Abstract
This paper presents a new approach for state estimation problems. It is based on the representation of random variables using stochastic functions. Its main idea is to expand the state variables in terms of the noise variables of the system, and then estimate the unnoisy value of the state variables by taking the mean value of the stochastic expansion. Moreover, it was shown that in some situations, the proposed approach may be adapted to the determination of the probability distribution of the state noise. For the determination of the coefficients of the expansions, we present three approaches: moment matching (MM), collocation (COL) and variational (VAR). In the numerical analysis section, three examples are analyzed including a discrete linear system, the Influenza in a boarding school and the state estimation problem in the Hodgkin–Huxley’s model. In all these examples, the proposed approach was able to estimate the values of the state variables with precision, i.e., with very low RMS values.
Similar content being viewed by others
References
Andrieu C, Doucet A, Singh SS, Tadic VB (2004) Particle methods for change detection, system identification, and control. Proc IEEE 92(3):423–438
Bailey NT (1975) The mathematical theory of infectious diseases. Hafner Press-MacMillian Pub. Co, New York
Blanchard ED, Sandu A, Sandu C (2010) A polynomial chaos-based Kalman filter approach for parameter estimation of mechanical systems. J Dyn Syst Meas Control 1326:061404
Bobrowski A (2005) Functional analysis for probability and stochastic processes: an introduction. Cambridge University Press, Cambridge
Carpenter JCP, Fearnhead P (2004) An improved particle filter for non-linear problems. IEE Proceedings Radar, Sonar and Navigation 146(1):2–7
de Cursi ES, Sampaio R (2015) Uncertainty quantification and stochastic modeling with Matlab. Elsevier, Amsterdam (ISTE, London)
Estumano D, Orlande H, Colaco M, Dulikravich G (2013) State estimation problem for Hodgkin–Huxley’s model: a comparison of particle filter algorithms. In: 4th inverse problems, design and optimization symposium (IPDO-2013)
Evensen G (2009) The ensemble Kalman filter for combined state and parameter estimation. IEEE Control Syst 29(3):83–104
Evensen G (1994) Sequential data assimilation with a nonlinear quasi-geostrophic model using monte carlo methods to forecast error statistics. J Geophys Res 99:10143–10162
Ghanem RG, Spanos PD (2003) Stochastic finite elements: a spectral approach. Dover Publications Inc., Mineola
Gordon NJ, Salmond DJ, Smith AFM (1993) Novel approach to nonlinear/non-Gaussian Bayesian state estimation. IEE Proc F (Radar Signal Process) 140(6):107–113
Grooms I, Lee Y, Majda AJ (2014) Ensemble Kalman filters for dynamical systems with unresolved turbulence. J Comput Phys 273:435–452
Harlim J, Mahdi A, Majda AJ (2014) An ensemble Kalman filter for statistical estimation of physics constrained nonlinear regression models. J Comput Phys 257(Part A):782–812
Hodgkin A, Huxley A (1952) A quantitative description of membrane current and it’s application to conduction and excitation in nerve. J Physiol 117:500–544
Kalman RE (1960) A new approach to linear filtering and prediction problems. Trans ASM J Basic Eng 82(Series D):35–45
Kalman RE, Bucy RS (1961) New results in linear filtering and prediction theory. J Basic Eng 83(1):95–108
Kotecha JH, Djuric PM (2003) Gaussian particle filtering. IEEE Trans Signal Process 51(10):2592–2601
Liu JS, Chen R (1998) Sequential monte carlo methods for dynamic systems. J Am Stat Assoc 93:1032–1044
Ljung L (1979) Asymptotic behavior of the extended Kalman filter as a parameter estimator for linear systems. IEEE Trans Autom Control 24(1):36–50
Long Q, Scavino M, Tempone R, Wang S (2013) Fast estimation of expected information gains for bayesian experimental designs based on Laplace approximations. Comput Methods Appl Mech Eng 259:24–39
Lopez RH, de Cursi ES, Lemosse D (2011) Approximating the probability density function of the optimal point of an optimization problem. Eng Optim 43(3):281–303
Lopez RH, Miguel LFF, de Cursi JES (2013) Uncertainty quantification for algebraic systems of equations. Comput Struct 128:189–202
Lopez RH, da Silva CR Jr (2015) A non-intrusive methodology for the representation of crack growth stochastic processes. Mech Res Commun 64:23–28
Lopez RH, Miguel LFF, Belo IM, Souza Cursi JE (2014) Advantages of employing a full characterization method over FORM in the reliability analysis of laminated composite plates. Compos Struct 107:635–642
Madankan R, Singla P, Singh T, Scott PD (2013) Polynomial-chaos-based Bayesian approach for state and parameter estimations. J Guidance Control Dyn W04417 36(4):1058–1074
Marzouk YM, Najm HN, Rahn LA (2007) Stochastic spectral methods for efficient Bayesian solution of inverse problems. J Comput Phys 224(2):560–586
Maybeck PS (1979) Stochastic models, estimation and control. Volume I, Academic Press, Cambridge
Nagel JB, Sudret B (2016) Spectral likelihood expansions for Bayesian inference. J Comput Phys 309:267–294
Nobile F, Tempone R, Webster CG (2008) A sparse grid stochastic collocation method for partial differential equations with random input data. SIAM J Numer Anal 46(5):2309–2345
Nouy A, Le Matre OP (2009) Generalized spectral decomposition for stochastic nonlinear problems. J Comput Phys 228(1):202–235
Saad GA (2007) Stochastic data assimilation with application to multi-phase flow and health monitoring problems. Comput Geosci 14(4):721–744
Saad G, Ghanem R (2009) Characterization of reservoir simulation models using a polynomial chaos-based ensemble Kalman filter. Water Resour Res 45(4). doi:10.1029/2008WR007148
Sorenson HW (1970) Least-squares estimation: from Gauss to Kalman. IEEE Spectr 7(7):63–68
Varon LAB, Orlande HRB, Eliabe GE (2015) Estimation of state variables in the hyperthermia therapy of cancer with heating imposed by radiofrequency electromagnetic waves. Int J Therm Sci 98:228–236
Wiener N (1938) The homogeneous chaos. Am J Math 60(23–26):897–936
Xiu D, Karniadakis GE (2002) The Wiener–Askey polynomial chaos for stochastic differential equations. SIAM J Sci Comput 24(2):619–644
Zeng L, Zhang D (2010) A stochastic collocation based Kalman filter for data assimilation. Comput Geosci 14(4):721–744
Zhen Y, Harlim J (2015) Adaptive error covariances estimation methods for ensemble Kalman filters. J Comput Phys 294:619–638
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by Dr. Jose Alberto Cuminato.
Rights and permissions
About this article
Cite this article
Lopez, R.H., Cursi, J.E.S. & Carlon, A.G. A state estimation approach based on stochastic expansions. Comp. Appl. Math. 37, 3399–3430 (2018). https://doi.org/10.1007/s40314-017-0515-0
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s40314-017-0515-0
Keywords
Mathematics Subject Classification
- 60G15 Gaussian processes
- 60G35 Signal detection and filtering
- 60H07 Stochastic calculus of variations and the Malliavin calculus
- 60H10 Stochastic ordinary differential equations
- 60H30 Applications of stochastic analysis (to PDE, etc.)
- 60H35 Computational methods for stochastic equations
- 65C30 Stochastic differential and integral equations