Year: 2007
Type: dissertation
Abstract: Bayesian surface smoothing using splines usually proceeds by \t\t\t\tchoosing the smoothness parameter through the use of data driven \t\t\t\tmethods like generalized cross validation. In this methodology, \t\t\t\tknots of the splines are assumed to lie at the data locations. \t\t\t\tWhen anisotropy is present in the data, modeling is done via \t\t\t\tparametric functions.\nIn the present thesis, we have proposed a non-parametric approach \t\t\t\tto Bayesian surface smoothing in the presence of anisotropy. We \t\t\t\tuse eigenfunctions generated by thin-plate splines as our basis \t\t\t\tfunctions. Using eigenfunctions does away with having to place \t\t\t\tknots arbitrarily, as is done customarily. The smoothing \t\t\t\tparameter, the anisotropy matrix, and other parameters are \t\t\t\tsimultaneously updated by a Reversible Jump Markov Chain Monte \t\t\t\tCarlo (RJMCMC) sampler. Unique in our implementation is model \t\t\t\tselection, which is again done concurrently with the parameter \t\t\t\tupdates.\nSince the posterior distribution of the coefficients of the basis \t\t\t\tfunctions for any given model order is available in closed form, \t\t\t\twe are able to simplify the sampling algorithm in the model \t\t\t\tselection step. This also helps us in isolating the parameters \t\t\t\twhich influence the model selection step.\nWe investigate the relationship between the number of basis \t\t\t\tfunctions used in the model and the smoothness parameter and \t\t\t\tfind that there is a delicate balance which exists between the \t\t\t\ttwo. Higher values of the smoothness parameter correspond to \t\t\t\tmore number of basis functions being selected.\nUse of a non-parametric approach to Bayesian surface smoothing \t\t\t\tprovides for more modeling flexibility. We are not constrained \t\t\t\tby the shape defined by a parametric shape of the covariance as \t\t\t\tused by earlier methods. A Bayesian approach also allows us to \t\t\t\tinclude the results obtained from previous analysis of the same \t\t\t\tdata, if any, as prior information. It also allows us to \t\t\t\tevaluate pointwise estimates of variability of the fitted \t\t\t\tsurface. We believe that our research also poses many questions \t\t\t\tfor future research.
Author: Subhashish Chakravarty
Language: English
Subfield: Statistics and Probability
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W135081964