Spatio-Temporal Models for Areal Data

Transcription

1 Spatio-Temporal Models for Areal Data Juan C. Vivar and Marco A. R. Ferreira Departamento de Métodos Estatísticos - IM Universidade Federal do Rio de Janeiro (UFRJ) 8o. Encontro Brasileiro de Estatística Bayesiana - EBEB 8 Búzios, RJ March 26-29, 2006

2 Introduction Our Model General Form Specific Models Inference Application Data Model Results Discussion Conclusions

3 Introduction We introduce a new class of spatio-temporal models for areal data. These models are built as Bayesian dynamic linear models (DLM) with observational and system equation errors following proper Gaussian Markov random field (PGMRF) processes. This new class of models allows several important spatio-temporal dynamics features such as local polynomial trends, seasonality and dispersion because it inherits the flexibility of the DLM framework. Moreover, the use of PGMRF errors allows time specific spatial dynamics for areal data. Our framework lends itself quite naturally to the important tasks of spatio-temporal prediction and smoothing. We illustrate the use of our spatio-temporal framework with an application to violence data in the state of Rio de Janeiro in the period.

4 Introduction We introduce a new class of spatio-temporal models for areal data. These models are built as Bayesian dynamic linear models (DLM) with observational and system equation errors following proper Gaussian Markov random field (PGMRF) processes. This new class of models allows several important spatio-temporal dynamics features such as local polynomial trends, seasonality and dispersion because it inherits the flexibility of the DLM framework. Moreover, the use of PGMRF errors allows time specific spatial dynamics for areal data. Our framework lends itself quite naturally to the important tasks of spatio-temporal prediction and smoothing. We illustrate the use of our spatio-temporal framework with an application to violence data in the state of Rio de Janeiro in the period.

5 Introduction We introduce a new class of spatio-temporal models for areal data. These models are built as Bayesian dynamic linear models (DLM) with observational and system equation errors following proper Gaussian Markov random field (PGMRF) processes. This new class of models allows several important spatio-temporal dynamics features such as local polynomial trends, seasonality and dispersion because it inherits the flexibility of the DLM framework. Moreover, the use of PGMRF errors allows time specific spatial dynamics for areal data. Our framework lends itself quite naturally to the important tasks of spatio-temporal prediction and smoothing. We illustrate the use of our spatio-temporal framework with an application to violence data in the state of Rio de Janeiro in the period.

6 General Form General Form y t = F tx t + ɛ t, ɛ t PGMRF (0, Vt 1 ), (1) x t = G t x t 1 + ω t, ω t PGMRF (0, Wt 1 ), (2) Z PGMRF (µ, P) means that ( p(z) exp 1 ) 2 (z µ) P(z µ) where P = τ(i n + φm), with I n the n n identity matrix and m k, k = l (M) kl = g kl, k N l 0, otherwise, where g kl > 0 is a measure of similarity between sites k and l, and m k = l N k g kl.

7 General Form General Form y t = F tx t + ɛ t, ɛ t PGMRF (0, Vt 1 ), (1) x t = G t x t 1 + ω t, ω t PGMRF (0, Wt 1 ), (2) Z PGMRF (µ, P) means that ( p(z) exp 1 ) 2 (z µ) P(z µ) where P = τ(i n + φm), with I n the n n identity matrix and m k, k = l (M) kl = g kl, k N l 0, otherwise, where g kl > 0 is a measure of similarity between sites k and l, and m k = l N k g kl.

8 General Form General Form y t = F tx t + ɛ t, ɛ t PGMRF (0, Vt 1 ), (1) x t = G t x t 1 + ω t, ω t PGMRF (0, Wt 1 ), (2) Z PGMRF (µ, P) means that ( p(z) exp 1 ) 2 (z µ) P(z µ) where P = τ(i n + φm), with I n the n n identity matrix and m k, k = l (M) kl = g kl, k N l 0, otherwise, where g kl > 0 is a measure of similarity between sites k and l, and m k = l N k g kl.

9 Specific Models Some Specific Models First-order polynomial model F t = I n and G t = ρi n ρ [ 1, 1] W 1 t = τ(i n + φm) Contamination model F t = I n 1 k = l G t = ρ 1+βh H {H} kl = β k N l 0 o.c. H is called the contamination matrix, ρ [ 1, 1]. W 1 t = τ(i n + φm)

10 Specific Models Some Specific Models First-order polynomial model F t = I n and G t = ρi n ρ [ 1, 1] W 1 t = τ(i n + φm) Contamination model F t = I n 1 k = l G t = ρ 1+βh H {H} kl = β k N l 0 o.c. H is called the contamination matrix, ρ [ 1, 1]. W 1 t = τ(i n + φm)

11 Specific Models Some Specific Models Second-order polynomial model F t = (I n, 0 n ) G t = ( G1t G 1t 0 n G 2t ( Wt 1 W 1 = 1t 0 n 0 n W2t 1 ), G it = ρ i I n, ρ i [ 1, 1], i = 1, 2 ), where W 1 it = τ i (I n + φ i M), i = 1, 2

12 Inference Bayesian Inference Parameters: Simulation is model specific. MCMC techniques including objective Bayesian analysis of PGMRF. Latent vector: Simulation of x t performed via forward information filter backward sampler (FIFBS). When Vt 1 and Wt 1 are constant in time the FIFBS have similar computational performance that the FFBS

13 Inference Bayesian Inference Parameters: Simulation is model specific. MCMC techniques including objective Bayesian analysis of PGMRF. Latent vector: Simulation of x t performed via forward information filter backward sampler (FIFBS). When Vt 1 and Wt 1 are constant in time the FIFBS have similar computational performance that the FFBS

14 Data Data Number of homicides in each county of Rio de Janeiro state from 1979 to Data kindly provided by Dr. Oswaldo G. Cruz (FIOCRUZ). We use the political map of 1979 (64 counties). For each county i and year t we have the estimated population n it and the observed number of homicides h it. Our interest is to understand the spatio-temporal dynamics of the standardized mortality ratio (SMR) per inhabitants SMR it = h it /n it

15 Data Data We assume that, conditional on the underlying risk λ it (i = 1,..., I = 64; t = 1,..., T = 20), h it are independent and h it Po(n it λ it ). As the homicides level is high, we can use the approximation: SMR it N(10 5 λ it, λ it n it ) In order to stabilize the variance, we use the squared root transformation and define y it = SMR it. Using the delta method, it is easy to show that y it a N( λ 0.5 it, n it )

16 Model Fitted Model An exploratory analysis showed an increasing trend and later stabilization of the SMR. We fit a second-order temporal trend model with ρ 1 = ρ 2 = 1 and x 1t = ( λ t1,..., λ ts ), that is y t = x 1t + ɛ t, ɛ t N ) (0 S, diag(n 1 t1,..., n 1 ts ),(3) x 1t = x 1,t 1 + x 2,t 1 + ω 1t, ω 1t PGMRF (0 S, W1t 1 ), (4) x 2t = x 2,t 1 + ω 2t, ω 2t PGMRF (0 S, W2t 1 ), (5) where (4) is the level equation and (5) is the velocity equation.

17 Results Results Parameter Mean Standard deviation 95% C.I. φ (0.868, 6.007) τ (0.258, 0.920) τ (28.08, 49.02) Table: Posterior summaries. τ 1 τ 2 φ 1 Figure: Marginal posterior densities of τ 1, τ 2 and φ 1.

18 Results Estimated latent vector x t

19 Results Estimated innovations vector ω t

20 Discussion Discussion Both level and velocity field present spatio-temporal dependence. The counties have a smooth variation within and between years. Level field: Metropolitan region of Rio de Janeiro had the higher levels of SMRs. Velocity field: During 80s Higher values Final 80s Decreasing During 90s Negative values That clearly shows that trend in 1998 was the stabilization or reduction of SMRs.

21 Discussion Discussion Both level and velocity field present spatio-temporal dependence. The counties have a smooth variation within and between years. Level field: Metropolitan region of Rio de Janeiro had the higher levels of SMRs. Velocity field: During 80s Higher values Final 80s Decreasing During 90s Negative values That clearly shows that trend in 1998 was the stabilization or reduction of SMRs.

22 Discussion Discussion Both level and velocity field present spatio-temporal dependence. The counties have a smooth variation within and between years. Level field: Metropolitan region of Rio de Janeiro had the higher levels of SMRs. Velocity field: During 80s Higher values Final 80s Decreasing During 90s Negative values That clearly shows that trend in 1998 was the stabilization or reduction of SMRs.

23 Conclusions New class of spatio-temporal models for areal data. This class allows various forms of space-time interactions and time specific spatially structured effects. Full Bayesian inference using MCMC with embedded FIFBS. Application: number of homicides in the State of Rio de Janeiro, using a non-stationary second order temporal trend model. The model has been able to capture several features of this process.