Gradients in Spatial Response Surfaces With Application to Urban Land Values

Abstract

For point-referenced spatial data, we often create explanatory models that introduce regression structure with error consisting of a spatial term and a white noise term. Here we consider more flexible regression structures that allow spatially varying regression coefficients. The resulting mean becomes a spatial response surface that is a linear combination of the components of the spatially varying coefficient vector. Of possible interest in this setting would be gradients associated with the coefficient surfaces as well as the mean surface. Gradients could be sought at arbitrary points and in arbitrary directions. Extending ideas developed in earlier work, we obtain a fully inferential approach within the Bayesian framework for examining such gradients. In particular, we can obtain posterior distributions for any such gradient, for the direction of maximal gradient, and for the magnitude of the maximal gradient. The motivation for our work is the desire to examine urban land value gradients. There is considerable literature in the real estate community on economic theory, modeling, and data analysis relating urban land values to distance from the city center. Here we focus on gradients to such surfaces. The flexibility of our approach allows for much richer insights into the behavior of such gradients than was available previously. We illustrate by fitting a portion of Olcott's classic Chicago land value data.

KEY WORDS:

• Bayesian framework
• Coregionalization
• Directional derivative
• Finite-directional difference
• Markov chain Monte Carlo
• Multivariate spatial process