Gravity Models - courses.cs.ut.ee · Gravity Models Metaphor of physical gravity M i, M...

Gravity Models Elena Nikolaeva, Egon Elbre,

Roland Pihlakas

MTAT.03.251 Graph Mining

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


FLOW

Traffic flow

http://www.mundi.net/maps/maps_014/ MTAT.03.251 Graph Mining

Characteristics Of The Network Flows


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


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

Routing Matrix


Routing Matrix Captures the manner in which traffic moves throughout the

network.

B - binary values

- fraction of flow in case multiple routes are possible

MTAT.03.251 Graph Mining Figures are from Lauri Eskor’s slides

AC AD BD BC CD

1 1 0 0 0 0

2 0 1 1 1 0

3 0 1 1 0 1

Be;ij=

B

C D

A 1

2

3

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


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

Traffic Matrix


Constructing Traffic Matrix

Copyright © 1998-2011, Dr. Jean-Paul Rodrigue, Dept. of Global Studies & Geography, Hofstra University.

A B C D E Ti

A 0 0 50 0 0 50

B 0 0 60 0 30 90

C 0 0 0 30 0 30

D 20 0 80 0 20 120

E 0 0 90 10 0 100

Tj 20 0 280 40 50 390

A

C

B

E D

20

20

10

30


Origin-Destination Matrix (Traffic Matrix)

Where Zij 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


Link Totals

,where Xe – the total flow over a given link e∈ E

,where Z-traffic matrix written as a vector


AC AD BD BC CD

1 0 0 0 0

0 1 1 1 0

0 1 1 0 1 B

C D

A 1

2

3 X

ZAC

ZAD

ZBD

ZBC

ZCD

=

X1

X2

X3

X B Z

€

Xe = Be;ij × Ziji, j∑

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


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

Cost


The “Four Ts” in International Trade


Transaction costs

Tariff and non-tariff costs Transport costs

Time costs


Total Logistics Costs Tradeoff


Cos

ts

Shipment Size or Number of Warehouses

Transport Costs

Total Logistics Costs

Warehousing Costs


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


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


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

Time


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 )


Characteristics Summary   Origin Destination

 B – Routing Matrix

 Z - Traffic Matrix

 C - Cost

 T - Time

FLOW


Flow analysis classification

Measurements Goal Method

OD flow volumes Zij Model observed flow volumes Zij

Gravity Models

Link volumes Xe Predict unobserved OD flow volumes Zij

Traffic matrix estimation(static, dynamic)

OD costs cij Predict unobserved OD and link costs

Estimation of network flow costs


Gravity Models

Metaphor of physical gravity

Mi,Mj -population size (mass)

Dij - measure of separation (distance, cost)

Applications: Social science (interaction between people of different populations), geography, economics, analysis of computer network traffic etc.

MTAT.03.251 Graph Mining http://www.lewishistoricalsociety.com/wiki/tiki-print_article.php?articleId=80

€

Tij =G Mi × MjDij 2

Application of an Elementary Spatial Interaction Equation

W X Y Z Ti

W 100,000 100,000

X 100,000 50,000 25,000 175,000

Y 50,000 50,000

Z 25,000 25,000

Tj 100,000 175,000 50,000 25,000 350,000

€

Tij = kPi ∗Pj

Dij


X

2,000,000

Y

Z W

800 km

400 km

2,000,000 1,000,000 k = 0.00001

(people per week)

2,000,000

Weight (P)

Distance (D)

Constant (k)

Centroid (i) Interaction (T)

Elementary Formulation


Relationship between Distance and Interactions


Distance

Interaction

A B

A C

A D

A B C D

T(B-A)

T(C-A)

T(D-A)


General Gravity Model Specifies that the traffic flows Zij to be in the form of counts, with

independent Poisson distributions and the mean function of the form of:

Tij=E(Zij)- expected value of interaction hO(i)=Pi - origin function hD(j)=Pj- destination function hS(cij)=Dij - separation function cij- vector of K separation attributes


€

Tij = kPi ∗Pj

Dij

,where

Pi

2,000,000

Pj

Dij=800 km

2,000,000

Extension of the Gravity Model.

X

2,000,000

Y

Z W

800 km

400 km

2,000,000 1,000,000 k = 0.00001

(people per week)

2,000,000

Weight (P)

Distance (D)

Constant (k)

Centroid (i) Interaction (T)

W X Y Z Ti

W 71,378 71,378

X 6,059 2,203 36 8,298

Y 19,420 19,420

Z 153,893 153,893

Tj 6,059 244,692 2,203 36 252,990

€

Tij = kPiα ∗Pj

β

Dijθ

Simple Formulation

Exponent

λ = 0.95 α = 1.05

λ = 1.03 α = 0.96

λ = 1.2 α = 0.7

λ = 1.00 α = 0.90



Extension of the Gravity Model. Power functions


€

hO (i) = (Pi)α = (πO ,i)α

€ Cij-scalar, θ ≥ 0

€

€

hD ( j) = (Pj)β = (πD j)β

€

hS (cij) = (Dij)−θ = (cij)−θ

,where

origin function

destination function

separation function

OR

flow

cost

~1/xa

~exp()

Gravity Models. Example Austrian Call Data

Need to understand the spatial structure of telecommunication interactions among populations between different geographical

regions




regions

WHY?




regions

  Regulation of the telecommunication sector   Anticipating the influence of telecommunication

technologies on regional development


Austrian Call Data


Number of districts - 32 Time -1 year Measurements - intensity zij, i≠j=1,…,32

πO,i is the GRP of origin i, πD, j is the GRP of destination j, ci j is the distance from origin i to destination j

Austrian Call Data

Scatter plots: Call flow volume versus each of Origin GRP Destination GRP Distance

nonparametric smoother

linear regression


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:


Alternative Representation. Destination Gravity Models

related to the counts of Zij from given origin i to all destinations j

Conditional destination probabilities:

In terms of components in general probabilities:


€

P(A |B) =P(A∩ B)P(B)

Inference For The Gravity Models

Zij – independent Poisson random variables with means:

General model specification:

,where


Poisson Distribution

The horizontal axis is the index k, the number of occurrences. The function is only defined at integer values of k.

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.)


Ma Given a sample of n measured values ki we wish to estimate the value of the parameter λ 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 λ and equate it to zero:

Solving for λ yields a stationary point, which if the second derivative is negative is the maximum-likelihood estimate of λ:

Checking the second derivative, it is found that it is negative for all λ and ki greater than zero, therefore this stationary point is indeed a maximum of the initial likelihood function: MTAT.03.251 Graph Mining

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 μ:

maximum likelihood for estimates:

for μi j satisfys the equations:

,where


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

Fitted Values versus Flow Volume Shows the fitted values μˆ i j versus observed flow volumes zij

•  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


Relative Error versus Flow Volume

light and dark points indicate under- and over- estimation, respectively


Shows the relative errors (zij－μˆij)/zij versus the flow volumes zij

•  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.

Thank you for your attention!


Gravity Models - courses.cs.ut.ee · Gravity Models Metaphor of physical gravity M i, M...

Documents

Transcript of Gravity Models - courses.cs.ut.ee · Gravity Models Metaphor of physical gravity M i, M...