IDEAS home Printed from https://ideas.repec.org/a/eee/apmaco/v430y2022ics0096300322003344.html
   My bibliography  Save this article

Identifiability analysis of linear ordinary differential equation systems with a single trajectory

Author

Listed:
  • Qiu, Xing
  • Xu, Tao
  • Soltanalizadeh, Babak
  • Wu, Hulin

Abstract

Ordinary differential equations (ODEs) are widely used to model dynamical behavior of systems. It is important to perform identifiability analysis prior to estimating unknown parameters in ODEs (a.k.a. inverse problem), because if a system is unidentifiable, the estimation procedure may fail or produce erroneous and misleading results. Although several qualitative identifiability measures have been proposed, much less effort has been given to developing quantitative (continuous) scores that are robust to uncertainties in the data, especially for those cases in which the data are presented as a single trajectory beginning with one initial value. In this paper, we first derived a closed-form representation of linear ODE systems that are not identifiable based on a single trajectory. This representation helps researchers design practical systems and choose the right prior structural information in practice. Next, we proposed several quantitative scores for identifiability analysis in practice. In simulation studies, the proposed measures outperformed the main competing method significantly, especially when noise was presented in the data. We also discussed the asymptotic properties of practical identifiability for high-dimensional ODE systems and conclude that, without additional prior information, many random ODE systems are practically unidentifiable when the dimension approaches infinity.

Suggested Citation

  • Qiu, Xing & Xu, Tao & Soltanalizadeh, Babak & Wu, Hulin, 2022. "Identifiability analysis of linear ordinary differential equation systems with a single trajectory," Applied Mathematics and Computation, Elsevier, vol. 430(C).
    • Handle: RePEc:eee:apmaco:v:430:y:2022:i:c:s0096300322003344
    DOI: 10.1016/j.amc.2022.127260
    as

    Download full text from publisher

    File URL: http://www.sciencedirect.com/science/article/pii/S0096300322003344
    Download Restriction: Full text for ScienceDirect subscribers only
    File URL: https://libkey.io/10.1016/j.amc.2022.127260?utm_source=ideas
    LibKey link: if access is restricted and if your library uses this service, LibKey will redirect you to where you can use your library subscription to access this item
    ---><---

    As the access to this document is restricted, you may want to search for a different version of it.

    References listed on IDEAS

    as
    1. Leqin Wu & Xing Qiu & Ya-xiang Yuan & Hulin Wu, 2019. "Parameter Estimation and Variable Selection for Big Systems of Linear Ordinary Differential Equations: A Matrix-Based Approach," Journal of the American Statistical Association, Taylor & Francis Journals, vol. 114(526), pages 657-667, April.
    2. Commenges, D. & Jolly, D. & Drylewicz, J. & Putter, H. & Thiébaut, R., 2011. "Inference in HIV dynamics models via hierarchical likelihood," Computational Statistics & Data Analysis, Elsevier, vol. 55(1), pages 446-456, January.
    3. Marc Lavielle & Adeline Samson & Ana Karina Fermin & France Mentré, 2011. "Maximum Likelihood Estimation of Long-Term HIV Dynamic Models and Antiviral Response," Biometrics, The International Biometric Society, vol. 67(1), pages 250-259, March.
    4. Yangxin Huang & Dacheng Liu & Hulin Wu, 2006. "Hierarchical Bayesian Methods for Estimation of Parameters in a Longitudinal HIV Dynamic System," Biometrics, The International Biometric Society, vol. 62(2), pages 413-423, June.
    5. Yangxin Huang & Hulin Wu, 2006. "A Bayesian approach for estimating antiviral efficacy in HIV dynamic models," Journal of Applied Statistics, Taylor & Francis Journals, vol. 33(2), pages 155-174.
    6. J. O. Ramsay & G. Hooker & D. Campbell & J. Cao, 2007. "Parameter estimation for differential equations: a generalized smoothing approach," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 69(5), pages 741-796, November.
    Full references (including those not matched with items on IDEAS)

    Citations

    Citations are extracted by the CitEc Project, subscribe to its RSS feed for this item.
    as


    Cited by:

    1. Soradi-Zeid, Samaneh & Mesrizadeh, Mehdi, 2023. "On the convergence of finite integration method for system of ordinary differential equations," Chaos, Solitons & Fractals, Elsevier, vol. 167(C).

    Most related items

    These are the items that most often cite the same works as this one and are cited by the same works as this one.
    1. Hulin Wu & Hongqi Xue & Arun Kumar, 2012. "Numerical Discretization-Based Estimation Methods for Ordinary Differential Equation Models via Penalized Spline Smoothing with Applications in Biomedical Research," Biometrics, The International Biometric Society, vol. 68(2), pages 344-352, June.
    2. Liu, Baisen & Wang, Liangliang & Nie, Yunlong & Cao, Jiguo, 2019. "Bayesian inference of mixed-effects ordinary differential equations models using heavy-tailed distributions," Computational Statistics & Data Analysis, Elsevier, vol. 137(C), pages 233-246.
    3. Hanwen Huang, 2022. "Bayesian multi‐level mixed‐effects model for influenza dynamics," Journal of the Royal Statistical Society Series C, Royal Statistical Society, vol. 71(5), pages 1978-1995, November.
    4. Zhang, Tingting & Sun, Yinge & Li, Huazhang & Yan, Guofen & Tanabe, Seiji & Miao, Ruizhong & Wang, Yaotian & Caffo, Brian S. & Quigg, Mark S., 2020. "Bayesian inference of a directional brain network model for intracranial EEG data," Computational Statistics & Data Analysis, Elsevier, vol. 144(C).
    5. Hanwen Huang & Andreas Handel & Xiao Song, 2020. "A Bayesian approach to estimate parameters of ordinary differential equation," Computational Statistics, Springer, vol. 35(3), pages 1481-1499, September.
    6. Qianwen Tan & Subhashis Ghosal, 2021. "Bayesian Analysis of Mixed-effect Regression Models Driven by Ordinary Differential Equations," Sankhya B: The Indian Journal of Statistics, Springer;Indian Statistical Institute, vol. 83(1), pages 3-29, May.
    7. J. O. Ramsay & G. Hooker & D. Campbell & J. Cao, 2007. "Parameter estimation for differential equations: a generalized smoothing approach," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 69(5), pages 741-796, November.
    8. Xinyu Zhang & Jiguo Cao & Raymond J. Carroll, 2017. "Estimating varying coefficients for partial differential equation models," Biometrics, The International Biometric Society, vol. 73(3), pages 949-959, September.
    9. Nanshan, Muye & Zhang, Nan & Xun, Xiaolei & Cao, Jiguo, 2022. "Dynamical modeling for non-Gaussian data with high-dimensional sparse ordinary differential equations," Computational Statistics & Data Analysis, Elsevier, vol. 173(C).
    10. Baisen Liu & Liangliang Wang & Yunlong Nie & Jiguo Cao, 2021. "Semiparametric Mixed-Effects Ordinary Differential Equation Models with Heavy-Tailed Distributions," Journal of Agricultural, Biological and Environmental Statistics, Springer;The International Biometric Society;American Statistical Association, vol. 26(3), pages 428-445, September.
    11. Zhou, Jie & Han, Lu & Liu, Sanyang, 2013. "Nonlinear mixed-effects state space models with applications to HIV dynamics," Statistics & Probability Letters, Elsevier, vol. 83(5), pages 1448-1456.
    12. Lee, Kyoungjae & Lee, Jaeyong & Dass, Sarat C., 2018. "Inference for differential equation models using relaxation via dynamical systems," Computational Statistics & Data Analysis, Elsevier, vol. 127(C), pages 116-134.
    13. Giles Hooker, 2010. "Comments on: Dynamic relations for sparsely sampled Gaussian processes," TEST: An Official Journal of the Spanish Society of Statistics and Operations Research, Springer;Sociedad de Estadística e Investigación Operativa, vol. 19(1), pages 50-53, May.
    14. Commenges, D. & Jolly, D. & Drylewicz, J. & Putter, H. & Thiébaut, R., 2011. "Inference in HIV dynamics models via hierarchical likelihood," Computational Statistics & Data Analysis, Elsevier, vol. 55(1), pages 446-456, January.
    15. Liu Baisen & Wang Liangliang & Cao Jiguo, 2018. "Bayesian estimation of ordinary differential equation models when the likelihood has multiple local modes," Monte Carlo Methods and Applications, De Gruyter, vol. 24(2), pages 117-127, June.
    16. Giles Hooker, 2009. "Forcing Function Diagnostics for Nonlinear Dynamics," Biometrics, The International Biometric Society, vol. 65(3), pages 928-936, September.
    17. Jaeger, Jonathan & Lambert, Philippe, 2012. "Bayesian penalized smoothing approaches in models specified using affine differential equations with unknown error distributions," LIDAM Discussion Papers ISBA 2012017, Université catholique de Louvain, Institute of Statistics, Biostatistics and Actuarial Sciences (ISBA).
    18. Gianluca Frasso & Jonathan Jaeger & Philippe Lambert, 2016. "Parameter estimation and inference in dynamic systems described by linear partial differential equations," AStA Advances in Statistical Analysis, Springer;German Statistical Society, vol. 100(3), pages 259-287, July.
    19. Damla Şentürk, 2010. "Comments on: Dynamic relations for sparsely sampled Gaussian processes," TEST: An Official Journal of the Spanish Society of Statistics and Operations Research, Springer;Sociedad de Estadística e Investigación Operativa, vol. 19(1), pages 54-55, May.
    20. Valdemar Melicher & Tom Haber & Wim Vanroose, 2017. "Fast derivatives of likelihood functionals for ODE based models using adjoint-state method," Computational Statistics, Springer, vol. 32(4), pages 1621-1643, December.

    Corrections

    All material on this site has been provided by the respective publishers and authors. You can help correct errors and omissions. When requesting a correction, please mention this item's handle: RePEc:eee:apmaco:v:430:y:2022:i:c:s0096300322003344. See general information about how to correct material in RePEc.

    If you have authored this item and are not yet registered with RePEc, we encourage you to do it here. This allows to link your profile to this item. It also allows you to accept potential citations to this item that we are uncertain about.

    If CitEc recognized a bibliographic reference but did not link an item in RePEc to it, you can help with this form .

    If you know of missing items citing this one, you can help us creating those links by adding the relevant references in the same way as above, for each refering item. If you are a registered author of this item, you may also want to check the "citations" tab in your RePEc Author Service profile, as there may be some citations waiting for confirmation.

    For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: Catherine Liu (email available below). General contact details of provider: https://www.journals.elsevier.com/applied-mathematics-and-computation .

    Please note that corrections may take a couple of weeks to filter through the various RePEc services.

    IDEAS is a RePEc service. RePEc uses bibliographic data supplied by the respective publishers.