A spatial editing and validation process for short count traffic data : final report  Page 27 
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Hughes Oliver, Heo, McDonald July 2006 A Spatial Editing and Validation Process for Short Count Traffic Data — 27 — conduct a preliminary investigation of our prior assumption that Euclidean distances would not be acceptable for building the covariance model. Fitted theoretical variograms obtained using both Euclidean and road distances resulted in estimates as shown in Table 5. Using Euclidean distance, both the partial sill and effective range are estimated very small. As previously discussed, small partial sill and effective range indicate weak dependence or possibly independence. Both the partial sill and effective range are much larger when obtained from road distances. The summary is that Euclidean distances obscure correlation structure because of the manner in which they group pairs of stations. Table 5. Parameter estimates for the exponential variogram fitted by nonlinear ordinary least squares to interstate stations in the test area based on both Euclidean and road distances. Distance Partial Sill Effective Range Nugget Euclidean 0.0701 0.9243 miles 0.2281 Road 0.3821 3.7778 miles 0.249 To understand how this happens, consider Figure 1 where we display locations of all 82 interstate stations of the test area. More specifically, consider station pairs ( 1,2), ( 2,3) and ( 1,3) on the I 440 beltline. Existing evidence supports the belief that because station pairs ( 1,2) and ( 2,3) are about the same driving distance of x meters apart, their values of 2 1 2 ( e − e ) and 2 2 3 ( e − e ) will be similar. As the same time, the driving distance between stations 1 and 3 is about twice that for ( 1,2) and ( 2,3), so evidence suggests that 2 1 3 ( e − e ) will be much larger than both 2 1 2 ( e − e ) and 2 2 3 ( e − e ) . Recall that the computational formula for the empirical variogram requires grouping pairs of stations by their distances then averaging the values of ( ) 2 i j e − e within these groups. Based on road distances, station pairs ( 1,2) and ( 2,3) get grouped together while ( 1,3) is placed in a separate group. As seen in the hypothetical variogram cloud of Figure 2, these road distance groupings lead to a steadily increasing empirical variogram, which is suggestive of correlation. On the other hand, Euclidean distances group all three stations pairs ( 1,2), ( 2,3) and ( 1,3) together, so that the resulting averages lead to a variogram that increases at a slower rate. This, in turn, suggests there is reduced correlation than is indicated by road distances.
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Title  A spatial editing and validation process for short count traffic data : final report  Page 27 
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Full Text  Hughes Oliver, Heo, McDonald July 2006 A Spatial Editing and Validation Process for Short Count Traffic Data — 27 — conduct a preliminary investigation of our prior assumption that Euclidean distances would not be acceptable for building the covariance model. Fitted theoretical variograms obtained using both Euclidean and road distances resulted in estimates as shown in Table 5. Using Euclidean distance, both the partial sill and effective range are estimated very small. As previously discussed, small partial sill and effective range indicate weak dependence or possibly independence. Both the partial sill and effective range are much larger when obtained from road distances. The summary is that Euclidean distances obscure correlation structure because of the manner in which they group pairs of stations. Table 5. Parameter estimates for the exponential variogram fitted by nonlinear ordinary least squares to interstate stations in the test area based on both Euclidean and road distances. Distance Partial Sill Effective Range Nugget Euclidean 0.0701 0.9243 miles 0.2281 Road 0.3821 3.7778 miles 0.249 To understand how this happens, consider Figure 1 where we display locations of all 82 interstate stations of the test area. More specifically, consider station pairs ( 1,2), ( 2,3) and ( 1,3) on the I 440 beltline. Existing evidence supports the belief that because station pairs ( 1,2) and ( 2,3) are about the same driving distance of x meters apart, their values of 2 1 2 ( e − e ) and 2 2 3 ( e − e ) will be similar. As the same time, the driving distance between stations 1 and 3 is about twice that for ( 1,2) and ( 2,3), so evidence suggests that 2 1 3 ( e − e ) will be much larger than both 2 1 2 ( e − e ) and 2 2 3 ( e − e ) . Recall that the computational formula for the empirical variogram requires grouping pairs of stations by their distances then averaging the values of ( ) 2 i j e − e within these groups. Based on road distances, station pairs ( 1,2) and ( 2,3) get grouped together while ( 1,3) is placed in a separate group. As seen in the hypothetical variogram cloud of Figure 2, these road distance groupings lead to a steadily increasing empirical variogram, which is suggestive of correlation. On the other hand, Euclidean distances group all three stations pairs ( 1,2), ( 2,3) and ( 1,3) together, so that the resulting averages lead to a variogram that increases at a slower rate. This, in turn, suggests there is reduced correlation than is indicated by road distances. 