Analysis of Economic Data with Multiscale Spatio-Temporal Models Marco A. R. Ferreira (University of Missouri - Columbia) Adelmo Bertolde (Federal University of Esp´ ırito Santo, Brazil) Scott Holan (University of Missouri, Columbia)
Outline Motivation Introduction Multiscale factorization Exploratory Multiscale Data Analysis The multiscale spatio-temporal model Empirical Bayes estimation Posterior exploration Agricultural Production in Esp´ ırito Santo Conclusions
Outline Motivation Introduction Multiscale factorization Exploratory Multiscale Data Analysis The multiscale spatio-temporal model Empirical Bayes estimation Posterior exploration Agricultural Production in Esp´ ırito Santo Conclusions
Motivation Introduction Factorization Exploratory Analysis Model EB estimation MCMC Application Conclusions Esp´ ırito Santo: Log of agriculture production per county Observed - 1990 Estimated -1990 Gaussian Multiscale Spatio-temporal Models Marco A. R. Ferreira
Motivation Introduction Factorization Exploratory Analysis Model EB estimation MCMC Application Conclusions Esp´ ırito Santo: Log of agriculture production per county Observed - 1993 Estimated - 1993 Gaussian Multiscale Spatio-temporal Models Marco A. R. Ferreira
Motivation Introduction Factorization Exploratory Analysis Model EB estimation MCMC Application Conclusions Esp´ ırito Santo: Log of agriculture production per county Observed - 1996 Estimated - 1996 Gaussian Multiscale Spatio-temporal Models Marco A. R. Ferreira
Motivation Introduction Factorization Exploratory Analysis Model EB estimation MCMC Application Conclusions Esp´ ırito Santo: Log of agriculture production per county Observed - 1999 Estimated - 1999 Gaussian Multiscale Spatio-temporal Models Marco A. R. Ferreira
Motivation Introduction Factorization Exploratory Analysis Model EB estimation MCMC Application Conclusions Esp´ ırito Santo: Log of agriculture production per county Observed - 2002 Estimated - 2002 Gaussian Multiscale Spatio-temporal Models Marco A. R. Ferreira
Motivation Introduction Factorization Exploratory Analysis Model EB estimation MCMC Application Conclusions Esp´ ırito Santo: Log of agriculture production per county Observed - 2005 Estimated - 2005 Gaussian Multiscale Spatio-temporal Models Marco A. R. Ferreira
Outline Motivation Introduction Multiscale factorization Exploratory Multiscale Data Analysis The multiscale spatio-temporal model Empirical Bayes estimation Posterior exploration Agricultural Production in Esp´ ırito Santo Conclusions
Motivation Introduction Factorization Exploratory Analysis Model EB estimation MCMC Application Conclusions Some background ◮ Many processes of interest are naturally spatio-temporal. ◮ Frequently, quantities related to these processes are available as areal data. ◮ These processes may often be considered at several different levels of spatial resolution. ◮ Related work on dynamic spatio-temporal multiscale modeling: Berliner, Wikle and Milliff (1999), Johannesson, Cressie and Huang (2007). Gaussian Multiscale Spatio-temporal Models Marco A. R. Ferreira
Motivation Introduction Factorization Exploratory Analysis Model EB estimation MCMC Application Conclusions Data Structure Here, the region of interest is divided in geographic subregions or blocks, and the data may be averages or sums over these subregions. Each state in Brazil is divided into counties, microregions and macroregions; counties are then grouped into microregions, which are then grouped into macroregions, according to their socioeconomic similarity. Thus, our analysis considers three levels of resolution: county, microregion, and macroregion. Gaussian Multiscale Spatio-temporal Models Marco A. R. Ferreira
Motivation Introduction Factorization Exploratory Analysis Model EB estimation MCMC Application Conclusions Geopolitical organization (a) (b) (c) Figure: Geopolitical organization of Esp´ ırito Santo State by (a) counties, (b) microregions, and (c) macroregions. Gaussian Multiscale Spatio-temporal Models Marco A. R. Ferreira
Outline Motivation Introduction Multiscale factorization Exploratory Multiscale Data Analysis The multiscale spatio-temporal model Empirical Bayes estimation Posterior exploration Agricultural Production in Esp´ ırito Santo Conclusions
Motivation Introduction Factorization Exploratory Analysis Model EB estimation MCMC Application Conclusions Multiscale factorization At each time point we decompose the data into empirical multiscale coefficients using the spatial multiscale modeling framework of Kolaczyk and Huang (2001). See also Chapter 9 of Ferreira and Lee (2007). Interest lies in agricultural production observed at the county level, which we assume is the L th level of resolution (i.e. the finest level of resolution), on a partition of a domain S ⊂ R 2 . For the j th county, let y Lj , µ Lj = E ( y Lj ) , and σ 2 Lj = V ( y Lj ) respectively denote agricultural production, its latent expected value and variance. Gaussian Multiscale Spatio-temporal Models Marco A. R. Ferreira
Motivation Introduction Factorization Exploratory Analysis Model EB estimation MCMC Application Conclusions Let D lj be the set of descendants of subregion ( l , j ). The aggregated measurements at the l th level of resolution are recursively defined by � y lj = y l +1 , j ′ . ( l +1 , j ′ ) ∈ D lj Analogously, the aggregated mean process is defined by � µ lj = µ l +1 , j ′ . ( l +1 , j ′ ) ∈ D lj Assuming conditional independence, σ 2 � σ 2 lj = l +1 , j ′ . ( l +1 , j ′ ) ∈ D lj Gaussian Multiscale Spatio-temporal Models Marco A. R. Ferreira
Motivation Introduction Factorization Exploratory Analysis Model EB estimation MCMC Application Conclusions Then � � y lj , µ L , σ 2 y D lj L ∼ N ( ν lj y lj + θ lj , Ω lj ) , � with σ 2 D lj /σ 2 = lj , ν lj θ lj = µ D lj − ν lj µ lj , and � ′ � Σ D lj − σ − 2 lj σ 2 σ 2 Ω lj = . D lj D lj Gaussian Multiscale Spatio-temporal Models Marco A. R. Ferreira
Motivation Introduction Factorization Exploratory Analysis Model EB estimation MCMC Application Conclusions Consider θ e lj = y D lj − ν lj y lj , which is an empirical estimate of θ lj . Then θ e lj | y lj , µ L , σ 2 L ∼ N ( θ lj , Ω lj ) , which is a singular Gaussian distribution (Anderson, 1984). Gaussian Multiscale Spatio-temporal Models Marco A. R. Ferreira
Outline Motivation Introduction Multiscale factorization Exploratory Multiscale Data Analysis The multiscale spatio-temporal model Empirical Bayes estimation Posterior exploration Agricultural Production in Esp´ ırito Santo Conclusions
Motivation Introduction Factorization Exploratory Analysis Model EB estimation MCMC Application Conclusions Exploratory Multiscale Data Analysis Macroregion 1 Disaggregated Empirical total by microregion multiscale coefficient 300 Micro−region 1 Micro−region 1 Micro−region 2 Micro−region 2 Micro−region 3 Micro−region 3 800 800 Micro−region 4 Micro−region 4 200 Micro−region 5 Micro−region 5 600 600 100 0 400 400 −100 200 200 −200 −300 0 0 1990 1995 2000 2005 1990 1995 2000 2005 1990 1995 2000 2005 year year year Esp´ ırito Santo: Agriculture production of Macroregion 1. Gaussian Multiscale Spatio-temporal Models Marco A. R. Ferreira
Motivation Introduction Factorization Exploratory Analysis Model EB estimation MCMC Application Conclusions Macroregion 2 Disaggregated Empirical total by microregion multiscale coefficient 300 Micro−region 6 Micro−region 6 Micro−region 7 Micro−region 7 800 800 200 600 600 100 0 400 400 −100 200 200 −200 −300 0 0 1990 1995 2000 2005 1990 1995 2000 2005 1990 1995 2000 2005 year year year Esp´ ırito Santo: Agriculture production of Macroregion 2. Gaussian Multiscale Spatio-temporal Models Marco A. R. Ferreira
Motivation Introduction Factorization Exploratory Analysis Model EB estimation MCMC Application Conclusions Macroregion 3 Disaggregated Empirical total by microregion multiscale coefficient 300 Micro−region 8 Micro−region 8 Micro−region 9 Micro−region 9 Micro−region 10 Micro−region 10 800 800 200 600 600 100 0 400 400 −100 200 200 −200 −300 0 0 1990 1995 2000 2005 1990 1995 2000 2005 1990 1995 2000 2005 year year year Esp´ ırito Santo: Agriculture production of Macroregion 3. Gaussian Multiscale Spatio-temporal Models Marco A. R. Ferreira
Motivation Introduction Factorization Exploratory Analysis Model EB estimation MCMC Application Conclusions Macroregion 4 Disaggregated Empirical total by microregion multiscale coefficient 300 Micro−region 11 Micro−region 11 Micro−region 12 Micro−region 12 800 800 200 600 600 100 0 400 400 −100 200 200 −200 −300 0 0 1990 1995 2000 2005 1990 1995 2000 2005 1990 1995 2000 2005 year year year Esp´ ırito Santo: Agriculture production of Macroregion 4. Gaussian Multiscale Spatio-temporal Models Marco A. R. Ferreira
Recommend
More recommend
