Spatial Smoothing in Stan: Conditional Auto-Regressive Models

Transcription

1 Spatial Smoothing in Stan: Conditional Auto-Regressive Models Charles DiMaggio, PhD, NYU School of Medicine Stephen J. Mooney, PhD, University of Washington Mitzi Morris, Columbia University Dan Simpson, PhD, University of Toronto Katherine Wheeler-Martin, NYU School of Medicine 1

2 Outline Data events: New York City pedestrian traffic accidents areal unit: Census Tract Models BYM, BYM2 model - Poisson GLM plus separate components for spatial and non-spatial variance ICAR model - component for spatial variance Stan Implementations ICAR model BYM2 model 2

3 Spatial Smoothing for Areal Data Counts of rare events in small-population regions are noisy. Conditional Auto-Regressive (CAR) models smooth noise by pooling information from neighboring regions. CAR models use binary, symmetric neighbor relationship: for map over N regions, neighbor relationship i j, (i j), holds if and only if regions n i and n j are neighbors CAR models defined by encoding neighbor relationships as an adjacency matrix: for map over N regions use N N matrix entries {i,j} and {j,i} are 1 when i j (i.e., regions n i and n j are neighbors), 0 otherwise entries {i,i} are always 0 3

4 NYC Study Data map regions: census tracts (2095) red points: accidents (17,193) median accidents per tract: 6 median population per tract: 510 4

5 NYC Census Tract Neighbors 5

6 NYC Census Tract Neighbors, Zoom black lines: census tract boundaries blue lines: neighbors median neighbors: 6 min neighbors: 1 (6 regions) max neighbors: 12 (5 regions) 6

7 Besag (1974): Conditional Auto-Regressive (CAR) Model Conditional specification of CAR model: multivariate normal - N-length normal random vector φ = (φ 1,..., φ n ) T where each φ i is conditional on the values of its neighbors and with unknown variance. n φ i φ j,j i Normal( w ij φ j, σ) j=1 w ij are entries in a N N weights matrix W W is constructed from the adjacency matrix and a parameter α which controls the amount of spatial correlation 0 < α < 1. φ is a Gaussian Markov Random Field (GMRF). 7

8 CAR to ICAR Joint specification of CAR model φ Normal(0, Q 1 ) φ is centered at 0, precision matrix Q Q is constructed from adjacency matrix and α to be positive definite. Log probability density of φ is proportional to 1 2 log(det(q)) 1 2 φt Qφ Computing the determinant of a N N matrix requires N 3 operations Intrinsic Conditional Auto-Regressive Model: set α = 1 precision matrix Q is constant, term 1 2 log(det(q)) drops out Q is not positive definite, log prob density is improper ICAR can only be used as a prior 8

9 ICAR Model - Pairwise Difference Formulation Joint specification rewrites to Pairwise Difference: p(φ) exp 1 (φ i φ j ) 2 2 i j centered at 0, assuming common variance for all elements of φ. Each (φ i φ j ) 2 contributes a penalty term based on the distance between the values of neighboring regions φ is non-identifiable, constant added to φ washes out of φ i φ j Sum-to-zero constraint centers φ 9

10 Stan ICAR Model Implementation p(φ) exp 1 (φ i φ j ) 2 2 Use Stan s vectorized operations to compute log probability density: i j target += -0.5 * dot_self(phi[node1] - phi[node2]); Encode neighbor information as graph edgeset, i.e. pairs of indices for neighbors i, j: int<lower=0> N; int<lower=0> N_edges; int<lower=1, upper=n> node1[n_edges]; int<lower=1, upper=n> node2[n_edges]; 10

11 Stan ICAR Model Implementation: Soft-centering φ is non-identifiable, sum-to-zero constraint centers φ Soft sum-to-zero constraint when vector values sum to zero, vector mean is zero as well. add prior to keep mean as close to zero as possible: mean(phi) ~ normal(0, 0.001); mean(phi) equivalent to 1/N * sum(phi) rewrite constraint to avoid division operation: sum(phi) ~ normal(0, * N); May allow for faster sampling than hard sum-to-zero constraint. 11

12 Stan ICAR Model Implementation functions { real icar_normal_lpdf(vector phi, int N, int[] node1, int[] node2) { return -0.5 * dot_self(phi[node1] - phi[node2]); } } data { int<lower=0> N; int<lower=0> N_edges; int<lower=1, upper=n> node1[n_edges]; // node1[i], node2[i] neighbors int<lower=1, upper=n> node2[n_edges]; // node1[i] < node2[i] } parameters { vector[n] phi; } model { phi ~ icar_normal_lpdf(n, node1, node2); sum(phi) ~ normal(0, * N); } 12

13 Stan ICAR Model: Visualize Covariance 13

14 Besag York Mollié (1991) BYM Model Lognormal Poisson model developed for disease risk mapping η i = µ + xβ + φ + θ where: µ is the fixed intercept. x is the design matrix, β is vector of regression coefficients. φ is an ICAR spatial component θ is an vector of ordinary random-effects components. MCMC samplers require strong hyperpriors on φ and θ. Bernardinelli et. al. (1995): hyperpriors depend on average number of neighbors, i.e., hyperpriors depend on data. Difficult to fit; difficult to interpret parameters. 14

15 Riebler et. al. (2016): BYM2 Model Reparameterize BYM model; rewrite sum of the spatial and non-spatial components φ + θ as ( ( (ρ/s) ) φ + ( ) 1 ρ) θ σ where: σ 0 is the overall standard deviation. ρ [0, 1] - proportion of spatial variance. φ is the ICAR component. θ N(0, 1) is the vector of ordinary random effects s is a scaling factor s.t. Var(φ i ) 1; s is data. 15

16 BYM2 Model Stan Implementation parameters { real beta0; vector[k] betas; vector[n] theta; vector[n] phi; real<lower=0> sigma; real<lower=0, upper=1> rho; } transformed parameters { vector[n] convolved_re; convolved_re = sqrt(rho / scaling_factor) * phi + sqrt(1 - rho) * theta; } model { y ~ poisson_log(log_e + beta0 + x * betas + convolved_re * sigma); beta0 ~ normal(0.0, 1.0); betas ~ normal(0.0, 1.0); sigma ~ normal(0, 1.0); rho ~ beta(0.5, 0.5); theta ~ normal(0.0, 1.0); phi ~ icar_normal_lpdf(n, node1, node2); sum(phi) ~ normal(0, * N); // sum-to-zero constraint } 16

17 Results 17