Spatial_Diffusion_Talk_Tony_Smith

Spatial_Diffusion_Talk_Tony_Smith - SPATIAL DIFFUSION...

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Unformatted text preview: 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 r 1 { ,.., } R r r r = R Regions: Adoptions: ( : 0,1,.., ) n r n N = Mixture Distribution Adoption Frequencies: [ ] ( ) : n n f f r r = R 1 1 ( | , ,.., ) ( | ) (1 ) ( ) n n c n p r r r r p r f p r- = +- Contact Model Intrinsic Model exp( ) ( | ) ( ) exp( ) R R r sr c n n s v sv v M c p r f f s M c - =- exp( ) ( ) exp( ) R r r s s s M x p r M x - = - STEADY STATE ANALYSIS State Probability Mapping ( ) (1 ) c p f P f p = +- Fixed Point Property ( ) (1 ) c f p f P p = = +- * 1 (1 )( ) c f I P p- =- - Convergence to Steady State ( 29 * Pr lim 1 n n f f = = Rate of Convergence ( 29 ( 1) * | | exp n t n f f O -- = ( 29 1 n n m m t = = MAXIMUM LIKELIHOOD Observed Data: 1 ( , ,.., ) N y y y y = Log Likelihood Function 1 ( , , | ) log ( ) log ( | ) N n n n n L y p y p y f = = + where: log ( | ) log ( | ) (1 ) ( ) n n n n n n p y f p y f p y = +- Problem: Can have Example: 18, 2, 200, rs rs R J N c d = = = = ( | ) ( ) , 1,.., n n n p y f p y n N < = 1 2 0.99999-2838.631 10.0 0.00000-1805646 0.30 0.00000-2.17340-2.0 0.00002 1.00038 1.0 P-value Estimate Value Param BAYESIAN ESTIMATION Prior Distributions: 0.5 1 0.2 0.4 0.6 0.8 1 0.5 1 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 0.99999 153.963 10.0 0.92034 0.00001 0.30 0.00000-2.17165-2.0 0.00002 0.99939 1.0 P-value Estimate Value Param FULL BAYES MODEL Prior Distributions: Posterior Distributions: ( 29 / 2 0, ( ) v N vI e - : 1 ( , ) ( ) b c b c e-- : Conditional Probability Model: 1 ( | , , ) ( | ) ( | , , , ) N n n n p y p y p y f = = ( , , | ) ( | , , ) ( ) ( ) ( ) p y p y ( | , , ) ( , , | ) ( | , , ) ( , , | ) ( | , , ) ( , , | ) p y p y p y p y p y p y BAYES MONTE CARLO Gibbs Sampling Procedure: Start with any initial values ( , , ) Sample new 1 ~ ( | , , ) p y Sample new Sample new 1 1 ~ ( | , , ) p y 1 1 1 ~ ( | , , ) p y Now start with and continue 1 1 1 ( , , ) Save final values [ ] 1 ( , , ) : ,.., m m m m M M = Plot marginal...
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Spatial_Diffusion_Talk_Tony_Smith - SPATIAL DIFFUSION...

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