Adaption of Akaike information criterion under least squares frameworks for comparison of stochastic models
Authors:
H. T. Banks and Michele L. Joyner
Journal:
Quart. Appl. Math. 77 (2019), 831-859
MSC (2010):
Primary 93E03, 37L55, 37H10
DOI:
https://doi.org/10.1090/qam/1542
Published electronically:
May 31, 2019
MathSciNet review:
4009334
Full-text PDF
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Additional Information
Abstract: In this paper, we examine the feasibility of extending the Akaike information criterion (AIC) for deterministic systems as a potential model selection criteria for stochastic models. We discuss the implementation method for three different classes of stochastic models: continuous time Markov chains (CTMC), stochastic differential equations (SDE), and random differential equations (RDE). The effectiveness and limitations of implementing the AIC for comparison of stochastic models is demonstrated using simulated data from the three types of models and then applied to experimental longitudinal growth data for algae.
References
- Kaska Adoteye, H. T. Banks, Karissa Cross, Stephanie Eytcheson, Kevin B. Flores, Gerald A. LeBlanc, Timothy Nguyen, Chelsea Ross, Emmaline Smith, Michael Stemkovski, and Sarah Stokely, Statistical validation of structured population models for Daphnia magna, Math. Biosci. 266 (2015), 73–84. MR 3367563, DOI https://doi.org/10.1016/j.mbs.2015.06.003
- Hirotugu Akaike, Information measures and model selection, Proceedings of the 44th session of the International Statistical Institute, Vol. 1 (Madrid, 1983), 1983, pp. 277–290 (English, with French summary). With a discussion in Vol. 3, pp. 209–219. MR 820726, DOI https://doi.org/10.1086/phos.50.4.187553
- Linda J. S. Allen, An introduction to stochastic processes with applications to biology, 2nd ed., CRC Press, Boca Raton, FL, 2011. MR 2560499
- L. Babák, P. Šupinová, R. Burdychová et. al., Growth models of thermus aquaticus and thermus scotoductus, Acta Universitatis Agriculturae et Silviculturae Mendelianae Brunensis, 60(5):19–26, 2013.
- H. T. Banks, K. Bekele-Maxwell, J. Canner, A. Mayhall, J. Menda, and M. Noorman, The effect of statistical error model formulation on the fit and selection of mathematical models of tumor growth for small sample sizes, CRSC-TR17-26, N. C. State University, Raleigh, NC, November, 2017; Int. J. Pure and Applied Mathematics, 117:203–234, 2017.
- H. Thomas Banks, Jared Catenacci, and Shuhua Hu, Use of difference-based methods to explore statistical and mathematical model discrepancy in inverse problems, J. Inverse Ill-Posed Probl. 24 (2016), no. 4, 413–433. MR 3530556, DOI https://doi.org/10.1515/jiip-2015-0090
- H. T. Banks, Shuhua Hu, and W. Clayton Thompson, Modeling and inverse problems in the presence of uncertainty, Monographs and Research Notes in Mathematics, CRC Press, Boca Raton, FL, 2014. MR 3203115
- H. T. Banks and Michele L. Joyner, AIC under the framework of least squares estimation, Appl. Math. Lett. 74 (2017), 33–45. MR 3677839, DOI https://doi.org/10.1016/j.aml.2017.05.005
- H. T. Banks, E. Collins, K. Flores, P. Pershad, M. Stemkovski, and L. Stephenson. Standard and proportional error model comparison for logistic growth of green algae (Raphidocelis subcapiala), Applied Mathematical Letters, 64:213–222, 2017.
- H. T. Banks and B. G. Fitzpatrick, Statistical methods for model comparison in parameter estimation problems for distributed systems, J. Math. Biol. 28 (1990), no. 5, 501–527. MR 1069009, DOI https://doi.org/10.1007/BF00164161
- H. T. Banks and M. L. Joyner, Determinisitc methodology for comparison of nested stochastic models, CRSC-TR16-13, NCSU, Raleigh, NC, November 2016; Communication in Applied Analysis, 21:15–50, 2017.
- H. T. Banks and M. L. Joyner, Parameter estimation for random differential equation models, CRSC-TR16-15, NCSU, Raleigh, NC, December,2016; Eurasian Journal of Mathematical and Computer Applications(EJMCA), 5:5–44, 2017.
- H. T. Banks and H. T. Tran, Mathematical and Experimental Modeling of Physical and Biological Processes, CRC Press, Chapman & Hall, Boca Raton, FL, 2009.
- Hamparsum Bozdogan, Model selection and Akaike’s information criterion (AIC): the general theory and its analytical extensions, Psychometrika 52 (1987), no. 3, 345–370. MR 914460, DOI https://doi.org/10.1007/BF02294361
- Hamparsum Bozdogan, Akaike’s information criterion and recent developments in information complexity, J. Math. Psych. 44 (2000), no. 1, 62–91. MR 1770002, DOI https://doi.org/10.1006/jmps.1999.1277
- Kenneth P. Burnham and David R. Anderson, Model selection and multimodel inference, 2nd ed., Springer-Verlag, New York, 2002. A practical information-theoretic approach. MR 1919620
- Stewart N. Ethier and Thomas G. Kurtz, Markov processes, Wiley Series in Probability and Mathematical Statistics: Probability and Mathematical Statistics, John Wiley & Sons, Inc., New York, 1986. Characterization and convergence. MR 838085
- P. Gerlee, The model muddle: in search of tumor growth laws, Cancer Research, 73(8):2407–2411, 2013.
- F. Helidoniotis, M. Haddon, G. Tuck, and D. Tarbath, The relative suitability of the von Bertalanffy, Gompertz and inverse Logistic models for describing growth in blacklip abalone populations (haliotis rubra) in Tasmania, Australia, Fisheries Research, 112(1-2):13–21, 2011.
- C. M. Jones, Fitting growth curves to retrospective size-at-age data, Fisheries Research, 46(1-3):123–129, 2000.
- M. Joyner and T. Robacker, Development of the MCR method for estimation of parameters in continuous time Markov Chain models, International J. of Pure and Applied Mathematics, 112(2):381–416, 2017.
- S. Kullback and R. A. Leibler, On information and sufficiency, Ann. Math. Statistics 22 (1951), 79–86. MR 39968, DOI https://doi.org/10.1214/aoms/1177729694
- Thomas G. Kurtz, Solutions of ordinary differential equations as limits of pure jump Markov processes, J. Appl. Probability 7 (1970), 49–58. MR 254917, DOI https://doi.org/10.2307/3212147
- A. R. Ortiz, H. T. Banks, C. Castillo-Chavez, G. Chowell, and X. Wang, A deterministic methodology for estimation of parameters in dynamic Markov chain models, J. Biol. Systems 19 (2011), no. 1, 71–100. MR 2785108, DOI https://doi.org/10.1142/S0218339011003798
- M. Stemkovski, R. Baraldi, K. B. Flores, and H. T. Banks, Validation of a mathematical model for green algae (Raphidocelis subcapitata) growth and implications for a coupled dynamical system with Daphnia magna, Applied Sciences, 6(5):155–173, 2016.
- L. von Bertalanffy, Problems of organic growth, Nature, 163(4135):156–158, 1949.
- E.-J. Wagenmakers and S. Farrell, AIC model selection using Akaike weights, Psychonomic Bulletin & Review, 11(1):192–196, 2004.
References
- Kaska Adoteye, H. T. Banks, Karissa Cross, Stephanie Eytcheson, Kevin B. Flores, Gerald A. LeBlanc, Timothy Nguyen, Chelsea Ross, Emmaline Smith, Michael Stemkovski, and Sarah Stokely, Statistical validation of structured population models for Daphnia magna, Math. Biosci. 266 (2015), 73–84. MR 3367563, DOI https://doi.org/10.1016/j.mbs.2015.06.003
- Hirotugu Akaike, Information measures and model selection, Proceedings of the 44th session of the International Statistical Institute, Vol. 1 (Madrid, 1983), 1983, pp. 277–290 (English, with French summary). With a discussion in Vol. 3, pp. 209–219. MR 820726, DOI https://doi.org/10.1086/phos.50.4.187553
- Linda J. S. Allen, An introduction to stochastic processes with applications to biology, 2nd ed., CRC Press, Boca Raton, FL, 2011. MR 2560499
- L. Babák, P. Šupinová, R. Burdychová et. al., Growth models of thermus aquaticus and thermus scotoductus, Acta Universitatis Agriculturae et Silviculturae Mendelianae Brunensis, 60(5):19–26, 2013.
- H. T. Banks, K. Bekele-Maxwell, J. Canner, A. Mayhall, J. Menda, and M. Noorman, The effect of statistical error model formulation on the fit and selection of mathematical models of tumor growth for small sample sizes, CRSC-TR17-26, N. C. State University, Raleigh, NC, November, 2017; Int. J. Pure and Applied Mathematics, 117:203–234, 2017.
- H. Thomas Banks, Jared Catenacci, and Shuhua Hu, Use of difference-based methods to explore statistical and mathematical model discrepancy in inverse problems, J. Inverse Ill-Posed Probl. 24 (2016), no. 4, 413–433. MR 3530556, DOI https://doi.org/10.1515/jiip-2015-0090
- H. T. Banks, Shuhua Hu, and W. Clayton Thompson, Modeling and inverse problems in the presence of uncertainty, Monographs and Research Notes in Mathematics, CRC Press, Boca Raton, FL, 2014. MR 3203115
- H. T. Banks and Michele L. Joyner, AIC under the framework of least squares estimation, Appl. Math. Lett. 74 (2017), 33–45. MR 3677839, DOI https://doi.org/10.1016/j.aml.2017.05.005
- H. T. Banks, E. Collins, K. Flores, P. Pershad, M. Stemkovski, and L. Stephenson. Standard and proportional error model comparison for logistic growth of green algae (Raphidocelis subcapiala), Applied Mathematical Letters, 64:213–222, 2017.
- H. T. Banks and B. G. Fitzpatrick, Statistical methods for model comparison in parameter estimation problems for distributed systems, J. Math. Biol. 28 (1990), no. 5, 501–527. MR 1069009, DOI https://doi.org/10.1007/BF00164161
- H. T. Banks and M. L. Joyner, Determinisitc methodology for comparison of nested stochastic models, CRSC-TR16-13, NCSU, Raleigh, NC, November 2016; Communication in Applied Analysis, 21:15–50, 2017.
- H. T. Banks and M. L. Joyner, Parameter estimation for random differential equation models, CRSC-TR16-15, NCSU, Raleigh, NC, December,2016; Eurasian Journal of Mathematical and Computer Applications(EJMCA), 5:5–44, 2017.
- H. T. Banks and H. T. Tran, Mathematical and Experimental Modeling of Physical and Biological Processes, CRC Press, Chapman & Hall, Boca Raton, FL, 2009.
- Hamparsum Bozdogan, Model selection and Akaike’s information criterion (AIC): the general theory and its analytical extensions, Psychometrika 52 (1987), no. 3, 345–370. MR 914460, DOI https://doi.org/10.1007/BF02294361
- Hamparsum Bozdogan, Akaike’s information criterion and recent developments in information complexity, J. Math. Psych. 44 (2000), no. 1, 62–91. MR 1770002, DOI https://doi.org/10.1006/jmps.1999.1277
- Kenneth P. Burnham and David R. Anderson, Model selection and multimodel inference, 2nd ed., Springer-Verlag, New York, 2002. A practical information-theoretic approach. MR 1919620
- Stewart N. Ethier and Thomas G. Kurtz, Markov processes: Characterization and convergence, Wiley Series in Probability and Mathematical Statistics: Probability and Mathematical Statistics, John Wiley & Sons, Inc., New York, 1986. MR 838085
- P. Gerlee, The model muddle: in search of tumor growth laws, Cancer Research, 73(8):2407–2411, 2013.
- F. Helidoniotis, M. Haddon, G. Tuck, and D. Tarbath, The relative suitability of the von Bertalanffy, Gompertz and inverse Logistic models for describing growth in blacklip abalone populations (haliotis rubra) in Tasmania, Australia, Fisheries Research, 112(1-2):13–21, 2011.
- C. M. Jones, Fitting growth curves to retrospective size-at-age data, Fisheries Research, 46(1-3):123–129, 2000.
- M. Joyner and T. Robacker, Development of the MCR method for estimation of parameters in continuous time Markov Chain models, International J. of Pure and Applied Mathematics, 112(2):381–416, 2017.
- S. Kullback and R. A. Leibler, On information and sufficiency, Ann. Math. Statistics 22 (1951), 79–86. MR 0039968, DOI https://doi.org/10.1214/aoms/1177729694
- Thomas G. Kurtz, Solutions of ordinary differential equations as limits of pure jump Markov processes, J. Appl. Probability 7 (1970), 49–58. MR 0254917, DOI https://doi.org/10.2307/3212147
- A. R. Ortiz, H. T. Banks, C. Castillo-Chavez, G. Chowell, and X. Wang, A deterministic methodology for estimation of parameters in dynamic Markov chain models, J. Biol. Systems 19 (2011), no. 1, 71–100. MR 2785108, DOI https://doi.org/10.1142/S0218339011003798
- M. Stemkovski, R. Baraldi, K. B. Flores, and H. T. Banks, Validation of a mathematical model for green algae (Raphidocelis subcapitata) growth and implications for a coupled dynamical system with Daphnia magna, Applied Sciences, 6(5):155–173, 2016.
- L. von Bertalanffy, Problems of organic growth, Nature, 163(4135):156–158, 1949.
- E.-J. Wagenmakers and S. Farrell, AIC model selection using Akaike weights, Psychonomic Bulletin & Review, 11(1):192–196, 2004.
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Additional Information
H. T. Banks
Affiliation:
Center for Research in Scientific Computation, North Carolina State University, Raleigh, North Carolina 27695
MR Author ID:
194993
Michele L. Joyner
Affiliation:
Department of Mathematics and Statistics, East Tennessee State University, Johnson City, Tennessee 37614
MR Author ID:
666534
Keywords:
Continuous time Markov chain models,
CTMC,
stochastic differential equations,
SDE,
random differential equations,
RDE,
inverse problems,
model comparison techniques,
Akaike information criterion,
AIC
Received by editor(s):
April 16, 2019
Published electronically:
May 31, 2019
Additional Notes:
This research was supported for both authors in part by the Air Force Office of Scientific Research under grant number AFOSR FA9550-18-1-0457
Article copyright:
© Copyright 2019
Brown University
