h s s h s h s h h

a b c d Figure 6. Density of Angle Between Ray of Maximum Gradient and CBD in Degrees and the Absolute Difference Between Values of Maximal Gradient and Directional Gradient From CBD for SEC. a and b 1985; c and d 1990. See text for details. with spatially varying coefficient models. The theory was built through dependent spatial coefficient and intercept surfaces and yields a resulting mean surface. Gradients at arbitrary locations in arbitrary directions can be studied for each of these surfaces. Also, at a point, for these surfaces the distribution of the direc- tion of maximum gradient and the magnitude of the maximum gradient can be obtained. Working within a Bayesian frame- work allows this full range of inference. Our motivation was to study land value gradients, but other potential applications where gradient analysis of the mean surface would be of inter- est include weather and pollution data modeling. We have focused on explaining land value as a function of distance from the CBD as well as from other “externalities.” Using various distances as explanatory variables, when applied to an urban area like Chicago, we have shown that the simple economic theory asserting exponential decay in distance is not valid. In fact, we have provided a collection of tables and fig- ures that reveal the more subtle nature of expected land value surfaces. At a global level, we see departure from the theoret- ical suggestion, but at a local point level, we see even more subtle departure. Some cities are known to be multicentric Los Angeles, for example. The distance-based regressors model of the previ- ous section is again appropriate here. Furthermore, we have clarified how non–distance-based regressors and, more gener- ally, regressors that need not be continuous can be introduced, noting that suitable spatially varying coefficient models can still be fitted. In this regard, we have indicated their effect on the gradient analysis. Finally, the Olcott data are available for 13 time points, roughly equally spaced. Space–time modeling for Ys, t, with t indicating the time point, extending 16 can be developed. This problem will be pursued in future work. APPENDIX A: DISTRIBUTION THEORY Here, continuing from Section 4, we provide the basic dis- tribution theory for directional finite difference processes and directional derivative processes. If Eβs = 0 for all s ∈ R d , then clearly Eβ

u,h

s = 0 and ED u βs = 0. Let C h u

s, s

′ and C u

s, s

′ denote the covariance functions associated with the process β

u,h

s and D u βs . If = s − s ′ and βs is weakly stationary, then we immediately have C h u

s, s

′ = 2K − K + hu − K − hu h 2 , A.1 where varβ

u,h

s = 2K0 − Khuh 2 . If βs is isotropic, then we obtain C h u

s, s

′ = 2 K − K + hu − K − hu h 2 . A.2 Expression A.2 shows that even if βs is isotropic, β

u,h

s is only stationary. In addition, varβ

u,h

s = 2 K 0 − Kh Downloaded by [Universitas Maritim Raja Ali Haji] at 23:23 12 January 2016 h 2 = γ hh 2 , where γ h is the familiar variogram of the βs