Date post: | 21-Dec-2015 |
Category: |
Documents |
View: | 220 times |
Download: | 0 times |
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
!(
!(
!(
!(
!(
!(
!(
!(
!(
!(
!(
!(!(
!(
!(
!(
!(
!(!(
!(
!(
!(
!(
!(
!(
!( !(!(
!(
!(
!(
!(
!( !(
!(
!(
!(
!(
!(
!(
!(
!(
!(
!(
!(
!(
!(!(!(
!(!( !(
!(!(
!(!(
!(
!(!(!(
!(
!(
!(
!(
!(!(
!(
!(
!(
!(!(
!(
!(
!(
!(
!(
!(
!(
!(
!(!(
!(
!(
!(
!(
!(
!(
!(
!(!(!(!(
!(
!(
!(
!(
!(
!(
!(
!(
!(
!(
!(
!(
!(
!(
!(
!(
!(
!(
!(
!(
!(
!(
!(!(
!(
!(
!(
!(
!(
!(!(
!(
!(
!(
!(
!(
!(
!(
!(!(
!(
!(
!(
!(!(
!(
!(
!(
!(
!(!(
!(
!(!(
!(
!(
!(
!(!(
!(
!(
!(
!(
!(
!(!(
!(
!(
!(
!(
!(
!(
!(!(
!(
!(!(
!(
Pennathur Village (India)
Individual > Mosquito > Individual
!(
!(
!(
!(
!(
!(
!(
!(
!(
!(
!(
!(!(
!(
!(
!(
!(
!(!(
!(
!(
!(
!(
!(
!(
!( !(!(
!(
!(
!(
!(
!( !(
!(
!(
!(
!(
!(
!(
!(
!(
!(
!(
!(
!(
!(!(!(
!(!( !(
!(!(
!(!(
!(
!(!(!(
!(
!(
!(
!(
!(!(
!(
!(
!(
!(!(
!(
!(
!(
!(
!(
!(
!(
!(
!(!(
!(
!(
!(
!(
!(
!(
!(
!(!(!(!(
!(
!(
!(
!(
!(
!(
!(
!(
!(
!(
!(
!(
!(
!(
!(
!(
!(
!(
!(
!(
!(
!(
!(!(
!(
!(
!(
!(
!(
!(!(
!(
!(
!(
!(
!(
!(
!(
!(!(
!(
!(
!(
!(!(
!(
!(
!(
!(
!(!(
!(
!(!(
!(
!(
!(
!(!(
!(
!(
!(
!(
!(
!(!(
!(
!(
!(
!(
!(
!(
!(!(
!(
!(!(
!(
((
((((((
!(!((
(((((
((
!(!(((((((((!((((
((
((((
(((
(((!(
(
((((!(
((
(
(((((((((((
(((
((
(((((((((
(( (
((((
((((((((((
(((((((
((!((( ((
((
((
(((((((
((
(((((((((( ((
(((((
!((!((((
(((((((!(!(((((((((((((
(((((
((((((((
(( ((((((((
(((((((((((
(((((((
((((((((
(
((((((
(((
((((((((
(((((
(((!(
(((((
(((((((
(((
((
((((
((((((((
(
(
(((((
((((
(((((((((((
WEEK 1
!(
!(
!(
!(
!(
!(
!(
!(
!(
!(
!(
!(!(
!(
!(
!(
!(
!(!(
!(
!(
!(
!(
!(
!(
!( !(!(
!(
!(
!(
!(
!( !(
!(
!(
!(
!(
!(
!(
!(
!(
!(
!(
!(
!(
!(!(!(
!(!( !(
!(!(
!(!(
!(
!(!(!(
!(
!(
!(
!(
!(!(
!(
!(
!(
!(!(
!(
!(
!(
!(
!(
!(
!(
!(
!(!(
!(
!(
!(
!(
!(
!(
!(
!(!(!(!(
!(
!(
!(
!(
!(
!(
!(
!(
!(
!(
!(
!(
!(
!(
!(
!(
!(
!(
!(
!(
!(
!(
!(!(
!(
!(
!(
!(
!(
!(!(
!(
!(
!(
!(
!(
!(
!(
!(!(
!(
!(
!(
!(!(
!(
!(
!(
!(
!(!(
!(
!(!(
!(
!(
!(
!(!(
!(
!(
!(
!(
!(
!(!(
!(
!(
!(
!(
!(
!(
!(!(
!(
!(!(
!(
((
((((((
!(!((
(((!((
((
!(!(!(!(((((((!((((
((
((((
(((
(((!(
(
((((!(
(!(
(
(((((((((((
(((
((
(((((((((
(( (
((((
((((((((((
(((((((
((!((( ((
((
((
(((((((
((
((((!(((((( ((
(((((
!((!((((
(((((((!(!(((((!(!(((((((
(((((
((((((((
(( ((((((((
(((((((((((
!(((((((
((((((((
(
((((((
(((
((((((((
(((((
(!((!(
(((((
(((((((
(((
((
((((
((((((((
(
(
(((((
((((
(((((((((((
WEEK 2
!(
!(
!(
!(
!(
!(
!(
!(
!(
!(
!(
!(!(
!(
!(
!(
!(
!(!(
!(
!(
!(
!(
!(
!(
!( !(!(
!(
!(
!(
!(
!( !(
!(
!(
!(
!(
!(
!(
!(
!(
!(
!(
!(
!(
!(!(!(
!(!( !(
!(!(
!(!(
!(
!(!(!(
!(
!(
!(
!(
!(!(
!(
!(
!(
!(!(
!(
!(
!(
!(
!(
!(
!(
!(
!(!(
!(
!(
!(
!(
!(
!(
!(
!(!(!(!(
!(
!(
!(
!(
!(
!(
!(
!(
!(
!(
!(
!(
!(
!(
!(
!(
!(
!(
!(
!(
!(
!(
!(!(
!(
!(
!(
!(
!(
!(!(
!(
!(
!(
!(
!(
!(
!(
!(!(
!(
!(
!(
!(!(
!(
!(
!(
!(
!(!(
!(
!(!(
!(
!(
!(
!(!(
!(
!(
!(
!(
!(
!(!(
!(
!(
!(
!(
!(
!(
!(!(
!(
!(!(
!(
((
((((((
!(!((
(((!((
(!(
!(!(!(!(((((((!((((
((
((((
(((
(!((!(
(
((((!(
!(!(
(
(((((!(!(((((
(((
((
(((((((((
(( (
((((
((((((((((
(((((!((
(!(!((( !(!(
((
((
(((((((
((
((((!((((((!(((((((
!((!((((
(((((((!(!((((!(!(!(((((((
(((((
((((((((
(( ((((((((
(((((((((((
!(((((((
((((((((
(
(((!(((
(((
(((((!(!((
(((((
(!(!(!(
(((((
((((!(((
(((
((
(!(((
((((((((
(
(
(((((
((((
((((((((!(!(!(
WEEK 3
!(
!(
!(
!(
!(
!(
!(
!(
!(
!(
!(
!(!(
!(
!(
!(
!(
!(!(
!(
!(
!(
!(
!(
!(
!( !(!(
!(
!(
!(
!(
!( !(
!(
!(
!(
!(
!(
!(
!(
!(
!(
!(
!(
!(
!(!(!(
!(!( !(
!(!(
!(!(
!(
!(!(!(
!(
!(
!(
!(
!(!(
!(
!(
!(
!(!(
!(
!(
!(
!(
!(
!(
!(
!(
!(!(
!(
!(
!(
!(
!(
!(
!(
!(!(!(!(
!(
!(
!(
!(
!(
!(
!(
!(
!(
!(
!(
!(
!(
!(
!(
!(
!(
!(
!(
!(
!(
!(
!(!(
!(
!(
!(
!(
!(
!(!(
!(
!(
!(
!(
!(
!(
!(
!(!(
!(
!(
!(
!(!(
!(
!(
!(
!(
!(!(
!(
!(!(
!(
!(
!(
!(!(
!(
!(
!(
!(
!(
!(!(
!(
!(
!(
!(
!(
!(
!(!(
!(
!(!(
!(
!((
!(!(((((
!(!((
(((!((
(!(
!(!(!(!((!(!((!(!(!(((!(
!(!(
(!(((
!(((
(!(!(!(
!(
(!(((!(
!(!(
(
(((((!(!(((((
(((
((
((((((!(!((
(( (
((((
(!((!(!((((((
(((((!((
!(!(!((( !(!(
((
!((
!(!((((!(!(
(!(
((((!((((((!((((!(((
!((!(!(!(!(
!(!(!(((!(!(!(!((!((!(!(!((((!(!((
(!((!((
(((((!(((
(( ((((!((((
!(!((((((!(((!(
!(!(!((!((!(
!(!(!(!((((!(
(
(((!((!(
((!(
(!((((!(!((
(((((
(!(!(!(
(((!((
((((!(!((
(((
!(!(
(!((!(
(!((!((!((!(
!(
!(
((!(!((
((((
(!(((((!(!(!(!(!(
WEEK 4
!(
!(
!(
!(
!(
!(
!(
!(
!(
!(
!(
!(!(
!(
!(
!(
!(
!(!(
!(
!(
!(
!(
!(
!(
!( !(!(
!(
!(
!(
!(
!( !(
!(
!(
!(
!(
!(
!(
!(
!(
!(
!(
!(
!(
!(!(!(
!(!( !(
!(!(
!(!(
!(
!(!(!(
!(
!(
!(
!(
!(!(
!(
!(
!(
!(!(
!(
!(
!(
!(
!(
!(
!(
!(
!(!(
!(
!(
!(
!(
!(
!(
!(
!(!(!(!(
!(
!(
!(
!(
!(
!(
!(
!(
!(
!(
!(
!(
!(
!(
!(
!(
!(
!(
!(
!(
!(
!(
!(!(
!(
!(
!(
!(
!(
!(!(
!(
!(
!(
!(
!(
!(
!(
!(!(
!(
!(
!(
!(!(
!(
!(
!(
!(
!(!(
!(
!(!(
!(
!(
!(
!(!(
!(
!(
!(
!(
!(
!(!(
!(
!(
!(
!(
!(
!(
!(!(
!(
!(!(
!(
!((
!(!((!(!(!(
!(!(!(
!(!((!((
(!(
!(!(!(!(!(!(!(!(!(!(!(!(!(!(
!(!(
!(!(!(!(
!(!(!(
!(!(!(!(
!(
(!(!(!(!(
!(!(
!(
!(!(!((!(!(!(!(!(((
(!(!(
!((
!(!(!(!(!(!(!(!(!(
!(!( !(
!(!(!(!(
!(!(!(!(!(!(!(!(!(!(
!(!(!(!(!(!(!(
!(!(!(!(!( !(!(
!(!(
!((
!(!(!((!(!(!(
!(!(
!((!(!(!((!((!(!(!(!(!((!(((
!(!(!(!(!(!(
!(!(!(!(!(!(!(!(!(!(!(!(!(!(!(!(!((!(!(!(
!(!(!(!((
!(!(!(!(!(!(!(!(
!(!( !(!(!(!(!(!(!(!(
!(!(!(!(!(!(!(!((!(!(
!(!(!(!(!(!(!(
!(!(!(!(!(!(!(!(
(
!(!(!(!((!(
((!(
(!(!((!(!(!(!(
!(!(!(!(!(
!(!(!(!(
!(!((!(!(
!(!(((!(!((
!(!(!(
!(!(
!(!(!(!(
!(!(!(!(!(!(!(!(
!(
!(
(!(!(!(!(
!((((
!(!(!((((!(!(!(!(!(
WEEK 5
!(
!(
!(
!(
!(
!(
!(
!(
!(
!(
!(
!(!(
!(
!(
!(
!(
!(!(
!(
!(
!(
!(
!(
!(
!( !(!(
!(
!(
!(
!(
!( !(
!(
!(
!(
!(
!(
!(
!(
!(
!(
!(
!(
!(
!(!(!(
!(!( !(
!(!(
!(!(
!(
!(!(!(
!(
!(
!(
!(
!(!(
!(
!(
!(
!(!(
!(
!(
!(
!(
!(
!(
!(
!(
!(!(
!(
!(
!(
!(
!(
!(
!(
!(!(!(!(
!(
!(
!(
!(
!(
!(
!(
!(
!(
!(
!(
!(
!(
!(
!(
!(
!(
!(
!(
!(
!(
!(
!(!(
!(
!(
!(
!(
!(
!(!(
!(
!(
!(
!(
!(
!(
!(
!(!(
!(
!(
!(
!(!(
!(
!(
!(
!(
!(!(
!(
!(!(
!(
!(
!(
!(!(
!(
!(
!(
!(
!(
!(!(
!(
!(
!(
!(
!(
!(
!(!(
!(
!(!(
!(
!(!(
!(!(!(!(!(!(
!(!(!(
!(!(!(!(!(
!(!(
!(!(!(!(!(!(!(!(!(!(!(!(!(!(
!(!(
!(!(!(!(
!(!(!(
!(!(!(!(
!(
!(!(!(!(!(
!(!(
!(
!(!(!(!(!(!(!(!(!(!(!(
!(!(!(
!(!(
!(!(!(!(!(!(!(!(!(
!(!( !(
!(!(!(!(
!(!(!(!(!(!(!(!(!(!(
!(!(!(!(!(!(!(
!(!(!(!(!( !(!(
!(!(
!(!(
!(!(!(!(!(!(!(
!(!(
!(!(!(!(!(!(!(!(!(!(!(!(!(!(!(!(!(
!(!(!(!(!(!(
!(!(!(!(!(!(!(!(!(!(!(!(!(!(!(!(!(!(!(!(!(
!(!(!(!(!(
!(!(!(!(!(!(!(!(
!(!( !(!(!(!(!(!(!(!(
!(!(!(!(!(!(!(!(!(!(!(
!(!(!(!(!(!(!(
!(!(!(!(!(!(!(!(
!(
!(!(!(!(!(!(
!(!(!(
!(!(!(!(!(!(!(!(
!(!(!(!(!(
!(!(!(!(
!(!(!(!(!(
!(!(!(!(!(!(!(
!(!(!(
!(!(
!(!(!(!(
!(!(!(!(!(!(!(!(
!(
!(
!(!(!(!(!(
!(!(!(!(
!(!(!(!(!(!(!(!(!(!(!(
WEEK 6
!(
!(
!(
!(
!(
!(
!(
!(
!(
!(
!(
!(!(
!(
!(
!(
!(
!(!(
!(
!(
!(
!(
!(
!(
!( !(!(
!(
!(
!(
!(
!( !(
!(
!(
!(
!(
!(
!(
!(
!(
!(
!(
!(
!(
!(!(!(
!(!( !(
!(!(
!(!(
!(
!(!(!(
!(
!(
!(
!(
!(!(
!(
!(
!(
!(!(
!(
!(
!(
!(
!(
!(
!(
!(
!(!(
!(
!(
!(
!(
!(
!(
!(
!(!(!(!(
!(
!(
!(
!(
!(
!(
!(
!(
!(
!(
!(
!(
!(
!(
!(
!(
!(
!(
!(
!(
!(
!(
!(!(
!(
!(
!(
!(
!(
!(!(
!(
!(
!(
!(
!(
!(
!(
!(!(
!(
!(
!(
!(!(
!(
!(
!(
!(
!(!(
!(
!(!(
!(
!(
!(
!(!(
!(
!(
!(
!(
!(
!(!(
!(
!(
!(
!(
!(
!(
!(!(
!(
!(!(
!(
!(!(
!(!(!(!(!(!(
!(!(!(
!(!(!(!(!(
!(!(
!(!(!(!(!(!(!(!(!(!(!(!(!(!(
!(!(
!(!(!(!(
!(!(!(
!(!(!(!(
!(
!(!(!(!(!(
!(!(
!(
!(!(!(!(!(!(!(!(!(!(!(
!(!(!(
!(!(
!(!(!(!(!(!(!(!(!(
!(!( !(
!(!(!(!(
!(!(!(!(!(!(!(!(!(!(
!(!(!(!(!(!(!(
!(!(!(!(!( !(!(
!(!(
!(!(
!(!(!(!(!(!(!(
!(!(
!(!(!(!(!(!(!(!(!(!(!(!(!(!(!(!(!(
!(!(!(!(!(!(
!(!(!(!(!(!(!(!(!(!(!(!(!(!(!(!(!(!(!(!(!(
!(!(!(!(!(
!(!(!(!(!(!(!(!(
!(!( !(!(!(!(!(!(!(!(
!(!(!(!(!(!(!(!(!(!(!(
!(!(!(!(!(!(!(
!(!(!(!(!(!(!(!(
!(
!(!(!(!(!(!(
!(!(!(
!(!(!(!(!(!(!(!(
!(!(!(!(!(
!(!(!(!(
!(!(!(!(!(
!(!(!(!(!(!(!(
!(!(!(
!(!(
!(!(!(!(
!(!(!(!(!(!(!(!(
!(
!(
!(!(!(!(!(
!(!(!(!(
!(!(!(!(!(!(!(!(!(!(!(
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