Gravity Models. Elena Nikolaeva, Egon Elbre, Roland Pihlakas. MTAT Graph Mining

Transcription

1 Gravity Models Elena Nikolaeva, Egon Elbre, Roland Pihlakas

2 G=(V,C) network graph Introduction Origin Destination Flow as traffic FLOW

3 Traffic flow

4 Characteristics Of The Network Flows

5 Characteristics Of The Network Flows How does the traffic move throughout the network?

6 Characteristics Of The Network Flows How does the traffic move throughout the network? Routing Matrix

7 Routing Matrix Captures the manner in which traffic moves throughout the network. B - binary values - fraction of flow in case multiple routes are possible A 1 B 2 C 3 D B e;ij = AC AD BD BC CD Figures are from Lauri Eskor s slides

8 Characteristics Of The Network Flows How much traffic flows from point A to point B?

9 Characteristics Of The Network Flows How much traffic flows from point A to point B? Traffic Matrix

10 Constructing Traffic Matrix A B A B C D E Ti A C 30 B C D D E E Tj Copyright , Dr. Jean-Paul Rodrigue, Dept. of Global Studies & Geography, Hofstra University.

11 Origin-Destination Matrix (Traffic Matrix) Where Z ij is the total volume of traffic flowing from origin vertex i to a destination vertex j in a given period of time. Net out-flow corresponding to vertices i Net in-flow corresponding to vertices j

12 Link Totals,where X e the total flow over a given link e E,where Z-traffic matrix written as a vector A 1 B 2 Xe = C i, j X 1 AC AD BD BC CD X 2 = X D Be;ij Zij X X B Z AC Z AD Z BD Z BC Z CD Z

13 Characteristics Of The Network Flows How much will it cost us?

14 Characteristics Of The Network Flows How much will it cost us? Cost

15 The Four Ts in International Trade Transaction costs Tariff and non-tariff costs Transport costs Time costs Copyright , Dr. Jean-Paul Rodrigue, Dept. of Global Studies & Geography, Hofstra University.

16 Total Logistics Costs Tradeoff Total Logistics Costs Costs Warehousing Costs Transport Costs Shipment Size or Number of Warehouses Copyright , Dr. Jean-Paul Rodrigue, Dept. of Global Studies & Geography, Hofstra University.

17 Additional Measurements Of Traffic Volume C cost associated with paths or links. i.e. generalized cost (in transport economics)-is the sum of the monetary and non-monetary costs of a journey Costs associated with QoS - quality of service (in computer and telecommunication networks ) is the ability to provide different priority to different applications, users, or data flows, or to guarantee a certain level of performance to a data flow

18 Characteristics Of The Network Flows How will traffic change over time?

19 Characteristics Of The Network Flows How will traffic change over time? Time

20 Time-Varying Perspective Flows have dynamic nature Z (t) - time dependent traffic matrix B - fixed (changes in routing occur in longer time than those associated with the scale )

21 Characteristics Summary Origin FLOW Destination B Routing Matrix Z - Traffic Matrix C - Cost T - Time

22 Flow analysis classification Measurements Goal Method OD flow volumes Z ij Link volumes X e OD costs c ij Model observed flow volumes Z ij Predict unobserved OD flow volumes Z ij Predict unobserved OD and link costs Gravity Models Traffic matrix estimation(static, dynamic) Estimation of network flow costs

23 Gravity Models Metaphor of physical gravity Tij = G Mi Mj Dij 2 M i, M j -population size (mass) D ij - measure of separation (distance, cost) Applications: Social science (interaction between people of different populations), geography, economics, analysis of computer network traffic etc.

24 Application of an Elementary Spatial Interaction Equation 2,000,000 X 800 km 2,000,000 Y Elementary Formulation T ij = k P i P j D ij W X Y Z Ti 400 km W 100, ,000 W Z X 100,000 50,000 25, ,000 2,000,000 k = (people per week) 1,000,000 Y 50,000 50,000 Z 25,000 25,000 Centroid (i) Weight (P) Distance (D) Constant (k) Interaction (T) Tj 100, ,000 50,000 25, ,000 Copyright , Dr. Jean-Paul Rodrigue, Dept. of Global Studies & Geography, Hofstra University.

25 Relationship between Distance and Interactions T(B-A) A B T(C-A) A C Interaction T(D-A) A D A B C D Copyright , Dr. Jean-Paul Rodrigue, Dept. of Global Studies & Geography, Hofstra University. Distance

26 General Gravity Model Specifies that the traffic flows Z ij to be in the form of counts, with independent Poisson distributions and the mean function of the form of: T ij = k P i P j D ij 2,000,000 P i D ij =800 km 2,000,000 P j,where T ij =E(Z ij )- expected value of interaction h O (i)=p i - origin function h D (j)=p j - destination function h S (c ij )=D ij - separation function c ij - vector of K separation attributes

27 Extension of the Gravity Model. 2,000,000 X λ = 0.95 α = km 2,000,000 Y Simple Formulation T ij = k P α β i P j θ D ij W 2,000, km λ = 1.03 α = 0.96 k = (people per week) Z 1,000,000 λ = 1.00 α = 0.90 λ = 1.2 α = 0.7 W X Y Z Ti W 71,378 71,378 X 6,059 2, ,298 Y 19,420 19,420 Z 153, ,893 Centroid (i) Weight (P) Distance (D) Constant (k) Interaction (T) Exponent Tj 6, ,692 2, ,990 Copyright , Dr. Jean-Paul Rodrigue, Dept. of Global Studies & Geography, Hofstra University.

28 Extension of the Gravity Model. Power functions,where origin function h O (i) = (Pi) α = (π O,i) α destination function h D ( j) = (Pj) β = (π D j) β flow ~1/x a separation function h S (cij) = (Dij) θ = (cij) θ C ij -scalar, 0 ~exp() OR cost

29 Gravity Models. Example Austrian Call Data Need to understand the spatial structure of telecommunication interactions among populations between different geographical regions

30 Gravity Models. Example Austrian Call Data Need to understand the spatial structure of telecommunication interactions among populations between different geographical regions WHY?

31 Gravity Models. Example Austrian Call Data Need to understand the spatial structure of telecommunication interactions among populations between different geographical regions Regulation of the telecommunication sector Anticipating the influence of telecommunication technologies on regional development

32 Number of districts - 32 Time -1 year Measurements - intensity z ij, i j=1,,32 Austrian Call Data π O, i is the GRP of origin i, π D, j is the GRP of destination j, c i j is the distance from origin i to destination j

33 Austrian Call Data Scatter plots: Call flow volume versus each of Origin GRP Destination GRP Distance nonparametric smoother linear regression

34 Alternative Representation. Interaction Probabilities represent the expected relative frequency at which interactions are specifically ij-interactions,where Under the general gravity model specification they can be expressed as:

35 Alternative Representation. Destination Gravity Models related to the counts of Z ij from given origin i to all destinations j Conditional destination probabilities: P(A B) = P(A B) P(B) In terms of components in general probabilities:

36 Inference For The Gravity Models Z ij independent Poisson random variables with means: General model specification:,where

37 Poisson Distribution is a discrete probability distribution that expresses the probability of a number of events occurring in a fixed period of time if these events occur with a known average rate and independently of the time since the last event. (The Poisson distribution can also be used for the number of events in other specified intervals such as distance, area or volume.) The horizontal axis is the index k, the number of occurrences. The function is only defined at integer values of k.

38 Ma Given a sample of n measured values ki we wish to estimate the value of the parameter of the Poisson population from which the sample was drawn. To calculate the maximum likelihood value, we form the log-likelihood function Take the derivative of L with respect to and equate it to zero: Solving for yields a stationary point, which if the second derivative is negative is the maximum-likelihood estimate of : Checking the second derivative, it is found that it is negative for all and ki greater than zero, therefore this stationary point is indeed a maximum of the initial likelihood function:

39 Inference For The Gravity Models. Maximum Likelihood Z = z be an (IJ) 1 vector of observations of the flows Zij, ordered by origin i, and by destination j within origin i Poisson log-likelihood for : maximum likelihood for estimates: for i j satisfys the equations:,where

40 Example. Analysis Of The Austrian Call Data consider two models Fitted using generic iteratively weighted least- squares method for generalized linear models Model arguments are considered significant at the 0,05 level

41 Fitted Values versus Flow Volume Shows the fitted values ˆ i j versus observed flow volumes z ij The relationship between the two quantities is found to be fairly linear for both models, and the variation around their linear trend, fairly uniform The standard model tends to over-estimate in somewhat greater frequency than the general model, particularly for medium- and low-volume flows

42 Relative Error versus Flow Volume Shows the relative errors (z ij ˆij )/z ij versus the flow volumes z ij light and dark points indicate under- and over- estimation, respectively For both models the relative error varies widely in magnitude. The relative error decreases with volume. For low volumes both models are inclined to over-estimate, while for higher volumes, they are increasingly inclined to under-estimate.

43 Thank you for your attention!