The Gravity Model: Example Calibration

Transcription

1 The Gravity Model: Example Calibration Philip A. Viton October 28, 2014 Gravity Example October 28, / 22

2 Introduction This summarizes the example calibration (not the prediction) from the Gravity Model: Derivation and Calibration handout, minus some of the detailed commentary. You may find it useful when you do gravity-model calculations, since you don t have to page through the longer handout. Note that this doesn t include the Appendix from the longer handout. If you find yourself in trouble when you apply the gravity model you should probably review the Appendix. Gravity Example October 28, / 22

3 Example Data Trip-interchange matrix: T 0 = Interzonal travel-times matrix: t 0 = We shall use a convergence criterion of 5% in all iterative steps (α L = 0.95, α H = 1.05). Gravity Example October 28, / 22

4 Step 0 In our example we shall take δ = 5 minutes. This defines the following superzones: with superzonal trip totals: Superzone Zonal Pairs 1 I 1 = {1, 1}, {2, 2}, {3, 3} 2 I 2 = {1, 2}, {2, 1} 3 I 3 = {1, 3}, {2, 3}, {3, 1}, {3, 2} O S = [450, 590, 490] Gravity Example October 28, / 22

5 Step 0 In our example, we will be calculating 3 distinct travel-time factors. It is a simple matter to parlay these three into a full set F of factors. Formally, we define a mapping (rule) φ that takes our 3 estimates of the travel time factors (F 1, F 2, F 3 ) and produce an estimate of the full F ij matrix. In our case this rule is: F 1 F 2 F 3 φ(f 1, F 2, F 3 ) = F 2 F 1 F 3. F 3 F 3 F 1 Finally we set K 1 = (This amounts to ignoring the K s for now, since they enter the gravity model multiplicatively). Gravity Example October 28, / 22

6 Step 1, Iteration 1 Initial estimate: F 0 i = (1, 1, 1). Apply φ and get F 0 = Gravity Example October 28, / 22

7 Step 1, Iteration 1 Apply the gravity model and find: T 1 = Tij 1 = O0 i A0 j F ij 0 m A 0 mf 0 im = Gravity Example October 28, / 22

8 Step 1, Iteration 1 Convergence check; against the superzone totals Target Actual Error ratio Ei Note that the targets are the superzonal trip totals. We see that we have not converged. Gravity Example October 28, / 22

9 Step 1, Iteration 2 New superzonal factors Fi 1 ratios: are the previous F s corrected by the error Fi 1 = Fi 0 Ei 1 = [1.0, 1.0, 1.0] [ , , ] = [ , , ] Apply φ to derive the full set of travel-time factors: F 1 = Gravity Example October 28, / 22

10 Step 1, Iteration 2 T 2 = Tij 2 = O0 i A0 j F ij 1 m A 0 mf 1 im = Gravity Example October 28, / 22

11 Step 1, Iteration 2 Convergence test: Still no convergence Target Actual Error ratio Ei Gravity Example October 28, / 22

12 Step 1, Iteration 3 New interzonal F i : Fi 2 = Fi 1 Ei 2 = [ , , ] [ , , ] = [ , , ] Apply φ : F 2 = Gravity Example October 28, / 22

13 Step 1, Iteration 3 Apply the gravity model: T 3 = Gravity Example October 28, / 22

14 Step 1, Iteration 3 Convergence check: These have all converged. Target Actual Error ratio Gravity Example October 28, / 22

15 Step 1, Final Result At the end of Step 1 we have our final estimate of the travel-time factors ˆF ˆF ij, which is the F matrix we obtained at the convergent iteration of our step-1 computations, ie ˆF = Gravity Example October 28, / 22

16 Step 2, Setup (I) Continuing our example, based on ˆF from Step 1 and K 1 we have an estimate of the travel-time matrix we re trying to reproduce, namely the final T matrix from the previous step: T 3 = Gravity Example October 28, / 22

17 Step 2, Setup (II) Is this good enough? We need only check the columns (conservation of attractions) for convergence: Target Actual Error ratio Note that we check against the actual A 0 we have no more use for the superzones. Clearly not good enough. So we need to do the balancing step. Gravity Example October 28, / 22

18 Step 2, Iteration 1, Column Factoring The column factors are the previous error ratios, so , , We multiply the elements in each column by its column factor giving T 1a = Example: the (1, 2) element of T 1 (= ) is in column 2, so we use the second column factor, giving = = T 1a 12. Note that this satisfies conservation of attractions, but no longer satisfies conservation of origins. Gravity Example October 28, / 22

19 Step 2, Iteration 1, Row Factoring The row factors are the row-wise error factors of T 1a : Target Actual Error ratio and multiplying each row by its row factor gives T 1b = Example: the (1, 2) element of T 1a, (= ) is in row 1, so we use the first row factor, giving = = T 1b 12. Gravity Example October 28, / 22

20 Step 2, Iteration 1, Convergence Check We need to check the columns only, since the row factoring assures conservation of origins. We have converged. Target Actual Error ratio Gravity Example October 28, / 22

21 Step 3 In order to reproduce the individual elements of T 0 we use the zonal adjustment factors, which we have ignored up to now. The zonal adjustment (K ) factors are now computed as ˆK ij = Tij 0 Tij 1b (ie on the last (convergent) T matrix from step 2) and where means element-by-element division. Then ˆK = = Gravity Example October 28, / 22

22 BPR Method: Summary of Example Results For our calibration of the gravity model using the BPR method we have found: ˆF = (from the final result of Step 1); and ˆK = (from Step 3). Gravity Example October 28, / 22