By: Dr Gary Deng (A) (B) (C)
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1 By: Dr Gary Deng (A) (B) (C)
2 Observed as a cross section in a single point in time, the three spatial landscapes are virtually indistinguishable. And yet they evolved from completely different spatial processes. In spatial data visualization, there is a strong need to take into account the dynamic nature of the data. What we see in a single cross section could be misleading. = + + Where: 1 cross sectional observations of spatial data. heterogeneous explanatory variables. spatial weights matrix that specifies spatial relationships. 1 random residuals. is the spatial autocorrelation coefficient. Thus: captures the extent of spatial autocorrelation. captures spatial heterogeneity. The underlying (strong) assumption is that the spatial process has reached an equilibrium, which can be seen from this reduced form. = Ι +
3 = The current value of is determined by: : persistence of its past values in the form of temporal autocorrelation. Typically 0< <1. : diffusion of neighbouring values in the form of spatial autocorrelation. Typically 0< <1. : exogenous factors in the form of spatial heterogeneity and/or deterministic trends. (A) Spatial Heterogeneity = α+ where =1 diagonal units and 0 otherwise. vs (B) Spatial Trend = + where =1 diagonal units and 0 otherwise. (A) (B)
4 (B) Spatial Trend = + where =1 diagonal units and 0 otherwise. vs (C) Asymmetric Spatial Diffusion = (B) (C) = The current value of is determined by: =0: assuming a priori there is NO spatial heterogeneity/trend. Everyone was created equal. : persistence of its past values in the form of temporal autocorrelation. : captures the general spatial diffusion process. : allows for asymmetries in the spatial diffusion process.
5 = + + =0.5: significant temporal autocorrelation. =0.4: significant general spatial diffusion. Results in significant spatial clusters. But the locations of the clusters are not constant. = =0.5: significant temporal autocorrelation. =0.1: weak general spatial diffusion. =0.3: in certain directions (as specified by ) spatial diffusion is much stronger. The location of the cluster is always the same, as predetermined by the asymmetric spatial structure. Most significantly, without imposing any spatial heterogeneity a priori, we are able to generate spatially heterogeneous spatial landscapes via an asymmetric diffusion process.
6 = =0.5: significant temporal autocorrelation. =0.0: NO general spatial diffusion. =0.5: Strong localized diffusion. Non-stationarity as a combined result of temporal and spatial autocorrelation. Importantly the non-stationarity is localized. = + + =0.7: significant temporal autocorrelation. We all know that Zombies are hard to kill. =0.7: We all know that the zombie virus is highly contagious. One bite and you are toast. Non-stationarity everywhere!
7 (A) Spatial Heterogeneity (B) Spatial Trend (C) Asymmetric Spatial Diffusion Heterogeneity and/or Trend: = + General Diffusion: = + + Standard statistical tests are available
8 General Diffusion: = Asymmetric Diffusion: = + + This is far more challenging In practice, spatial structures are assumed known a priori But how do we know that we have specified spatial structures correctly? In a single cross section, we do not have enough information to distinguish between individual spatial fixed effects (i.e., spatial heterogeneity) and asymmetric spatial structures. But we can borrow information from the time dimension Bayesian Variable Selection in a Spatial VAR Testing misspecification of the spatial weights matrix
9 Spatial Heterogeneity Spatial Trend
10 General Spatial Diffusion Asymmetric Spatial Diffusion
11 Localized Non-stationarity Global Non-stationarity
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