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Hughes Oliver, Heo, McDonald July 2006 A Spatial Editing and Validation Process for Short Count Traffic Data — 26 — i ranges from one to the sample size of n = 3,431 for the test area. Then the empirical variogram is a function of the distance between two stations and is obtained as Σ + = − + + ( ) ( ) 2  ( )  1 2 ˆ ( ) N h i j e e N h γ h , where the sum is over all ( i, j) pairs of stations that are h units apart and  N+ ( h)  is the number of such pairs. For irregularly spaced locations, as is the case for the PTC stations, the variogram estimator is usually smoothed to address the fact that stations typically are not separated by a small number of distinct distances. The smoothed empirical variogram estimator, which is our default estimator, is defined as a function of h∈ T( h( l)) , where the region T( h( l)) is some specified “ tolerance” or buffer region around h( l) , for l = 1,…, k . The equation is = Σ − ( ) ( ) 2  ( )  1 2 ˆ ( ) N h i j e e N h γ h , where N( h) represents the set of ( i, j) pairs falling in T( h( l)) . The empirical variogram estimates the true variogram 2 ( ) var( ) i j γ h = e − e for stations i and j that have distance h units between them, where “ var” represents variance. Having obtained empirical variograms, we then used both unweighted ( ordinary) and weighted nonlinear least squares to fit several theoretical variograms, from which we selected the best fit. All programming was done using the SAS software. The effective range of a theoretical variogram model is defined as the distance at which the variogram increases 95% from its minimum nonzero value to its maximum possible or asymptotic value. This may be interpreted as the distance at which spatial correlation is ignorable or even zero. In other words, the effective range is the distance at which spatially recorded measurements may reasonably be regarded as nearly, if not completely, independent. As such, effective range is an important parameter to be estimated for any theoretical variogram model. Partial sill is the change in variogram between its maximum ( or asymptotic) value and minimum non zero value; a small partial sill indicates weak dependence or possibly independence. The nugget is the minimum non zero value of the variogram, with variance being the sum of nugget and partial sill. Under an assumption of weak stationarity, the covariance structure may be determined from the variogram using 2 ( ) 2 var( ) 2cov( , ) i i j γ h = e − e e , where “ cov” represents covariance. Weak stationarity assumes that any pair of sites separated by distance h will have the same covariance, irrespective of their actual locations. But is this a reasonable assumption? Two stations five miles apart on an interstate might well be more correlated than two stations five miles apart on a non interstate road segment. If nonstationarity is an issue, can it be adequately addressed by choice of distance metric? Tae Young and Jacqueline Hughes Oliver investigated several options of distance metrics. Euclidean Distances Are Not Appropriate. In lieu of the complete set of distances along most likely traveled paths ( which were not yet available), Tae Young Heo obtained sub optimal road distances for subsets of stations and used these distances in modeling spatial structure. This work was conducted during 3rd quarter 2002. We classified our “ road distances” as sub optimal because they were based solely on route lengths, not likelihood of route selection. By taking this simplified approach, we were able to
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Full Text  Hughes Oliver, Heo, McDonald July 2006 A Spatial Editing and Validation Process for Short Count Traffic Data — 26 — i ranges from one to the sample size of n = 3,431 for the test area. Then the empirical variogram is a function of the distance between two stations and is obtained as Σ + = − + + ( ) ( ) 2  ( )  1 2 ˆ ( ) N h i j e e N h γ h , where the sum is over all ( i, j) pairs of stations that are h units apart and  N+ ( h)  is the number of such pairs. For irregularly spaced locations, as is the case for the PTC stations, the variogram estimator is usually smoothed to address the fact that stations typically are not separated by a small number of distinct distances. The smoothed empirical variogram estimator, which is our default estimator, is defined as a function of h∈ T( h( l)) , where the region T( h( l)) is some specified “ tolerance” or buffer region around h( l) , for l = 1,…, k . The equation is = Σ − ( ) ( ) 2  ( )  1 2 ˆ ( ) N h i j e e N h γ h , where N( h) represents the set of ( i, j) pairs falling in T( h( l)) . The empirical variogram estimates the true variogram 2 ( ) var( ) i j γ h = e − e for stations i and j that have distance h units between them, where “ var” represents variance. Having obtained empirical variograms, we then used both unweighted ( ordinary) and weighted nonlinear least squares to fit several theoretical variograms, from which we selected the best fit. All programming was done using the SAS software. The effective range of a theoretical variogram model is defined as the distance at which the variogram increases 95% from its minimum nonzero value to its maximum possible or asymptotic value. This may be interpreted as the distance at which spatial correlation is ignorable or even zero. In other words, the effective range is the distance at which spatially recorded measurements may reasonably be regarded as nearly, if not completely, independent. As such, effective range is an important parameter to be estimated for any theoretical variogram model. Partial sill is the change in variogram between its maximum ( or asymptotic) value and minimum non zero value; a small partial sill indicates weak dependence or possibly independence. The nugget is the minimum non zero value of the variogram, with variance being the sum of nugget and partial sill. Under an assumption of weak stationarity, the covariance structure may be determined from the variogram using 2 ( ) 2 var( ) 2cov( , ) i i j γ h = e − e e , where “ cov” represents covariance. Weak stationarity assumes that any pair of sites separated by distance h will have the same covariance, irrespective of their actual locations. But is this a reasonable assumption? Two stations five miles apart on an interstate might well be more correlated than two stations five miles apart on a non interstate road segment. If nonstationarity is an issue, can it be adequately addressed by choice of distance metric? Tae Young and Jacqueline Hughes Oliver investigated several options of distance metrics. Euclidean Distances Are Not Appropriate. In lieu of the complete set of distances along most likely traveled paths ( which were not yet available), Tae Young Heo obtained sub optimal road distances for subsets of stations and used these distances in modeling spatial structure. This work was conducted during 3rd quarter 2002. We classified our “ road distances” as sub optimal because they were based solely on route lengths, not likelihood of route selection. By taking this simplified approach, we were able to 