  Inference methods in traditional statistics, machine learning and data mining
assume that data is generated from an independent and identically distributed
(iid) process. Spatial data exhibits behavior for which the iid assumption must
be relaxed. For example, the standard approach in spatial regression is to
assume the existence of a contiguity matrix which captures the spatial
autoregressive properties of the data. However all spatial methods, till now,
have assumed that the contiguity matrix is given apriori or can be estimated by
using a spatial similarity function. In this paper we propose a convex
optimization formulation to solve the spatial autoregressive regression (SAR)
model in which both the contiguity matrix and the non-spatial regression
parameters are unknown and inferred from the data. We solve the problem using
the alternating direction method of multipliers (ADMM) which provides a
solution which is both robust and efficient. While our approach is general we
use data from housing markets of Boston and Sydney to both guide the analysis
and validate our results. A novel side effect of our approach is the automatic
discovery of spatial clusters which translate to submarkets in the housing data
sets.
