The Annals of Applied Statistics

Order selection in nonlinear time series models with application to the study of cell memory

Ying Hung

Full-text: Open access

Abstract

Cell adhesion experiments are biomechanical experiments studying the binding of a cell to another cell at the level of single molecules. Such a study plays an important role in tumor metastasis in cancer study. Motivated by analyzing a repeated cell adhesion experiment, a new class of nonlinear time series models with an order selection procedure is developed in this paper. Due to the nonlinearity, there are two types of overfitting. Therefore, a double penalized approach is introduced for order selection. To implement this approach, a global optimization algorithm using mixed integer programming is discussed. The procedure is shown to be asymptotically consistent in estimating both the order and parameters of the proposed model. Simulations show that the new order selection approach outperforms standard methods. The finite-sample performance of the estimator is also examined via a simulation study. The application of the proposed methodology to a T-cell experiment provides a better understanding of the kinetics and mechanics of cell adhesion, including quantifying the memory effect on a repeated unbinding force experiment and identifying the order of the memory.

Article information

Source
Ann. Appl. Stat., Volume 6, Number 3 (2012), 1256-1279.

Dates
First available in Project Euclid: 31 August 2012

Permanent link to this document
https://projecteuclid.org/euclid.aoas/1346418582

Digital Object Identifier
doi:10.1214/12-AOAS546

Mathematical Reviews number (MathSciNet)
MR3012529

Zentralblatt MATH identifier
06096530

Keywords
Consistency micropipette experiment order selection single molecule threshold autoregressive model

Citation

Hung, Ying. Order selection in nonlinear time series models with application to the study of cell memory. Ann. Appl. Stat. 6 (2012), no. 3, 1256--1279. doi:10.1214/12-AOAS546. https://projecteuclid.org/euclid.aoas/1346418582


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References

  • Akaike, H. (1973). Information theory and an extension of the maximum likelihood principle. In Second International Symposium on Information Theory (Tsahkadsor, 1971) (B. N. Petrov and F. Csaki, eds.) 267–281. Akad. Kiadó, Budapest.
  • Burman, P., Chow, E. and Nolan, D. (1994). A cross-validatory method for dependent data. Biometrika 81 351–358.
  • Chan, K. S. (1993). Consistency and limiting distribution of the least squares estimator of a threshold autoregressive model. Ann. Statist. 21 520–533.
  • Chen, J. and Khalili, A. (2008). Order selection in finite mixture models with a nonsmooth penalty. J. Amer. Statist. Assoc. 103 1674–1683.
  • Chen, W., Evans, E. A., McEver, R. P. and Zhu, C. (2008). Monitoring receptor-ligand interactions between surfaces by thermal fluctuations. Biophys. J. 94 694–701.
  • Craven, P. and Wahba, G. (1979). Smoothing noisy data with spline functions. Estimating the correct degree of smoothing by the method of generalized cross-validation. Numer. Math. 31 377–403.
  • Diggle, P. J., Heagerty, P. J., Liang, K.-Y. and Zeger, S. L. (2002). Analysis of Longitudinal Data, 2nd ed. Oxford Statistical Science Series 25. Oxford Univ. Press, Oxford.
  • Donoho, D. L. and Johnstone, I. M. (1994). Ideal spatial adaptation by wavelet shrinkage. Biometrika 81 425–455.
  • Fan, J. and Gijbels, I. (1996). Local Polynomial Modelling and Its Applications. Monographs on Statistics and Applied Probability 66. Chapman and Hall, London.
  • Fan, J. and Li, R. (2001). Variable selection via nonconcave penalized likelihood and its oracle properties. J. Amer. Statist. Assoc. 96 1348–1360.
  • Fan, J. and Yao, Q. (2003). Nonlinear Time Series: Nonparametric and Parametric Methods. Springer, New York.
  • Hall, P. and Heyde, C. C. (1980). Martingale Limit Theory and Its Application. Academic Press, New York.
  • Hansen, B. E. (2000). Sample splitting and threshold estimation. Econometrica 68 575–603.
  • Huang, J., Zarnitsyna, V. I., Liu, B., Edwards, L. J., Chien, Y. H., Jiang, N., Evavold, B. D. and Zhu, C. (2010). The kinetics of two-dimensional TCR and pMHC interactions determine T-cell responsiveness. Nature 464 932–936.
  • Hung, Y. (2011). Maximum likelihood estimation of nonlinear time series models. Technical report, Dept. Statistics and Biostatistics, Rutgers Univ., Piscataway, NJ.
  • Hung, Y., Zarnitsyna, V., Zhang, Y., Zhu, C. and Wu, C. F. J. (2008). Binary time series modeling with application to adhesion frequency experiments. J. Amer. Statist. Assoc. 103 1248–1259.
  • Hyman, J. M. and LaForce, T. (2003). Modeling the spread of influenza among cities. In Biomathematical Modeling Applications for Homeland Security (T. Banks and C. Castillo-Chavez, eds.). SIAM, Philadelphia, PA.
  • Jiang, J. (1996). REML estimation: Asymptotic behavior and related topics. Ann. Statist. 24 255–286.
  • Liu, Y. and Wu, Y. (2007). Variable selection via a combination of the $L_{0}$ and $L_{1}$ penalties. J. Comput. Graph. Statist. 16 782–798.
  • Marshall, B. T., Long, M., Piper, J. W., Yago, T., McEver, R. P. and Zhu, C. (2003). Direct observation of catch bonds involving cell-adhesion molecules. Nature 423 190–193.
  • Marshall, B. T., Sarangapani, K. K., Lou, J., McEver, R. P. and Zhu, C. (2005). Force history dependence of receptor-ligand dissociation. Biophys. J. 88 1458–1466.
  • McCulloch, C. E. and Searle, S. R. (2008). Generalized, Linear, and Mixed Models, 2nd ed. Wiley, New York.
  • Nemhauser, G. and Wolsey, L. (1999). Integer and Combinatorial Optimization. Wiley, New York.
  • Nie, L. (2006). Strong consistency of the maximum likelihood estimator in generalized linear and nonlinear mixed-effects models. Metrika 63 123–143.
  • Pollard, D. (1984). Convergence of Stochastic Processes. Springer, New York.
  • Racine, J. (2000). Consistent cross-validatory model-selection for dependent data: hv-block cross-validation. J. Econometrics 99 39–61.
  • Samia, N. I. and Chan, K.-S. (2011). Maximum likelihood estimation of a generalized threshold stochastic regression model. Biometrika 98 433–448.
  • Samia, N. I., Chan, K.-S. and Stenseth, N. C. (2007). A generalized threshold mixed model for analyzing nonnormal nonlinear time series, with application to plague in Kazakhstan. Biometrika 94 101–118.
  • Schwarz, G. (1978). Estimating the dimension of a model. Ann. Statist. 6 461–464.
  • Stone, M. (1974). Cross-validatory choice and assessment of statistical predictions (with discussion). J. Roy. Statist. Soc. Ser. B 36 111–147.
  • Tibshirani, R. (1996). Regression shrinkage and selection via the lasso. J. Roy. Statist. Soc. Ser. B 58 267–288.
  • Tibshirani, R. J. (1997). The lasso method for variable selection in the Cox model. Stat. Med. 16 385–395.
  • Tong, H. (1980). Threshold autoregression, limit cycles and cyclical data. J. Roy. Statist. Soc. Ser. B 42 245–292.
  • Tong, H. (1983). Threshold Models in Nonlinear Time Series Analysis. Lecture Notes in Statistics 21. Springer, New York.
  • Tong, H. (1990). Nonlinear Time Series: A Dynamical System Approach. Oxford Statistical Science Series 6. Oxford Univ. Press, New York.
  • Tong, H. (2007). Birth of the threshold time series model. Statist. Sinica 17 8–14.
  • Tong, H. and Lim, K. S. (1980). Threshold autoregression, limit cycles and cyclical data (with discussion). J. Roy. Statist. Soc. Ser. B 42 245–292.
  • Tsay, R. S. (1989). Testing and modeling threshold autoregressive processes. J. Amer. Statist. Assoc. 84 231–240.
  • Zarnitsyna, V. I., Huang, J., Zhang, F., Chien, Y. H., Leckband, D. and Zhu, C. (2007). Memory in receptor-ligand mediated cell adhesion. Proc. Natl. Acad. Sci. USA 104 18037–18042.
  • Zou, H. (2006). The adaptive lasso and its oracle properties. J. Amer. Statist. Assoc. 101 1418–1429.