Generalized nonlinear modeling with multivariate free-knot regression splines

A Bayesian method is presented for the nonparametric modeling of univariate and multivariate non-Gaussian response data. Data-adaptive multivariate regression splines are used where the number and location of the knot points are treated as random. The posterior model space is explored using a reversible-jump Markov chain Monte Carlo sampler. Computational difficulties are partly alleviated by introducing a random residual effect in the model that leaves many of the posterior conditional distributions of the model parameters in standard form. The use of the latent residual effect provides a convenient vehicle for modeling correlation in multivariate response data, and as such our method can be seen to generalize the seemingly unrelated regression model to non-Gaussian data. We illustrate the method on a number of examples, including two previously unpublished datasets relating to the spatial smoothing of multivariate accident data in Texas and the modeling of credit card use across multiple retail sectors.