SPATIAL DIFFUSION ANALYSIS

Example Application Areas

• Diffusion of Information

• Diffusion of Toxic Wastes

• Spread of Infectious Diseases

Product Adoption Example

http://www.seas.upenn.edu~tesmith

• Basic Model

• Steady State Analysis

• Parameter Estimation

• Philadelphia Application

Tony E. Smith and Sanyoung Song

PHILADELPHIA APPLICATION

First purchases at Netgrocer.com

( N = 1288 over 3 yrs., R = 46 zipcode areas )

1997 1998 1999

Concentrated in University Area

BASIC MODEL

r1{ ,.., }Rr r r RRegions:

Adoptions: ( : 0,1,.., )nr n N

Mixture Distribution

Adoption Frequencies: ( ) :n nf f r r R

0 1 1 0( | , ,.., ) ( | ) (1 ) ( )n n c np r r r r p r f p r

Contact Model

Intrinsic Model

exp( )( | ) ( )

exp( )RR

r src n ns

v svv

M cp r f f s

M c

0

exp( )( )

exp( )R

r r

s ss

M xp r

M x

STEADY STATE ANALYSIS

State Probability Mapping

0( ) (1 )cp f P f p

Fixed Point Property

0( ) (1 )cf p f P p

* 10(1 )( )cf I P p

Convergence to Steady State

*Pr lim 1n nf f

Rate of Convergence

( 1)*| | exp ntnf f O 1

0

n

n mmt

MAXIMUM LIKELIHOOD

Observed Data: 0 1( , ,.., )Ny y y y

Log Likelihood Function

0 0 1( , , | ) log ( ) log ( | )

N

n n nnL y p y p y f

where:

log ( | ) log ( | ) (1 ) ( )n n n n n np y f p y f p y

Problem: Can have

Example: 18, 2, 200, rs rsR J N c d

( | ) ( ) , 1,..,n n np y f p y n N

1

2

0.99999-2838.63110.0

0.00000-18056460.30

0.00000-2.17340-2.0

0.000021.000381.0

P-valueEstimateValue Param

BAYESIAN ESTIMATION

Prior Distributions:

0 0.5 10

0.2

0.4

0.6

0.8

1

0 0.5 10

0.2

0.4

0.6

0.8

1

1 1( ) (1 )a a ( ), ( ) 1

Maximum Aposteriori (MAP) Estimates

( , , | ) ( , , | ) ( 1) log log(1 )y L y a

a = 1.01 a = 2.00

1

2

0.99999153.96310.0

0.920340.000010.30

0.00000-2.17165-2.0

0.000020.999391.0

P-valueEstimateValue Param

FULL BAYES MODEL

Prior Distributions:

Posterior Distributions:

/ 20, ( ) vN vI e

1( , ) ( ) b cb c e

Conditional Probability Model:

0 1( | , , ) ( | ) ( | , , , )

N

n nnp y p y p y f

( , , | ) ( | , , ) ( ) ( ) ( )p y p y

0 0 0 0

0 0 0 0

0 0 0 0

( | , , ) ( , , | )

( | , , ) ( , , | )

( | , , ) ( , , | )

p y p y

p y p y

p y p y

BAYES MONTE CARLO

Gibbs Sampling Procedure:

• Start with any initial values 0 0 0( , , )

• Sample new 1 0 0~ ( | , , )p y

• Sample new

• Sample new

1 1 0~ ( | , , )p y

1 1 1~ ( | , , )p y

• Now start with and continue1 1 1( , , )

Save final values 0 1( , , ) : ,..,m m m m M M

Plot marginal sampling distributions

-3 -2 -1 0 1 2 3 4 50

20

40

60

80

100

120

140

-6 -5 -4 -3 -2 -1 0 1 20

20

40

60

80

100

120

140

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.90

20

40

60

80

100

120

140

2 4 6 8 10 12 14 16 18 20 220

20

40

60

80

100

120

140

0 1 0.3 10

-21

1 2

SIMULATION RESULTS

Size Mean Medn Stdev % < 0100 1.173 1.057 0.615 0200 1.102 1.029 0.413 0500 1.077 1.017 0.372 01000 1.012 1.002 0.159 02000 1.008 0.999 0.098 0

Size Mean Medn Stdev % < .01100 0.264 0.266 0.130 0.029200 0.259 0.261 0.106 0.005500 0.269 0.263 0.096 0.0011000 0.274 0.274 0.071 02000 0.280 0.275 0.061 0

Size Mean Medn Stdev % < 0100 16.38 9.67 169.2 0.041200 11.12 9.13 123.7 0.020500 5.11 9.07 130.4 0.0021000 9.31 9.42 2.21 02000 9.40 9.48 1.78 0

BETA 1

LAMBDA

THETA

PHILADELPHIA APPLICATION

First purchases at Netgrocer.com

( N = 1288 over 3 yrs., R = 46 zipcode areas )

1997 1998 1999

Concentrated in University Area

PHILADELPHIA DATA

Variable Description

BDR5 % of Housing units with more than 5 Bedrooms

COLDEG % of over 25 year-olds with College Degrees

DIWK % of Households with both parents working.

ELDERLY % of population over 65 years old.

FAMLARG % of Households with more that 5 members

SOLO % of Households with exactly one member

SUPMAS Number of Supermarkets per person

INTRINSIC VARIABLES

CONTACT COSTS = Centroid Distances

ESTIMATION RESULTS

Variable Estimate P-Value

BDR5 6.045 0.1302

COLDEG -3.952 0.1637

DIWK -0.467 0.5611

ELDERLY -7.728 0.0182

FAMLARG -13.58 0.0041

SOLO 7.781 0.0094

SUPMAS -158.74 0.6186

LAMBDA 0.678 < .0000

THETA 1195.9 0.9996

All significant values are consistent with the student populations where adoptions are concentrated.

( P-value not meaningful )

Lambda+Theta shows strong local contacts

DENGUE FEVER EXAMPLE

Tony E. Smith and Shimrit Keddem

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!(!(

!(!(!(!(

!(!(!(!(!(!(!(!(

!(

!(

!(!(!(!(!(

!(!(!(!(

!(!(!(!(!(!(!(!(!(!(!(

TOTAL

NEW MODEL

ii IIndividuals:

Infections: ( : 0,1,.., )nj n N

Mixture Distribution

Non-Infected Population:

0 1 1{ , ,.., }n nJ j j j

0( | ) ( | ) (1 ) ( ) ,n n nc n n np i J p i J p i i I

Contact Model

Intrinsic Model

( )( | ) ( | )

( )n

n

ijnc n nj J

vjv I

a dp i J p j J

a d

0

exp( )( )

exp( )n

in

vv I

xp i

x

Infected Population:

n nI I J