DISCRETE_AUTOREGRESSIVE_MODELS

DISCRETE_AUTOREGRESSIVE_MODELS - ESE 502 Tony E. Smith _...

Info iconThis preview shows pages 1–3. Sign up to view the full content.

View Full Document Right Arrow Icon
ESE 502 Tony E. Smith ________________________________________________________________________ 1 DISCRETE SPATIAL AUTOREGRESSIVE MODELS The standard logistic, binomial and Poisson regression models of discrete counting data have natural spatial generalizations in a manner similar to the conditional autoregressive (CAR) model. We consider each in turn. 1. Autologistic Model Suppose one considers the voting outcomes for Pennsylvania counties during a given Presidential election. Let 1 i v = if county i voted Democratic and 0 i v = otherwise. To model this voting behavior, one may consider a number of county covariates, 1 ( ,.., ) iii k x xx = , say with 1 i x = “per capita income” in county i , 2 i x = “average years of education” for voters in county i , and so on. A standard logistic model of voting behavior would then take the form: (1) Pr( 1) 1 i i i v µ == + where (2) () 1 exp k ij i j j x µαβ = =+ However, it may well be that there are voting similarities among adjacent counties. Hence, if we let 1 ij w = if county j is a contiguous neighbor of county i (with 0 ii w = ) and if we let 11 1 ( ,. ., , ,.., ) ii i n vv v −− + = denote the voting outcomes of all counties other than i , then a natural spatial generalization of this model would be to hypothesize that the conditional distribution of i v given the voting outcomes, i v , of all other counties depends only on the voting behavior of its neighbors, say, (3) Pr( 1| ) 1( ) v v + with (4) 1 ()e x p k j i j i j j jj i vx w v µα β ρ =≠ + ∑∑ To estimate this model, we start with the standard logistic model that has a likelihood function of the form: (5) 1 1 (, |) P r ( 1 )[ 1P r ( 1 ) ] n i Lv v v αβ = =
Background image of page 1

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
ESE 502 Tony E. Smith ________________________________________________________________________ 2 {} 1 log ( , | ) logPr( 1) (1 )log[1 Pr( 1)] n ii i i i Lv v v v v αβ = ⇒= = + = where 1 ( ,. ., ) n vvv = and where Pr( i v = is given by (1) and (2). This function is quite easy to maximize, and yields well behaved maximum-likelihood estimates of α and β .
Background image of page 2
Image of page 3
This is the end of the preview. Sign up to access the rest of the document.

This note was uploaded on 09/14/2011 for the course ESE 502 taught by Professor Ese502 during the Fall '08 term at UPenn.

Page1 / 6

DISCRETE_AUTOREGRESSIVE_MODELS - ESE 502 Tony E. Smith _...

This preview shows document pages 1 - 3. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online