Estimating varying coefficients for partial differential equation models. (English) Zbl 1522.62290
Summary: Partial differential equations (PDEs) are used to model complex dynamical systems in multiple dimensions, and their parameters often have important scientific interpretations. In some applications, PDE parameters are not constant but can change depending on the values of covariates, a feature that we call varying coefficients. We propose a parameter cascading method to estimate varying coefficients in PDE models from noisy data. Our estimates of the varying coefficients are shown to be consistent and asymptotically normally distributed. The performance of our method is evaluated by a simulation study and by an empirical study estimating three varying coefficients in a PDE model arising from LIDAR data.
{© 2017, The International Biometric Society}
{© 2017, The International Biometric Society}
MSC:
| 62P10 | Applications of statistics to biology and medical sciences; meta analysis |
Keywords:
B-splines; dynamical models; inverse problems; LIDAR data; parameter cascading; system identificationSoftware:
fda (R)References:
| [1] | Bär, M., Hegger, R., and Kantz, H. (1999). Fitting partial differential equations to space‐time dynamics. Physical Review E59, 337-342. |
| [2] | Bard, Y. (1974). Nonlinear Parameter Estimation. New York, NY: Academic Press. · Zbl 0345.62045 |
| [3] | Biegler, L., Damiano, J. J., and Blau, G. E. (1986). Nonlinear parameter estimation: A case study comparison. AIChE Journal32, 29-45. |
| [4] | Brown, M. B. and Forsythe, A. B. (1974). Robust tests for the equality of variances. Journal of the American Statistical Association69, 364-367. · Zbl 0291.62063 |
| [5] | Brunel, N. J. (2008). Parameter estimation of ODE’s via nonparametric estimators. Electronic Journal of Statistics2, 1242-1267. · Zbl 1320.62063 |
| [6] | Cao, J., Fussmann, G., and Ramsay, J. O. (2008). Estimating a predator‐prey dynamical model with the parameter cascades method. Biometrics64, 959-967. · Zbl 1147.92042 |
| [7] | Cao, J., Huang, J. Z., and Wu, H. (2012). Penalized nonlinear least squares estimation of time‐varying parameters in ordinary differential equations. Journal of Computational and Graphical Statistics21, 42-56. |
| [8] | Cao, J. and Ramsay, J. (2007). Parameter cascades and profiling in functional data analysis. Computational Statistics22, 335-351. · Zbl 1194.62043 |
| [9] | Cao, J. and Ramsay, J. (2010). Linear mixed effects modeling by parameter cascading. Journal of the American Statistical Association105, 365-374. · Zbl 1397.65093 |
| [10] | Cao, J., Wang, L., and Xu, J. (2011). Robust estimation for ordinary differential equation models. Biometrics67, 1305-1313. · Zbl 1274.62739 |
| [11] | Chen, J. and Wu, H. (2008). Efficient local estimation for time‐varying coefficients in deterministic dynamic models with applications to HIV‐1 dynamics. Journal of the American Statistical Association103, 369-383. · Zbl 1469.62365 |
| [12] | Chkrebtii, O. A., Campbell, D. A., Calderhead, B., and Girolami, M. A. (2013). Bayesian solution uncertainty quantification for differential equations. Preprint: arXiv:1306.2365. |
| [13] | Coca, D. and Billings, S. (2000). Direct parameter identification of distributed parameter systems. International Journal of Systems Science31, 11-17. · Zbl 1080.93531 |
| [14] | Dattner, I. and Klaassen, C. A. J. (2015). Optimal rate of direct estimators in systems of ordinary differential equations linear in functions of the parameters. Electronic Journal of Statistics9, 1939-1973. · Zbl 1327.62120 |
| [15] | Fan, J. and Gijbels, I. (1996). Local Polynomial Modelling and Its Applications. New York: CRC Press. · Zbl 0873.62037 |
| [16] | Frasso, G., Jaeger, J., and Lambert, P. (2016). Parameter estimation and inference in dynamic systems described by linear partial differential equations. Advances in Statistical Analysis100, 259-287. doi:10.1007/s10182‐015‐0257‐5 · Zbl 1443.62075 |
| [17] | Gelman, A., Bois, F., and Jiang, J. (1996). Physiological pharmacokinetic analysis using population modeling and informative prior distributions. Journal of the American Statistical Association91, 1400-1412. · Zbl 0882.62103 |
| [18] | Gugushvili, S. and Klaassen, C. A. J. (2012). \( \sqrt{n} \)‐consistent parameter estimation for systems of ordinary differential equations: Bypassing numerical integration via smoothing. Bernoulli18, 1061-1098. · Zbl 1257.49033 |
| [19] | Huang, Y., Liu, D., and Wu, H. (2006). Hierachical bayesian methods for estimation of parameters in a longitudinal HIV dynamic system. Biometrics62, 413-423. · Zbl 1097.62128 |
| [20] | Jost, J. (2002). Partial Differential Equations. New York: Springer‐Verlag. · Zbl 1034.35001 |
| [21] | Levene, H. (1960). In Contributions to Probability and Statistics: Essays in Honor of Harold Hotelling, I. Olkin, S.Ghurye, W.Hoeffding, W. G.Madow and H. B.Mann (eds), Stanford, California: Stanford University Press. · Zbl 0094.31604 |
| [22] | Liang, H., Miao, H., and Wu, H. (2010). Estimation of constant and time‐varying dynamic parameters of HIV infection in a nonlinear differential equation model. Annals of Applied Statistics4, 460-483. · Zbl 1189.62171 |
| [23] | Lu, T., Liang, H., Li, H., and Wu, H. (2011). High dimensional odes coupled with mixed‐effects modeling techniques for dynamic gene regulatory network identification. Journal of the American Statistical Association106, 1242-1258. · Zbl 1234.62146 |
| [24] | Magnus, J. R. and Neudecker, H. (2007). Matrix Differential Calculus with Applications in Statistics and Econometrics. New York: John Wiley & Sons. |
| [25] | Marx, B. and Eilers, P. (2005). Multidimensional penalized signal regression. Technometrics47, 13-22. |
| [26] | Miao, H., Wu, H., and Xue, H. (2014). Generalized ordinary differential equation models. Journal of the American Statistical Association109, 1672-1682. · Zbl 1368.62055 |
| [27] | Müller, T. G. and Timmer, J. (2002). Fitting parameters in partial differential equations from partially observed noisy data. Physica D171, 1-7. · Zbl 1009.65062 |
| [28] | Müller, T. G. and Timmer, J. (2004). Parameter identification techniques for partial differential equations. International Journal of Bifurcation and Chaos14, 2053-2060. · Zbl 1062.35177 |
| [29] | Parlitz, U. and Merkwirth, C. (2000). Prediction of spatiotemporal time series based on reconstructed local states. Physical Review Letters84, 1890-1893. |
| [30] | Qi, X. and Zhao, H. (2010). Asymptotic efficiency and finite‐sample properties of the generalized profiling estimation of parameters in ordinary differential equations. The Annals of Statistics38, 435-481. · Zbl 1181.62156 |
| [31] | Ramsay, J. O., Hooker, G., Campbell, D., and Cao, J. (2007). Parameter estimation for differential equations: a generalized smoothing approach (with discussion). Journal of the Royal Statistical Society, Series B69, 741-796. · Zbl 07555374 |
| [32] | Ramsay, J. O. and Silverman, B. W. (2005). Functional Data Analysis. New York: Springer. · Zbl 1079.62006 |
| [33] | Sangalli, L., Ramsay, J., and Ramsay, T. (2013). Spatial spline regression models. Journal of the Royal Statistical Society, Series B Statistical Methodology75, 681-703. · Zbl 1411.62134 |
| [34] | Sauer, T. (2006). Numerical Analysis. New York: Pearson Addison‐Wesley. |
| [35] | Vallette, D., Jacobs, G., and Gollub, J. (1997). Oscillations and spatiotemporal chaos of one‐dimensional fluid fronts. Physical Review E55, 4274-4287. |
| [36] | Voss, H. U., Kolodner, P., Abel, M., and Kurths, J. (1999). Amplitude equations from spatiotemporal binary‐fluid convection data. Physical Review Letters83, 3422-3425. |
| [37] | Vujacic, I., Mahmoudi, S. M., and Wit, E. (2016). Generalized Tikhonov regularization in estimation of ordinary differential equations models. Stat5, 132-143. · Zbl 07848557 |
| [38] | Wu, H., Lu, T., Xue, H., and Liang, H. (2014). Sparse additive ODEs for dynamic gene regulatory network modeling. JASA109, 700-716. · Zbl 1367.62221 |
| [39] | Xiao, L., Li, Y., and Ruppert, D. (2013). Fast bivariate p‐splines: The sandwich smoother. Journal of the Royal Statistical Society, Series B75, 577-599. · Zbl 1411.62109 |
| [40] | Xue, H., Miao, H., and Wu, H. (2010). Sieve estimation of constant and time‐varying coefficients in nonlinear ordinary differential equation models by considering both numerical error and measurement error. Annals of Statistics38, 2351-2387. · Zbl 1203.62049 |
| [41] | Xun, X., Cao, J., Mallick, B., Maity, A., and Carroll, R. J. (2013). Parameter estimation of partial differential equation models. JASA108, 1009-1020. · Zbl 06224983 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
