Abstract—In time-series analysis, state-space models (SSMs)have been widely used in estimating the conditional probabilitydistributions of hidden variables and parameter values, and inelucidating the structure that can generate the observation data.For example, Kalman filter is utilized to analytically calculatethe conditional probability distribution on linear SSMs in termsof minimizing the variance, and several extensions, such asunscented Kalman filter and particle filter, have been appliedto approximately calculate the distribution on nonlinear SSMs.Recently, approximate Bayesian computation (ABC) has beenapplied to such time-series filtering to handle intractable noises;however, it remains problems in (i) reducing estimation bias,(ii) applying to replicated observations, and (iii) evaluating thevalidity of models. In this paper, to address such problems,we propose a novel method combined with the kernel ABCfor the filtering problem in SSMs. Simulation studies showedthat the proposed method can reduce the estimation bias on theconditional probability distributions comparing to the previousmethods irrespective to the tolerance parameter. In addition, weevaluated the performance of the model selection ability usingtrue and competitive models on synthetic data from nonlinearSSMs. Finally, we applied the method to real observation datain rat circadian oscillation and demonstrated the usefulness inpractical situations.Index Terms—Nonlinear State Space Model, ApproximateBayesian Computation, Time-series Analysis, Likelihood-free InferenceI. INTRODUCTIONFOR the analysis of time-series data, such as economictime-series and intracellular biological observations,state-space models (SSMs) have been widely utilizedto estimate the conditional probability distributions of thehidden variables and the parameter values, and select thebest fitting model to the data. For linear SSMs, Kalmanfilter 1, 2 can analytically calculate the proper conditionalprobability distributions given observation data in terms ofminimizing variances. On the other hand, for models includingnonlinear structures, several approaches, such as ensembleKalman filter 3, unscented Kalman filter 4–6, and particlefilter 7, 8, have been proposed to approximately estimatethe conditional probability distributions. In particular, the particlefilter and its extensions have been widely used to emulateT. Hasegawa is with Health Intelligence Center, The Institute of MedicalScience, The University of Tokyo 4-6-1 Shirokanedai, Minato-ku, Tokyo,Japan (e-mail: [email protected]).K. Kojima, Y. Kawai and M. Nagasaki are with Department ofIntegrative Genomics, Tohoku Medical Megabank Organization, TohokuUniversity, 2-1 Seiryomachi, Aoba-ku, Sendai, Miyagi, Japan (email:[email protected]; [email protected]; [email protected]).Manuscript received XX xx, 20xx; revised XX xx, 20xx.the dynamic behavior of models with nonlinear differentialequations because these can estimate the theoretically validprobability distributions when using an infinite number ofparticles.Although particle filter can estimate the theoretically validconditional distributions, there exist some drawbacks in applyingit to practical problems: (i) the number of uniqueparticles decreases rapidly when the data or the parametervector are high dimension, and (ii) it cannot be applied whenthe likelihood function is analytically intractable. Due to theseproblems, it is difficult to estimate the valid conditional probabilitydistributions with the particle filter in many practicalsituations.To overcome these problems, previous researches haveproposed methods utilizing the approximate Bayesian computation(ABC) to estimate the conditional distributions ofhidden state and parameter variables in frameworks of theSSM. Let y and be the observation data and a particularparameter vector. The original ABC was developed to estimatethe posterior distribution of the parameter vector p(jy) whenthe likelihood function p(yj) is analytically intractable ordifficult to calculate 9, 10. Several extensions have beenproposed that mainly utilize either one of Markov chain MonteCarlo and sequential Monte Carlo (SMC) approaches 11–13. Recently, SMC-ABC has been further utilized to calculatethe conditional probability distributions in the filteringsteps of nonlinear SSM as a framework of ABC filtering 14,15. However, it still remains problems that (i) the consistencyproperty is retained only when the tolerance parameter ?is zero but this compromises the acceptance rate, and (ii)there is no general consensus on how to handle replicatedobservation. In addition, (iii) the previous method cannotcompare the model validity because ? is adaptively selectedon each time-point in order to retain the predefined numberof unique particles 15, 16. To overcome these remainingproblems, we develop a novel filtering method utilizing thekernel Bayes’ rule 17, 18, which was applied previouslyas kernel ABC in the framework of ABC. The utilizationof kernel ABC can resolve the first and second problemssince the consistency property is retained irrespective to atolerance parameter, and we also show that we can handlereplicated observation with retaining the consistency propertyby selecting appropriate kernel functions. Then, by fixingthe tolerance parameter value, Widely-applicable BayesianInformation Criterion (WBIC) 19 is suggested as a way toselect the most suitable model for our proposed procedure.At first, we demonstrate the usefulness of kernel ABC
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