Generalized nonlinear modeling with multivariate free-knot regression splines

Journal of the American Statistical Association

Volumn 98, Issue 462, 2003, Pages 352-368

  Holmes, C C ^a   Mallick, B K ^b  

^a IMPERIAL COLLEGE LONDON   (United Kingdom)

^b TEXAS A AND M UNIVERSITY   (United States)

Author keywords

Bayesian nonlinear regression; Generalized nonlinear regression; Local linear regression; Multivariate nonlinear regression; Multivariate spline; Piecewise linear; Seemingly unrelated regression

Indexed keywords

EID: 0041743086     PISSN: 01621459     EISSN: None     Source Type: Journal    
DOI: 10.1198/016214503000143     Document Type: Article

References (46)

1. Albert, J. H., and Chib, S. (1993), "Bayesian Analysis of Binary and Polychotomous Response Data," Journal of the American Statistical Association, 88, 669-679.
2. Anderson, E. (1995), LAPACK Users' Guide (2nd ed.), Philadelphia: Society for Industrial and Applied Mathematics.
3. Bernardo, J. M., and Smith, A. F. M. (1994), Bayesian Theory, Chichester, U.K.: Wiley.
4. Biller, C. (2000), "Adaptive Bayesian Regression Splines in Semiparametric Generalized Linear Models," Journal of Computational and Graphical Statistics, 9, 122-140.
5. Breiman, L. (1993), "Hinging Hyperplanes for Regression, Classification and Function Approximation," IEEE Transactions on Information Theory, 3, 999-1013.
6. Breiman, L., Friedman, J. H., Olshen, R., and Stone, C. J. (1984), Classification and Regression Trees, Belmont, CA: Wadsworth.
7. Breslow, N. E., and Clayton, D. G. (1993), "Approximate Inference in Generalized Linear Mixed Models," Journal of the American Statistical Association, 88, 9-25.
8. Chib, S., and Greenberg, E. (1998), "Analysis of Multivariate Probit Models," Biometrika, 85, 347-361.
9. Chipman, H., George, E. I., and McCulloch, R. E. (1998), "Bayesian CART Model Search" (with discussion), Journal of the American Statistical Association, 93, 935-960.
10. _ (2000), "Hierarchical Priors for Bayesian CART Shrinkage," Statistics and Computation, 10, 17-24.
11. Clayton, D. G. (1996), "Generalised Linear Mixed Models," in Markov Chain Monte Carlo in Practice, eds. W. R. Gilks, S. Richardson, and D. J. Spiegel-halter, New York: Chapman and Hall.
12. Cressie, N. A. C. (1993), Statistics for Spatial Data (2nd ed.), London: Chapman and Hall.
13. Denison, D. G. T., Holmes, C. C., Mallick, B. K., and Smith, A. F. M. (2002), Bayesian Methods for Nonlinear Classification and Regression, Chichester, U.K.: Wiley.
14. Denison, D. G. T., Mallick, B. K. and Smith, A. F. M. (1988a), "Automatic Bayesian Curve Fitting," Journal of the Royal Statistical Society, Ser. B, 60, 333-350.
15. _ (1998b), "A Bayesian CART Algorithm," Biometrika, 85, 363-377.
16. Dey, D. K., Ghosh, S. K., and Mallick, B. K. (eds.) (2000), Generalised Linear Models: A Bayesian Perspective, New York: Marcel Decker.
17. Diggle, P., Morris, S., Elliott, P., and Shaddick, G. (1997), "Regression Modeling of Disease Risk in Relation to Point Sources," Journal of the Royal Statistical Society, Ser. A, 160, 491-505.
18. Diggle, P., Tawn, J. A., and Moyeed, R. A. (1998), "Model-Based Geostatistics" (with discussion), Applied Statistics, 47, 299-350.
19. Franke, R. (1982), "Scattered Data Interpolation: Tests of Some Methods," Mathematical Computation, 38, 181-200.
20. Friedman, J. H. (1991), "Multivariate Adaptive Regression Splines" (with discussion), The Annals of Statistics, 19, 1-141.
21. Gao, F., Wahba, G., Klein, R., and Klein, B. (2001), "Smoothing Spline ANOVA for Multivariate Bernoulli Observations, With Applications to Opthalmology Data," Journal of the American Statistical Association, 96, 127-160.
22. Gelfand, A. E., and Smith, A. F. M. (1990), "Sampling-Based Approaches to Calculating Marginal Densities," Journal of the American Statistical Association, 85, 398-409.
23. Gelman, A., Carlin, J. B., Stern, H. S., and Rubin, D. B. (1995), Bayesian Data Analysis, New York: Chapman and Hall.
24. Green, P. J. (1995), "Reversible Jump Markov Chain Monte Carlo Computation and Bayesian Model Determination," Biometrika, 82, 711-732.
25. Green, P. J., and Silverman, B. W. (1994), Nonparametric Regression and Generalised Linear Models, London: Chapman and Hall.
26. Hand, D. J., and Henley, W. E. (1997), "Statistical Classification Methods in Consumer Credit Scoring: A Review," Journal of the Royal Statistical Society, Ser. A, 160, 523-541.
27. Hansen, M. H., and Kooperberg, C. (2002), "Spline Adaptation in Extended Linear Models," Statistical Science, 17, 2-51.
28. Hastie, T. J., and Tibshirani, R. J. (1990), Generalized Additive Models, London: Chapman and Hall.
29. Holmes, C. C., and Denison, D. G. T. (1999), "Bayesian Wavelet Analysis With a Model Complexity Prior," in Bayesian Statistics 6, eds. J. M. Bernardo, J. O. Berger, A. P. Dawid, and A. F. M. Smith, Oxford, U.K.: Clarendon Press, pp. 769-776.
30. Holmes, C. C., Denison, D. G. T., and Mallick, B. K. (1999), "Accounting for Model Uncertainty in Seemingly Unrelated Regressions," Journal of Computational and Graphical Statistics, 11, 533-551.
31. Holmes, C. C., and Mallick, B. K. (2001), "Bayesian Regression With Multivariate Linear Splines," Journal of the Royal Statistical Society, Ser. B, 63, 3-17.
32. Lin, X., and Zhang, D. (1999), "Inference in Generalized Additive Mixed Models by Using Smoothing Splines," Journal of the Royal Statistical Society, Ser. B, 61, 381-400.
33. McCullagh, P., and Nelder, J. A. (1989), Generalized Linear Models (2nd ed.), London: Chapman and Hall.
34. Metropolis, N., Rosenbluth, A. W., Rosenbluth, M. N., Teller, A. H., and Teller, E. (1953), "Equations of State Calculations by Fast Computing Machines," Journal of Chemical Physics, 21, 1087-1091.
35. Mitchell, T. M. (1997), Machine Learning, New York: McGraw-Hill.
36. Müller, P., and Rios Insua, D. (1998), "Issues in Bayesian Analysis of Neural Network Models," Neural Computation, 10, 571-592.
37. Nychka, D. (1999), "Spatial Process Estimates as Smoothers," in Smoothing and Regression, ed. M. G. Schimek, New York: Wiley.
38. Percy, D. F. (1996), "Zellner's Influence on Multivariate Linear Models," in Bayesian Analysis in Statistics and Econometrics, eds. D. A. Berry, K. M. Chaloner and J. K. Geweke, New York: Wiley, pp. 203-213.
39. Ripley, B. D. (1994), "Neural Networks and Related Methods for Classification" (with discussion), Journal of the Royal Statistical Society, Ser. B, 56, 409-456.
40. Robert, C. P. (1995), "Simulation of Truncated Normal Variables," Statistics and Computation, 5, 121-125.
41. Smith, M., and Kohn, R. (1996), "Nonparametric Regression Using Bayesian Variable Selection," Journal of Econometrics, 75, 317-344.
42. _ (2000), "Nonparametric Seemingly Unrelated Regression," Journal of Econometrics, 98, 257-281.
43. Sun, D., Speckman, P. L., and Tsutakawa, R. K. (2000), "Random Effects in Generalized Linear Mixed Models," in Generalised Linear Models: A Bayesian Perspective, eds. D. K. Dey, S. K. Ghosh and B. K. Mallick, New York: Marcel Decker.
44. Wood, S., and Kohn, R. (1998), "A Bayesian Approach to Robust Binary Nonparametric Regression," Journal of the American Statistical Association, 93, 203-213.
45. Zeger, S. L., and Karim (1991), "Generalized Linear Models With Random Effects: A Gibbs Sampling Approach," Journal of the American Statistical Association. 86, 79-86.
46. Zellner, A. (1962), "An Efficient Method of Estimating Seemingly Unrelated Regressions and Tests of Aggregation Bias," Journal of the American Statistical Association, 57, 500-509.