Extending the Lattice-Based Smoother using a generalized additive model

Master's Project (M.S.) University of Alaska Fairbanks, 2017The Lattice Based Smoother was introduced by McIntyre and Barry (2017) to estimate a surface defined over an irregularly-shaped region. In this paper we consider extending their method to allow for additional covariates and non-continuous responses. We describe our extension which utilizes the framework of generalized additive models. A simulation study shows that our method is comparable to the Soap film smoother of Wood et al. (2008), under a number of different conditions. Finally we illustrate the method's practical use by applying it to a real data set

Oracally Efficient Two-Step Estimation of Generalized Additive Model

Generalized additive models (GAM) are multivariate nonparametric regressions for non-Gaussian responses including binary and count data. We propose a spline-backfitted kernel (SBK) estimator for the component functions. Our results are for weakly dependent data and we prove oracle efficiency. The SBK techniques is both computational expedient and theoretically reliable, thus usable for analyzing high-dimensional time series. Inference can be made on component functions based on asymptotic normality. Simulation evidence strongly corroborates with the asymptotic theory.Bandwidths, B spline, knots, link function, mixing, Nadaraya-Watson estimator

Introducing COZIGAM: An R Package for Unconstrained and Constrained Zero-Inflated Generalized Additive Model Analysis

Zero-inflation problem is very common in ecological studies as well as other areas. Nonparametric regression with zero-inflated data may be studied via the zero-inflated generalized additive model (ZIGAM), which assumes that the zero-inflated responses come from a probabilistic mixture of zero and a regular component whose distribution belongs to the 1-parameter exponential family. With the further assumption that the probability of non-zero-inflation is some monotonic function of the mean of the regular component, we propose the constrained zero-inflated generalized additive model (COZIGAM) for analyzingzero-inflated data. When the hypothesized constraint obtains, the new approach provides a unified framework for modeling zero-inflated data, which is more parsimonious and efficient than the unconstrained ZIGAM. We have developed an R package COZIGAM which contains functions that implement an iterative algorithm for fitting ZIGAMs and COZIGAMs to zero-inflated data basedon the penalized likelihood approach. Other functions included in the packageare useful for model prediction and model selection. We demonstrate the use ofthe COZIGAM package via some simulation studies and a real application.

Variable Selection and Model Averaging in Semiparametric Overdispersed Generalized Linear Models

We express the mean and variance terms in a double exponential regression model as additive functions of the predictors and use Bayesian variable selection to determine which predictors enter the model, and whether they enter linearly or flexibly. When the variance term is null we obtain a generalized additive model, which becomes a generalized linear model if the predictors enter the mean linearly. The model is estimated using Markov chain Monte Carlo simulation and the methodology is illustrated using real and simulated data sets.Comment: 8 graphs 35 page

Inferences About the Components of a Generalized Additive Model

A method for making inferences about the components of a generalized additive model is described. It is found that a variation of the method, based on means, performs well in simulations. Unlike many other inferential methods, switching from a mean to a 20% trimmed mean was found to offer little or no advantage in terms of both power and controlling the probability of a Type I error

Fitting Log-Gaussian Cox Processes Using Generalized Additive Model Software

While log-Gaussian Cox process regression models are useful tools for modeling point patterns, they can be technically difficult to fit and require users to learn/adopt bespoke software. We show that, for suitably formatted data, we can actually fit these models using generalized additive model software, via a simple line of code, demonstrated on R by the popular mgcv package. We are able to do this because a common and computationally efficient way to fit a log-Gaussian Cox process model is to use a basis function expansion to approximate the Gaussian random field, as is provided by a generic bivariate smoother over geographic space. We further show that if basis functions are parameterized appropriately then we can estimate parameters in the spatial covariance function for the latent random field using a generalized additive model. We use simulation to show that this approach leads to model fits of comparable quality to state-of-the-art software, often more quickly. But we see the main advance from this work as lowering the technology barrier to spatial statistics for applied researchers, many of whom are already familiar with generalized additive model software

Generalized Spatial Regression with Differential Regularization

We aim at analyzing geostatistical and areal data observed over irregularly shaped spatial domains and having a distribution within the exponential family. We propose a generalized additive model that allows to account for spatially-varying covariate information. The model is fitted by maximizing a penalized log-likelihood function, with a roughness penalty term that involves a differential quantity of the spatial field, computed over the domain of interest. Efficient estimation of the spatial field is achieved resorting to the finite element method, which provides a basis for piecewise polynomial surfaces. The proposed model is illustrated by an application to the study of criminality in the city of Portland, Oregon, USA