Efficient Computation of the Continuous-Discrete Extended Kalman Filter Sensitivities Applied to Maximum Likelihood Estimation

In this paper, we present and compare different methods for computing the likelihood function and its gradient. We consider nonlinear continuous-discrete models described by a system of stochastic differential equations (SDEs) with discrete-time measurements. The problem of maximum likelihood estimation (MLE) is formulated as a nonlinear program (NLP) and it is solved numerically using a gradient-based single shooting algorithm. The estimates of the mean and its covariance are computed using a continuous-discrete extended Kalman filter (CDEKF). We derive analytical expressions for the gradient of the likelihood function. We discuss some aspects of the implementation of MLE for non-stiff systems. In particular, we present an efficient way of computing the state covariance matrix and its gradient using explicit Runge-Kutta schemes. We verify our implementation using a numerical example related to type 1 diabetes and demonstrate how to apply it for nonlinear parameter estimation.

[1]  Henrik Madsen,et al.  ctsmr - Continuous Time Stochastic Modeling in R , 2016, 1606.00242.

[2]  Peter A. Zadrozny.  Analytic Derivatives for Estimation of Linear Dynamic Models , 1988 .

[3]  S. Bay Jørgensen,et al.  Comparison of Prediction-Error-Modelling Criteria , 2007, ACC.

[4]  Christos Georgakis,et al.  How To NOT Make the Extended Kalman Filter Fail , 2013 .

[5]  Dimitri Boiroux,et al.  Nonlinear Model Predictive Control and Artificial Pancreas Technologies , 2018, 2018 IEEE Conference on Decision and Control (CDC).

[6]  Garry M. Steil,et al.  Identification of Intraday Metabolic Profiles during Closed-Loop Glucose Control in Individuals with Type 1 Diabetes , 2009, Journal of diabetes science and technology.

[7]  H. Bock.  Numerical Treatment of Inverse Problems in Chemical Reaction Kinetics , 1981 .

[8]  Niels Kjølstad Poulsen,et al.  Model Identification using Continuous Glucose Monitoring Data for Type 1 Diabetes , 2016 .

[9]  Jonathan D. Cryer,et al.  Time Series Analysis , 1986 .

[10]  J. Timmer,et al.  Parameter estimation in ordinary differential equations for biochemical processes using the method of multiple shooting. , 2007, IET systems biology.

[11]  R. Bhushan Gopaluni,et al.  Particle filtering without tears: A primer for beginners , 2016, Comput. Chem. Eng..

[12]  Florian Nadel,et al.  Stochastic Processes And Filtering Theory , 2016 .

[13]  N. Sandell,et al.  MAXIMUM LIKELIHOOD IDENTIFICATION OF STATE SPACE MODELS FOR LINEAR DYNAMIC SYSTEMS , 1978 .

[14]  Henrik Madsen,et al.  An efficient UD-based algorithm for the computation of maximum likelihood sensitivity of continuous-discrete systems , 2016, 2016 IEEE 55th Conference on Decision and Control (CDC).

[15]  Simo Särkkä,et al.  On Unscented Kalman Filtering for State Estimation of Continuous-Time Nonlinear Systems , 2007, IEEE Trans. Autom. Control..

[16]  Heinz Neudecker,et al.  A direct derivation of the exact Fisher information matrix of Gaussian vector state space models , 2000 .

[17]  R. B. Gopaluni.  A particle filter approach to identification of nonlinear processes under missing observations , 2008 .

[18]  Hermann Singer,et al.  Continuous-Discrete Unscented Kalman Filtering , 2005 .

[19]  Dimitri Boiroux,et al.  A Nonlinear Model Predictive Control Strategy for Glucose Control in People with Type 1 Diabetes , 2018 .

[20]  Benjamin Friedlander,et al.  Computation of the exact information matrix of Gaussian time series with stationary random components , 1985, 1985 24th IEEE Conference on Decision and Control.

[21]  Maria V. Kulikova,et al.  State Sensitivity Evaluation Within UD Based Array Covariance Filters , 2013, IEEE Transactions on Automatic Control.

[22]  J. B. Jørgensen.  Adjoint sensitivity results for predictive control, state- and parameter-estimation with nonlinear models , 2007, 2007 European Control Conference (ECC).

[23]  Xiao Liang,et al.  Maximum likelihood estimation of inflation factors on error covariance matrices for ensemble Kalman filter assimilation , 2012 .

[24]  H. Madsen,et al.  A Computationally Efficient and Robust Implementation of the Continuous-Discrete Extended Kalman Filter , 2007, 2007 American Control Conference.

[25]  J. Dormand,et al.  A family of embedded Runge-Kutta formulae , 1980 .

[26]  Dimitri Boiroux,et al.  Parameter Estimation in Type 1 Diabetes Models for Model-Based Control Applications , 2019, 2019 American Control Conference (ACC).

[27]  Kjell Gustafsson,et al.  Control of Error and Convergence in ODE Solvers , 1992 .

[28]  Karl Johan Åström.  Maximum likelihood and prediction error methods , 1980, Autom..

[29]  Sten Bay Jørgensen,et al.  Parameter estimation in stochastic grey-box models , 2004, Autom..

[30]  H. Unbehauen,et al.  Sensitivity models for nonlinear filters with application to recursive parameter estimation for nonlinear state-space models , 2001 .

[31]  W. Kenneth Ward,et al.  Modeling the Glucose Sensor Error , 2014, IEEE Transactions on Biomedical Engineering.

[32]  John Bagterp Jørgensen,et al.  Modeling and Prediction Using Stochastic Differential Equations , 2016 .