1. Spatial Dynamic Factor Analysis Hedibert Freitas Lopes, Esther Salazar, Dani Gamerman Presented by Zhengming Xing Jan 29,2010 * tables and figures are directly copied from the original paper.

2. Outline Introduction Spatial Dynamic Factor Analysis Model Inference and Application Experiment Future direction

3. Basic factor analysis Spatial dynamic FA model Locations:Times: key idea: Temporal dependence is modeled by latent factor score and spatial dependence is modeled by the factor loadings Factor loading matrix Factor score observations introduction

4. Spatial Dynamic Factor Analysis Model:

5. Covariate effects 1.Constant mean 2.Regression model 3.Dynamic coefficient model 1. 2. 3. Mean level of the spatio-time process Mean level of Gaussian process

6. Prior information Recall: Priors:

7. Seasonal dynamic factors Goal: capture the periodic or cyclical behavior Example : p=52 for weekly data and annual cycle

8. Spatio-temporal separability Random process indexed by space and time Assume if thenseparable Choose for convenience rather than for the ability to fit the data SDFA model m=1 m>1

9. MCMC Inference Assume: Posterior distribution: Full conditional distribution of all parameters can be found in appendix Model in matrix notation

10. Number of factors Reverse jump MCMC accept With probability Collect samples Proposal distribution:

11. Applications Prediction Interpolation

12. Experiment Sulfur dioxide concentration in eastern US 24 stations 342 observations (from the first week of 1998 to the 30th week of 2004) 2 station left out for interpolation and the last 30 weeks left out for prediction Dataset available online: http://www.epa.gov/castnet/data.html Data description

13. Experiment Spatial dynamic factor models Benchmark model

14. Experiment

19. Future direction Time varying factor loadings Allow binomial and Poisson response Non-diagonal covariance matrix and more general dynamic structure.