Frasso, Gianluca
Jaeger, Jonathan
Lambert, Philippe
[UCL]
Differential equations (DEs) are commonly used to describe dynamic systems evolving in one (ordinary differential equations or ODEs) or in more than one dimensions (partial differential equations or PDEs). In real data applications, the parameters involved in the DE models are usually unknown and need to be estimated from the available measurements together with the state function. In this paper, we present frequentist and Bayesian approaches for the joint estimation of the parameters and of the state functions involved in linear PDEs.We also propose two strategies to include state (initial and/or boundary) conditions in the estimation procedure.We evaluate the performances of the proposed strategy through simulated examples and a real data analysis involving (known and necessary) state conditions.
- Biegler L. T., Damiano J. J., Blau G. E., Nonlinear parameter estimation: A case study comparison, 10.1002/aic.690320105
- Biller, C.: Adaptive bayesian regression splines in semiparametric generalized linear models. J. Comp. Graph. Stat. 9(1), 122–140 (2000)
- Black Fischer, Scholes Myron, The Pricing of Options and Corporate Liabilities, 10.1086/260062
- Botella O, On a collocation B-spline method for the solution of the Navier–Stokes equations, 10.1016/s0045-7930(01)00058-5
- Carmona R.: Statistical Analysis of Financial Data in S-Plus. Springer Text in Statistics (2004)
- Cox John C., Ross Stephen A., Rubinstein Mark, Option pricing: A simplified approach, 10.1016/0304-405x(79)90015-1
- Currie Iain D, Smoothing constrained generalized linear models with an application to the Lee-Carter model, 10.1177/1471082x12471373
- Denison D. G. T., Mallick B. K., Smith A. F. M., Automatic Bayesian curve fitting, 10.1111/1467-9868.00128
- Dierckx, P.: Curve and Surface Fitting with Splines. Oxford University Press (1995)
- Eilers Paul H. C., Marx Brian D., Flexible smoothing with B -splines and penalties, 10.1214/ss/1038425655
- Eilers Paul H.C., Marx Brian D., Multivariate calibration with temperature interaction using two-dimensional penalized signal regression, 10.1016/s0169-7439(03)00029-7
- Eilers Paul H. C., Marx Brian D., Splines, knots, and penalties, 10.1002/wics.125
- Epperson James F., On the Runge Example, 10.2307/2323093
- Friedman Jerome H., Silverman Bernard W., Flexible Parsimonious Smoothing and Additive Modeling, 10.1080/00401706.1989.10488470
- Golub, G.H., Ortega, J.M.: Scientific computing and differential equations. Academic Press, New York and London (1992)
- Grenander, U.: Abstract Inference. Probability and Statistics Series. Wiley (1981)
- Hastie, T.J., Tibshirani, R.J.: Generalized additive models. Chapman & Hall, London (1990)
- Heston Steven L., A Closed-Form Solution for Options with Stochastic Volatility with Applications to Bond and Currency Options, 10.1093/rfs/6.2.327
- Holmes C. C., Mallick B. K., Bayesian Radial Basis Functions of Variable Dimension, 10.1162/089976698300017421
- Jaeger Jonathan, Lambert Philippe, Bayesian P-spline estimation in hierarchical models specified by systems of affine differential equations, 10.1177/1471082x12471371
- Jaeger, J., Lambert, P.: Bayesian penalized smoothing approaches in models specified using differential equations with unknown error distributions. J. App Stat. 1–18 (2014)
- Ma Shuangge, Kosorok Michael R., Robust semiparametric M-estimation and the weighted bootstrap, 10.1016/j.jmva.2004.09.008
- Piché, R., Kanniainen, J.: Solving financial differential equations using differentiation matrices. Lecture Notes in Engineering and Computer Science. Newswood Limited, In World Congress on Engineering (2007)
- Ramsay J. O., Hooker G., Campbell D., Cao J., Parameter estimation for differential equations: a generalized smoothing approach : Parameter Estimation for Differential Equations, 10.1111/j.1467-9868.2007.00610.x
- Rodriguez-Fernandez Maria, Egea Jose A, Banga Julio R, 10.1186/1471-2105-7-483
- Ruppert, D., Wand, M.P. and Carroll, R.J.: Semiparametric Regression. Cambridge Series in Statistical and Probabilistic Mathematics 12, (2003)
- SCHALL ROBERT, Estimation in generalized linear models with random effects, 10.1093/biomet/78.4.719
- Schultz M. H., Varga R. S., L-Splines, 10.1007/bf02162033
- Shen, X.: On methods of sieves and penalization. Ann. Statist. 25(6),2555–2591, 12 (1997)
- Smith Michael, Yau Paul, Shively Thomas, Kohn Robert, Estimating Long-term Trends in Tropospheric Ozone Levels, 10.1111/j.1751-5823.2002.tb00351.x
- Wu, Z.: Compactly supported positive definite radial functions. Advances in Computational Mathematics, 4(1),283–292, (1995) (ISSN 1019-7168)
- Xue Hongqi, Miao Hongyu, Wu Hulin, Sieve estimation of constant and time-varying coefficients in nonlinear ordinary differential equation models by considering both numerical error and measurement error, 10.1214/09-aos784
- Xun Xiaolei, Cao Jiguo, Mallick Bani, Maity Arnab, Carroll Raymond J., Parameter Estimation of Partial Differential Equation Models, 10.1080/01621459.2013.794730
Bibliographic reference |
Frasso, Gianluca ; Jaeger, Jonathan ; Lambert, Philippe. Parameter estimation and inference in dynamic systems described by linear partial differential equations. In: A St A - Advances in Statistical Analysis, Vol. 100, no. 3, p. 259-287 (2016) |
Permanent URL |
http://hdl.handle.net/2078.1/168110 |