Spatial Modelling of Economic Data for Southwest Germany

Transcription

1 Spatial Modelling of Economic Data for Southwest Germany September 5, 2014

2 Background Spatial Statistics Gaussian Markov Random Fields (GMRFs) Data Exploratory Analysis Model Model Gibbs Sampling Analysis Model Verification Results Prediction Discussion

3 Spatial Statistics What is Spatial Statistics?

4 Spatial Statistics What is Spatial Statistics? Spatial information contributes directly to the statistical model Lattice data - we have data for each sub-region of a certain region

5 Spatial Statistics What is Spatial Statistics? Spatial information contributes directly to the statistical model Lattice data - we have data for each sub-region of a certain region Spatial dependency - violates independence of errors assumption

6 Gaussian Markov Random Fields (GMRFs) Gaussian Markov Random Fields (GMRFs) Gaussian distributed random vector which satifies the Markov properties Undirected graph, edges correspond to non-zero entries in precision matrix

7 Gaussian Markov Random Fields (GMRFs) Gaussian Markov Random Fields (GMRFs) Gaussian distributed random vector which satifies the Markov properties Undirected graph, edges correspond to non-zero entries in precision matrix Full conditionals π(x i x i) are Normal

8 Background Data Model Analysis Discussion Exploratory Analysis Exploratory Analysis I Economic data for the region Baden in southern Germany for 2008 I Wish to formulate a statistical model for GDP per person I Incorporating the spatial information through additive spatial random effects

9 Exploratory Analysis Correlation between GDP per person and type of district

10 Exploratory Analysis Correlation between GDP per person and type of district Some positive correlation between GDP per person, unemployment and no. of employees per company

11 Exploratory Analysis Correlation between GDP per person and type of district Some positive correlation between GDP per person, unemployment and no. of employees per company Clusters of similar GDPs in the south west and centre to south east

12 Exploratory Analysis Correlation between GDP per person and type of district Some positive correlation between GDP per person, unemployment and no. of employees per company Clusters of similar GDPs in the south west and centre to south east Covariates Employees per Company and Type contribute most to the GDP per person

13 Exploratory Analysis (a) GDP per person 2008 (b) Simple linear model

14 Model The Spatial Model For each district k our model is Y k N(β 0 + β T X k + U k, η 1 ) U 1,..., U K MVN(0, Σ) where Y is the GDP per person, X are the covariates (Area type and Employees per company) with corresponding parameters β, u 1,..., u K are the spatial random effects and η is the precision.

15 Model The Spatial Model For each district k our model is Y k N(β 0 + β T X k + U k, η 1 ) U 1,..., U K MVN(0, Σ) ( j k U k U j k N ω ) kju j 1 j k ω, kj τ CAR j k ω kj where Y is the GDP per person, X are the covariates (Area type and Employees per company) with corresponding parameters β, u 1,..., u K are the spatial random effects and η is the precision.

16 Model The Spatial Model For each district k our model is Y k N(β 0 + β T X k + U k, η 1 ) U 1,..., U K MVN(0, Σ) ( j k U k U j k N u j, n k 1 τn k where Y is the GDP per person, X are the covariates (Area type and Employees per company) with corresponding parameters β, u 1,..., u K are the spatial random effects and η is the precision. )

17 Model The Spatial Model For each district k our model is Y k N(β 0 + β T X k + U k, η 1 ) U 1,..., U K MVN(0, Σ) ( j k U k U j k N u j, n k 1 τn k where Y is the GDP per person, X are the covariates (Area type and Employees per company) with corresponding parameters β, u 1,..., u K are the spatial random effects and η is the precision. We have priors; τ CAR Gamma(a τ, b τ ), β 0 N(0, η0 1 ), β i N(0, η 1 i ) and η Gamma(a η, b η ) )

18 Gibbs Sampling Gibbs Sampling

19 Gibbs Sampling Gibbs Sampling Algorithm 1. Choose arbitrary initial sample 2. Repeatedly sample from full conditional distributions for each parameter, updating parameters each time

20 Gibbs Sampling Gibbs Sampling Algorithm 1. Choose arbitrary initial sample 2. Repeatedly sample from full conditional distributions for each parameter, updating parameters each time At some time influence of starting point has gone, call samples up to this time burn-in and discard

21 Gibbs Sampling Gibbs Sampling Algorithm 1. Choose arbitrary initial sample 2. Repeatedly sample from full conditional distributions for each parameter, updating parameters each time At some time influence of starting point has gone, call samples up to this time burn-in and discard Samples not independent so we thin by using only every kth sample

22 Model Verification Model Verification Fit of the model Distribution of the white noise Sensitivity analysis Mixing and convergence Sample beta_ ACF No. of samples Lag

23 Model Verification Model Verification Fit of the model Distribution of the white noise Sensitivity analysis Mixing and convergence Sample beta_ ACF No. of samples Lag

24 Results Interpretation GDP per person 2008 Spatial Model with all covariates

25 Results Interpretation GDP per person 2008 Spatial Model with Type

26 Prediction Prediction Using our parameter estimates we predict GDP per person in 2012, Ŷ k,2012 = ˆβX k, Ûk

27 Prediction Prediction Using our parameter estimates we predict GDP per person in 2012, Ŷ k,2012 = ˆβX k, Ûk Comparison to 2012 data - minimising sum of squared errors 22 k=1 (Ŷ k,2012 Y k,2012 ) 2

28 Prediction Prediction Using our parameter estimates we predict GDP per person in 2012, Ŷ k,2012 = ˆβX k, Ûk Comparison to 2012 data - minimising sum of squared errors 22 k=1 (Ŷ k,2012 Y k,2012 ) 2 For 16 out of the 22 districts the true values are within the central 90% prediction interval

29 Discussion 1 covariate ( Type ) provides best fit for the 2008 data but full covariate model gives better predictions

30 Discussion 1 covariate ( Type ) provides best fit for the 2008 data but full covariate model gives better predictions We expect a change in GDP with time

31 Discussion 1 covariate ( Type ) provides best fit for the 2008 data but full covariate model gives better predictions We expect a change in GDP with time Other weighting system may be more appropriate

32 Discussion 1 covariate ( Type ) provides best fit for the 2008 data but full covariate model gives better predictions We expect a change in GDP with time Other weighting system may be more appropriate Covariates included vs. no. of districts

33 Discussion 1 covariate ( Type ) provides best fit for the 2008 data but full covariate model gives better predictions We expect a change in GDP with time Other weighting system may be more appropriate Covariates included vs. no. of districts Non-gaussian distribution?

34 Any Questions? References Gelfand, A.E., Diggle, P.J., Fuentes, M. and Guttorp, P. (2010). Handbook of Spatial statistics, Chapters 12 and 14 Duncan Lee (Nov 2013). Journal of Statistic Software, Volume 55, Issue 13. CARBayes: An R Package for Bayesian Spatial Modeling with Conditional Autoregressive Priors