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Hughes Oliver, Heo, McDonald July 2006 A Spatial Editing and Validation Process for Short Count Traffic Data — 37 — For purposes of obtaining predictions and prediction intervals, categories 2 and 3 will both be classified as “ new” stations— their predictions and prediction intervals will be determined in the same manner. Category 1 will be referred to as “ old” stations. Predictions and prediction intervals for old stations were obtained and delivered by Tae Young Heo in February 2004 for the network of all 34,944 PTC stations and in March 2004 for the network of stations on primary road segments. Procedures for obtaining predictions and prediction intervals for new stations were provided to TSU staff and Shannon McDonald on November 19, 2004. Old Stations. Universal kriging is the most popular approach for obtaining predictions and prediction uncertainties from spatial models where the mean structure has to be estimated. Under an assumption of normality, universal kriging yields the best linear unbiased predictors under squared error loss. Unfortunately, because universal kriging is a perfect interpolator, using it on the old stations would result in predictions exactly equal to the observed AADT values used to build the model and prediction uncertainties would all equal zero. This would be entirely uninformative, so an alternative approach was necessary. The formulas we used to obtain predictions yield best linear unbiased predictors under squared error loss only if the covariance structure is known without error. Because we had to estimate the covariance structure, there is clearly a disconnect with the assumptions, but this was the best possible approach for meeting project needs. The consequence is that we may underestimate the prediction uncertainty, which might lead to more flags than necessary. This conservative approach was deemed acceptable. Let Y be the vector of observed ( AADT) 0.15 for all old stations and X be the matrix of attributes such that each column represents a variable in the mean model and each row corresponds to a station. The spatial model is Mean model: Y = Xβ + ε , E( Y) = Xβ Covariance model: var( Y) = Σ , Σ is matrix from estimated statewide Gaussian covariance Distributional assumption: ε is distributed as normal with E( ε ) = 0, var( ε ) = Σ . Predictions were obtained as Y X X T X X T Y pred ˆ = ( Σ− 1 )− 1 Σ− 1 with matrix of prediction uncertainties determined by ignoring cov( , ˆ ) pred Y Y to get T T pred var( Y − Y ˆ ) = Σ + X ( X Σ− 1X )− 1 X and ( 1− α ) 100% prediction intervals obtained as ˆ [ var( ˆ )] pred / 2 pred Y ± zα diag Y − Y , where 1.96 / 2 zα = for 95% prediction intervals, 2.58 / 2 zα = for 99% prediction intervals, diag[ A] represents the vector formed by extracting the diagonal elements of matrix A, and diag[ A] is the elementwise square root. Because these prediction intervals were for ( AADT) 0.15, we had to transform them back to the original scale. The final prediction intervals were lower endpoints: { [ ]} 1/ 0.15 / 2 ˆ var( ˆ ) pred pred Y − zα diag Y − Y upper endpoints: { [ ]} 1/ 0.15 / 2 ˆ var( ˆ ) pred pred Y + zα diag Y − Y . Although the formulas given above are analytically attractive, they are computationally inconvenient. We actually converted the spatial model above to become free of a covariance model as follows:
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Title  A spatial editing and validation process for short count traffic data : final report  Page 37 
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Full Text  Hughes Oliver, Heo, McDonald July 2006 A Spatial Editing and Validation Process for Short Count Traffic Data — 37 — For purposes of obtaining predictions and prediction intervals, categories 2 and 3 will both be classified as “ new” stations— their predictions and prediction intervals will be determined in the same manner. Category 1 will be referred to as “ old” stations. Predictions and prediction intervals for old stations were obtained and delivered by Tae Young Heo in February 2004 for the network of all 34,944 PTC stations and in March 2004 for the network of stations on primary road segments. Procedures for obtaining predictions and prediction intervals for new stations were provided to TSU staff and Shannon McDonald on November 19, 2004. Old Stations. Universal kriging is the most popular approach for obtaining predictions and prediction uncertainties from spatial models where the mean structure has to be estimated. Under an assumption of normality, universal kriging yields the best linear unbiased predictors under squared error loss. Unfortunately, because universal kriging is a perfect interpolator, using it on the old stations would result in predictions exactly equal to the observed AADT values used to build the model and prediction uncertainties would all equal zero. This would be entirely uninformative, so an alternative approach was necessary. The formulas we used to obtain predictions yield best linear unbiased predictors under squared error loss only if the covariance structure is known without error. Because we had to estimate the covariance structure, there is clearly a disconnect with the assumptions, but this was the best possible approach for meeting project needs. The consequence is that we may underestimate the prediction uncertainty, which might lead to more flags than necessary. This conservative approach was deemed acceptable. Let Y be the vector of observed ( AADT) 0.15 for all old stations and X be the matrix of attributes such that each column represents a variable in the mean model and each row corresponds to a station. The spatial model is Mean model: Y = Xβ + ε , E( Y) = Xβ Covariance model: var( Y) = Σ , Σ is matrix from estimated statewide Gaussian covariance Distributional assumption: ε is distributed as normal with E( ε ) = 0, var( ε ) = Σ . Predictions were obtained as Y X X T X X T Y pred ˆ = ( Σ− 1 )− 1 Σ− 1 with matrix of prediction uncertainties determined by ignoring cov( , ˆ ) pred Y Y to get T T pred var( Y − Y ˆ ) = Σ + X ( X Σ− 1X )− 1 X and ( 1− α ) 100% prediction intervals obtained as ˆ [ var( ˆ )] pred / 2 pred Y ± zα diag Y − Y , where 1.96 / 2 zα = for 95% prediction intervals, 2.58 / 2 zα = for 99% prediction intervals, diag[ A] represents the vector formed by extracting the diagonal elements of matrix A, and diag[ A] is the elementwise square root. Because these prediction intervals were for ( AADT) 0.15, we had to transform them back to the original scale. The final prediction intervals were lower endpoints: { [ ]} 1/ 0.15 / 2 ˆ var( ˆ ) pred pred Y − zα diag Y − Y upper endpoints: { [ ]} 1/ 0.15 / 2 ˆ var( ˆ ) pred pred Y + zα diag Y − Y . Although the formulas given above are analytically attractive, they are computationally inconvenient. We actually converted the spatial model above to become free of a covariance model as follows: 