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Theory and Methods

Kernel Ordinary Differential Equations

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Pages 1711-1725 | Received 06 Aug 2020, Accepted 23 Jan 2021, Published online: 27 Apr 2021

References

  • Bachoc, F., Leeb, H., and Pötscher, B. M. (2019), “Valid Confidence Intervals for Post-Model-Selection Predictors,” The Annals of Statistics, 47, 1475–1504. DOI: 10.1214/18-AOS1721.
  • Berk, R., Brown, L., Buja, A., Zhang, K., and Zhao, L. (2013), “Valid Post-Selection Inference,” The Annals of Statistics, 41, 802–837. DOI: 10.1214/12-AOS1077.
  • Brunton, S. L., Proctor, J. L., and Kutz, J. N. (2016), “Discovering Governing Equations From Data by Sparse Identification of Nonlinear Dynamical Systems,” Proceedings of the National Academy of Sciences of the United States of America, 113, 3932–3937. DOI: 10.1073/pnas.1517384113.
  • Buxton, R. B., Uludağ, K., Dubowitz, D. J., and Liu, T. T. (2004), “Modeling the Hemodynamic Response to Brain Activation,” Neuroimage, 23, S220–S233. DOI: 10.1016/j.neuroimage.2004.07.013.
  • Cao, J., and Zhao, H. (2008), “Estimating Dynamic Models for Gene Regulation Networks,” Bioinformatics, 24, 1619–1624. DOI: 10.1093/bioinformatics/btn246.
  • Cao, X., Sandstede, B., and Luo, X. (2019), “A Functional Data Method for Causal Dynamic Network Modeling of Task-Related fMRI,” Frontiers in Neuroscience, 13, 127. DOI: 10.3389/fnins.2019.00127.
  • Chen, S., Shojaie, A., and Witten, D. M. (2017), “Network Reconstruction From High-Dimensional Ordinary Differential Equations,” Journal of the American Statistical Association, 112, 1697–1707. DOI: 10.1080/01621459.2016.1229197.
  • Chernozhukov, V., Hansen, C., and Spindler, M. (2015), “Valid Post-Selection and Post-Regularization Inference: An Elementary, General Approach,” Annual Review of Economics, 7, 649–688. DOI: 10.1146/annurev-economics-012315-015826.
  • Chou, I.-C., and Voit, E. O. (2009), “Recent Developments in Parameter Estimation and Structure Identification of Biochemical and Genomic Systems,” Mathematical Biosciences, 219, 57–83. DOI: 10.1016/j.mbs.2009.03.002.
  • Cox, D. D. (1983), “Asymptotics for M-Type Smoothing Splines,” The Annals of Statistics, 11, 530–551. DOI: 10.1214/aos/1176346159.
  • Dattner, I., and Klaassen, C. A. J. (2015), “Optimal Rate of Direct Estimators in Systems of Ordinary Differential Equations Linear in Functions of the Parameters,” Electronic Journal of Statistics, 9, 1939–1973. DOI: 10.1214/15-EJS1053.
  • Friston, K. J., Harrison, L., and Penny, W. (2003), “Dynamic Causal Modelling,” Neuroimage, 19, 1273–1302. DOI: 10.1016/S1053-8119(03)00202-7.
  • Friston, K. J., Preller, K. H., Mathys, C., Cagnan, H., Heinzle, J., Razi, A., and Zeidman, P. (2019), “Dynamic Causal Modelling Revisited,” Neuroimage, 199, 730–744. DOI: 10.1016/j.neuroimage.2017.02.045.
  • González, J., Vujačić, I., and Wit, E. (2014), “Reproducing Kernel Hilbert Space Based Estimation of Systems of Ordinary Differential Equations,” Pattern Recognition Letters, 45, 26–32. DOI: 10.1016/j.patrec.2014.02.019.
  • Gu, C. (2013), Smoothing Spline ANOVA Models, New York: Springer-Verlag.
  • Henderson, J., and Michailidis, G. (2014), “Network Reconstruction Using Nonparametric Additive ODE Models,” PLOS ONE, 9, 1–15. DOI: 10.1371/journal.pone.0094003.
  • Huang, J. Z. (1998), “Projection Estimation in Multiple Regression With Application to Functional ANOVA Models,” The Annals of Statistics, 26, 242–272. DOI: 10.1214/aos/1030563984.
  • Izhikevich, E. (2007), Dynamical Systems in Neuroscience, Cambridge, MA: MIT Press.
  • Javanmard, A., and Montanari, A. (2014), “Confidence Intervals and Hypothesis Testing for High-Dimensional Regression,” Journal of Machine Learning Research, 15, 2869–2909.
  • Koltchinskii, V., and Yuan, M. (2010), “Sparsity in Multiple Kernel Learning,” The Annals of Statistics, 38, 3660–3695. DOI: 10.1214/10-AOS825.
  • Liang, H., and Wu, H. (2008), “Parameter Estimation for Differential Equation Models Using a Framework of Measurement Error in Regression Models,” Journal of the American Statistical Association, 103, 1570–1583. DOI: 10.1198/016214508000000797.
  • Lin, Y. (2000), “Tensor Product Space ANOVA Models,” The Annals of Statistics, 28, 734–755. DOI: 10.1214/aos/1015951996.
  • Lin, Y., and Zhang, H. H. (2006), “Component Selection and Smoothing in Multivariate Nonparametric Regression,” The Annals of Statistics, 34, 2272–2297. DOI: 10.1214/009053606000000722.
  • Loh, P.-L., and Wainwright, M. J. (2012), “High-Dimensional Regression With Noisy and Missing Data: Provable Guarantees With Nonconvexity,” The Annals of Statistics, 40, 1637–1664. DOI: 10.1214/12-AOS1018.
  • Lu, J., Kolar, M., and Liu, H. (2020), “Kernel Meets Sieve: Post-Regularization Confidence Bands for Sparse Additive Model,” Journal of the American Statistical Association, 115, 2084–2099. DOI: 10.1080/01621459.2019.1689984.
  • Lu, T., Liang, H., Li, H., and Wu, H. (2011), “High-Dimensional ODEs Coupled With Mixed-Effects Modeling Techniques for Dynamic Gene Regulatory Network Identification,” Journal of the American Statistical Association, 106, 1242–1258. DOI: 10.1198/jasa.2011.ap10194.
  • Ma, W., Trusina, A., El-Samad, H., Lim, W. A., and Tang, C. (2009), “Defining Network Topologies That Can Achieve Biochemical Adaptation,” Cell, 138, 760–773. DOI: 10.1016/j.cell.2009.06.013.
  • Marbach, D., Prill, R. J., Schaffter, T., Mattiussi, C., Floreano, D., and Stolovitzky, G. (2010), “Revealing Strengths and Weaknesses of Methods for Gene Network Inference,” Proceedings of the National Academy of Sciences of the United States of America, 107, 6286–6291. DOI: 10.1073/pnas.0913357107.
  • Marbach, D., Schaffter, T., Mattiussi, C., and Floreano, D. (2009), “Generating Realistic in Silico Gene Networks for Performance Assessment of Reverse Engineering Methods,” Journal of Computational Biology, 16, 229–239. DOI: 10.1089/cmb.2008.09TT.
  • Mikkelsen, F. V., and Hansen, N. R. (2017), “Learning Large Scale Ordinary Differential Equation Systems,” arXiv no. 1710.09308.
  • Opsomer, J. D., and Ruppert, D. (1997), “Fitting a Bivariate Additive Model by Local Polynomial Regression,” The Annals of Statistics, 25, 186–211. DOI: 10.1214/aos/1034276626.
  • Pfister, N., Bauer, S., and Peters, J. (2019), “Learning Stable and Predictive Structures in Kinetic Systems,” Proceedings of the National Academy of Sciences of the United States of America, 116, 25405–25411. DOI: 10.1073/pnas.1905688116.
  • Raskutti, G., Wainwright, M. J., and Yu, B. (2011), “Minimax Rates of Estimation for High-Dimensional Linear Regression Over lq -Balls,” IEEE Transactions on Information Theory, 57, 6976–6994.
  • Ravikumar, P., Wainwright, M. J., and Lafferty, J. (2010), “High-Dimensional Ising Model Selection Using l1-Regularized Logistic Regression,” The Annals of Statistics, 38, 1287–1319. DOI: 10.1214/09-AOS691.
  • Rubenstein, P. K., Bongers, S., Schölkopf, B., and Mooij, J. M. (2018), “From Deterministic ODEs to Dynamic Structural Causal Models,” in Proceedings of the 34th Conference Annual Conference on Uncertainty in Artificial Intelligence (UAI).
  • Schaffter, T., Marbach, D., and Floreano, D. (2011), “GeneNetWeaver: In Silico Benchmark Generation and Performance Profiling of Network Inference Methods,” Bioinformatics, 27, 2263–2270. DOI: 10.1093/bioinformatics/btr373.
  • Talagrand, M. (1996), “New Concentration Inequalities in Product Spaces,” Inventiones Mathematicae, 126, 505–563. DOI: 10.1007/s002220050108.
  • Tzafriri, A. R. (2003), “Michaelis–Menten Kinetics at High Enzyme Concentrations,” Bulletin of Mathematical Biology, 65, 1111–1129. DOI: 10.1016/S0092-8240(03)00059-4.
  • van der Vaart, A. W., and Wellner, J. A. (1996), Weak Convergence and Empirical Processes, New York: Springer-Verlag.
  • Varah, J. M. (1982), “A Spline Least Squares Method for Numerical Parameter Estimation in Differential Equations,” SIAM Journal on Scientific and Statistical Computing, 3, 28–46. DOI: 10.1137/0903003.
  • Volterra, V. (1928), “Variations and Fluctuations of the Number of Individuals in Animal Species Living Together,” ICES Journal of Marine Science, 3, 3–51. DOI: 10.1093/icesjms/3.1.3.
  • Wahba, G. (1983), “Bayesian ‘Confidence Intervals’ for the Cross-Validated Smoothing Spline,” Journal of the Royal Statistical Society, Series B, 45, 133–150. DOI: 10.1111/j.2517-6161.1983.tb01239.x.
  • Wahba, G. (1990), Spline Models for Observational Data, Philadelphia, PA: SIAM.
  • Wahba, G., Wang, Y., Gu, C., Klein, R., and Klein, B. (1995), “Smoothing Spline ANOVA for Exponential Families, With Application to the Wisconsin Epidemiological Study of Diabetic Retinopathy,” The Annals of Statistics, 23, 1865–1895. DOI: 10.1214/aos/1034713638.
  • Wang, S., Nan, B., Zhu, N., and Zhu, J. (2009), “Hierarchically Penalized Cox Regression With Grouped Variables,” Biometrika, 96, 307–322. DOI: 10.1093/biomet/asp016.
  • Wu, H., Lu, T., Xue, H., and Liang, H. (2014), “Sparse Additive Ordinary Differential Equations for Dynamic Gene Regulatory Network Modeling,” Journal of the American Statistical Association, 109, 700–716. DOI: 10.1080/01621459.2013.859617.
  • Yuan, M., and Zhou, D.-X. (2016), “Minimax Optimal Rates of Estimation in High Dimensional Additive Models,” The Annals of Statistics, 44, 2564–2593. DOI: 10.1214/15-AOS1422.
  • Zhang, C.-H., and Zhang, S. S. (2014), “Confidence Intervals for Low Dimensional Parameters in High Dimensional Linear Models,” Journal of the Royal Statistical Society, Series B, 76, 217–242. DOI: 10.1111/rssb.12026.
  • Zhang, T., Wu, J., Li, F., Caffo, B., and Boatman-Reich, D. (2015), “A Dynamic Directional Model for Effective Brain Connectivity Using Electrocorticographic (ECoG) Time Series,” Journal of the American Statistical Association, 110, 93–106. DOI: 10.1080/01621459.2014.988213.
  • Zhang, T., Yin, Q., Caffo, B., Sun, Y., and Boatman-Reich, D. (2017), “Bayesian Inference of High-Dimensional, Cluster-Structured Ordinary Differential Equation Models With Applications to Brain Connectivity Studies,” The Annals of Applied Statistics, 11, 868–897. DOI: 10.1214/17-AOAS1021.
  • Zhang, X., Cao, J., and Carroll, R. J. (2015), “On the Selection of Ordinary Differential Equation Models With Application to Predator-Prey Dynamical Models,” Biometrics, 71, 131–138. DOI: 10.1111/biom.12243.
  • Zhao, P., and Yu, B. (2006), “On Model Selection Consistency of Lasso,” Journal of Machine Learning Research, 7, 2541–2563.
  • Zhu, H., Yao, F., and Zhang, H. H. (2014), “Structured Functional Additive Regression in Reproducing Kernel Hilbert Spaces,” Journal of the Royal Statistical Society, Series B, 76, 581–603. DOI: 10.1111/rssb.12036.

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