Standard error computations for uncertainty quantification in inverse problems: asymptotic theory vs. bootstrapping. (English) Zbl 1205.93170
Summary: We computationally investigate two approaches for uncertainty quantification in inverse problems for nonlinear parameter dependent dynamical systems. We compare the bootstrapping and asymptotic theory approaches for problems involving data with several noise forms and levels. We consider both constant variance absolute error data and relative error, which produce non-constant variance data in our parameter estimation formulations. We compare and contrast parameter estimates, standard errors, confidence intervals, and computational times for both bootstrapping and asymptotic theory methods.
MSC:
| 93E24 | Least squares and related methods for stochastic control systems |
| 62F40 | Bootstrap, jackknife and other resampling methods |
References:
| [1] | Anger, G., Inverse Problems in Differential Equations (1990), Plenum Press: Plenum Press New York · Zbl 0752.35083 |
| [2] | Banks, H. T.; Buksas, M. W.; Lin, T., Electromagnetic Material Interrogation Using Conductive Interfaces and Acoustic Wavefronts (2002), SIAM: SIAM Philadelphia · Zbl 1008.78500 |
| [3] | (Chowell, Gerardo; Hyman, Mac; Hengartner, Nick; Bettencourt, Luis M. A.; Castillo-Chavez, Carlos, Statistical Estimation Approaches in Epidemiology (2009), Springer: Springer Berlin, Heidelberg, New York), 249-302, (Chapter 11) · Zbl 1345.92006 |
| [4] | Banks, H. T.; Kunsich, K., Estimation Techniques for Distributed Parameter Systems (1989), Birkhaüser: Birkhaüser Boston · Zbl 0695.93020 |
| [5] | Banks, H. T.; Tran, H. T., Mathematical and Experimental Modeling of Physical and Biological Processes (2009), CRC Press: CRC Press Boca Raton · Zbl 1026.93514 |
| [6] | Baumeister, J., Stable Solutions of Inverse Problems (1987), Vieweg: Vieweg Braunschweig · Zbl 0623.35008 |
| [7] | (Engl, H. W.; Groetsch, C. W., Inverse and Ill-posed Problems (1987), Academic Press: Academic Press Orlando) · Zbl 0623.00010 |
| [8] | Groetsch, C. W., Inverse Problems in the Mathematical Sciences (1993), Vieweg: Vieweg Braunschweig · Zbl 1266.34019 |
| [9] | Groetsch, C. W., The Theory of Tikhonov Regularization for Fredholm Equations of the First Kind (1984), Pitman: Pitman London · Zbl 0545.65034 |
| [10] | Hoffman, B., Regularization for Applied Inverse and Ill-posed Problems (1986), Teubner: Teubner Leipzig · Zbl 0606.65038 |
| [11] | Tikhonov, A. N.; Arsenin, V. Y., Solutions of Ill-posed Problems (1977), Winston and Sons: Winston and Sons Washington · Zbl 0354.65028 |
| [12] | Vogel, C. R., Computational Methods for Inverse Problems (2002), SIAM: SIAM Philadelphia · Zbl 1008.65103 |
| [13] | Banks, H. T.; Dediu, S.; Ernstberger, S. E., Sensitivity functions and their uses in inverse problems, J. Inverse Ill-Posed Probl., 15, 683-708 (2007) · Zbl 1141.34049 |
| [14] | Banks, H. T.; Ernstberger, S. L.; Grove, S. L., Standard errors and confidence intervals in inverse problems: sensitivity and associated pitfalls, J. Inverse Ill-Posed Probl., 15, 1-18 (2007) · Zbl 1124.34002 |
| [15] | Casella, G.; Berger, R. L., Statistical Inference (2002), Duxbury: Duxbury California |
| [16] | Davidian, M.; Giltinan, D. M., Nonlinear Models for Repeated Measurement Data (1995), Chapman & Hall: Chapman & Hall London |
| [17] | Seber, G. A.F.; Wild, C. J., Nonlinear Regression (2003), Wiley-Interscience: Wiley-Interscience Hoboken, NJ · Zbl 0721.62062 |
| [18] | Kot, M., Elements of Mathematical Ecology (2001), Cambridge University Press: Cambridge University Press Cambridge |
| [19] | Carroll, R. J.; Ruppert, D., Transformation and Weighting in Regression (1988), Chapman & Hall: Chapman & Hall London · Zbl 0666.62062 |
| [20] | Davidian, M., (Nonlinear Models for Univariate and Multivariate Response. Nonlinear Models for Univariate and Multivariate Response, Lecture Notes, vol. 762 (2007)), (Chapters 9 and 11) |
| [21] | Matloff, N.; Rose, R.; Tai, R., A comparison of two methods for estimating optimal weights in regression analysis, J. Stat. Comput. Simul., 19, 265-274 (1984) |
| [22] | Carroll, R. J.; Wu, C. F.J.; Ruppert, D., The effect of estimating weights in Weighted Least Squares, J. Amer. Statist. Assoc., 83, 1045-1054 (1988) · Zbl 0691.62061 |
| [23] | Efron, B.; Tibshirani, R. J., An Introduction to the Bootstrap (1998), Chapman & Hall, CRC Press: Chapman & Hall, CRC Press Boca Raton |
| [24] | Rothenberg, T. J., Approximate normality of generalized least squares estimates, Econometrica, 52, 4, 811-825 (1984) · Zbl 0557.62064 |
| [25] | Bates, D. M., Nonlinear Regression Analysis and its Applications (1988), J. Wiley & Sons: J. Wiley & Sons Somerset, NJ · Zbl 0728.62062 |
| [26] | H.T. Banks, S. Dediu, S.L. Ernstberger, F. Kappel, A new approach to optimal design problems, CRSC Technical Report CRSC-TR08-12, NCSU, September, 2008.; H.T. Banks, S. Dediu, S.L. Ernstberger, F. Kappel, A new approach to optimal design problems, CRSC Technical Report CRSC-TR08-12, NCSU, September, 2008. · Zbl 1279.62089 |
| [27] | Thomaseth, K.; Cobelli, C., Generalized sensitivity functions in physiological system identification, Ann. Biomed. Eng., 27, 607-616 (1999) |
| [28] | Bickel, P.; Freedman, D., Some asymptotic theory for the bootstrap, Ann. Statist., 9, 6, 1196-1217 (1981) · Zbl 0449.62034 |
| [29] | Shao, J.; Tu, D., The Jackknife and Bootstrap (1995), Springer-Verlag: Springer-Verlag New York · Zbl 0947.62501 |
| [30] | Shi, X.; Chen, J.; Wu, C. F.J., Weak and strong representations for quantile processes from finite populations with applications to simulation size in resampling methods, Canad. J. Statist., 18, 141-148 (1990) · Zbl 0706.62051 |
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