Spatial autoregressive methods Nr245 Austin Troy Based on Spatial Analysis by Fortin and Dale, Chapter 5

Autcorrelation types None: independence Spatial independence, functional dependence True autocorrelation>> inherent autoregressive Functional autocorr>> induced autoregressive

Autocorrelation types Double autoregressive Notice there are now two autocorrelation parameters  - x and  -z

Effects? Standard test statistics become “too liberal”—more significant results than the data justify Because observations are not totally independent have lower actual degrees of freedom, or lower “effective sample size”: n’ instead of n; since t stat denominator = s/n, if n is too big it inflates the t statistic

What to do? Non-effective Why not just adjust up the significance level? E.g. 99% instead of 95%? Because don’t how by how much to adjust without further information. Could end up with a test that is way too conservative Why not just adjust sampling to only include “independent samples?” Because wasteful of data and because easy to mistake “critical distance to independence”

Best approach: Adjust effective sample size For large sample sizes –So for instance n=1000 and ro=.4 means n’=429 Problem is that, to be useful, autoregressive model (ro parameter) has to be an effective descriptor of the structure of autocorrelation of the data

Moving average models How calculated depends on “order” A simple model for adjusting sample size: first order autoregressive model, only immediate (first order) neighbors are correlated with ro>0. All other pairs are zero. In such a model x i is a function of x i+1 and x i-1 Hence half the info for x i is in each neighbor; produce ro=.5 for large n and n’=n/2. An n order model can take form Translates into generalized matrix form With variance covariance matrix

Moving average When you increase the order, calculating sample size gets complicated; e.g. second order model, where two ro parameters now Important point: If there are several different levels of autocorrelation (  k ), each  k must be incorporated even if non-significant; this can have a huge impact on the calculation of effective sample size Fortin and Dale recommend not using moving average approach because very sensitive to irregularities in the data and can produce a wide range of estimates

Two dimensional approaches Problem with MA approach as it was just presented is assumes one-dimensionality In spatial data, xi depends on all neighbors most likely Two best ways for dealing with this: –Simultaneous autoregressive models (SAR) –Conditional autoregressive models (CAR)

SAR Based on concept of set of simultaneous equations to be solved. In this x i and x i-1 are each defined by their own equations Where x is a vector and is linearly dependent on a vector of underlying variables z 1, z 2 z 3 …. Given as matrix Z, u is a vector non-independent error terms with mean zero and var-covar matrix C Spatial autocorrelation enters via u where Here e is independent error term and W is neighbor weights standardized to row totals of 1. W is not necessarily symmetrical, allowing for inclusion of anisotropy. W ij is >0 if values at location i is not independent of value at location j

SAR This yields the model With variance covariance matrix (from u) Note how similar to MA—difference is no inverse in formula The elements of C are variances From Fortin and Dale p. 231

CAR More commonly used in spatial statistics Not based on spatial dependence per se; instead probability of a certain value is conditional on neighbor values Similar to SAR, but requires that weight matrix be symmetrical Here Where  is the autocorrelation parameter and V is a symmetrical weight matrix Any SAR process is a CAR process if V= W + W T – W T W