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How Temperatures . . . A Model (cont-d) Model: Final Formulas How Time Variability of Testing the Model Current Map with 8 . . . Testing the Model . . . Temperature Depends on a Testing the Model: . . . Testing the Model: . . . Spatial Location: Home Page A Model and Preliminary Title Page Results of Its Testing ◭◭ ◮◮ ◭ ◮ Christian Servin and Vladik Kreinovich Page 1 of 9 Cyber-ShARE Center Go Back University of Texas at El Paso El Paso, TX 79968, USA Full Screen christians@miners.utep.edu Close vladik@utep.edu Quit

How Temperatures . . . 1. How Temperatures Change from One Spatial A Model (cont-d) Location to Another: A Model Model: Final Formulas Testing the Model • Each environmental characteristic q changes from one Current Map with 8 . . . spatial location to another. Testing the Model . . . • A large part of this change is unpredictable (i.e., ran- Testing the Model: . . . dom). Testing the Model: . . . • A reasonable value to describe the random component Home Page of the difference q ( x ) − q ( x ′ ) is the variance Title Page def V ( x, x ′ ) = E [(( q ( x ) − E [ q ( x )]) − ( q ( x ′ ) − E [ q ( x ′ )])) 2 ] . ◭◭ ◮◮ ◭ ◮ • Comment: we assume that averages are equal. Page 2 of 9 • Locally, processes should not change much with shift x → x + s : V ( x + s, x ′ + s ) = V ( x, x ′ ). Go Back • For s = − x ′ , we get V ( x, x ′ ) = C ( x − x ′ ) for Full Screen def Close C ( x ) = V ( x, 0) . Quit

How Temperatures . . . 2. A Model (cont-d) A Model (cont-d) Model: Final Formulas • In general, the further away the points x and x ′ , the Testing the Model larger the difference C ( x − x ′ ). Current Map with 8 . . . • In the isotropic case, C ( x − x ′ ) depends only on the Testing the Model . . . distance D = | x − x ′ | 2 = ( x 1 − x ′ 1 ) 2 + ( x 2 − x ′ 2 ) 2 . Testing the Model: . . . • It is reasonable to consider a scale-invariant depen- Testing the Model: . . . dence C ( x ) = A · D α . Home Page Title Page • In practice, we may have more changes in one direction and less change in another direction. ◭◭ ◮◮ • E.g., 1 km in x is approximately the same change as 2 ◭ ◮ km in y . Page 3 of 9 • The change can also be mostly in some other direction, Go Back not just x - and y -directions. Full Screen • Thus, in general, in appropriate coordinates ( u, v ), we have C = A · D α for D = ( u − u ′ ) 2 + ( v − v ′ ) 2 . Close Quit

How Temperatures . . . 3. Model: Final Formulas A Model (cont-d) Model: Final Formulas • In general, C = A · D α , for D = ( u − u ′ ) 2 + ( v − v ′ ) 2 in Testing the Model appropriate coordinates ( u, v ). Current Map with 8 . . . • In the original coordinates x 1 and x 2 , we get: Testing the Model . . . Testing the Model: . . . C ( x − x ′ ) = A · D α , where Testing the Model: . . . 2 2 Home Page � � g ij · ( x i − x ′ i ) · ( x j − x ′ D = j ) = Title Page i =1 j =1 g 11 · ( x 1 − x ′ 1 ) 2 +2 g 12 · ( x 1 − x ′ 1 ) · ( x 2 − x ′ 2 )+ g 22 · ( x 2 − x ′ 2 ) 2 . ◭◭ ◮◮ ◭ ◮ • From the computational viewpoint, we can include A into g ij if we replace g ij with A 1 /α · g ij , then Page 4 of 9 C ( x − x ′ ) = Go Back Full Screen 1 ) 2 + 2 g 12 · ( x 1 − x ′ 2 ) 2 � α g 11 · ( x 1 − x ′ 1 ) · ( x 2 − x ′ 2 ) + g 22 · ( x 2 − x ′ � Close Quit

How Temperatures . . . 4. Testing the Model A Model (cont-d) Model: Final Formulas • The above model describes between-station variance Testing the Model T C ( x − x ′ ) = 1 Current Map with 8 . . . � (( q ( x, t ) − q ( x )) − ( q ( x ′ , t ) − q ( x ′ ))) 2 . T · Testing the Model . . . t =1 Testing the Model: . . . • According to the model, Testing the Model: . . . Home Page C ( x − x ′ ) ≈ Title Page 1 ) 2 + 2 g 12 · ( x 1 − x ′ 2 ) 2 � α g 11 · ( x 1 − x ′ 1 ) · ( x 2 − x ′ 2 ) + g 22 · ( x 2 − x ′ � ◭◭ ◮◮ • For several stations close to El Paso: ◭ ◮ – we estimated C ( x − x ′ ) and then Page 5 of 9 – we used Least Squares to find the best fit values α Go Back and g ij . Full Screen • The dependence on α is non-linear, so we tried all val- Close ues α = 0 . 25, 0.3, 0.35, . . . , 1.25. Quit

How Temperatures . . . 5. Current Map with 8 stations A Model (cont-d) Model: Final Formulas Testing the Model Current Map with 8 . . . Testing the Model . . . Testing the Model: . . . Testing the Model: . . . Home Page Title Page ◭◭ ◮◮ ◭ ◮ Page 6 of 9 Go Back Full Screen Close Quit

How Temperatures . . . 6. Testing the Model (cont-d) A Model (cont-d) Model: Final Formulas • For each α , we applied Least Squares to Testing the Model C 1 /α ( x − x ′ ) ≈ Current Map with 8 . . . Testing the Model . . . 1 ) 2 + 2 g 12 · ( x 1 − x ′ g 11 · ( x 1 − x ′ 1 ) · ( x 2 − x ′ 2 ) + g 22 · ( x 2 − x ′ 2 ) 2 . Testing the Model: . . . • Then, we chose α for which the resulting mean square Testing the Model: . . . error is the smallest. Home Page • To check whether the model works, we compared: Title Page – the residual mean squared error with ◭◭ ◮◮ – the original mean squared value of C ( x − x ′ ) (which ◭ ◮ correspond to g ij ≡ 0). Page 7 of 9 • When we considered all 8 stations, the error reduced Go Back from 5.2 to only 3.9 ( ≈ 25%). Full Screen • When we separated the stations into E and to the W Close of the mountains, we got a better decrease in error. Quit

How Temperatures . . . 7. Testing the Model: Results A Model (cont-d) Model: Final Formulas E W Both Testing the Model Initial 2.4 6.2 5.2 Current Map with 8 . . . Error Testing the Model . . . Residual 0.23 4.3 3.9 Testing the Model: . . . Error Testing the Model: . . . Decrease 90% 34% 25% Home Page 0.80 1.00 0.35 α 2.5 0.6 12.4 g 11 Title Page g 12 1.7 1.7 31.0 ◭◭ ◮◮ 1.5 9.4 88.0 g 22 ◭ ◮ Page 8 of 9 Go Back Full Screen Close Quit

8. Testing the Model: Analysis of the Results How Temperatures . . . A Model (cont-d) • When we considered all eight stations, the model did Model: Final Formulas Testing the Model not work: the error reduced from 5.2 to only 3.9. Current Map with 8 . . . Testing the Model . . . • When we separated the stations into E and to the W Testing the Model: . . . of the mountains, we got a good decrease in error. Testing the Model: . . . • The resulting values g ij show that the dependence on Home Page spatial locations is different in two areas: Title Page – in E, we have g 11 ≥ g 12 , g 22 , so the main change is ◭◭ ◮◮ in E-W direction; ◭ ◮ – in W, we have g 22 ≫ g 11 , g 12 , so the main change is Page 9 of 9 in the S-N direction. Go Back Full Screen Close Quit