Abstract
In many application areas, data are collected on a count or binary response with spatial covariate information. In this article, we introduce a new class of generalized geoadditive models (GGAMs) for spatial data distributed over complex domains. Through a link function, the proposed GGAM assumes that the mean of the discrete response variable depends on additive univariate functions of explanatory variables and a bivariate function to adjust for the spatial effect. We propose a two-stage approach for estimating and making inferences of the components in the GGAM. In the first stage, the univariate components and the geographical component in the model are approximated via univariate polynomial splines and bivariate penalized splines over triangulation, respectively. In the second stage, local polynomial smoothing is applied to the cleaned univariate data to average out the variation of the first-stage estimators. We investigate the consistency of the proposed estimators and the asymptotic normality of the univariate components. We also establish the simultaneous confidence band for each of the univariate components. The performance of the proposed method is evaluated by two simulation studies. We apply the proposed method to analyze the crash counts data in the Tampa-St. Petersburg urbanized area in Florida. Supplementary materials for this article are available online.
Highlights
First we introduce the general notations that we use in the following proof
J = {1, . . . , Jn} as the index set of univariate spline basis functions
Properties of Penalized Quasi-likelihood Estimators Recall that ukj(xk), j ∈ J, are the original B-spline basis functions for the kth covariate, where J is the index set of the basis functions
Summary
First we introduce the general notations that we use in the following proof. a2i and its supremum norm as |a| = max1≤i≤n |ai|. For any real symmetric matrix A = (aij)mi=,1n,j=1, denote by λmin (A) and λmax (A) its smallest and largest eigenvalues, and its L2 norm as A 2 =. Jn} as the index set of univariate spline basis functions. We define the norm on the space G. For functions ψ1, ψ2 ∈ G, define their theoretical inner product as ψ1, ψ2 =. For the quasi-likelihood function {g−1(x), y}, let q1(x, y) =. Properties of Penalized Quasi-likelihood Estimators Recall that ukj(xk), j ∈ J , are the original B-spline basis functions for the kth covariate, where J is the index set of the basis functions. In the following we define their centered basis u0kj(xk) and. We define the standardized Bernstein basis polynomials as Bm∗ (s) = Bm(s)/ Bm , m ∈ M, where M is the index set of Bernstein basis functions. ΓmBm∗ (s), xk ∈ [0, 1], s ∈ Ω, θkj, γm ∈ R
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