Abstract
Normalized differences of several adjacent observations, referred to as pseudo-measurement errors in this paper, are used in so-called difference-based estimation methods as building blocks for the variance estimate of measurement errors. Numerical results demonstrate that pseudo-measurement errors can be used to serve the role of measurement errors. Based on this information, we propose the use of pseudo-measurement errors to determine an appropriate statistical model and then to subsequently investigate whether there is a mathematical model misspecification or error. We also propose to use the information provided by pseudo-measurement errors to quantify uncertainty in parameter estimation by bootstrapping methods. A number of numerical examples are given to illustrate the effectiveness of these proposed methods.
Funding source: Air Force Office of Scientific Research
Award Identifier / Grant number: AFOSR FA9550-12-1-0188
Funding statement: This research was supported in part by the Air Force Office of Scientific Research under grant number AFOSR FA9550-12-1-0188, and in part by the US Department of Education Graduate Assistance in Areas of National Need (GAANN) under grant number P200A120047.
The authors wish to gratefully acknowledge the efforts of two referees of an earlier version of this manuscript whose questions and suggestions resulted in significant improvements of the current manuscript.
References
[1] Adoteye K., Banks H. T., Flores K. B. and LeBlanc G. A., Estimation of time-varying mortality rates using continuous models for Daphnia Magna, Appl. Math. Lett. 44 (2015), 12–16; CRSC-TR14-17, Center for Research in Scientific Computation, North Carolina State University, 2014. 10.1016/j.aml.2014.12.014Suche in Google Scholar
[2] Arendt P. D., Apley D. W. and Chen W., Quantification of model uncertainty: Calibration, model discrepancy and identifiability, J. Mech. Design 134 (2012), Article ID 100908. 10.1115/1.4007390Suche in Google Scholar
[3] Banks H. T., Banks J. E., Murad N., Rosenheim J. A. and Tillman K., Modelling pesticide treatment effects on Lygus hesperus in cotton fields, CRSC-TR15-09, Center for Research in Scientific Computation, North Carolina State University, Raleigh, 2015. 10.1007/978-3-319-55795-3_8Suche in Google Scholar
[4] Banks H. T., Banks J. E., Rosenheim J. and Tillman K., Modeling populations of Lygus Hesperus on cotton fields in the San Joaquin Valley of California: The importance of statistical and mathematical model choice, J. Biol. Dyn. (2016), 10.1080/17513758.2016.1143533; CRSC-TR15-04, Center for Research in Scientific Computation, North Carolina State University, Raleigh, 2015. 10.1080/17513758.2016.1143533Suche in Google Scholar PubMed
[5] Banks H. T., Baraldi R., Cross K., Flores K., McChesney C., Poag L. and Thorpe E., Uncertainty quantification in modeling HIV viral mechanics, Math. Biosciences Engr. 12 (2015), 937–964; CRSC-TR13-16, North Carolina State University, Raleigh, 2013. 10.3934/mbe.2015.12.937Suche in Google Scholar PubMed
[6] Banks H. T., Catenacci J. and Hu S., Asymptotic properties of probability measure estimators in a nonparametric model, SIAM/ASA J. Uncertainty Quantification 3 (2015), no. 1, 417–433. 10.1137/140972639Suche in Google Scholar
[7] Banks H. T., Doumic M., Kruse C., Prigent S. and Rezaei H., Information content in data sets for a nucleated-polymerization model, J. Biol. Dyn. 9 (2015), no. 1, 172–197; CRSC-TR14-15, North Carolina State University, Raleigh, 2014. 10.1080/17513758.2015.1050465Suche in Google Scholar PubMed PubMed Central
[8] Banks H. T., Hu S. and Thompson W. C., Modeling and Inverse Problems in the Presence of Uncertainty, Chapman & Hall/CRC Press, Boca Raton, 2014. 10.1201/b16760Suche in Google Scholar
[9] Banks H. T., Kapraun D. F., Link K. G., Thompson W. C., Peligero C., Argilaguet J. and Meyerhans A., Analysis of variability in estimates of cell proliferation parameters for cyton-based models using CFSE-based flow cytometry data, J. Inverse Ill-Posed Probl. 23 (2014), 135–171; CRSC-TR13-14, North Carolina State University, Raleigh, 2013. 10.1515/jiip-2013-0065Suche in Google Scholar
[10] Kapraun H. T. Banks D. F., Peligero C., Argilaguet J. and Meyerhans A., Evaluating the importance of mitotic asymmetry in cyton-based models for CFSE-based flow cytometry data, Int. J. Pure Appl. Math. 100 (2015), no. 1, 131–156; CRSC-TR15-02, North Carolina State University, Raleigh, 2015. 10.12732/ijpam.v100i1.12Suche in Google Scholar
[11] Banks H. T., Kapraun D. F., Thompson W. C., Peligero C., Argilaguet J. and Meyerhans A., A novel statistical analysis and interpretation of flow cytometry data, J. Biol. Dyn. 7 (2013), 96–132; CRSC-TR12-23, North Carolina State University, Raleigh, 2013. 10.1080/17513758.2013.812753Suche in Google Scholar PubMed PubMed Central
[12] Banks H. T. and Tran H. T., Mathematical and Experimental Modeling of Physical and Biological Processes, CRC Press, Boca Raton, 2009. 10.1201/b17175Suche in Google Scholar
[13] Bocharov G., Luzyanina T., Cupovic J. and Ludewig B., Asymmetry of cell division in CFSE-based lymphocyte proliferation analysis, Front. Immunol. 4 (2013), 10.3389/fimmu.2013.00264. 10.3389/fimmu.2013.00264Suche in Google Scholar PubMed PubMed Central
[14] Brown L. D. and Levine M., Variance estimation in nonparametric regression via the difference sequence method, Ann. Statist. 35 (2007), 2219–2232. 10.1214/009053607000000145Suche in Google Scholar
[15] Brynjarsdóttir J. and O’Hagan A., Learning about physical parameters: The importance of model discrepancy, Inverse Problems 30 (2014), Article ID 114007. 10.1088/0266-5611/30/11/114007Suche in Google Scholar
[16] Carroll R. J. and Ruppert D., Transformation and Weighting in Regression, Chapman & Hall, New York, 1988. 10.1007/978-1-4899-2873-3Suche in Google Scholar
[17] Davidian M., Nonlinear Models for Univariate and Multivariate Response. Chapters 9 and 11, lecture notes 2007, http://www4.stat.ncsu.edu/~davidian/courses.html. Suche in Google Scholar
[18] Davidian M. and Giltinan D., Nonlinear Models for Repeated Measurement Data, Chapman & Hall, London, 1998. Suche in Google Scholar
[19] Dette H., Munk A. and Wagner T., Estimating the variance in nonparametric regression – What is a reasonable choice?, J. R. Statist. Soc. B 60 (1998), 751–764. 10.1111/1467-9868.00152Suche in Google Scholar
[20] Doherty J. and Welter D., A short exploration of structural noise, Water Resource Research 46 (2010), Article ID W05525. 10.1029/2009WR008377Suche in Google Scholar
[21] Efron B. and Tibshirani R. J., An Introduction to the Bootstrap, Chapman & Hall/CRC Press, Boca Raton, 1998. Suche in Google Scholar
[22] Kapraun D. F., Cell proliferation models, CFSE-based flow cytometry data, and quantification of uncertainty, Ph.D. thesis, North Carolina State University, Raleigh, 2014. Suche in Google Scholar
[23] Levine M., Bandwidth selection for a class of difference-based variance estimators in the nonparametric regression: A possible approach, Comput. Statist. Data Anal. 50 (2006), 3405–3431. 10.1016/j.csda.2005.08.001Suche in Google Scholar
[24] Luzyanina T., Cupovic J., Ludewig B. and Bocharov G., Mathematical models for CFSE labelled lymphocyte dynamics: Asymmetry and time-lag in division, J. Math. Biol. 69 (2014), 1547–1583. 10.1007/s00285-013-0741-zSuche in Google Scholar PubMed
[25] Müller H.-G. and Stadtmüller U., Estimation of heteroscedasticity in regression analysis, Ann. Statist. 15 (1987), 610–625. 10.1214/aos/1176350364Suche in Google Scholar
[26] Seber G. A. F. and Wild C. J., Nonlinear Regression, John Wiley & Sons, Hoboken, 2003. 10.1002/9780471722199Suche in Google Scholar
[27] Thompson W. C., Partial differential equation modeling of flow cytometry data from CFSE-based proliferation assays, Ph.D. thesis, North Carolina State University, Raleigh, 2011. Suche in Google Scholar
[28] Tong T., Liu A. and Wang Y., Relative errors of difference-based variance estimators in nonparametric regression, Comm. Statist. Theory Methods 37 (2008), 2890–2902. 10.1080/03610920802162656Suche in Google Scholar
[29] Vrugt J. A., Diks C. G. H., Gupta H. V., Bouten W. and Verstraten J. M., Improved treatment of uncertainty in hydrologic modeling: Combining the strengths of global optimization and data assimilation, Water Resources Research 41 (2005), Article ID W01017. 10.1029/2004WR003059Suche in Google Scholar
© 2016 by De Gruyter
Artikel in diesem Heft
- Frontmatter
- Inversions of the windowed ray transform
- Inverse problems of demand analysis and their applications to computation of positively-homogeneous Konüs–Divisia indices and forecasting
- Some generalizations for Landweber iteration for nonlinear ill-posed problems in Hilbert scales
- An inverse problem for Sturm–Liouville operators with non-separated boundary conditions containing the spectral parameter
- Use of difference-based methods to explore statistical and mathematical model discrepancy in inverse problems
- Computing quasisolutions of nonlinear inverse problems via efficient minimization of trust region problems
- Accuracy estimates of Gauss–Newton-type iterative regularization methods for nonlinear equations with operators having normally solvable derivative at the solution
- On convergence rates for asymptotic discrepancy principle
- Identification of an unknown coefficient in KdV equation from final time measurement
- Improved asymptotic analysis for dynamical probe method
Artikel in diesem Heft
- Frontmatter
- Inversions of the windowed ray transform
- Inverse problems of demand analysis and their applications to computation of positively-homogeneous Konüs–Divisia indices and forecasting
- Some generalizations for Landweber iteration for nonlinear ill-posed problems in Hilbert scales
- An inverse problem for Sturm–Liouville operators with non-separated boundary conditions containing the spectral parameter
- Use of difference-based methods to explore statistical and mathematical model discrepancy in inverse problems
- Computing quasisolutions of nonlinear inverse problems via efficient minimization of trust region problems
- Accuracy estimates of Gauss–Newton-type iterative regularization methods for nonlinear equations with operators having normally solvable derivative at the solution
- On convergence rates for asymptotic discrepancy principle
- Identification of an unknown coefficient in KdV equation from final time measurement
- Improved asymptotic analysis for dynamical probe method