Aditya Tulsyan

A Particle Filter Approach to Approximate Posterior Cramer-Rao Lower Bound: The Case of Hidden States

The posterior Cramer-Rao lower bound (PCRLB) derived in [1] provides a bound on the mean square error (MSE) obtained with any nonlinear state filter. Computing the PCRLB involves solving complex, multi-dimensional expectations, which do not lend themselves to an easy analytical solution. Furthermore, any attempt to approximate it using numerical or simulation-based approaches require a priori access to the true states, which may not be available, except in simulations or in carefully designed experiments. To allow recursive approximation of the PCRLB when the states are hidden or unmeasured, a new approach based on sequential Monte-Carlo (SMC) or particle filters (PFs) is proposed. The approach uses SMC methods to estimate the hidden states using a sequence of the available sensor measurements. The developed method is general and can be used to approximate the PCRLB in nonlinear systems with non-Gaussian state and sensor noise. The efficacy of the developed method is illustrated on two simulation examples, including a practical problem of ballistic target tracking at reentry phase.

Particle filtering without tears: A primer for beginners

Abstract The main purpose of this primer is to systematically introduce the theory of particle filters to readers with limited or no prior understanding of the subject. The primer is written for beginners and practitioners interested in learning about the theory and implementation of particle filtering methods. Throughout this primer we highlight the common mistakes that beginners and first-time researchers make in understanding and implementing the theory of particle filtering. We also discuss and demonstrate the use of particle filtering in nonlinear state estimation applications. We conclude the primer by providing an implementable version of MATLAB code for particle filters. The code not only aids in improving the understanding of particle filters, it also serves as a template for building and implementing advanced nonlinear state estimation routines.

Bayesian identification of non-linear state-space models: Part II- Error analysis

Abstract In the last two decades, several methods based on sequential Monte Carlo (SMC) and Markov chain Monte Carlo (MCMC) have been proposed for Bayesian identification of stochastic non-linear state-space models (SSMs). It is well known that the performance of these simulation based identification methods depends on the numerical approximations used in their design. We propose the use of posterior Cramer-Rao lower bound (PCRLB) as a mean square error (MSE) bound. Using PCRLB, a systematic procedure is developed to analyse the estimates delivered by Bayesian identification methods in terms of bias, MSE, and efficiency. The efficacy and utility of the proposed approach is illustrated through a numerical example.

Assessment of type II diabetes mellitus using irregularly sampled measurements with missing data

Diabetes mellitus is one of the leading diseases in the developed world. In order to better regulate blood glucose in a diabetic patient, improved modelling of insulin-glucose dynamics is a key factor in the treatment of diabetes mellitus. In the current work, the insulin-glucose dynamics in type II diabetes mellitus can be modelled by using a stochastic nonlinear state-space model. Estimating the parameters of such a model is difficult as only a few blood glucose and insulin measurements per day are available in a non-clinical setting. Therefore, developing a predictive model of the blood glucose of a person with type II diabetes mellitus is important when the glucose and insulin concentrations are only available at irregular intervals. To overcome these difficulties, we resort to online sequential Monte Carlo (SMC) estimation of states and parameters of the state-space model for type II diabetic patients under various levels of randomly missing clinical data. Our results show that this method is efficient in monitoring and estimating the dynamics of the peripheral glucose, insulin and incretins concentration when 10, 25 and 50xa0% of the simulated clinical data were randomly removed.

PERKS: Software for Parameter Estimation in Reaction Kinetic Systems

Abstract To bring forth the recent developments in global dynamic optimization methods to practitioners, we present a software implementation- PERKS : Parameter Estimation in Reaction Kinetic Systems. PERKS performs parameter estimation in complex reaction-network models using a state-of-the-art branch-and-bound global dynamic optimization method. The built-in global optimization method in PERKS guarantees a best possible least-squares fit of experimental data by the kinetic model. PERKS is equipped with a graphical-user-interface (GUI) to streamline user interaction and reduce training-time on the software. Further, PERKS is designed keeping in mind the diverse background of our possible end-users. Example problems are included with the software to get practitioners started with minimal effort and limited background knowledge in optimization methods. In this paper, we introduce PERKS and discuss some of its key features.

Robust model-based delay timer alarm for non-linear processes

We consider the problem of robust alarm design for chemical processes represented by stochastic state-space models (SSMs). Alarm design for such complex processes faces three important challenges: 1) processes exhibit highly non-linear behavior; 2) state variables are not precisely known (modeling error) and often unmeasured (hidden); and 3) process signals are not necessarily Gaussian, stationary or uncorrelated. We propose the design of a delay timer alarm for monitoring unmeasured latent variables. The proposed design reduces the false and missed alarm rates. To deal with non-stationary process behavior in normal and abnormal operations, we propose an average-case delay timer alarm design. The efficacy of the proposed robust alarm design is illustrated on a non-stationary chemical reactor.

Bayesian identification of non-linear state-space models: Part I- Input design

Abstract We propose an algorithm for designing optimal inputs for on-line Bayesian identification of stochastic non-linear state-space models. The proposed method relies on minimization of the posterior Cramer Rao lower bound derived for the model parameters, with respect to the input sequence. To render the optimization problem computationally tractable, the inputs are parametrized as a multi-dimensional Markov chain in the input space. The proposed approach is illustrated through a simulation example.

Performance Assessment of Nonlinear State Filters

Abstract Nonlinear state filters of different approximations and capabilities have been developed in the last decade. The quality of different nonlinear filters, in terms of the mean squared error (MSE) of the estimates, depends on the approximations used in the filtering algorithm; however, there are no known methods for effectively evaluating the relative performance of these filters. A new method which measures the performance of different state filters against the theoretical posterior Cramer-Rao lower bound (PCRLB) is proposed. The complex high-dimensional integrals in PCRLB are approximated using sequential Monte-Carlo (SMC) methods. Efficacy of the proposed method is illustrated through a simulation example.

A switching strategy for adaptive state estimation

Abstract This paper develops a switching strategy for adaptive state estimation in systems represented by nonlinear, stochastic, discrete-time state space models (SSMs). The developed strategy is motivated by the fact that there is no single Bayesian estimator that is guaranteed to perform optimally for a given nonlinear system and under all operating conditions. The proposed strategy considers a bank of plausible Bayesian estimators for adaptive state estimation, and then switches between them based on their performance. The performance of a Bayesian estimator is assessed using a performance measure derived from the posterior Cramer-Rao lower bound (PCRLB). It is shown that the switching strategy is stable, and yields estimates that are at least as good as any individual estimator in the bank. The efficacy of the switching strategy is illustrated on a practical simulation example.