Boolean Kalman filter and smoother under model uncertainty☆
Introduction
Partially-Observed Boolean dynamical systems (POBDS) are a general class of nonlinear state-space models consisting of a hidden Boolean state process observed through an arbitrary noisy mapping to a measurement space. This signal model has many applications in fields such as genomics (Kauffman, 1969), robotics (Imani and Braga-Neto, 2017a, Roli et al., 2011), and digital communication systems (Messerschmitt, 1990). The optimal minimum mean square error (MMSE) state estimators for POBDS are the Boolean Kalman filter (BKF) (Imani, 2019, Imani and Braga-Neto, 2017a, McClenny et al., 2017c) and Boolean Kalman smoother (BKS) (Imani and Braga-Neto, 2015b, Imani and Braga-Neto, 2017a), respectively. In Imani and Braga-Neto (2018c) and McClenny, Imani, and Braga-Neto (2017a), optimal state estimators for POBDS with correlated measurement noise are introduced.
Due to the structure of the multivariate Boolean lattice, the BKF and BKS have the desirable property of yielding both the optimal maximum a posteriori (MAP) and MMSE solutions for each state vector component (Imani and Braga-Neto, 2017a, Imani and Braga-Neto, 2018e, McClenny et al., 2017b), which is not the case for the general multivariate MAP estimator, in general. In addition, exact algorithms are available for the computation of the BKF and BKS (Braga-Neto, 2011, Imani and Braga-Neto, 2017a), which is not the case for the optimal MMSE solution in general nonlinear state-space models, in which case approximate solutions employing sequential Monte-Carlo techniques (also known as particle filters) (Doucet, De Freitas et al., 2001, Doucet et al., 2000, Imani and Braga-Neto, 2018e, Imani et al., 2019, Kantas et al., 2015), the Extended Kalman filter (EKF) (Jazwinski, 1970), the Unscented Kalman filter (UKF) (Julier, Uhlmann, & Durrant-Whyte, 1995), and the Sigma-Point Kalman filter (SPKF) (Van Der Merwe, 2004) must be used. It should be noted that the exact algorithms for the computation of the optimal MMSE estimator are available in the case of a linear-Gaussian state space through the classical Kalman Filter and Smoother (Kalman, 1960), and for POBDS through BKF and BKS.
Exact calculation of the aforementioned optimal estimators requires complete information about the system model; however, in many real-world applications, the system parameters are not fully known and must be estimated. Several techniques have been developed for approximate estimation of general nonlinear non-Gaussian state space models with unknown parameters. The methods can be divided into two main categories of Maximum-Likelihood (ML) and Bayesian techniques. The class of ML techniques includes: (1) direct gradient-based ML techniques, where the idea is to maximize the log-likelihood function using gradient-ascent or quasi-Newton techniques (DeJong et al., 2012, Ionides et al., 2006, Johansen et al., 2008, Malik and Pitt, 2011), (2) Expectation–Maximization (EM) techniques (Schön et al., 2011, Wills et al., 2013), where the idea is to maximize the “complete” log-likelihood function, as opposed to ML-based techniques which maximize the “incomplete” log-likelihood function, using the fact that maximizing the complete log-likelihood is easier than maximizing the incomplete one. There are several particle-based Bayesian techniques for the inference of general nonlinear state-space models (Lindsten et al., 2014, Urteaga et al., 2016, Whiteley et al., 2010). An important representative is the Particle Marginal Metropolis–Hastings (PMMH) method (Andrieu, Doucet, & Holenstein, 2010). Several online particle-based techniques have also been developed for applications when fully-recursive estimation is desired (Crisan, Miguez, et al., 2018).
For POBDS under model uncertainty, maximum-likelihood (ML) and maximum a posteriori (MAP) adaptive estimators were proposed in Imani and Braga-Neto, 2015a, Imani and Braga-Neto, 2017a, Imani and Braga-Neto, 2017b, respectively. These techniques are built on ML and MAP point-based estimators for unknown parameters combined with optimal MMSE state estimators for the state. The drawback of these approaches is their sensitivity to initialization and the requirement of large amount of data for good performance.
We propose in this paper instead an optimal Bayesian filter (OBF) approach to the problem of POBDS recursive estimation. The basic principle is that the unknown true model belongs to an uncertainty class of models and the OBF minimizes the expected cost over the uncertainty class. The idea has roots going back to the 1960s in control theory (Martin, 1967, Silver, 1963), but has more recently applied in a fully optimized form with intrinsically Bayesian optimal (IBR) filters, in which optimization is relative to a prior distribution (Dalton & Dougherty, 2014), and with optimal Bayesian filters, in particular, regression, where optimization is relative to a posterior distribution (Qian & Dougherty, 2016). These concepts have also been recently applied to classification in the form of optimal Bayesian classifiers (Dalton & Dougherty, 2013a) and IBR classifiers (Dalton & Dougherty, 2013b). Directly relevant to the developments in the current paper is their application in recursive linear filtering: the IBR Kalman filter (Dehghannasiri, Esfahani, & Dougherty, 2017), the optimal Bayesian Kalman filter, which uses the data to update the prior, thereby producing superior filtering to the IBR Kalman filter (Dehghannasiri, Esfahani, Qian, & Dougherty, 2018), and the optimal Bayesian Kalman smoother (Dehghannasiri & Dougherty, 2018).
Here, we extend the BKF and BKS to the cases where POBDS is under model uncertainty. The methods are optimal relative to the posterior distribution of the parameters. When the parameter space is discrete and finite, exact algorithms based on an efficient, recursive matrix-based implementation are introduced. These algorithms contain a bank of BKFs/BKSs in parallel, which is reminiscent of the multiple model adaptive estimation (MMAE) procedure for linear systems (Magill, 1965, Maybeck, 1982). These algorithms can be seen as generalizations of the regular BKF and BKS. For general parameter spaces, an approximate Markov-Chain Monte-Carlo (MCMC) implementation of the optimal Bayesian estimators is described. Via numerical examples, the performances of these filters are compared to that of the ML, MAP, and IBR estimators introduced in Dalton and Dougherty (2014) and Imani and Braga-Neto, 2017a, Imani and Braga-Neto, 2017b, respectively.
The article is organized as follows. In Section 2, the POBDS signal model is introduced and its optimal MMSE estimators are briefly described. In Section 3, first the POBDS under model uncertainty is introduced, followed by the exact algorithms for computation of optimal Bayesian estimators in the case of discrete parameter space and the approximate MCMC solution for continuous parameter space. Section 4 contains numerical examples using POBDS models of gene regulatory networks observed through noisy measurements. Finally, Section 5 contains concluding remarks.
Section snippets
POBDS signal model
The POBDS model consists of a state model that describes the evolution of the Boolean dynamical system and an observation model that relates the state to the system output (measurements).
The state model is defined as: where represents the state of Boolean state variables of the system at time , is the input at time step which is assumed to be deterministic and known, and the process noise is i.i.d. with arbitrary distribution, which is independent
POBDS under model uncertainty
In many practical problems, full information about the system model is not available. There might be uncertainty about the transition or observation functions or noise statistics. We assume the uncertainty is parameterized by a vector of unknown parameters, where takes a value in a set , called the uncertainty class. The POBDS model can then be expressed as:
Direct application of the algorithms described in the previous section is not
Numerical results and performance analysis
In this section, we present the results of numerical experiments using a gene regulatory network model, which compare the performance of the proposed framework with four other approaches: (1) the optimal model-specific BKF/BKS (Braga-Neto, 2011, Imani and Braga-Neto, 2017a); (2) the maximum-likelihood (ML) adaptive BKF/BKS (Imani & Braga-Neto, 2017a); (3) the maximum a posteriori (MAP) adaptive BKF/BKS (Imani & Braga-Neto, 2017b); and (4) the intrinsically Bayesian robust (IBR) estimator (
Conclusion
This paper introduces an optimal Bayesian framework for joint state and parameter estimation for a class of partially-observed Boolean dynamical systems (POBDS) under model uncertainty. The proposed framework provides the optimal expected MSE solution relative to the posterior distribution over the parameter space. For a discrete (finite) parameter space, we introduce exact optimal filter and smoother algorithms, called the BKF-DMU and BKS-DMU respectively. These two estimators can be seen as
Acknowledgment
The authors acknowledge the support of the National Science Foundation, USA , through NSF award CCF-1718924.
Mahdi Imani received his Ph.D. degree in Electrical and Computer Engineering from Texas A&M University, College Station, TX in 2019. He is an Assistant Professor in the Department of Electrical and Computer Engineering at George Washington University, Washington, DC, USA. His research interests include Machine Learning, Bayesian Statistics and Decision Theory, with a wide range of applications from computational biology to cyber–physical systems. He is the recipient of the Association of Former
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Mahdi Imani received his Ph.D. degree in Electrical and Computer Engineering from Texas A&M University, College Station, TX in 2019. He is an Assistant Professor in the Department of Electrical and Computer Engineering at George Washington University, Washington, DC, USA. His research interests include Machine Learning, Bayesian Statistics and Decision Theory, with a wide range of applications from computational biology to cyber–physical systems. He is the recipient of the Association of Former Students Distinguished Graduate Student Award for Excellence in Research-Doctoral in 2019, the Best PhD Student Award in ECE department at Texas A&M University in 2015, and a single finalist nominee of ECE department for the Outstanding Graduate Student Award in college of engineering at Texas A&M University in 2018. He is also recipient of the best paper finalist award from the 49th Asilomar Conference on Signals, Systems, and Computers, 2015.
Edward R. Dougherty received the M.Sc. degree in computer science from Stevens Institute of Technology, the Ph.D. degree in mathematics from Rutgers University, and has been awarded the Doctor Honoris Causa by the Tampere University of Technology, Finland. He is a Distinguished Professor in the Department of Electrical and Computer Engineering, Texas A&M University, College Station, TX, USA, where he holds the Robert M. Kennedy ’26 Chair in electrical engineering and is Scientific Director of the Center for Bioinformatics and Genomic Systems Engineering. He is the author of 16 books, the editor of 5 others, and author of more than 300 journal papers. Dr. Dougherty is a Fellow of SPIE, has received the SPIE President’s Award, and served as the Editor of the SPIE/IS&T Journal of Electronic Imaging. At Texas A&M University, he received the Association of Former Students Distinguished Achievement Award in Research, and was named Fellow of the Texas Engineering Experiment Station, and Halliburton Professor of the Dwight Look College of Engineering.
Ulisses M. Braga-Neto received the Ph.D. degree in electrical and computer engineering from The Johns Hopkins University, Baltimore, MD, USA. He is an Associate Professor in the Department of Electrical and Computer Engineering and a member of the Center for Bioinformatics and Genomic Systems Engineering, Texas A&M University, College Station, TX, USA. He has held postdoctoral positions at the University of Texas M.D. Anderson Cancer Center, Houston, TX, and at the Oswaldo Cruz Foundation, Recife, Brazil. His research interests include pattern recognition and statistical signal processing. He is the author of the textbook Error Estimation for Pattern Recognition (IEEE-Wiley, 2015) and has received the NSF CAREER Award for his work in this area.
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The material in this paper was not presented at any conference. This paper was recommended for publication in revised form by Associate Editor Thomas Bo Schön under the direction of Editor Torsten Sóderstróm