear Gaussian state space model.

[0287] [Hardware structure]

[0288] The information processing device 100 of the above-described embodiment is composed of, for example, Fig. 20 The structure shown is implemented by a computer 1000. Fig. 201 is a hardware configuration diagram showing an example of a computer that realizes the functions of the information processing device 100. The computer 1000 includes a CPU 1100, a RAM 1200, a read-only memory (ROM) 1300, an auxiliary storage device 1400, a communication interface (I / F) 1500, and an input / output I / F (interface) 1600 connected via a bus 1800.

[0289] The CPU 1100 operates based on the programs stored in the ROM 1300 or the auxiliary storage device 1400 to control each unit. The ROM 1300 stores a boot program executed by the CPU 1100 when the computer 1000 is started, a program depending on the hardware of the computer 1000, and the like.

[0290] 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 transmits the data to the CPU 1100, and transmits data generated by the CPU 1100 to other devices via a predetermined communication network. The CPU 1100 controls output devices such as a display or a printer, and input and output devices 1700 such as a keyboard or a mouse via the input and output I / F 1600. The CPU 1100 obtains data from the input and output devices 1700 via the input and output I / F 1600. In addition, the CPU 1100 outputs the generated data to the input and output devices 1700 via the input and output I / F 1600.

[0291] For example, when the computer 1000 functions as the information processing apparatus 100 according to the present embodiment, the CPU 1100 of the computer 1000 implements the function of the control unit 120 by executing a program loaded on the RAM 1200 .

[0292] [other]

[0293] All or part of the processes described in the embodiments and modifications as automatically performed processes may be performed manually, or all or part of the processes described as manually performed processes may be performed automatically by known methods. In addition, the information including the processing sequence, specific names, various data or parameters shown in the documents or drawings may be changed arbitrarily except for the cases specifically described. For example, the various information shown in the drawings is not limited to the information shown in the drawings.

[0294] In addition, the components of each device shown in the figure are conceptual functions and do not necessarily need to be physically configured as shown in the figure. That is, the specific form of the distribution and integration of each device is not limited to the figure, and all or part of them can be functionally or physically distributed and integrated in arbitrary units according to various loads or usage conditions.

[0295] The above-mentioned components include components that can be easily assumed by a person skilled in the art, components that are substantially the same, and components within the so-called equivalent scope. Furthermore, the above-mentioned embodiments and modifications can be appropriately combined within the scope that the processing contents do not conflict.

[0296] In addition, the "section (part (section), module (module), unit (unit)" can be replaced by "means" or "circuit" etc. For example, the control section can be replaced by control means or control circuit.

[0297] Although several embodiments of the present invention have been described in detail above based on the drawings, these are merely examples, and the present invention can be implemented in other forms that are variously modified and improved based on the knowledge of those skilled in the art, starting from the forms described in the disclosure column of the invention.

Claims

1. An information processing device, characterized in that have: a generating unit that generates a Kalman filter for a target process of time series data input to a state-space model, calculates a prediction distribution using the Kalman filter, and generates a sample of a parameter of the state-space model based on the prediction distribution; as well as An updating unit calculates a prediction distribution using the Kalman filter generated by the generating unit, and updates a weight of a sample of the parameter generated by the generating unit based on the prediction distribution.

2. The information processing device according to claim 1, characterized in that The generator generates a Kalman filter for a target process of time series data input to a nonlinear non-Gaussian state-space model, calculates a prediction distribution using the Kalman filter, and generates a sample of a parameter of the nonlinear non-Gaussian state-space model based on the prediction distribution.

3. The information processing device according to claim 1, characterized in that The generation unit generates samples of parameters of the state-space model using a transition kernel designed so that parameter distribution of the state-space model becomes a stable distribution.

4. The information processing device according to claim 3, characterized in that: The generation unit generates a sample of a parameter of the state-space model using a transition kernel designed to satisfy a detailed balance condition.

5. The information processing device according to claim 1, characterized in that The updating unit accumulates the weights of the samples of the parameters generated by the generating unit based on the predicted distribution, and updates the weights of the samples of the parameters generated by the generating unit.

6. An information processing method, which is an information processing method executed by an information processing device, The information processing method is characterized by comprising: a generation step of generating a Kalman filter for a target process of time series data input to a state-space model, calculating a prediction distribution using the Kalman filter, and generating a sample of a parameter of the state-space model based on the prediction distribution; as well as The updating step calculates a prediction distribution using the Kalman filter generated in the generating step, and updates the weights of the samples of the parameters generated in the generating step based on the prediction distribution.

7. A computer-readable storage medium storing an information processing program, characterized in that: Causes the computer to execute: A generating step of generating a Kalman filter for the target process of the time series data input to the state-space model, calculating a prediction distribution using the Kalman filter, and generating a sample of the parameters of the state-space model based on the prediction distribution; as well as An updating step of calculating a prediction distribution using the Kalman filter generated by the generating step, and updating the weights of the samples of the parameters generated by the generating step based on the prediction distribution.

Images