Ecological Prediction With Nonlinear Multivariate Time-Frequency Functional Data Models

  • Wen-Hsi Yang
  • Christopher K. Wikle
  • Scott H. Holan
  • Mark L. Wildhaber


Time-frequency analysis has become a fundamental component of many scientific inquiries. Due to improvements in technology, the amount of high-frequency signals that are collected for ecological and other scientific processes is increasing at a dramatic rate. In order to facilitate the use of these data in ecological prediction, we introduce a class of nonlinear multivariate time-frequency functional models that can identify important features of each signal as well as the interaction of signals corresponding to the response variable of interest. Our methodology is of independent interest and utilizes stochastic search variable selection to improve model selection and performs model averaging to enhance prediction. We illustrate the effectiveness of our approach through simulation and by application to predicting spawning success of shovelnose sturgeon in the Lower Missouri River.

Key Words

Bayesian model averaging Dimension reduction Empirical orthogonal functions Nonlinearity Shovelnose sturgeon Spectrogram Stochastic search variable selection 


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  1. Albert, J. H., and Chib, S. (1993), “Bayesian Analysis of Binary and Polychotomous Response Data,” Journal of the American Statistical Association, 88 (422), 669–679. MathSciNetCrossRefMATHGoogle Scholar
  2. Ansari-Asl, K., Bellanger, J., Bartolomei, F., Wendling, F., and Senhadji, L. (2005), “Time-Frequency Characterization of Interdependencies in Nonstationary Signals: Application to Epileptic EEG,” IEEE Transactions on Biomedical Engineering, 52 (7), 1218–1226. CrossRefGoogle Scholar
  3. Chipman, H. (1996), “Bayesian Variable Selection With Related Predictors,” Canadian Journal of Statistics, 24 (1), 17–36. MathSciNetCrossRefMATHGoogle Scholar
  4. Crainiceanu, C. M., Staicu, A., and Di, C. (2009), “Generalized Multilevel Functional Regression,” Journal of the American Statistical Association, 104 (488), 1550–1561. MathSciNetCrossRefMATHGoogle Scholar
  5. Cranstoun, S. D., Ombao, H. C., von Sachs, R., Guo, W., and Litt, B. (2002), “Time-Frequency Spectral Estimation of Multichannel EEG Using the Auto-SLEX Method,” IEEE Transactions on Biomedical Engineering, 49 (9), 988–996. CrossRefGoogle Scholar
  6. Cressie, N., and Wikle, C. K. (2011), Statistics for Spatio-Temporal Data, New York: Wiley. MATHGoogle Scholar
  7. DeLonay, A. J., Papoulias, D. M., Wildhaber, M. L., Annis, M., Bryan, J. L., Griffith, S. A., Holan, S. H., and Tillit, D. E. (2007), “Use of Behavioral and Physiological Indicators to Evaluate Scaphirhynchus Sturgeon Spawning Success,” Journal of Applied Ichthyology, 23, 428–435. CrossRefGoogle Scholar
  8. Draper, D. (1995), “Assessment and Propagation of Model Uncertainty” (with discussion), Journal of the Royal Statistical Society. Series B. Methodological, 57 (1), 45–97. MathSciNetMATHGoogle Scholar
  9. Feichtinger, H. G., and Strohmer, T. (1998), Gabor Analysis and Algorithms: Theory and Applications, Basel: Birkhäuser. CrossRefMATHGoogle Scholar
  10. Funk, J. L., and Robinson, J. W. (1974), Changes in the Channel of the Lower Missouri River and Effects on Fish and Wildlife, Jefferson City: Missouri Department of Conservation. Google Scholar
  11. Galat, D. L., and Lipkin, R. (2000), “Restoring Ecological Integrity of Great Rivers: Historical Hydrographs Aid in Defining Reference Conditions for the Missouri River,” Hydrobiologia, 422, 29–48. CrossRefGoogle Scholar
  12. Gelfand, A. E., and Ghosh, S. K. (1998), “Model Choice: A Minimum Posterior Predictive Loss Approach,” Biometrika, 85 (1), 1–11. MathSciNetCrossRefMATHGoogle Scholar
  13. George, E. I. (2000), “The Variable Selection Problem,” Journal of the American Statistical Association, 95 (452), 1304–1308. MathSciNetCrossRefMATHGoogle Scholar
  14. George, E. I., and McCulloch, R. E. (1993), “Variable Selection Via Gibbs Sampling,” Journal of the American Statistical Association, 88 (423), 881–889. CrossRefGoogle Scholar
  15. — (1997), “Approaches for Bayesian Variable Selection,” Statistica Sinica, 7, 339–374. MATHGoogle Scholar
  16. Geweke, J. (1992), “Variable Selection and Model Comparison in Regression,” in Bayesian Statistics 4, eds. J. M. Bernardo, J. O. Berger, A. P. Dawid, and A. F. M. Smith, Oxford: Oxford Press, pp. 609–620. Google Scholar
  17. Gröchenig, K. (2001), Foundations of Time-Frequency Analysis, Basel: Birkhäuser. CrossRefMATHGoogle Scholar
  18. Hesse, L. W., and Sheets, W. (1993), “The Missouri River Hydrosystem,” Fisheries, 18 (5), 5–14. CrossRefGoogle Scholar
  19. Hoeting, J. A., Madigan, D., Rafferty, A. E., and Volinsky, C. T. (1999), “Bayesian Model Averaging: A Tutorial” (with discussion), Statistical Science, 14 (4), 382–417. MathSciNetCrossRefMATHGoogle Scholar
  20. Holan, S. H., Davis, G. M., Wildhaber, M. L., DeLonay, A. J., and Papoulias, D. M. (2009), “Hierarchical Bayesian Markov Switching Models With Application to Predicting Spawning Success of Shovelnose Sturgeon,” Journal of the Royal Statistical Society. Series C. Applied Statistics, 58 (1), 47–64. MathSciNetCrossRefGoogle Scholar
  21. Holan, S. H., Wikle, C. K., Sullivan-Beckers, L. E., and Cocroft, R. B. (2010), “Modeling Complex Phenotypes: Generalized Linear Models Using Spectrogram Predictors of Animal Communication Signals,” Biometrics, 66 (3), 914–924. MathSciNetCrossRefMATHGoogle Scholar
  22. Holan, S. H., Yang, W. H., Matteson, D. S., and Wikle, C. K. (2012), “An approach for identifying and predicting economic recessions in real-time using time-frequency functional models,” Applied Stochastic Models in Business and Industry, 28, 485–499. MathSciNetCrossRefGoogle Scholar
  23. Hosmer, D., and Lemeshow, S. (2000), Applied Logistic Regression, New York: Wiley. CrossRefMATHGoogle Scholar
  24. James, G. M. (2002), “Generalized Linear Models with Functional Predictors,” Journal of the Royal Statistical Society. Series B. Statistical Methodology, 64 (3), 411–432. MathSciNetCrossRefMATHGoogle Scholar
  25. Jolliffe, I. T. (2010), Principal Component Analysis, Berlin: Springer. Google Scholar
  26. Kestin, T. S., Karoly, D. J., Yano, J.-I., and Rayner, N. A. (1998), “Time-Frequency Variability of ENSO and Stochastic Simulations,” Journal of Climate, 11 (9), 2258–2272. CrossRefGoogle Scholar
  27. Martinez, J. G., Bohn, K. M., Carroll, R. J., and Morris, J. S. (2013), “A Study of Mexican Free-Tailed Bat Chirp Syllables: Bayesian Functional Mixed Models for Nonstationary Acoustic Time Series,” UT MD Anderson Cancer Center Department of Biostatistics Working Paper Series, Working Paper 79. Google Scholar
  28. Morris, J. S., Baladandayuthapani, V., Herrick, R. C., Sanna, P., and Gutstein, H. B. (2011), “Automated Analysis of Quantitative Image Data Using Isomorphic Functional Mixed Models, With Application to Proteomics Data,” Annals of Applied Statistics, 5 (2A). Google Scholar
  29. Müller, H. G., and Stadtmüller, U. (2005), “Generalized Functional Linear Models,” The Annals of Statistics, 33 (2), 774–805. MathSciNetCrossRefMATHGoogle Scholar
  30. O’Hara, R. B., and Sillanpää, M. J. (2009), “A Review of Bayesian Variable Selection Methods: What, How and Which,” Bayesian Analysis, 4 (1), 85–118. MathSciNetCrossRefGoogle Scholar
  31. Ombao, H., Raz, J., Von Sachs, R., and Guo, W. (2002), “The SLEX Model of a Non-Stationary Random Process,” Annals of the Institute of Statistical Mathematics, 54 (1), 171–200. MathSciNetCrossRefMATHGoogle Scholar
  32. Ombao, H., Von Sachs, R., and Guo, W. (2005), “SLEX Analysis of Multivariate Nonstationary Time Series,” Journal of the American Statistical Association, 100 (470), 519–531. MathSciNetCrossRefMATHGoogle Scholar
  33. Oppenheim, A. V., and Schafer, R. W. (2009), Discrete-Time Signal Processing, Prentice Hall Signal Processing. Google Scholar
  34. Qin, L., Guo, W., and Litt, B. (2009), “A Time-Frequency Functional Model for Locally Stationary Time Series Data,” Journal of Computational and Graphical Statistics, 18 (3), 675–693. MathSciNetCrossRefGoogle Scholar
  35. Reiss, P. T., and Ogden, R. T. (2010), “Functional Generalized Linear Models With Images as Predictors,” Biometrics, 66 (1), 61–69. MathSciNetCrossRefMATHGoogle Scholar
  36. Spiegelhalter, D. J., Best, N. G., Carlin, B. P., and Van Der Linde, A. (2002), “Bayesian Measures of Model Complexity and Fit,” Journal of the Royal Statistical Society. Series B. Statistical Methodology, 64 (4), 583–639. MathSciNetCrossRefMATHGoogle Scholar
  37. Stingo, F. C., Vannucci, M., and Downey, G. (2012), “Bayesian Wavelet-Based Curve Classification Via Distribution Analysis With Markov Random Tree Priors,” Statistica Sinica, 22, 465–488. MathSciNetCrossRefMATHGoogle Scholar
  38. U.S. Fish and Wildlife Service (2000), Biological Opinion on the Operation of the Missouri River Main Stem Reservoir System, Operation and Maintenance of the Missouri River Bank Stabilization and Navigation Project, and Operation of the Kansas River Reservoir System, Bismarck: US Fish and Wildlife Service. Google Scholar
  39. Vannucci, M., and Stingo, F. C. (2010), “Bayesian Models for Variable Selection That Incorporate Biological Information,” in Bayesian Statistics 9, eds. J. M. Bernardo, M. J. Bayarri, J. O. Berger, A. P. Dawid, D. Heckerman, A. F. M. Smith, and M. West, Oxford: Oxford University Press. Google Scholar
  40. Wahba, G. (1983), “Bayesian Confidence Intervals for the Cross-Validated Smoothing Spline,” Journal of the Royal Statistical Society. Series B. Methodological, 45 (1), 133–150. MathSciNetMATHGoogle Scholar
  41. Wikle, C. K. (2010), “Low Rank Representations as Models for Spatial Processes,” in Handbook of Spatial Statistics, eds. A. Gelfand, P. Diggle, M. Fuentes, and P. Guttorp, London: Chapman and Hall/CRC, pp. 107–118. CrossRefGoogle Scholar
  42. Wikle, C. K., and Cressie, N. (1999), “A Dimension-Reduced Approach to Space-Time Kalman Filtering,” Biometrika, 86 (4), 815. MathSciNetCrossRefMATHGoogle Scholar
  43. Wikle, C. K., and Holan, S. H. (2011), “Polynomial Nonlinear Spatio-Temporal Integro-Difference Equation Models,” Journal of Time Series Analysis, 32 (4), 339–350. MathSciNetCrossRefGoogle Scholar
  44. Wikle, C. K., and Hooten, M. B. (2010), “A General Science-Based Framework for Spatio-Temporal Dynamical Models,” Test, 19 (3), 417–451. MathSciNetCrossRefMATHGoogle Scholar
  45. Wildhaber, M. L., DeLonay, A. J., Papoulias, D. M., Galat, D. L., Jacobson, R. B., Simpkins, D. G., Braaten, P. J., Korschegen, C. E., and Mac, M. J. (2007), “A Conceptual Life-History Model for Pallid and Shovelnose Sturgeon,” Tech. rep., USGS Circular 1315. Google Scholar
  46. — (2011a), “Identifying Structural Elements Needed for Development of a Predictive Life-History Model for Pallid and Shovelnose Sturgeons,” Journal of Applied Ichthyology, 27, 462–469. CrossRefGoogle Scholar
  47. Wildhaber, M. L., Holan, C. H., Davis, G. M., Gladish, D. W., DeLonay, A. J., Papoulias, D. M., and Sommerhauser, D. K. (2011b), “Evaluating Spawning Migration Patterns and Predicting Spawning Success of Shovelnose Sturgeon in the Lower Missouri River,” Journal of Applied Ichthyology, 27, 301–308. CrossRefGoogle Scholar
  48. Wolfe, P. J., Godsill, S. J., and Ng, W.-J. (2004), “Bayesian Variable Selection and Regularization for Time–Frequency Surface Estimation,” Journal of the Royal Statistical Society. Series B. Statistical Methodology, 66 (3), 575–589. MathSciNetCrossRefMATHGoogle Scholar
  49. Yao, F., and Müller, H. G. (2010), “Functional Quadratic Regression,” Biometrika, 97 (1), 49–64. MathSciNetCrossRefMATHGoogle Scholar

Copyright information

© International Biometric Society 2013

Authors and Affiliations

  • Wen-Hsi Yang
    • 1
  • Christopher K. Wikle
    • 2
  • Scott H. Holan
    • 2
  • Mark L. Wildhaber
    • 3
  1. 1.Department of StatisticsUniversity of MissouriColumbiaUSA
  2. 2.Department of StatisticsUniversity of MissouriColumbiaUSA
  3. 3.US Geological SurveyColumbiaUSA

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