D . . . , D β β F. S

B r is the rth school district and I is an indicator function. So for such a regressor, the contribution to the gradient in 10 is 0. To clarify, the foregoing regressors provide an explanation for the mean surface but will appear only partially or not at all in the gradient analysis. From Section 4.1, the direction in which the directional derivative is maximized is ∇ T EYs = D 10 EYs, D 01 EYs D 10 EYs, D 01 EYs . 11 It then follows that the directional derivative in that direction becomes D ∇ EYs ∇ EYs EYs = D 1,0 EYs 2 + D 0,1 EYs 2 12 . 12 For posterior inference, we could to generate the predictive sur- face of the mean response process EYs and the associated predictive variance. For gradient behavior, we could make pos- terior predictive inference for the finite-difference process at various resolutions, h, as well as for the directional derivative process. For the latter, at location s we need to sample the pre- dictive joint distribution of the 2p × 1 vector, Zs T = D 1,0 β 1

s, D

0,1 β 1

s, . . . ,

D 1,0 β p

s, D

0,1 β p s . Using usual composition, this requires the conditional distribu- tion of Zs given βs i , i = 1, . . . , n, and θ . Similarly, for finite differences, for each h and each u, we need the 2p × 1 vector ˜ Z ⋆ s T = β 1

s, β

1 s + hu, . . . , β p

s, β

p s + hu , given βs i , i = 1, . . . , n, and θ . Indeed, following the development in Section 4.2, both of these distributions are Gaussian with mean 0 and covariance structures given in Appendix B. 6.3 A Useful Result Suppose that g is differentiable on R 1 . Then gx = g y + x − yg ′ y + Rx − y , where Rx − y = ox − y. Also, sup- pose that Vs has a directional derivative process, D u Vs . Then, from 3, Vs + hu = Vs + hu T ∇ V s + rs, hu , where E rs,hu h 2 → 0. Consider Ws = gVs. Now the question is whether D u Ws exists. We have Ws + hu − Ws = gVs + hu − gVs = Vs + hu − Vs g ′ Vs + R Vs + hu − Vs . Also, lim h→ E Ws + hu − Ws − hu T ∇ V sg ′ Vs h 2 = lim h→ E Rhu T ∇ V s + rs, hu h 2 . 13 We observe that as h → 0, h ∗ = hu T ∇ V s + rs, hu → 0, and so ERh ∗ h ∗ 2 → 0. Also, note that hu T ∇ V s + rs, huh → u T ∇ V s . It follows by the dominated con- vergence theorem that 13 → 0. Hence we can conclude that D u Ws exists and is given by D u Ws = D u gVs = g ′ VsD u Vs . The foregoing result can be useful in the context of land value gradients. In particular, suppose that we model Ys, the log land value at location s. So if we consider D u EYs we are obtaining gradients for the mean log land value sur- face. Instead, we might wish to consider gradients for the mean on a transformed scale e.g., the original scale, that is, D u gEYs , where, say, g· = exp·. Then D u gEYs = g ′ EYsD u EYs . Note that D u EgYs is not accessi- ble here. A new model for gYs is required; different spatial processes would be introduced. More generally, suppose that the model for Ws is not Gaussian, but rather an exponen- tial family model as is customary in generalized linear model specification, with suitable link function g. Then EWs = g −1 Vs T βs , and, using the result, D u EWs can be ob- tained. 7. ANALYSIS OF THE OLCOTT CHICAGO DATA We illustrate the use of our model working with the Olcott’s data for the years 1985 and 1990. For each year, we take p = 4 in the model in 2, introducing distance-based regressors sim- ilar to McMillen 1996. So, in addition to an intercept sur- face β s , we bring in a coefficient surface β 1 s for distance from the CBD, distance from Midway airport β 2 s , and for distance from a secondary employment center SEC, β 3 s , resulting in βs being a four-dimensional spatial process. Em- ployment centers were identified using data obtained from the Northern Illinois Planning Commission NIPC. NIPC reports the level of employment per 12 × 12 square mile a quar- ter section for the Chicago PMSA. Areas within the Chicago area were identified as employment centers if the total employ- ment within a 1-mile radius of a quarter section was 16,000. Within the employment subcenters identified, we arbitrarily se- lected one of these employment subcenters for our analysis. Our models were fitted using a MCMC algorithm, as outlined in Section 5. We assumed flat priors for the β l ’s the global re- gression parameters and a gamma G.0001, .0001 mean 1.0; variance 1.0E+8 prior for the precision 1τ 2 . For the parame- ters in the four-dimensional spatial process, we assumed weakly informative gamma priors for the four correlation decay pa- rameters scaled to have a mean range of about half of the maximum distance, whereas an inverted-Wishart IW4, .01I 4 prior with 4 degrees of freedom, and diagonal means = 100 for AA T induces the prior on A. Convergence was diagnosed by monitoring the mixing of two parallel chains with overdispersed starting values. With Gaussian likelihoods and weakly informa- tive priors, convergence was diagnosed within 2,000 iterations, with a further 1,000 iterations retained from each chain as pos- terior samples. In the interest of simple model comparison, we also fitted a version of 2 with only a spatially varying intercept i.e., a usual spatial random effect, that is, with constant coefficients for the distance-based regressors. We used the posterior pre- dictive model selection criterion of Gelfand and Ghosh 1998. Letting G denote the goodness-of-fit term, P denote the penalty Downloaded by [Universitas Maritim Raja Ali Haji] at 23:23 12 January 2016 Table 1. Posterior Estimates of Model Parameters for the Olcott Data, 1985 and 1990 Percentiles 1985 Percentiles 1990 Parameters 50 2.50, 97.5 50 2.50, 97.50 β 5.191 5.019, 5.363 5.675 5.346, 5.813 β 1 − 1.863 −1.970, −1.771 − 1.947 −2.117, −1.806 β 2 1.510 1.183, 1.675 1.371 1.106, 1.508 β 3 1.232 1.145, 1.338 1.241 1.174, 1.533 φ × 10 3 .532 .219, .810 .539 .273, .869 φ 1 × 10 3 .437 .162, .800 .451 .112, .792 φ 2 × 10 3 .668 .248, .819 .626 .238, .900 φ 3 × 10 3 .439 .157, .823 .466 .183, .846 τ 2 .785 .644, .930 .779 .533, 1.317 T 00 .363 .266, .599 .413 .271, .513 T 11 .498 .460, .632 .814 .606, .991 T 22 .853 .693, 1.151 1.321 .574, 1.896 T 33 .835 .473, 1.547 .973 .763, 1.089 T 01 T 00 ∗ T 11 − .277 −.393, −.107 − .412 −.495, −.163 T 02 T 00 ∗ T 22 − .323 −.360, −.186 − .126 −.173, .151 T 03 T 00 ∗ T 33 − .110 −.244, .029 − .176 −.364, .004 T 12 T 11 ∗ T 22 − .019 −.179, .141 .047 −.242, .121 T 13 T 11 ∗ T 33 − .084 −.291, .258 .025 −.190, .152 T 23 T 22 ∗ T 33 .203 −1.0E–4, .259 − .360 −.529, .005 term, and D denote the criterion value, for 1985, for the sim- pler model G = .202, P = .501, and D = .703, whereas for the spatially varying coefficient model, G = .168, P = .513, and D = .681. For 1990, we obtain G = .189, P = .511, and D = . 700 with G = .157, P = .521, and D = .678. So the full model is preferred, and hence we present posterior inference under this model including the gradient analysis. The posterior inferences for 1985 and 1990 are summarized in Table 1. The intercept for 1990 is slightly greater than that for 1985, indicating increased land value over time. The four global regression parameters, for both years, reveal a negative impact on land value for the distance from the CBD land value decreases for locations distant from the CBD and an oppo- site effect land value increasing with distance for the other two distances. The result related to Midway is not surprising when one considers that the sample is dominated by residential land values. Close proximity to areas of congestions and sig- nificant noise, such as Midway, leads to lower land values. At first, the positive effect on land values at greater distances from SEC seems counterintuitive until one considers the location of this employment subcenter. Its location is to the southeast of the CBD and approximately 1.6 km to the west of the shore of Lake Michigan. The positive parameter could reflect the proximity to the CBD, but as our analysis shows, it more likely is attributed to a lake effect found in this region. Also shown are estimates of the correlation decay para- meters, the measurement error τ 2 , and the spatial variance– covariance parameters as appearing in the matrix T with the covariances converted to correlations. The relatively large con- tributions of T 00 through T 33 toward the variability justifies a rather rich spatial model for the data. Also, the tendency toward negative correlation between the intercept and the regression parameters with some being significantly so is expected. Turning to Figures 1 and 2, we plot the coefficient process surfaces for 1985 and 1990. These are image plots with over- laid contour lines indicating the levels. The rather rich distribu- tion of contours seem to justify our use of the spatially varying coefficients. Figures 3 and 4 plot the posterior means of the mean surfaces for 1985 and 1990. In general, we find from the mean land value price surface that the surface’s maximum value corresponds to the location of the CBD and that land prices fall as one moves further from the lake in both time periods. However, the results do not support the idea of a constant gradient over the entire ur- ban area in either time period. In fact, the gradient’s magnitude not only varies with location, but also depends on the direction from the center of the city for which it is evaluated. The gra- dient is also found to be steepest close to the CBD and then flattens in all directions as distance increases. Both mean surfaces suggest several directions to examine. Note that the putative location of the CBD is at the intersection of the 447902.14038 Easting and 4636874.42216367 Northing near Lake Michigan and is at the conjunction of the four rays in each figure. The four rays denote the four directions in which we travel to understand gradient behavior. These directions are indicated in the figures as NLk Northern Lake vicinity, NW Northwest of the CBD, SW Southwest of the CBD, and SLk Southern Lake vicinity. Tables 2 and 3 examine several points at different distances indicated as .5, 1.5, 3.0, and 6.0 from the CBD in kilometer units. For each ray, two fundamental direc- tional gradients are evaluated: a moving away from the CBD along the ray, and b the direction normal or orthogonal to the ray. Note that moving toward the CBD along a ray would be the negative of that evaluated in a. A closer look at Tables 2 and 3 for the directional gradi- ents evaluated at points along the different rays reveals that a strong gradient exists in all directions from the CBD. This gra- dient is steepest close to the CBD and tends to diminish as dis- tance increases. This flattening seems to be more abrupt for the 1990 data than for 1985. Indeed, if we imagine an arc pass- ing through the .5 km points, then the directional gradients are significantly negative and quite similar in magnitude. But this slope steadily decreases in significance when the arc is extended to pass through the 1.5-, 3.0-, and 6.0-km points. For example, Downloaded by [Universitas Maritim Raja Ali Haji] at 23:23 12 January 2016 a b c d Figure 2. The Coefficient Process Surfaces for 1990. a The intercept process β s; b the CBD distance coefficient process β 1 s; c the Areal distance coefficient process β 2 s; d the Midway distance coefficient process β 3 s. for the 1985 data, significant gradients along the NW and SW rays are seen up to 3 km, whereas for the 1990 data they are seen only until the 1.5-km points. No significant gradients are seen in the orthogonal directions to the NW and SW rays. Traveling north and south of the CBD along the rays NLk and SLk, we come across a more interesting phenomenon. As before, the diminishing impact of the CBD is manifested with gradients along these rays away from the CBD becoming less pronounced. For the northern lake region, they are significant up to the 3-km point barely for 1985 and the .5-km point for 1990, whereas for southern region they are significant up to the 1.5-km point for 1985 and the .5-km point for 1990. For the or- thogonal direction, however, the gradients become significant for the 3- and 6-km points for 1985 and are quicker for 1990, when they are significant immediately after the .5-km point. Apparently, beyond a certain distance from the CBD, proxim- ity to Lake Michigan becomes a prominent determinant for land value gradients. To further illuminate the inferential ability of our approach, two locations anticipated to reveal differential gradient beha- vior—an SEC and a secondary population center SPC—were chosen employment and population centers are important parts of the urban geography. The agglomeration economies associ- ated with employment centers, as well as the clustering of in- Downloaded by [Universitas Maritim Raja Ali Haji] at 23:23 12 January 2016 Figure 3. Posterior Mean Surface and Locations for Directional Gra- dient Analysis in the 1985 Olcott Data. See text for details. dividuals to benefit from neighborhood amenities, should lead to interesting land value gradient patterns. The SEC is lo- cated at 450040.902 Easting, 4626786.843 Northing and was used in part of the foregoing analysis. The SPC is located at 441062.526 Easting, 4633147.015 Northing. Both are labeled in Figures 3 and 4. For each point, we obtained posterior distri- butions of the angle of the maximal gradient relative to the line Figure 4. Posterior Mean Surface and Locations for Directional Gra- dient Analysis in the 1990 Olcott Data. See text for details. from the point to the CBD for both 1985 and 1990, as well as the posterior distribution of the difference between the maximal gradient and the gradient in the direction away from the CBD. These plots are shown in Figures 5 and 6. Because these two points are rather far from the CBD, simple contour analysis or other descriptive methods are inadequate for formal evaluation of the foregoing geometric quantities. How- Table 2. Directional Derivative Gradients at Different Directions Related to 1985 Data Away from CBD Orthogonal to direction away from CBD Points Percentiles 50 2.5, 97.5 Percentiles 50 2.5, 97.50 NW.5 − 2.761 −4.004, −1.512 − .264 −1.203, 1.479 NW1.5 − 2.047 −3.243, −.858 .047 −1.091, 1.152 NW3.0 − 1.492 −2.782, −.285 .151 −1.277, 1.367 NW6.0 − .041 −1.564, 1.482 − .621 −2.038, .583 SW.5 − 2.727 −3.926, −1.104 − .142 −1.343, 1.205 SW1.5 − 2.014 −3.253, −.737 .045 −1.152, .929 SW3.0 − 1.210 −2.382, −.037 − .044 −1.169, 1.001 SW6.0 − .005 −1.459, 1.450 − .101 −1.09, .991 NLk.5 − 2.596 −4.001, −1.262 − .107 −1.129, 1.021 NLk1.5 − 1.481 −2.801, −.202 − .358 −1.948, 1.036 NLk3.0 − .947 −1.959, −.007 − 2.095 −3.036, −1.031 NLk6.0 − .922 −1.883, .306 − 1.434 −2.626, −.549 SLk.5 − 2.653 −3.851, −1.474 .118 −1.026, 1.002 SLk1.5 − 1.880 −2.793, −.843 .149 −.982, 1.223 SLk3.0 − 1.334 −2.492, .032 − 1.062 −2.291, −.021 SLk6.0 − .898 −1.994, .195 − 2.153 −3.278, −1.087 Downloaded by [Universitas Maritim Raja Ali Haji] at 23:23 12 January 2016 Table 3. Directional Derivative Gradients at Different Directions Related to 1990 Data Away from CBD Orthogonal to direction away from CBD Points Percentiles 50 2.5, 97.50 Percentiles 50 2.5, 97.50 NW.5 − 2.457 −4.166, −.776 .255 −1.296, 1.670 NW1.5 − 2.165 −3.818, −.094 .110 −1.386, 1.452 NW3.0 − 1.366 −3.179, .440 .135 −1.387, 1.539 NW6.0 − 1.221 −3.008, .292 .208 −1.284, 1.510 SW.5 − 2.307 −3.851, −.796 − .183 −1.584, 1.166 SW1.5 − 2.078 −3.536, −.381 − .081 −1.592, 1.663 SW3.0 − .492 −1.939, .836 1.033 −.492, 2.311 SW6.0 − 1.217 −2.836, .063 .502 −1.106, 1.996 NLk.5 − 2.212 −3.974, −.587 .324 −1.305, 2.073 NLk1.5 .088 −1.906, 1.603 − 1.333 −3.046, .560 NLk3.0 .034 −1.415, 1.280 − .967 −2.738, .802 NLk6.0 − 1.044 −2.371, .723 − 1.057 −2.502, .302 SLk.5 − 2.493 −4.197, −.868 .036 −1.631, 1.653 SLk1.5 − .170 −1.951, 1.025 − 2.248 −4.212, −.524 SLk3.0 − .054 −1.467, 1.657 − 2.586 −3.902, −.633 SLk6.0 − .150 −1.632, 1.332 − 2.084 −3.665, −.754 ever, using our sampling-based methods, we find that SPC has a maximal gradient direction not significantly different from the direction away from the CBD contains 0. On the other hand, SEC, being in the southeast along the lake has a much more sig- nificant difference between the maximal gradient direction and the direction from the CBD for both years. In support, Tables 2 and 3 show that in both 1985 and 1990, the gradient quickly becomes flat along a ray moving south along the lake away from the CBD, whereas the gradient orthogonal to this direc- tion stays significant. Thus the gradient is larger in a westerly direction 90 ◦ , that is, perpendicular to the lake. However, Fig- ure 6 shows that the direction of maximal gradient is expected to be roughly southwest 45 ◦ from the lake. 8. DISCUSSION AND EXTENSIONS We have developed a general theoretical approach, includ- ing full distribution theory, to examine gradients associated a b c d Figure 5. 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 SPC in the Olcott Data. a and b 1985; c and d 1990. See text for details. Downloaded by [Universitas Maritim Raja Ali Haji] at 23:23 12 January 2016 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.