GEOADDITIVE BAYESIAN MODEL FOR DATA WITH LIMITED SPATIAL INFORMATION

Abstract

Large area estimation has been mostly accomplished using Geoadditive Models (GM) which combines the ideas of Geostatistics and additive models. The GM relaxes the classical assumptions of traditional parametric model by simultaneously incorporating linear and nonlinear, nonparametric effects of covariates, nonlinear interactions and spatial e ects into a Geoadditive predictor. In the past, estimation of GM has been based on large area as a result of insu cient information in small areas. However, Bayesian approach allows out-of-sample information which can be used to augment the limited information in small areas. Hence, this study adopted the Geoadditive Bayesian model to estimate small areas with insu cient spatial information focusing on small district areas. The GM by Kamman and Wand was speci ed by using E ect Coding (EC) to capture the spatial e ect. The posterior was obtained by combining the likelihood (data) with the prior (out-of-sample) information. The likelihood and the prior information were assumed to be Gaussian and inverse gamma distribution respectively. The numerical solutions were obtained for the posterior distribution, which were not having a closed form solution, using Markov Chain Monte Carlo (MCMC) simulation technique. Finite di erence and partial derivative methods were used to estimate other components of the Geoadditive Bayesian model. Kane analyser was used to collect vehicular emission (carbondioxide, carbonmonoxide and hydrocarbon). Information were also collected on age of vehicles, vehicle types (car and buses), vehicle uses (private and commercial) from 9211 vehicles for 3 years (2008-2011) covering 4 locations: Abeokuta, Sagamu, Ijebu-Ode and Sango-Ota. Data were also collected on respiratory health records of 9211 individuals (18 years and below) in six di erent hospitals on number of visits (nv) and diagnosis within the locality of the collection point of pollutants. Exploratory DataAnalysis (EDA) was carried out on emitted pollutants and age of vehicles. Autocorrelation plot was used to determine model performance. The Geoadditive Bayesian model was : exp[g0(t) + 1 p2 2 e􀀀1 2 2 j p j=1zij + 1 p2 2 e 􀀀1 2 2 ( j )2 + 1 q2 2 j e􀀀1 22 ( spat)2 + 1 q2 2 j e􀀀1 22 ( gi)2 ]:exp Z 1 0 exp(g0(u) + p i=1gj(u)zij)du where zij ,gj , spat and j were non-linear time varying e ect, linear time varying e ect, spatial e ect, and random component, respectively. The MCMC simulation technique gave the posterior means and the standard errors. This revealed that nv, diagnosis, vehicle uses, vehicle types jointly determine the health e ect of pollutants on the individuals considered. Compared with Abeokuta individuals who lived in Sagamu (posterior mean = 0.036) were more likely to be a ected by emitted pollutants while those in Sango-Ota (posterior = - 0.002) and Ijebu-Ode (posterior = - 0.015) were less likely to be a ected. The EDA indicated non-linearity in the pollutants and age of vehicles. There were convergences of parameters at 250 Lag. A signi cant increase in the nonlinear e ects was observed for age of vehicle (5years - 12years), Carbondioxide (10100 - 14400 ppm), Carbonmonoxide (0 - 25000 ppm) and hydrocarbon (4953 - 19812 ppm). The derived Geoadditive Bayesian Model was found suitable and therefore recommended for estimating location e ect of small areas with limited spatial information