Hierarchical Modeling for Spatio-temporal Data
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1 Hierarchical Modeling for Spatio-temporal Data Sudipto Banerjee 1 and Andrew O. Finley 2 1 Biostatistics, School of Public Health, University of Minnesota, Minneapolis, Minnesota, U.S.A. 2 Department of Forestry & Department of Geography, Michigan State University, Lansing Michigan, U.S.A. March 4,
2 Specification: Again point-referenced vs. areal unit data Continuous time vs. discretized time association in space, association in time For point-referenced data, t continuous, Gaussian Y (s, t) = µ(s, t) + w(s, t) + ɛ(s, t) non-gaussian data, g(ey (s, t) = µ(s, t) + w(s, t) Don t treat time as a third coordinate (s, t) Cov(Y (s, t), Y (s, t )) = C(s s, t t ) 2 NEON Applied Bayesian Regression Spatio-temporal Workshop
3 Separable form: C(s s, t t ) = σ 2 ρ 1 (s s ; φ 1 )ρ 2 (t t ; φ 2 ) 3 NEON Applied Bayesian Regression Spatio-temporal Workshop
4 Separable form: C(s s, t t ) = σ 2 ρ 1 (s s ; φ 1 )ρ 2 (t t ; φ 2 ) Nonseparable form: Sum of independent separable processes Mixing of separable covariance functions Spectral domain approaches 3 NEON Applied Bayesian Regression Spatio-temporal Workshop
5
6 Type of data: time series or cross-sectional
7 Type of data: time series or cross-sectional For time series data, exploratory analysis:
8 Type of data: time series or cross-sectional For time series data, exploratory analysis: Arrange into an n T matrix Y with entries Y t (s i )
9 Type of data: time series or cross-sectional For time series data, exploratory analysis: Arrange into an n T matrix Y with entries Y t (s i ) Center by row averages of Y yields Y rows
10 Type of data: time series or cross-sectional For time series data, exploratory analysis: Arrange into an n T matrix Y with entries Y t (s i ) Center by row averages of Y yields Y rows Center by column averages of Y yields Y cols
11 Type of data: time series or cross-sectional For time series data, exploratory analysis: Arrange into an n T matrix Y with entries Y t (s i ) Center by row averages of Y yields Y rows Center by column averages of Y yields Y cols sample spatial covariance matrix: 1 T Y rowsy T rows
12 Type of data: time series or cross-sectional For time series data, exploratory analysis: Arrange into an n T matrix Y with entries Y t (s i ) Center by row averages of Y yields Y rows Center by column averages of Y yields Y cols sample spatial covariance matrix: 1 T Y rowsy T rows sample autocorrelation matrix: 1 n Y T cols Y cols
13 Type of data: time series or cross-sectional For time series data, exploratory analysis: Arrange into an n T matrix Y with entries Y t (s i ) Center by row averages of Y yields Y rows Center by column averages of Y yields Y cols sample spatial covariance matrix: 1 T Y rowsy T rows sample autocorrelation matrix: 1 n Y T cols Y cols E, residuals matrix after a regression fitting, Empirical orthogonal functions (EOF)
14 Modeling: Y t (s) = µ t (s) + w t (s) + ɛ t (s), or perhaps g(e(y t (s)) = µ t (s) + w t (s) 5 NEON Applied Bayesian Regression Spatio-temporal Workshop
15 Modeling: Y t (s) = µ t (s) + w t (s) + ɛ t (s), or perhaps g(e(y t (s)) = µ t (s) + w t (s) For ɛ t (s), i.i.d. N(0, τ 2 t ) 5 NEON Applied Bayesian Regression Spatio-temporal Workshop
16 Modeling: Y t (s) = µ t (s) + w t (s) + ɛ t (s), or perhaps g(e(y t (s)) = µ t (s) + w t (s) For ɛ t (s), i.i.d. N(0, τ 2 t ) For w t (s) w t (s) = α t + w(s) w t (s) independent for each t w t (s) = w t 1 (s) + η t (s), independent spatial process innovations 5 NEON Applied Bayesian Regression Spatio-temporal Workshop
17 Dynamic spatiotemporal models Measurement Equation Y (s, t) = µ(s, t) + ɛ(s, t); ɛ(s, t) ind N(0, σ 2 ɛ ). µ(s, t) = x(s, t) β(s, t). β(s, t) = β t + β(s, t) Transition Equation β t = β t 1 + η t, η t ind N p (0, Ση) β(s, t) = β(s, t 1) + w(s, t). 6 NEON Applied Bayesian Regression Spatio-temporal Workshop
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