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

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

3. Autocorrelation types
Double autoregressive
Notice there are now two autocorrelation parameters r-x and r-z

4. 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
Above: simulations yield 4x the type 1 errors for inherent AR than expected, induced AR model yields 2 x and double AR model is 8x

5. 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”

6. Best approach: Adjust effective sample size
In presence of SA, variance of mean of obs can be adjusted sing covariances of Xs Cov(Xi, Xj) becomes
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, but it’s a simplification
Next step therefore is factoring in correlation matrix, R, based on lag distances r(d)

7. Moving average models At 1st order we get a matrix like:
Half of info for Xi contained in Xi+1
Half contained in Xi-1
Hence only every other ob. Needed
So produce ro=.5 for large n and n’=n/2. n’=n/2
A k order model can take form
Translates into generalized matrix form
With variance covariance matrix

8. 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 (rk), each rk must be incorporated even if non-significant
Using only significant values can understate n’
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

9. Two dimensional approaches
Problem with MA approach as it was just presented is assumes one-dimensionality
In 2-d spatial data, xi depends on all neighbors most likely
Now must define what is “neighbor” in 2d (e.g. w=1/8 for 9 cell grid of neighbors, all else = 0)
Two best ways for dealing with this:
Simultaneous autoregressive models (SAR)
Conditional autoregressive models (CAR)
CAR’s neighborhood matrices specify relationship between lagged response values at each location and neighboring location
SAR’s specify relationship between lagged residuals
Both use nxn spatial weights matrix (W) composed of wij
Can be based on adjacency, number neighbors or distance
Zeros on diagonals, weights on off diagonals
In both SAR and CAR, SA tends to persist across long distances

10. 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
Here
Where j is the autocorrelation parameter and V is a symmetrical weight matrix
Symmetrical requirement means that directional processes can’t be modeled.

11. SAR
Based on concept of set of simultaneous equations to be solved. In this xi and xi-1 are each defined by their own equations containing other xs
Where x is a vector and is linearly dependent on a vector of underlying variables z1, z2 z3…. 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. Wij is >0 if values at location i is not independent of value at location j

12. 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

13. SAR
Advantages: doesn’t require weight matrix to be symmetrical, so can model anisotropic phenomena.
SAR can take three forms
Lagged response model: autoregressive process only occurs in the response variable
Lagged mixed model, where SA affects both response and predictors
Spatial error model: assumes SA process occurs only in error term and not in response or predictor