Copyright (c) 2009 All rights reserved. http://works.bepress.com/michael_smith Recent documents in Michael Stanley Smith en-us Sun, 05 Jul 2009 17:34:32 PDT 3600 http://works.bepress.com/michael_smith/16 http://works.bepress.com/michael_smith/16 Tue, 16 Jun 2009 18:37:54 PDT Michael S. Smith Bayesian Model Averaging and Semiparametric Regression Forecasting and Time Series http://works.bepress.com/michael_smith/15 http://works.bepress.com/michael_smith/15 Tue, 16 Jun 2009 17:51:22 PDT A new regression based approach is proposed for modeling marketing databases. The approach is Bayesian and provides a number of significant improvements over current methods. Independent variables can enter into the model in either a parametric or nonparametric manner, significant variables can be identified from a large number of potential regressors and an appropriate transformation of the dependent variable can be automatically selected from a discrete set of pre-specified candidate transformations. All these features are estimated simultaneously and automatically using a Bayesian hierarchical model coupled with a Gibbs sampling scheme. Being Bayesian, it is straightforward to introduce subjective information about the relative importance of each variable, or with regard to a suitable data transformation. The methodology is applied to print advertising Starch data collected from thirteen issues of an Australian women's monthly magazine. The empirical results highlight the complex and detailed relationships that can be uncovered using the methodology. Michael S. Smith Multivariate Models in Marketing Bayesian Model Averaging and Semiparametric Regression http://works.bepress.com/michael_smith/14 http://works.bepress.com/michael_smith/14 Thu, 19 Feb 2009 15:47:21 PST In this research we introduce a new class of multivariate probability models to the marketing literature. Known as "copula models", they have a number of attractive features. First, they permit the combination of any univariate marginal distributions that need not come from the same distributional family. Second, a particular class of copula models, called "elliptical copula", have the property that they increase in complexity at a much slower rate than existing multivariate probability models as the number of dimensions increase. Third, they are very general, encompassing a number of existing multivariate models, and provide a framework for generating many more. These advantages give copula models a greater potential for use in empirical analysis than existing probability models used in marketing. We exploit and extend recent developments in Bayesian estimation to propose an approach that allows reliable estimation of elliptical copula models in high dimensions. Rather than focusing on a single marketing problem, we demonstrate the versatility and accuracy of copula models with four examples to show the flexibility of the method. In every case, the copula model either handles a situation that could not be modeled previously, or gives improved accuracy compared with prior models. Peter J. Danaher Multivariate Models in Marketing http://works.bepress.com/michael_smith/13 http://works.bepress.com/michael_smith/13 Tue, 29 Jul 2008 00:07:28 PDT A method is presented for simultaneously estimating a system of nonparametric regressions which may seem unrelated, but where the errors are potentially correlated between equations. We show that the advantage of estimating such a `seemingly unrelated' system of nonparamet­ ric regressions is that less observations can be required to obtain reliable function estimates than if each of the regression equations is estimated separately and the correlation ignored. This increase in efficiency is investigated empirically using both simulated and real data. The method uses a Bayesian hierarchical framework where the regression function is represented as a linear combination of a large number of basis terms. All the regression coefficients, and the variance matrix of the errors, are estimated simultaneously by their posterior means. The computation is carried out using a Markov chain Monte Carlo sampling scheme that employs a `focused sampling' step to combat the high dimensional representation of the unknown regression functions. The methodology extends easily to other nonparametric multivariate regression models. Michael S. Smith Bayesian Model Averaging and Semiparametric Regression http://works.bepress.com/michael_smith/12 http://works.bepress.com/michael_smith/12 Tue, 29 Jul 2008 00:03:17 PDT Michael S. Smith Forecasting and Time Series http://works.bepress.com/michael_smith/11 http://works.bepress.com/michael_smith/11 Tue, 29 Jul 2008 00:00:23 PDT This paper discusses a Bayesian approach to nonparametric regression initially proposed by Smith and Kohn (1996. Journal of Econometrics 75: 317-344). In this approach the regression function is represented as a linear combination of basis terms. The basis terms can be univariate or multivariate functions and can include polynomials, natural splines and radial basis functions. A Bayesian hierarchical model is used such that the coefficient of each basis term can be zero with positive prior probability. The presence of basis terms in the model is determined by latent indicator variables. The posterior mean is estimated by Markov chain Monte Carlo simulation because it is computationally intractable to compute the posterior mean analytically unless a small number of basis terms is used. The present article updates the work of Smith and Kohn (1996. Journal of Econometrics 75: 317-344) to take account of work by us and others over the last three years. A careful discussion is given to all aspects of the model specification, function estimation and the use of sampling schemes. In particular, new sampling schemes are introduced to carry out the variable selection methodology Robert Kohn Bayesian Model Averaging and Semiparametric Regression http://works.bepress.com/michael_smith/10 http://works.bepress.com/michael_smith/10 Mon, 28 Jul 2008 23:56:45 PDT This paper develops methodology for estimating long ­term trends in the daily maxima of tropospheric ozone. The methods are then applied to study long­term trends in ozone at six monitoring sites in the state of Texas. The methodology controls for the effects of me­teorological variables because it is known that variables such as temperature, wind speed and humidity substantially affect the formation of tropospheric ozone. A semiparametric regression model is estimated in which a nonparametric trivariate surface is used to model the relationship between ozone and these meteorological variables because, while it is known the relationship is a complex nonlinear one, its functional form is unknown. The model also allows for the effects of wind direction and seasonality. The errors are modeled as an au­toregression, which is methodologically challenging because the observations are unequally spaced over time. Each function in the model is represented as a linear combination of basis functions located at all of the design points. We also estimate an appropriate data transfor­mation simultaneously with the functions. The functions are estimated nonparametrically by a Bayesian hierarchical model that uses indicator variables to allow a non­zero probability that the coefficient of each basis term is zero. The entire model, including the nonparametric surfaces, data transformation and autoregression for the unequally spaced errors, is esti­mated using a Markov chain Monte Carlo sampling scheme with a computationally efficient transition kernel for generating the indicator variables. The empirical results indicate that key meteorological variables explain most of the variation in daily ozone maxima through a nonlinear interaction and that their effects are consistent across the six sites. However, the estimated trends vary considerably from site to site, even within the same city. Michael S. Smith Forecasting and Time Series http://works.bepress.com/michael_smith/9 http://works.bepress.com/michael_smith/9 Mon, 28 Jul 2008 23:48:16 PDT This article proposes a data-driven method to identify parsimony in the covariance matrix of longitudinal data, and to exploit any such parsimony to produce a statistically efficient estimator of the covariance matrix. The approach parameterizes the covariance matrix through the Cholesky decomposition of its inverse. For longitudinal data this is a one step ahead predictive representation, and the Cholesky factor is likely to have off-diagonal elements that are zero, or close to zero. A hierarchical Bayesian model is employed to identify any such zeros in the Cholesky factor, similar to approaches that have been successful in Bayesian variable selection. The model is estimated using a Markov chain Monte Carlo sampling scheme that is computationally efficient and can be applied to covariance matrices of high dimension. It is demonstrated through simulations that the proposed method compares favorably in terms of statistical efficiency with a highly regarded competing approach. The estimator is applied to three real examples in which the dimension of the covariance matrix is large relative to the sample size. The first two are from biometry and electricity demand modeling and are longitudinal. The third is from finance and highlights the potential of our method for estimating cross-sectional covariance matrices. Michael S. Smith Bayesian Model Averaging and Semiparametric Regression http://works.bepress.com/michael_smith/8 http://works.bepress.com/michael_smith/8 Mon, 28 Jul 2008 23:35:05 PDT Statistical parametric mapping (SPM), relying on the general linear model and classical hypothesis testing, is a benchmark tool for assessing human brain activity using data from fMRI experiments. Friston et al. (Neuroimage 16 (2002a), 484) discuss some limitations of this frequentist approach and point out promising Bayesian perspectives. In particular, a Bayesian formulation allows explicit modeling and estimation of activation probabilities. In this study, we directly address this issue and develop a new regression based approach using spatial Bayesian variable selection. Our method has several advantages. First, spatial correlation is directly modeled for activation probabilities and indirectly for activation amplitudes. As a consequence, there is no need for spatial adjustment in a postprocessing step. Second, anatomical prior information, such as the distribution of grey matter or expert knowledge, can be included as part of the model. Third, the method has superior edge-preservation properties as well as being fast to compute. When applied to data from a simple visual experiment, the results demonstrate improved sensitivity for detecting activated cortical areas and for better preserving details of activated structures. Michael S. Smith Bayesian Spatial Smoothing http://works.bepress.com/michael_smith/7 http://works.bepress.com/michael_smith/7 Mon, 28 Jul 2008 23:24:15 PDT With the advent of wholesale electricity markets there has been renewed focus on intra-day electricity load forecasting. This paper employs a multi-equation regression model with a diagonal first order stationary vector autoregresson (VAR) for modeling and forecasting intra-day electricity load. The correlation structure of the disturbances to the VAR and the appropriate subset of regressors are explored using Bayesian model selection methodology. The full spectrum of finite sample inference is obtained using a Bayesian Markov chain Monte Carlo sampling scheme. This includes the predictive distribution of load and the distribution of the time and level of daily peak load, something that is difficult to obtain with other methods of inference. The method is applied to several multi-equation models of half-hourly total system load in New South Wales, Australia. A detailed model based on three years of data reveals trend, seasonal, bivariate temperature/humidity and serial correlation components that all vary intra-day, justifying the assumption of a multi-equation approach. Short-term forecasts from simple models highlight the gains that can be made if accurate temperature predictions are exploited. Bayesian predictive means for half-hourly load compare favourably with point forecasts obtained using iterated generalized least squares estimation of the same models. Remy Cottet Forecasting and Time Series