Gaussian Multiscale Spatio-temporal Models for Areal Data
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1 Gaussian Multiscale Spatio-temporal Models for Areal Data (University of Missouri) Scott H. Holan (University of Missouri) Adelmo I. Bertolde (UFES)
2 Outline Motivation Multiscale factorization The multiscale spatio-temporal model Bayesian analysis Application: Agricultural Production in Espírito Santo Unknown multiscale structure Conclusions
3 Outline Motivation Multiscale factorization The multiscale spatio-temporal model Bayesian analysis Application: Agricultural Production in Espírito Santo Unknown multiscale structure Conclusions
4 Missouri: square root of standardized mortality ratio per 100, 000 inhabitants 1990 under over 33
5 Missouri: square root of standardized mortality ratio per 100, 000 inhabitants 1991 under over 33
6 Missouri: square root of standardized mortality ratio per 100, 000 inhabitants 1992 under over 33
7 Missouri: square root of standardized mortality ratio per 100, 000 inhabitants 1993 under over 33
8 Missouri: square root of standardized mortality ratio per 100, 000 inhabitants 1994 under over 33
9 Missouri: square root of standardized mortality ratio per 100, 000 inhabitants 1995 under over 33
10 Missouri: square root of standardized mortality ratio per 100, 000 inhabitants 1996 under over 33
11 Missouri: square root of standardized mortality ratio per 100, 000 inhabitants 1997 under over 33
12 Missouri: square root of standardized mortality ratio per 100, 000 inhabitants 1998 under over 33
13 Missouri: square root of standardized mortality ratio per 100, 000 inhabitants 1999 under over 33
14 Missouri: square root of standardized mortality ratio per 100, 000 inhabitants 2000 under over 33
15 Missouri: square root of standardized mortality ratio per 100, 000 inhabitants 2001 under over 33
16 Missouri: square root of standardized mortality ratio per 100, 000 inhabitants 2002 under over 33
17 Missouri: square root of standardized mortality ratio per 100, 000 inhabitants 2003 under over 33
18 Missouri: square root of standardized mortality ratio per 100, 000 inhabitants 2004 under over 33
19 Missouri: square root of standardized mortality ratio per 100, 000 inhabitants 2005 under over 33
20 Missouri: square root of standardized mortality ratio per 100, 000 inhabitants 2006 under over 33
21 Some background Many processes of interest are naturally spatio-temporal. Frequently, quantities related to these processes are available as areal data. These processes may often be considered at several different levels of spatial resolution. Related work on dynamic spatio-temporal multiscale modeling: Berliner, Wikle and Milliff (1999), Johannesson, Cressie and Huang (2007).
22 Data Structure Here, the region of interest is divided in geographic subregions or blocks, and the data may be averages or sums over these subregions. Moreover, we assume the existence of a hierarchical multiscale structure. For example, each state in Brazil is divided into counties, microregions and macroregions.
23 Geopolitical organization (a) (b) (c) Figure: Geopolitical organization of Espírito Santo State by (a) counties, (b) microregions, and (c) macroregions.
24 Outline Motivation Multiscale factorization The multiscale spatio-temporal model Bayesian analysis Application: Agricultural Production in Espírito Santo Unknown multiscale structure Conclusions
25 Multiscale factorization At each time point we decompose the data into empirical multiscale coefficients using the spatial multiscale modeling framework of Kolaczyk and Huang (2001). See also Chapter 9 of Ferreira and Lee (2007). Interest lies in agricultural production observed at the county level, which we assume is the L th level of resolution (i.e. the finest level of resolution), on a partition of a domain S R 2. For the j th county, let y Lj, µ Lj = E(y Lj ), and σ 2 Lj = V (y Lj) respectively denote agricultural production, its latent expected value and variance.
26 Let D lj be the set of descendants of subregion (l, j). The aggregated measurements at the l th level of resolution are recursively defined by y lj = p l+1,j y l+1,j. (l+1,j ) D lj Analogously, the aggregated mean process is defined by µ lj = p l+1,j µ l+1,j. (l+1,j ) D lj Assuming conditional independence, σlj 2 = pl+1,j 2 σ2 l+1,j. (l+1,j ) D lj
27 Define σ 2 l = (σ 2 l1,..., σ2 ln l ), ρ l = p l σ 2 l, and Σ l = diag(σ 2 l ), where denotes the Hadamard product. Then with and y Dlj ylj, µ L, σ 2 L N(ν ljy lj + θ lj, Ω lj ), ν lj = ρ Dlj /σ 2 lj, θ lj = µ Dlj ν lj µ lj, Ω lj = Σ Dlj σ 2 lj ρ Dlj ρ D lj.
28 Consider θ e lj = y D lj ν lj y lj, which is an empirical estimate of θ lj. Then (Kolaczyk and Huang, 2001) θ e lj y lj, µ L, σ 2 L N(θ lj, Ω lj ), which is a singular Gaussian distribution (Anderson, 1984).
29 Exploratory Multiscale Data Analysis Macroregion 1 total Disaggregated by microregion Microregion 1 Microregion 2 Microregion 3 Microregion 4 Microregion year year
30 Empirical multiscale coefficient for Macroregion year year year θ e t111 θ e t112 θ e t year year θ e t114 θ e t115
31 Outline Motivation Multiscale factorization The multiscale spatio-temporal model Bayesian analysis Application: Agricultural Production in Espírito Santo Unknown multiscale structure Conclusions
32 The multiscale spatio-temporal model Observation equation: y tl = µ tl + v tl, v tl N(0, Σ L ) where Σ L = diag(σl1 2,..., σ2 Ln L ). Multiscale decomposition of the observation equation: y t1k µ t1k N(µ t1k, σ1k 2 ) θ e tlj θ tlj N(θ tlj, Ω lj )
33 System equations: µ t1k = µ t 1,1k + w t1k, w t1k N(0, ξ k σ 2 1k ) θ tlj = θ t 1,lj + ω tlj, ω tlj N(0, ψ lj Ω lj )
34 Priors µ 01k D 0 N(m 01k, c 01k σ 2 1k ), θ 0lj D 0 N(m 0lj, C 0lj Ω lj ), ξ k IG(0.5τ k, 0.5κ k ), ψ lj IG(0.5ϱ lj, 0.5ς lj ),
35 Outline Motivation Multiscale factorization The multiscale spatio-temporal model Bayesian analysis Application: Agricultural Production in Espírito Santo Unknown multiscale structure Conclusions
36 Empirical Bayes estimation of ν lj and Ω lj ν lj : vector of relative volatilities of the descendants of (l, j), Ω lj : singular covariance matrix of the empirical multiscale coefficient of subregion (l, j) In order to obtain an initial estimate of σlj 2, we perform a univariate time series analysis for each county using first-order dynamic linear models (West and Harrison, 1997). These analyses yield estimates σ Lj 2. Let ρ l = p l σ 2 l. We estimate ν lj and Ω lj by ν lj = ρ Dlj / σ 2 lj, Ω lj = Σ Dlj σ 2 lj ρ Dlj ρ D lj.
37 Posterior exploration Let θ lj = (θ 0lj,..., θ Tlj ), θ t j = (θ t1j,..., θ tlj ), θ = (θ 11,..., θ 1n 1, θ 21,..., θ 2n 2,..., θ L1,..., θ Ln L ), with analogous definitions for the other quantities in the model. It can be shown that, given σ 2, ξ, and ψ, the vectors µ 11,..., µ 1n1, θ 11,..., θ 1n1,..., θ L1,..., θ LnL, are conditionally independent a posteriori.
38 Gibbs sampler µ 1k : Forward Filter Backward Sampler (FFBS) (Carter and Kohn, 1994; Fruhwirth-Schnatter,1994). ξ k µ 1k, σ 2 1k, D T IG (0.5τ k, 0.5κ k ), where τ k = τ k + T and κ k = κ k + σ 2 1k T t=1 (µ t1k µ t 1,1k ) 2. ψ lj θ lj, D T IG(0.5ϱ lj, 0.5ς lj ), where ϱ lj = ϱ lj + T (m lj 1) and ςlj = ς lj + T t=1 (θ tlj θ t 1,lj ) Ω lj (θ tlj θ t 1,lj ), where Ω lj is a generalized inverse of Ω lj. θ lj : Singular FFBS.
39 Singular FFBS 1. Use the singular forward filter to obtain the mean and covariance matrix of f (θ 1lj σ 2, ψ lj, D 1 ),..., f (θ Tlj σ 2, ψ lj, D T ): posterior at t 1: θ t 1,lj D t 1 N (m t 1,lj, C t 1,lj Ω lj ) ; prior at t: θtlj D t 1 N (a tlj, R tlj Ω lj ), where a tlj = m t 1,lj and R tlj = C t 1,lj + ψ lj ; posterior at t: θtlj D t N (m tlj, C tlj ( Ω lj ), where ) C tlj = (1 + R 1 tlj ) 1 and m tlj = C tlj θ e tlj + R 1 tlj a tlj. 2. Simulate θ Tlj from θ Tlj σ 2, ψ lj, D T N(m Tlj, C Tlj Ω lj ). 3. Recursively simulate θ tlj, t = T 1,..., 0, from θ tlj θ t+1,lj,..., θ Tlj, D T θ tlj θ t+1,lj, D t N(h tlj, H tlj Ω lj ), ( ) 1 where H tlj = C 1 tlj + ψ 1 lj and ( ) h tlj = H tlj C 1 tlj m tlj + ψ 1 lj θ t+1,lj.
40 Reconstruction of the latent mean process One of the main interests of any multiscale analysis is the estimation of the latent mean process at each scale of resolution. From the g th draw from the posterior distribution, we can recursively compute the corresponding latent mean process at each level of resolution using the equation µ (g) t,d lj = θ (g) tlj + ν tlj µ (g) tlj, proceeding from the coarsest to the finest resolution level. With these draws, we can then compute the posterior mean, standard deviation and credible intervals for the latent mean process.
41 Outline Motivation Multiscale factorization The multiscale spatio-temporal model Bayesian analysis Application: Agricultural Production in Espírito Santo Unknown multiscale structure Conclusions
42 Posterior densities Density Density ξ 1 ξ 2 ξ 3 ξ 4 Density ψ 11 ψ 12 ψ 13 ψ σ 2 ξ 1,..., ξ 4 ψ 11,..., ψ 14
43 Multiscale coefficient for Macroregion year year year θ t111 θ t112 θ t year year θ t114 θ t115
44 Outline Motivation Multiscale factorization The multiscale spatio-temporal model Bayesian analysis Application: Agricultural Production in Espírito Santo Unknown multiscale structure Conclusions
45 Discrepancy measurement model Consider two subregions at the same level of resolution with observations y t1 and y t2 at time t, observational variances σ 2 1 and σ 2 2, and aggregation weights p 1 and p 2. We define the Discrepancy Measurement Model (DMM) by the dynamic linear model ( yt1 y t2 ( µt t ) ) = = ( p1 σ 2 1 /(p2 1 σ2 1 + p2 2 σ2 2 ) p 2 p 2 σ2 2/(p2 1 σ2 1 + p2 2 σ2 2 ) p 1 ) + w t, ( µt 1 t 1 ) ( µt t ) + v t, where v t N(0, V), w t N(0, W), V = diag(σ1 2, σ2 2 ), and W = diag(w µ, W ).
46 Relative discrepancy Consider the DMM model and two subregions at the same resolution level. We define the relative discrepancy between these two subregions as W /W µ.
47 Multiscale clustering algorithm 1. For each county, create a link to one of its neighbors which minimizes the estimated relative discrepancy. 2. Create intermediate-level clusters of counties. Counties are in the same cluster if and only if there is a path connecting them. 3. Obtain the data at the intermediate-level through aggregation. 4. Apply steps 1 and 2 to the intermediate-level clusters in order to obtain the coarse-level clusters. We illustrate the algorithm with a dataset on mortality in Missouri.
48 Estimated multiscale structure for the State of Missouri Kansas City Columbia Jefferson City Saint Louis Springfield
49 Missouri: square root of standardized mortality ratio per 100, 000 inhabitants 1990 Observed Fitted under over 33
50 Missouri: square root of standardized mortality ratio per 100, 000 inhabitants 1991 Observed Fitted under over 33
51 Missouri: square root of standardized mortality ratio per 100, 000 inhabitants 1992 Observed Fitted under over 33
52 Missouri: square root of standardized mortality ratio per 100, 000 inhabitants 1993 Observed Fitted under over 33
53 Missouri: square root of standardized mortality ratio per 100, 000 inhabitants 1994 Observed Fitted under over 33
54 Missouri: square root of standardized mortality ratio per 100, 000 inhabitants 1995 Observed Fitted under over 33
55 Missouri: square root of standardized mortality ratio per 100, 000 inhabitants 1996 Observed Fitted under over 33
56 Missouri: square root of standardized mortality ratio per 100, 000 inhabitants 1997 Observed Fitted under over 33
57 Missouri: square root of standardized mortality ratio per 100, 000 inhabitants 1998 Observed Fitted under over 33
58 Missouri: square root of standardized mortality ratio per 100, 000 inhabitants 1999 Observed Fitted under over 33
59 Missouri: square root of standardized mortality ratio per 100, 000 inhabitants 2000 Observed Fitted under over 33
60 Missouri: square root of standardized mortality ratio per 100, 000 inhabitants 2001 Observed Fitted under over 33
61 Missouri: square root of standardized mortality ratio per 100, 000 inhabitants 2002 Observed Fitted under over 33
62 Missouri: square root of standardized mortality ratio per 100, 000 inhabitants 2003 Observed Fitted under over 33
63 Missouri: square root of standardized mortality ratio per 100, 000 inhabitants 2004 Observed Fitted under over 33
64 Missouri: square root of standardized mortality ratio per 100, 000 inhabitants 2005 Observed Fitted under over 33
65 Missouri: square root of standardized mortality ratio per 100, 000 inhabitants 2006 Observed Fitted under over 33
66 State of Missouri application. Residuals check. Fitted Sample Quantiles Observed Theoretical Quantiles
67 Outline Motivation Multiscale factorization The multiscale spatio-temporal model Bayesian analysis Application: Agricultural Production in Espírito Santo Unknown multiscale structure Conclusions
68 Conclusions New multiscale spatio-temporal model for areal data. Estimated multiscale coefficients shed light on similarities and differences between regions within each scale of resolution. Modeling strategy naturally respects nonsmooth transitions between geographic subregions. Our divide-and-conquer modeling strategy leads to computational procedures that are scalable and fast.
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