The Gravity Model: Derivation and Calibration

Transcription

1 The Gravity Model: Derivation and Calibration Philip A. Viton October 28, 2014 Philip A. Viton CRP/CE 5700 () Gravity Model October 28, / 66

2 Introduction We turn now to the Gravity Model of trip distribution. As previously noted, this is most widely used model in current practice. From our perspective, it is based on two important ideas, one new and one old. 1. It is based on a parametrized theoretical model, meaning that the theory depends (unlike the Average Growth Factor model) on constants that need to be estimated. 2. It utilizes the same idea of improvement via iteration as the Average Growth Factor model, though, as we will see, in a slightly different context. Philip A. Viton CRP/CE 5700 () Gravity Model October 28, / 66

3 Derivation of the Gravity Model Physics The gravity model is based on Newton s gravitational theory from physics, interpreted in a transportation context. In physics, the attractive force between two bodies is directly proportional to their masses, and inversely proportional to the square of the distance between them: where: h ij = g m i m j d 2 ij h ij is the attractive force between bodies i and j. m i and m j are the bodies masses. d ij is the distance between them. g is Newton s gravitational (proportionality) constant, approximately in cgs (centimeters/grammes/seconds) units. Philip A. Viton CRP/CE 5700 () Gravity Model October 28, / 66

4 Derivation of the Gravity Model Transportation The derivation of the transportation version of the gravity model is a simple argument from analogy: The Newtonian attractive force term (h ij ) is analogized to directed interzonal trip-making: h ij T ij. The masses are analogized to total trips in and out of zones, so that m i O i and m j A j. The distance term is retained, but in an attempt to be more general (whether there is a serious theoretical basis for this is a question) the power-2 term is permitted to be arbitrary. There is no reason to believe that the Newtonian proportionality term applies here, so we replace it by a different one. The result is the first form of the (transportation) gravity model: T ij = θ O i A j d β ij Philip A. Viton CRP/CE 5700 () Gravity Model October 28, / 66

5 Geographical Justification Tobler s First Law of Geography states that: Everything is related to everything else, but closer things more so (W. Tobler, Cellular Geography in S. Gale and S. Olssohn, eds, Philosophy in Geography, 1979, pp ). The gravity model is consistent with the First Law, but is not implied by it (there are other possible functional forms that are equally consistent with the First Law). Philip A. Viton CRP/CE 5700 () Gravity Model October 28, / 66

6 Travel-Time Factors (I) In the first form of the gravity model, let s write T ij = θ O i A j d β ij 1/d β ij = F ij The F ij are called the travel-time factors. With this notation, the first form of the transportation gravity model becomes T ij = θo i A j F ij Philip A. Viton CRP/CE 5700 () Gravity Model October 28, / 66

7 Travel-Time Factors (II) There are several reasons for writing the model this way. It looks neater. Even if we have data on (say, centroid-to-centroid) distances, it will be hard to estimate the power-law β using linear techniques. More importantly, this can be considered a way to sneak other influences into the model. If travel between zones i and j depends on, say, the number of jobs present in zone j, we might want to write the denominator of the gravity model say telling us that trips depend on distance, jobs and a lot of other factors. Writing all those influences as F ij provides a way of doing this. Note that the F s are to be considered as unknowns. Philip A. Viton CRP/CE 5700 () Gravity Model October 28, / 66

8 Gravity Model with Conservation of Origins (I) We now show that by requiring the gravity model to satisfy conservation of origins, we can eliminate the proportionality constant θ. With the introduction of the travel-time factors, version 1 of the gravity model is: T ij = θo i A j F ij For fixed i, sum both sides over the destinations j: j T ij = θo i A j F ij j On the left, j T ij is just O i. On the right, we can take terms not involving j outside the summation: O i = θo i A j F ij j Philip A. Viton CRP/CE 5700 () Gravity Model October 28, / 66

9 Gravity Model with Conservation of Origins (II) Now divide both sides by O i 1 = θ A j F ij j and solve for θ : θ = 1 j A j F ij Philip A. Viton CRP/CE 5700 () Gravity Model October 28, / 66

10 Gravity Model with Conservation of Origins (III) Now insert this into the gravity model. Since we are interested in T ij (which already uses j) we re-label the summation index (as m) in the expression defining θ. The result is: T ij = 1 m A m F im O i A j F ij Then we have our second form of the transportation gravity model: T ij = O i A j F ij m A m F im Philip A. Viton CRP/CE 5700 () Gravity Model October 28, / 66

11 Gravity Model with Conservation of Origins (IV) There are two important things to note about this form of the model: 1. It is guaranteed to satisfy conservation of origins exactly. Therefore we will never need to check whether this condition is satisfied or not. Of course, the same does not apply to conservation of attractions: there is no reason to suppose that this condition will be satisfied; and in fact it usually will not be. 2. The travel-time factors F ij all depend on an unknown parameter (the parameter β). This means that the interpretation of these F s is not simply as a growth rate, which was something we could calculate given data on the present and some predictions about the future. Philip A. Viton CRP/CE 5700 () Gravity Model October 28, / 66

12 Gravity Model with Zonal Adjustment Factors As an empirical matter, early users of the gravity model observed that it did not appear to yield very accurate predictions. In order to improve the predictive accuracy, they introduced a set of zonal adjustment factors, to be denoted K ij. When these were incorporated, the transportation gravity model took the form T ij = As to the zonal adjustment factors, note that: O i A j F ij K ij m A m F im K im (1) 1. They have no basis whatever in theory, unlike the travel-time factors. They are introduced simply to improve predictions. 2. The matrix K {K ij } is another Z Z = Z 2 parameters which need to be estimated. It might reasonably be asked whether this will be asking too much of the available data. Philip A. Viton CRP/CE 5700 () Gravity Model October 28, / 66

13 Calibration (I) We turn now to the task of calibrating the gravity model of trip distribution specified as: T ij = O i A j F ij K ij m A m F im K im The standard method for doing this is an iterative method developed by the US Bureau of Public Roads (now the Federal Highway Administration), known as the BPR method. As we have seen, the transportation form of the gravity model requires us to estimate two sets of unknown parameters: the travel time factors (F {F ij }) and the zonal adjustment factors (K {K ij }). Note that both of these are Z Z matrices. Philip A. Viton CRP/CE 5700 () Gravity Model October 28, / 66

14 Calibration (II) The underlying idea of the BPR method is to choose these unknowns in order to make the gravity model work as well as possible for our present-day data. That is, unlike the growth factor model calibrations, the BPR calibration method makes no use of future data. What the BPR method does is find the F s and K s that reproduce T 0, the present-day (observed) trip matrix. Once we have calibrated the model (ie found estimates of the travel-time and zonal adjustment factors, which we will denote by ˆF { ˆF ij }) and ˆK { ˆK ij } respectively), we will assume that these are stable (ie unchanged) as between our current period and the future period for which we wish to distribute the trips. This makes prediction simply a matter of plugging in the future (predicted) data and utilizing our estimated ˆF and ˆK. Philip A. Viton CRP/CE 5700 () Gravity Model October 28, / 66

15 BPR Method : Summary (I) The BPR method involves the following steps: 0. (optional) reduce the dimensionality of the problem by imposing similarity assumptions on the different F -factors. We will assume that pairs of zones with similar interzonal travel times involve the same travel-time factor. Thus we will require additional data: the current interzonal travel times. These sets of similar F -factors define what we will call superzones. For now, set the K ij 1 (ie ignore them). 1. Iterative step: estimate the F s based on observed superzonal total trips. That is, we use the gravity model formulation (which will involve only the F s, since we are ignoring the K s) to reproduce as closely as we wish, the observed superzonal total trips. The result of this step (Step 1) is estimates of the travel-time factors, denoted ˆF. Philip A. Viton CRP/CE 5700 () Gravity Model October 28, / 66

16 BPR Method : Summary (II) 2. At the end of Step 1 we have a candidate trip matrix (an estimate of our target T 0 ). However, though it will satisfy conservation of origins, it will usually not satisfy conservation of attractions. So we perform another iterative step, called row-and column factoring, to approximately balance the trip matrix. This is Step At the end of Step 2 we have the F s and an approximately balanced trip matrix. But it still does not reproduce the original T 0. We reproduce T 0 by using the one set of parameters we have so far ignored, namely the zonal adjustment factors (the K matrix). This is Step 3. Philip A. Viton CRP/CE 5700 () Gravity Model October 28, / 66

17 BPR Method : Example In my view, the simplest way to understand the fairly complicated details of the BPR calibration method is to develop the formal expressions for the calibration steps alongside an actual example. So that is what we will do here. Note : I will post to the course website an abbreviated version of the example, containing just the computations. You may find it helpful to refer to this as you review the details. Philip A. Viton CRP/CE 5700 () Gravity Model October 28, / 66

18 Example Data Present-day (observed) trip-interchange matrix: T 0 = Present-day(observed) 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). Philip A. Viton CRP/CE 5700 () Gravity Model October 28, / 66

19 Step 0 (I) We will assume that pairs of zones with similar interzonal travel times involve the same travel-time factor. This will define a set of superzones. We operationalize this by assuming that all pairs of zones i and j satisfying 0 t 0 ij < δ for a given interzonal travel-time difference δ, utilize the same travel-time factor, F 1. And we take all zonal pairs satisfying δ t 0 ij < 2δ to utilize a second travel time factor, F 2. And then 2δ t 0 ij < 3δ are assumed to involve a third travel-time factor F 3. And so on. Philip A. Viton CRP/CE 5700 () Gravity Model October 28, / 66

20 Step 0 (II) In our example we shall take δ = 5 minutes. This defines the following 3 superzones: with superzonal trip totals: Superzone Zonal Pairs 1 {1, 1}, {2, 2}, {3, 3} 2 {1, 2}, {2, 1} 3 {1, 3}, {2, 3}, {3, 1}, {3, 2} O S = [450, 590, 490] Philip A. Viton CRP/CE 5700 () Gravity Model October 28, / 66

21 Step 0 (III) In our example, we will be calculating 3 distinct travel-time factors, one for each superzone. 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: Finally we set φ(f 1, F 2, F 3 ) = K 1 = F 1 F 2 F 3 F 2 F 1 F 3 F 3 F 3 F (This amounts to ignoring the K s for now). Philip A. Viton CRP/CE 5700 () Gravity Model October 28, / 66.

22 Step 1 (I) In Step 1 we will estimate the distinct superzonal travel-time factors F i, as set up in step 0. In this step we work exclusively with superzonal (aggregated) travel. But in that context, the method is very much like the way we calibrated the growth-factor models: we generate an estimate of T, and we compute error factors by comparing actual superzonal totals (ie those implied by T 0 ) with those implied by the current T. If we have not converged we update the F s (not T as in the average growth factor model) and try again. Philip A. Viton CRP/CE 5700 () Gravity Model October 28, / 66

23 Step 1 (II) An iteration in Step 1 consists of the following tasks: 1. Use the previously computed (or assumed) F s and the gravity model to generate a new T matrix. 2. Compare the total superzonal travel implied by this T matrix with the superzonal totals implied by T 0, via error ratios E i. 3. If convergence is not satisfied, update the F s and try again. 4. The updating rule generates a new set of superzonal F s by multiplying the old F s by the error ratios. In general F k i = F k 1 i E k i (where i indexes the superzones). Philip A. Viton CRP/CE 5700 () Gravity Model October 28, / 66

24 Step 1, Iteration 1 (I) Initial estimate: F 0 i = (1, 1, 1). Apply φ and get F 0 = Philip A. Viton CRP/CE 5700 () Gravity Model October 28, / 66

25 Step 1, Iteration 1 (II) Apply the gravity model and find: T 1 = Tij 1 = O0 i A0 j F ij 0 m A 0 mf 0 im = (Of course this is just T 1 ij = O 0 i A0 j m A 0 m = O 0 i A0 j S(T 0 ). Philip A. Viton CRP/CE 5700 () Gravity Model October 28, / 66

26 Step 1, Iteration 1 (III) Convergence check against the superzone totals (NB: not the origins or attractions): Target Actual Error ratio Ei Note that the targets are the actual superzonal trip totals. We see that we have not converged. Philip A. Viton CRP/CE 5700 () Gravity Model October 28, / 66

27 Step 1, Iteration 2 (I) The new superzonal factors Fi 1 error ratios: are the previous F s corrected by the Fi 1 = Fi 0 Ei 1 = [1.0, 1.0, 1.0] [ , , ] = [ , , ] Apply φ to obtain the full set of new travel-time factors: F 1 = Philip A. Viton CRP/CE 5700 () Gravity Model October 28, / 66

28 Step 1, Iteration 2 (II) Apply the gravity model: T 2 = Tij 2 = O0 i A0 j F ij 1 m A 0 mf 1 im = The Appendix, beginning at slide 57, contains the detailed computations behind this result. Philip A. Viton CRP/CE 5700 () Gravity Model October 28, / 66

29 Step 1, Iteration 2 (III) Convergence test: Still no convergence. Target Actual Error ratio Ei Philip A. Viton CRP/CE 5700 () Gravity Model October 28, / 66

30 Step 1, Iteration 3 (I) New interzonal F i : Fi 2 = Fi 1 Ei 2 = [ , , ] [ , , ] = [ , , ] Apply φ : F 2 = Philip A. Viton CRP/CE 5700 () Gravity Model October 28, / 66

31 Step 1, Iteration 3 (II) Apply the gravity model: T 3 = Philip A. Viton CRP/CE 5700 () Gravity Model October 28, / 66

32 Step 1, Iteration 3 (III) Convergence check: These have all converged. Target Actual Error ratio Philip A. Viton CRP/CE 5700 () Gravity Model October 28, / 66

33 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. For our example: ˆF = Philip A. Viton CRP/CE 5700 () Gravity Model October 28, / 66

34 Step 2 (I) At the end of step 1 we have estimated the F s. We also have, associated with those F s (and under the assumption that the K s are all 1) an estimate of the trip matrix. As compared with T 0 (the trip matrix we are trying to reproduce) the current estimate T satisfies conservation of origins (automatically) but: probably does not satisfy conservation of attractions (even approximately) probably does not reproduce the individual elements of T 0 Step 2 is an attempt to remedy the first problem (conservation of attractions). Note that it is possible that at the end of Step 1 we do satisfy conservation of attractions (approximately). In that case we omit Step 2. Philip A. Viton CRP/CE 5700 () Gravity Model October 28, / 66

35 Step 2 (II) If we need to do step 2 (ie if we don t approximately satisfy conservation of origins at the end of Step 1) then an iteration of Step 2 consists of: 1. A column factoring : this scales each column of the trip matrix to ensure conservation of attractions. But in doing this, we destroy conservation of origins; so 2. We perform a row factoring. This scales the rows of the result in part (1) to restore conservation of origins. But it will destroy conservation of attractions. At the end of a Step-2 iteration we check conservation of attractions. If it is not approximately satisfied, then we perform another iteration; and we continue until conservation of attractions is approximately satisfied. Philip A. Viton CRP/CE 5700 () Gravity Model October 28, / 66

36 Step 2, Setup (I) Continuing our example, based on ˆF fron 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 = Philip A. Viton CRP/CE 5700 () Gravity Model October 28, / 66

37 Step 2, Setup (II) Is this good enough? We need to check only the columns (conservation of attractions) for convergence: Target Actual Error ratio Note that we check against the actual attractions A 0 we have no more use for the superzones. Clearly not good enough. So we need to do Step 2. Philip A. Viton CRP/CE 5700 () Gravity Model October 28, / 66

38 Step 2, Setup (III) To simplify the notation, let s rename our starting point (T 3 ) to be T 1. At each iteration of step 2 (given that we need to do it at all, as we have to in this example) we must do: 1. a column factoring 2. a row factoring 3. a convergence check on the columns. Philip A. Viton CRP/CE 5700 () Gravity Model October 28, / 66

39 Step 2, Iteration 1, Column Factoring The column factors are the previous error ratios, so we have: , , 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. Philip A. Viton CRP/CE 5700 () Gravity Model October 28, / 66

40 Step 2, Iteration 1, Row Factoring The row factors are the row-wise (originations) error ratios 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. Philip A. Viton CRP/CE 5700 () Gravity Model October 28, / 66

41 Step 2, Iteration 1, Convergence Check We need to check only the columns (ie the attractions), since the row factoring assures conservation of origins. We have converged. Target Actual Error ratio Philip A. Viton CRP/CE 5700 () Gravity Model October 28, / 66

42 Step 3 (I) We now have an approximately balanced T matrix (ie T 1b ) derived on the hypothesis that K 1, namely T 1b = This satisfies conservation of origins, satisfies conservation of attractions to within our convergence criterion, but still does not reproduce the individual elements of T 0. Philip A. Viton CRP/CE 5700 () Gravity Model October 28, / 66

43 Step 3 (II) 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 = = Philip A. Viton CRP/CE 5700 () Gravity Model October 28, / 66

44 BPR Method: Example Results Summary 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). Philip A. Viton CRP/CE 5700 () Gravity Model October 28, / 66

45 Discussion : Step 2 (I) Remember that the motivation for the BPR method is to construct the F and K factors to allow us to exactly reproduce our observed trip matrix T 0. However, a quick calculation shows that we have not succeeded in this: we find, plugging in our calibrated results: T ij = O0 i A0 j ˆF ij ˆK ij m A 0 ˆF = m im ˆK im which does not reproduce our original observed trip matrix. Philip A. Viton CRP/CE 5700 () Gravity Model October 28, / 66

46 Discussion : Step 2 (II) If you think about this for a minute you ll realize that the problem seems to lie in Step 2, the row-and-column factoring. For one thing, this step has nothing to do with the gravity model at all: it s just an ad-hoc way of arriving at a balanced interim trip matrix. For another, you might reasonably say, why do it at all? If we can exactly reproduce T 0 by first doing Step 1 and then adjusting the results via the zonal adjustment factors in Step 3, won t we automatically have a balanced trip matrix? It seems to me that the answer to this is Yes; suggesting that Step 2 is dispensable, despite the BPR. Philip A. Viton CRP/CE 5700 () Gravity Model October 28, / 66

47 Calibration Without Step 2 (I) At the end of Step 1 we had an interim estimate of T 0 as: T 3 = implying that we could take: ˆK = = Philip A. Viton CRP/CE 5700 () Gravity Model October 28, / 66

48 Calibration Without Step 2 (II) And a simple calculation shows that if we plug these new estimates ˆF = ˆK = into the gravity model, we do indeed exactly reproduce our original trip matrix T 0. Philip A. Viton CRP/CE 5700 () Gravity Model October 28, / 66

49 Prediction (I) In order to predict a future trip matrix remember that this is the object of the whole exercise we need: predictions of the future origins by zone, O i predictions of the future attractions by zone, A j possibly, predictions of the future interzonal travel times, if we believe these have changed Given these, we assume that the ˆF and ˆK matrices are stable over time, and derive our predictions by plugging everything into the gravity model. That is,we take T ij = O i A j ˆF ij ˆK ij m A m ˆF im ˆK im Philip A. Viton CRP/CE 5700 () Gravity Model October 28, / 66

50 Prediction (II) Note that there is no guarantee that this T will satisfy conservation of attractions One possibility is to perform an artificial balancing step on the results even though this is not mandated by the gravity model. We could do this is via row-and-column factoring (Step 2) on T. (Remember that conservation of origins is guaranteed). So the suggestion is: first compute T according to the gravity model formula, and then if necessary go through Step 2 (row-and-column factoring) to arrive at an approximately balanced T. If we do this, we report T as our final prediction. Philip A. Viton CRP/CE 5700 () Gravity Model October 28, / 66

51 Prediction Example (I) We work with the conventionally-calibrated gravity model, ie the one in which we do Step 2, whose results are shown in slide 44. Suppose that we predict: O = (650, 590, 350) A = (350, 700, 540) So that total trip-making rises from 1530 to We assume that the travel-time matrix is unchanged. There is an interesting question of what to do when an interzonal travel time changes and falls into a superzonal category not covered by the calibration (eg, if in our example the 15-minute time became 16 minutes). About all we can reasonably do is to include this in the original superzone, even though it really doesn t belong there. Philip A. Viton CRP/CE 5700 () Gravity Model October 28, / 66

52 Prediction Example (II) To start, plug in O and A into the gravity model, using the calibrated ˆF and ˆK matrices. The result is: T = Philip A. Viton CRP/CE 5700 () Gravity Model October 28, / 66

53 Prediction Example (III) Is this good enough? We need to check (only) conservation of attractions, based on our predictions (A ): Target Actual Error ratio all of which are outside our 5% convergence criteria So if we are concerned about balance/consistency, it may be a good idea to do some row-and-column factoring. Let s do this. Philip A. Viton CRP/CE 5700 () Gravity Model October 28, / 66

54 Prediction Example (IV) The column factors are: ( , , ). The column-factored matrix is: The row factors are: ( , , ). And the row-factored matrix is: Philip A. Viton CRP/CE 5700 () Gravity Model October 28, / 66

55 Prediction Example (V) Good enough? We check: Target Actual Error ratio all of which are OK, so we take as our approximately-balanced predicted trip interchange matrix: T = Philip A. Viton CRP/CE 5700 () Gravity Model October 28, / 66

56 Appendix Appendix Philip A. Viton CRP/CE 5700 () Gravity Model October 28, / 66

57 Applying the Gravity Model (I) This appendix does the detailed calculations in which we apply the gravity model to the second iteration of Step 1. We have (from slide 27): F 1 = O 0 i = 550, 600, 380 A 0 j = 400, 620, We will compute the T 2 matrix, as shown in slide 28, by: ij = O0 i A0 j F ij 1 m A 0 mf 1 T 2 im Philip A. Viton CRP/CE 5700 () Gravity Model October 28, / 66

58 Applying the Gravity Model (II) Before we get started, one preliminary note. The denominator of the expression for T ij is of the form Z A m F im = A m F im m m=1 The important thing to note is that this does not involve j. In other words, we will use the same denominator for all elements of T in the same row: T 11, T 12 and T 13 (row 1) all involve the same denominator; as do T 21, T 22 and T 23 (row 2). For a Z Z problem, you have to compute only Z distinct denominators. Philip A. Viton CRP/CE 5700 () Gravity Model October 28, / 66

59 Applying the Gravity Model (III) For our sample problem, we have a Z = 3 zone region, so all summations run from m = 1 to m = 3. We begin by computing the three denominators, since they will be used repeatedly. i = 1 : 3 A 0 mf 1 3 im = A 0 mf1m 1 m=1 m=1 = A 0 1F A 0 2F A 0 3F 0 13 = ( ) + ( ) + ( ) = Philip A. Viton CRP/CE 5700 () Gravity Model October 28, / 66

60 Applying the Gravity Model (IV) Denominators, continued: i = 2 : 3 A 0 mf 1 3 im = A 0 mf2m 1 m=1 m=1 = A 0 1F A 0 2F A 0 3F 0 23 = ( ) + ( ) + ( ) = i = 3 : 3 A 0 mf 1 3 im = A 0 mf3m 1 m=1 m=1 = A 0 1F A 0 2F A 0 3F 0 33 = ( ) + ( ) + ( ) = Philip A. Viton CRP/CE 5700 () Gravity Model October 28, / 66

61 Applying the Gravity Model Now we begin computing the individual terms. Row 1: i = 1; j = 1 : T11 2 = O0 1 A0 1 F 11 1 m A 0 mf1m 1 = = = i = 1; j = 2 : T12 2 = O0 1 A0 2 F 12 1 m A 0 mf1m 1 = = = Philip A. Viton CRP/CE 5700 () Gravity Model October 28, / 66

62 Applying the Gravity Model (V) Final entry for Row 1: i = 1; j = 3 : T13 2 = O0 1 A0 3 F 13 1 m A 0 mf1m 1 = = = Note that the denominators for all the entries in row 1 (as they will be in each row) are the same. Philip A. Viton CRP/CE 5700 () Gravity Model October 28, / 66

63 Applying the Gravity Model (VI) Row 2: i = 2; j = 1 : T21 2 = O0 2 A0 1 F 21 1 m A 0 mf2m = = i = 2; j = 2 : T22 2 = O0 2 A0 2 F 22 1 m A 0 mf2m = = Philip A. Viton CRP/CE 5700 () Gravity Model October 28, / 66

64 Applying the Gravity Model (VII) Final entry for Row 2: Row 3: i = 2; j = 3 : T23 2 = O0 2 A0 3 F 23 1 m A 0 mf2m = = i = 3; j = 1 : T31 2 = O0 3 A0 1 F 31 1 m A 0 mf3m = = Philip A. Viton CRP/CE 5700 () Gravity Model October 28, / 66

65 Applying the Gravity Model (VIII) Row 3, concluded: i = 3; j = 2 : T32 2 = O0 3 A0 2 F 32 1 m A 0 mf3m = = i = 3; j = 3 : T33 2 = O0 3 A0 3 F 33 1 m A 0 mf3m = = Philip A. Viton CRP/CE 5700 () Gravity Model October 28, / 66

66 Applying the Gravity Model (IX) Putting all these together we get: T 2 = T 2 ij = which matches the result in slide 28. Philip A. Viton CRP/CE 5700 () Gravity Model October 28, / 66