Statistics Seminar
| Date/Time: | Monday, 02 Dec 2013 from 4:10 pm to 5:00 pm |
|---|---|
| Location: | Snedecor 3105 |
| Cost: | Free |
| URL: | www.stat.iastate.edu |
| Contact: | Jeanette La Grange |
| Phone: | 515-294-3440 |
| Channel: | College of Liberal Arts and Sciences |
| Categories: | Lectures |
| Actions: | Download iCal/vCal | Email Reminder |
The problem of spline smoothing has been thoroughly studied for univariate models. A number of authors have considered restricted versions of multivariate smoothing splines with multivariate dependent variables. Yee & Wild (1996) related multivariate spline model to generalized additive models and provided an algorithm for parameter estimation. Fessler (1991) proposed a GCV (generalized cross validation) estimate of the smoothing parameter of multivariate splines, and Wang et al. (2000) proposed GCV and GML estimate of smoothing parameters of a bivariate spline. In the literature, the multivariate signal processes are often assumed to be independent. In applications of multivariate models, the components of the multivariate function g(t) may be influenced by common factors. The smooth (signal) components are often correlated. For instance, economic theory suggests that the macroeconomic variables such as GDP and private investment have correlated trends. Restricting independence of the signal processes is inadvisable under such a scenario.
In this talk, joint smoothing is considered for multivariate models with correlated error components and correlated derivatives of the curves. Several issues raised. (a) Naive multivariate analog of the univariate smoothing parameter is not a symmetric matrix and is overparameterized. One central problem is to suitably define the multivariate smoothing parameter. (b) It is well known that the univariate smoothing spline is equivalent to the Bayes estimate with a generalized Gaussian prior (Kimeldorf & Wahba 1970). The Bayesian interpretation to the multivariate case can be verified in terms of multivariate linear mixed models. (c) Are there cases where the general multivariate smoothing spline that can be obtained by univariate smoothing splines? Bayesian Factor analysis for the multivariate smoothing gives the structures of the multivariate smoothing. (d) It is common practice to use objective priors for the error variance and certain terms in linear models. This practice can be generalized to multivariate smoothing splines. (e) Can faster computational algorithms be derived through the factor analysis?
Two applications are given. In estimating the trend of earth surface temperatures, the multivariate spline delivers the same answer as univariate splines. On the other hand, in estimating the trend in macroeconomic time series data, the multivariate spline produce vastly different and more reasonable result than the univariate splines. The multivariate smoothing spline produces more reasonable measurements of business cycles than univariate spline does.