Abstract
Current experimental design techniques for dynamical systems often only incorporate measurement noise, while dynamical systems also involve process noise. To construct experimental designs we need to quantify their information content. The Fisher information matrix is a popular tool to do so. Calculating the Fisher information matrix for linear dynamical systems with both process and measurement noise involves estimating the uncertain dynamical states using a Kalman filter. The Fisher information matrix, however, depends on the true but unknown model parameters. In this paper we combine two methods to solve this issue and develop a robust experimental design methodology. First, Bayesian experimental design averages the Fisher information matrix over a prior distribution of possible model parameter values. Second, adaptive experimental design allows for this information to be updated as measurements are being gathered. This updated information is then used to adapt the remainder of the design.
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References
Anisimov VV, Fedorov VV, Leonov SL (2007) Optimal design of pharmacokinetic studies described by stochastic differential equations. In: mODa 8-advances in model-oriented design and analysis: proceedings of the 8th international workshop in model-oriented design and analysis held in Almagro, Spain, June 4–8, 2007. Springer, pp 9–16
Atkinson A, Donev A, Tobias R (2007) Optimum experimental designs, with SAS, vol 34. Oxford University Press, Oxford
Bezanson J, Edelman A, Karpinski S, Shah VB (2017) Julia: a fresh approach to numerical computing. SIAM Rev 59.1:65–98
Cavanaugh JE, Shumway RH (1996) On computing the expected Fisher information matrix for state-space model parameters. Stat Probab Lett 26.4:347–355
Chaloner K, Isabella V (1995) Bayesian experimental design: a review. Stat Sci 273–304
Chen R, Liu JS (2000) Mixture Kalman filters. J R Stat Soc Ser B (Stat Methodol) 62.3:493–508
Efron B, Hinkley DV (1978) Assessing the accuracy of the maximum likelihood estimator: observed versus expected Fisher information. Biometrika 65.3:457–483
Elfving G (1952) Optimum allocation in linear regression theory. Ann Math Stat 255–262
Fedorov VV (1972) Theory of optimal experiments. Academic Press, New York
Fedorov VV, Leonov SL (2013) Optimal design for nonlinear response models. CRC Press, Boca Raton
Fedorov VV, Leonov SL, Vasiliev VA (2010) Pharmacokinetic studies described by stochastic differential equations: optimal design for systems with positive trajectories. In: mODa 9–advances in model-oriented design and analysis: proceedings of the 9th international workshop in model-oriented design and analysis held in Bertinoro, Italy, June 14–18, 2010. Springer, New York, pp 73–80
Findeisen R, Allgöwer F (2002) An introduction to nonlinear model predictive control. 21st Benelux meeting on systems and control, vol 11. Technische Universiteit Eindhoven Veldhoven Eindhoven, The Netherlands, pp 119–141
Franceschini G, Macchietto S (2008) Model-based design of experiments for parameter precision: state of the art. Chem Eng Sci 63(19):4846–4872
Goodwin GC, Payne RL (1977) Dynamic system identification, experiment design and data analysis, vol 136. Academic Press, London
Griewank A, Walther A (2008) Evaluating derivatives: principles and techniques of algorithmic differentiation. SIAM, Philadelphia
He J, Asma K, Peter S (2021) A Kalman particle filter for online parameter estimation with applications to affine models. Stat Inference Stoch Process 1–51
Hjalmarsson H (2005) From experiment design to closed-loop control. Automatica 41.3:393–438
Johnson SG (2014) The NLopt nonlinear-optimization package. https://nlopt.readthedocs.io/en/latest/. Accessed 31 Nov 2021
Kantas N, Arnaud D, Sindhu SS, Maciejowski JM (2009) An overview of sequential Monte Carlo methods for parameter estimation in general state-space models. IFAC Proc 42(10):774–785
Körkel S, Kostina E, Bock HG, Schlöder JP (2004) Numerical methods for optimal control problems in design of robust optimal experiments for nonlinear dynamic processes. Optim Methods Softw 19.3–4:327–338
Kraft D (1988) A software package for sequential quadratic programming. In: Deutsche Forschungs- und Versuchsanstalt für Luft- und Raumfahrt DFVLR-FB, pp 88–28
Kraft D (1994) Algorithm 733: TOMP-Fortran modules for optimal control calculations. ACM Trans Math Softw 20.3:262–281
Lane A (2017) Adaptive designs for optimal observed fisher information. arXiv:1712.08499
Pintelon R, Schoukens J (2012) System identification: a frequency domain approach, 2nd edn. Wiley, Hoboken
Pronzato L, Pázman A (2013) Design of experiments in nonlinear models, vol 212. Lecture notes in statistics. Springer, New York
Rawlings JB, David QM, Moritz D (2017) Model predictive control: theory, computation, and design, 2nd edn. Madison Publishing, Nob Hill
Revels J, Miles L, Theodore P (2016) Forward-mode automatic differentiation in Julia. arXiv:1607.07892
Ryan EG, Christopher CD, James MM, Anthony NP (2016) A review of modern computational algorithms for Bayesian optimal design. Int Stat Rev 84(1):128–154
Sagnol G, Harman R (2015) Optimal designs for steady-state Kalman filters. Springer, New York
Särkkä S (2013) Bayesian filtering and smoothing, vol 3. Cambridge University Press, Cambridge
Särkkä S, Solin A (2019) Applied stochastic differential equations, vol 10. Cambridge University Press, Cambridge
Stojanovic V, Nedic N, Prsic D, Dubonjic L (2016) Optimal experiment design for identification of ARX models with constrained output in non-Gaussian noise. Appl Math Model 40.13–14:6676–6689
Telen D, Houska B, Logist F, Van Derlinden E, Diehl M, Van Impe J (2013) Optimal experiment design under process noise using Riccati differential equations. J Process Control 23.4:613–629
Telen D, Vercammen D, Logist F, Van Impe J (2014) Robustifying optimal experiment design for nonlinear, dynamic (bio) chemical systems. Comput Chem Eng 71:415–425
Teymur O, Jackson G, Marina R, Chris O (2021) Optimal quantisation of probability measures using maximum mean discrepancy. In: International conference on artificial intelligence and statistics. PMLR, pp 1027–1035
Titterington DM (1980) Aspects of optimal design in dynamic systems. Technometrics 22.3:287–299
Von Richard M (2014) Mathematical theory of probability and statistics. Academic Press, New York
Wong W-K (1992) A unified approach to the construction of minimax designs. Biometrika 79.3:611–619
Acknowledgements
The authors would like to thank the KU Leuven for financial support (Project C16/16/002). Author Arno Strouwen thanks the Fund for Scientific Research, Flanders (FWO), Project 1S58717N.
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Strouwen, A., Nicolaï, B.M. & Goos, P. Adaptive and robust experimental design for linear dynamical models using Kalman filter. Stat Papers 64, 1209–1231 (2023). https://doi.org/10.1007/s00362-023-01438-9
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DOI: https://doi.org/10.1007/s00362-023-01438-9