Bayesian dynamic modeling for large space-time weather datasets using Gaussian predictive processes
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1 Bayesian dynamic modeling for large space-time weather datasets using Gaussian predictive processes 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. July 25,
2 Collaborators Collaborators Sudipto Banerjee, School of Public Health, University of Minnesota. Alan Gelfand, Department of Statistical Science, Duke University. This talk draws from: Finley, Banerjee, Gelfand (2012) Bayesian dynamic modeling for large space-time datasets using Gaussian predictive processes. Journal of Geographical Systems. 14: Special Issue: In Honor of James P. LeSage. 2 Graduate Workshop on Environmental Data Analytics 2014
3 Outline Collaborators 1 Opportunities and challenges in spatio-temporal data analysis 2 Typical examples 3 Dynamic space-time modeling 4 Dimension reduction and predictive spatial process models 5 Data analysis 6 Extensions to multivariate setting 3 Graduate Workshop on Environmental Data Analytics 2014
4 Opportunities and challenges in spatio-temporal data analysis Challenges in spatio-temporal data analysis Data sets often exhibit: missingness and misalignment among outcomes space- and time-varying impact of covariates complex residual dependence structures nonstationarity among multiple outcomes across locations unknown time and perhaps space lags between outcomes and covariates 4 Graduate Workshop on Environmental Data Analytics 2014
5 Opportunities and challenges in spatio-temporal data analysis Challenges in spatio-temporal data analysis Data sets often exhibit: missingness and misalignment among outcomes space- and time-varying impact of covariates complex residual dependence structures nonstationarity among multiple outcomes across locations unknown time and perhaps space lags between outcomes and covariates We seek modeling frameworks that: incorporate many sources of space and time indexed data accommodate structured residual dependence propagate uncertainty through to predictions scale to effectively exploit information in massive datasets 4 Graduate Workshop on Environmental Data Analytics 2014
6 Typical examples Discrete-time, continuous-space settings Data is viewed as arising from a time series of spatial processes. For example: US Environmental Protection Agency s Air Quality System which reports pollutants mean, minimum, and maximum at 8 and 24 hour intervals soil sensor array that measures moisture and CO 2 at 1 minute intervals climate model outputs of weather variables generated on hourly or daily intervals remotely sensed landuse/landcover change recorded at annual or decadal time steps 5 Graduate Workshop on Environmental Data Analytics 2014
7 Typical examples NCDC weather station data from North-eastern US 1530 stations that recorded temperature between 1/1/2000 1/1/2005. The outcome variable is mean monthly temperature. Northing (km) Elevation (m) Easting (km) 6 Graduate Workshop on Environmental Data Analytics 2014
8 Typical examples NCDC weather station data from North-eastern US 1530 stations that recorded temperature between 1/1/2000 1/1/2005. The outcome variable is mean monthly temperature. Northing (km) Jan 2000 Temperature degrees C Graduate Workshop on Environmental Data Analytics 2014
9 Typical examples NCDC weather station data from North-eastern US 1530 stations that recorded temperature between 1/1/2000 1/1/2005. The outcome variable is mean monthly temperature. Northing (km) Feb 2000 Temperature degrees C Graduate Workshop on Environmental Data Analytics 2014
10 Typical examples NCDC weather station data from North-eastern US 1530 stations that recorded temperature between 1/1/2000 1/1/2005. The outcome variable is mean monthly temperature. Northing (km) Mar 2000 Temperature degrees C Graduate Workshop on Environmental Data Analytics 2014
11 Typical examples NCDC weather station data from North-eastern US 1530 stations that recorded temperature between 1/1/2000 1/1/2005. The outcome variable is mean monthly temperature. Northing (km) Apr 2000 Temperature degrees C Graduate Workshop on Environmental Data Analytics 2014
12 Typical examples NCDC weather station data from North-eastern US 1530 stations that recorded temperature between 1/1/2000 1/1/2005. The outcome variable is mean monthly temperature. Northing (km) May 2000 Temperature degrees C Graduate Workshop on Environmental Data Analytics 2014
13 Typical examples NCDC weather station data from North-eastern US 1530 stations that recorded temperature between 1/1/2000 1/1/2005. The outcome variable is mean monthly temperature. Northing (km) Jun 2000 Temperature degrees C Graduate Workshop on Environmental Data Analytics 2014
14 Typical examples NCDC weather station data from North-eastern US 1530 stations that recorded temperature between 1/1/2000 1/1/2005. The outcome variable is mean monthly temperature. Northing (km) Jul 2000 Temperature degrees C Graduate Workshop on Environmental Data Analytics 2014
15 Typical examples NCDC weather station data from North-eastern US 1530 stations that recorded temperature between 1/1/2000 1/1/2005. The outcome variable is mean monthly temperature. Northing (km) Aug 2000 Temperature degrees C Graduate Workshop on Environmental Data Analytics 2014
16 Typical examples NCDC weather station data from North-eastern US Of the n N t = 1530 station 61 months = 93,330 possible observations, 15,107 had missing outcomes. Station count Percent missing temperature observations 7 Graduate Workshop on Environmental Data Analytics 2014
17 Dynamic space-time modeling Dynamic space-time modeling West and Harrison (1997); Tonellato; (1997); Stroud et al. (2001); Gelfand et al. (2005);... Measurement equation: Regression specification + space-time process + serially and spatially uncorrelated zero-centered Gaussian disturbances as measurement error Transition equation: Markovian dependence in time temporal component: akin to usual state-space model space-time component: modeled using Gaussian processes 8 Graduate Workshop on Environmental Data Analytics 2014
18 Dynamic space-time modeling Dynamic space-time model y t (s) = x t (s) β t + u t (s) + ɛ t (s), t = 1, 2,..., N t 9 Graduate Workshop on Environmental Data Analytics 2014
19 Dynamic space-time modeling Dynamic space-time model y t (s) = x t (s) β t + u t (s) + ɛ t (s), β t = β t 1 + η t, η t iid N(0, Σ η ) ; t = 1, 2,..., N t 9 Graduate Workshop on Environmental Data Analytics 2014
20 Dynamic space-time modeling Dynamic space-time model y t (s) = x t (s) β t + u t (s) + ɛ t (s), β t = β t 1 + η t, u t (s) = u t 1 (s) + w t (s), η t iid N(0, Σ η ) ; t = 1, 2,..., N t w t (s) ind GP (0, C t (, θ t )), 9 Graduate Workshop on Environmental Data Analytics 2014
21 Dynamic space-time modeling Dynamic space-time model y t (s) = x t (s) β t + u t (s) + ɛ t (s), β t = β t 1 + η t, u t (s) = u t 1 (s) + w t (s), ɛ t (s) ind N(0, τ 2 t ). η t iid N(0, Σ η ) ; t = 1, 2,..., N t w t (s) ind GP (0, C t (, θ t )), Assume β 0 N(m 0, Σ 0 ) and u 0 (s) 0 9 Graduate Workshop on Environmental Data Analytics 2014
22 Dynamic space-time modeling Consider settings where the number of locations is large but the number of time points is manageable. For any S = {s 1, s 2,..., s n } R d. Then, w t = (w t (s 1 ), w t (s 2 ),..., w t (s n )) N(0, C t (θ t )), where C t (θ t ) = {C t (s i, s j ; θ t )}. 10 Graduate Workshop on Environmental Data Analytics 2014
23 Dynamic space-time modeling Consider settings where the number of locations is large but the number of time points is manageable. For any S = {s 1, s 2,..., s n } R d. Then, w t = (w t (s 1 ), w t (s 2 ),..., w t (s n )) N(0, C t (θ t )), where C t (θ t ) = {C t (s i, s j ; θ t )}. Model parameter estimation requires iterative matrix decomposition for C t (θ t ) 1 and determinant, i.e., computation of order O(n 3 ) for every MCMC iteration! Conducting full Bayesian inference is computationally onerous! 10 Graduate Workshop on Environmental Data Analytics 2014
24 Dimension reduction and predictive spatial process models Modeling large spatial datasets Covariance tapering (Furrer et al. 2006; Zhang and Du 2007; Du et al. 2009; Kaufman et al., 2009) Spectral domain: (Fuentes 2007; Paciorek 2007) Approximate using MRFs: INLA (Rue et al. 2009) Approximations using cond. indep. (Vecchia 1988; Stein et al. 2004) Low-rank approaches (Wahba 1990; Higdon 2002; Lin et al. 2000; Kamman & Wand, 2003; Paciorek 2007; Rasmussen & Williams 2006; Tokdar et al. 2007, 2011; Stein 2007, 2008; Cressie & Johannesson 2008; Banerjee et al. 2008, 2011; Finley et al ) 11 Graduate Workshop on Environmental Data Analytics 2014
25 Dimension reduction and predictive spatial process models Predictive process models Wahba (1990), Lin et al. (2000), Rasmussen and Williams (2006), Tokdar (2007, 2011) and Banerjee, Finley et al. ( ). Parent process: w t (s) GP (0, C t ( ; θ t )) 12 Graduate Workshop on Environmental Data Analytics 2014
26 Dimension reduction and predictive spatial process models Predictive process models Wahba (1990), Lin et al. (2000), Rasmussen and Williams (2006), Tokdar (2007, 2011) and Banerjee, Finley et al. ( ). Parent process: w t (s) GP (0, C t ( ; θ t )) Knots : S = {s 1, s 2,..., s n} with n << n w t = (w t (s { 1 ), w t(s 2 ),... }, w t(s n )) N(0, C t (θ t )), where C t (θ t ) = C t (s i, s j ; θ t) w t (s) = E[w{ t (s) w t ] = c} t (s; θ t ) C t (θ t ) 1 w t, where c t (s; θ t ) = C t (s, s j ; θ t) Replace u t (s) by ũ t (s) = ũ t 1 (s) + w t (s). Now we only have order O(n 3 ) complexity for every MCMC iteration. 12 Graduate Workshop on Environmental Data Analytics 2014
27 Dimension reduction and predictive spatial process models y y Illustration of biased estimates of variance parameters from the predictive process (synthetic data). Parent process surface Predictive process surface x x 13 Graduate Workshop on Environmental Data Analytics 2014
28 Dimension reduction and predictive spatial process models Predictive process model estimates (synthetic data) σ 2 τ 2 True parameter values σ 2 =5 and τ 2 =5. Increasing knot intensity reduces bias. 14 Graduate Workshop on Environmental Data Analytics 2014
29 Dimension reduction and predictive spatial process models Bias-adjusted predictive process w ɛ (s) ind N( w(s), C(s, s; θ) c(s; θ) C (θ) 1 c(s; θ)). 15 Graduate Workshop on Environmental Data Analytics 2014
30 Dimension reduction and predictive spatial process models Bias-adjusted predictive process w ɛ (s) ind N( w(s), C(s, s; θ) c(s; θ) C (θ) 1 c(s; θ)). Rectified predictive process model estimates (synthetic data) σ 2 τ 2 15 Graduate Workshop on Environmental Data Analytics 2014
31 Dimension reduction and predictive spatial process models Bias-adjusted predictive process w ɛ (s) ind N( w(s), C(s, s; θ) c(s; θ) C (θ) 1 c(s; θ)). Rectified predictive process model estimates (synthetic data) σ 2 τ 2 So, in our current setting replace u t (s) by ũ t (s) = ũ t 1 (s) + w t (s) + ɛ t (s), where again, ɛ(s) ind N(0, C t (s, s; θ) c t (s; θ t ) C t (θ t ) 1 c t (s; θ t )). 15 Graduate Workshop on Environmental Data Analytics 2014
32 Dimension reduction and predictive spatial process models Hierarchical dynamic predictive process models N t N t p (θ t ) IG ( τt 2 ) a τ, b τ N(β0 m 0, Σ 0 ) IW (Σ η r η, Υ η ) t=1 t=1 N t N t N(β t β t 1, Σ η ) N(w t 0, C t (θ t )) t=1 t=1 N t N(ũ t ũ t 1 + c t (θ t ) C t (θ t ) 1 w t, D t ) t=1 N t N(y t X t β t + ũ t, τt 2 I n ). t=1 16 Graduate Workshop on Environmental Data Analytics 2014
33 Dimension reduction and predictive spatial process models Selection of knots How do we choose S Knot selection Regular grid? More knots near locations we have sampled more? formal spatial design paradigm: maximize information metrics geometric considerations: space-filling designs; various clustering algorithms adaptive modeling of knots using point processes, see, e.g., Guhaniyogi et al. (2011) 17 Graduate Workshop on Environmental Data Analytics 2014
34 Dimension reduction and predictive spatial process models Selection of knots How do we choose S Knot selection Regular grid? More knots near locations we have sampled more? formal spatial design paradigm: maximize information metrics geometric considerations: space-filling designs; various clustering algorithms adaptive modeling of knots using point processes, see, e.g., Guhaniyogi et al. (2011) In practice: compare performance of estimation of range and smoothness by varying knot intensity usually inference is quite robust to S. more important to capture loss of variability due to low rank 17 Graduate Workshop on Environmental Data Analytics 2014
35 Data analysis Candidate models: NCDC weather station data analysis non-spatial, i.e., ũ t s = 0, but temporally dependent β t s full model with spatial predictive process using 5, 10, 25, and 50 knot intensities Assessment: model fit minimum posterior predictive loss approach predictive performance based on 1000 holdout observations RMSPE and CI coverage rate 18 Graduate Workshop on Environmental Data Analytics 2014
36 Data analysis Elevation covariate Northing (km) Elevation (m) Easting (km) 19 Graduate Workshop on Environmental Data Analytics 2014
37 Data analysis Knot locations Northing (km) Stations Predictive process knots Easting (km) 20 Graduate Workshop on Environmental Data Analytics 2014
38 Data analysis Dynamic estimation of the intercept β 0,t (25 knot model) Beta Years 21 Graduate Workshop on Environmental Data Analytics 2014
39 Data analysis Dynamic estimation of the impact of elevation β 1,t (25 knot model) Beta Years 22 Graduate Workshop on Environmental Data Analytics 2014
40 Data analysis Non-spatial Predictive process knots 25 Σ η [1,1] (17.97, 36.65) (16.83, 35.45) Σ η [2,1] (-0.050, 0.054) (-0.050, 0.055) Σ η [2,2] (0.001, 0.002) (0.001, 0.002) Σ η parameter credible intervals, 50 (2.5, 97.5) percentiles, for the non-spatial and 25 knot predictive process models. 23 Graduate Workshop on Environmental Data Analytics 2014
41 Data analysis Dynamic estimation of spatial range θ 1 (25 knot model) Spatial range (km) Years 24 Graduate Workshop on Environmental Data Analytics 2014
42 Data analysis Dynamic estimation of partial sill θ 2 (25 knot model) sigmaˆ Years 25 Graduate Workshop on Environmental Data Analytics 2014
43 Data analysis Dynamic estimation of nugget τ 2 t (25 knot model) tauˆ Years 26 Graduate Workshop on Environmental Data Analytics 2014
44 Data analysis Fitted values Mean Non-spatial model Mean Full model (25 knots) Observed Temp. Observed Temp. 27 Graduate Workshop on Environmental Data Analytics 2014
45 Data analysis Predictive process knots Non-spatial G P D RMSPE CI cover % Using a squared error loss function, G lack of fit, P predictive variance, D = G + P lower is better (Gelfand and Ghosh, 1998). 28 Graduate Workshop on Environmental Data Analytics 2014
46 Data analysis Cross-validation and temporal imputation Temp Years Temp Years 29 Graduate Workshop on Environmental Data Analytics 2014
47 Data analysis Graduate Workshop on Environmental Data Analytics 2014
48 Extensions to multivariate setting Extending this model to consider m outcome variables, e.g., temperature and precipitation, at each location. Simply choose your favorite approach to modeling w t (s) = (w t,1 (s), w t,2 (s),..., w t,m (s)). LMC w t (s) Multivariate Matérn Constructive Kernel Convolution Characterize Latent Dimension 30 Graduate Workshop on Environmental Data Analytics 2014
49 Summary Extensions to multivariate setting Challenge to meet modeling needs: predictive process models for large datasets use model-based adjustment to compensate for over-smoothing Future work: extend to multivariate response and spatially-varying coefficients make these models available in the R package spbayes 31 Graduate Workshop on Environmental Data Analytics 2014
Bayesian dynamic modeling for large space-time weather datasets using Gaussian predictive processes
Bayesian dynamic modeling for large space-time weather datasets using Gaussian predictive processes Andrew O. Finley Department of Forestry & Department of Geography, Michigan State University, Lansing
Bayesian dynamic modeling for large space-time weather datasets using Gaussian predictive processes
Bayesian dynamic modeling for large space-time weather datasets using Gaussian predictive processes Andrew O. Finley 1 and Sudipto Banerjee 2 1 Department of Forestry & Department of Geography, Michigan
Bayesian dynamic modeling for large space-time weather datasets using Gaussian predictive processes
Bayesian dynamic modeling for large space-time weather datasets using Gaussian predictive processes Alan Gelfand 1 and Andrew O. Finley 2 1 Department of Statistical Science, Duke University, Durham, North
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