Bayesian dynamic modeling for large space-time weather datasets using Gaussian predictive processes
|
|
- Rafe Perry
- 5 years ago
- Views:
Transcription
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
More informationBayesian 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
More informationBayesian 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
More informationOn Gaussian Process Models for High-Dimensional Geostatistical Datasets
On Gaussian Process Models for High-Dimensional Geostatistical Datasets Sudipto Banerjee Joint work with Abhirup Datta, Andrew O. Finley and Alan E. Gelfand University of California, Los Angeles, USA May
More informationHierarchical Modeling for Spatio-temporal Data
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
More informationBayesian Modeling and Inference for High-Dimensional Spatiotemporal Datasets
Bayesian Modeling and Inference for High-Dimensional Spatiotemporal Datasets Sudipto Banerjee University of California, Los Angeles, USA Based upon projects involving: Abhirup Datta (Johns Hopkins University)
More informationHierarchical Nearest-Neighbor Gaussian Process Models for Large Geo-statistical Datasets
Hierarchical Nearest-Neighbor Gaussian Process Models for Large Geo-statistical Datasets Abhirup Datta 1 Sudipto Banerjee 1 Andrew O. Finley 2 Alan E. Gelfand 3 1 University of Minnesota, Minneapolis,
More informationNearest Neighbor Gaussian Processes for Large Spatial Data
Nearest Neighbor Gaussian Processes for Large Spatial Data Abhi Datta 1, Sudipto Banerjee 2 and Andrew O. Finley 3 July 31, 2017 1 Department of Biostatistics, Bloomberg School of Public Health, Johns
More informationHierarchical Modelling for Univariate Spatial Data
Hierarchical Modelling for Univariate Spatial 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
More informationModelling Multivariate Spatial Data
Modelling Multivariate Spatial Data Sudipto Banerjee 1 Biostatistics, School of Public Health, University of Minnesota, Minneapolis, Minnesota, U.S.A. June 20th, 2014 1 Point-referenced spatial data often
More informationHierarchical Modeling for Multivariate Spatial Data
Hierarchical Modeling for Multivariate Spatial 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
More informationHierarchical Modelling for Univariate Spatial Data
Spatial omain Hierarchical Modelling for Univariate Spatial ata Sudipto Banerjee 1 and Andrew O. Finley 2 1 Biostatistics, School of Public Health, University of Minnesota, Minneapolis, Minnesota, U.S.A.
More informationBayesian Dynamic Modeling for Space-time Data in R
Bayesian Dynamic Modeling for Space-time Data in R Andrew O. Finley and Sudipto Banerjee September 5, 2014 We make use of several libraries in the following example session, including: ˆ library(fields)
More informationIntroduction to Geostatistics
Introduction to Geostatistics Abhi Datta 1, Sudipto Banerjee 2 and Andrew O. Finley 3 July 31, 2017 1 Department of Biostatistics, Bloomberg School of Public Health, Johns Hopkins University, Baltimore,
More informationPrinciples of Bayesian Inference
Principles of Bayesian Inference 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
More informationBayesian Linear Models
Bayesian Linear Models 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
More informationFusing point and areal level space-time data. data with application to wet deposition
Fusing point and areal level space-time data with application to wet deposition Alan Gelfand Duke University Joint work with Sujit Sahu and David Holland Chemical Deposition Combustion of fossil fuel produces
More informationModel Assessment and Comparisons
Model Assessment and Comparisons 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
More informationSpatial bias modeling with application to assessing remotely-sensed aerosol as a proxy for particulate matter
Spatial bias modeling with application to assessing remotely-sensed aerosol as a proxy for particulate matter Chris Paciorek Department of Biostatistics Harvard School of Public Health application joint
More informationPrinciples of Bayesian Inference
Principles of Bayesian Inference Sudipto Banerjee and Andrew O. Finley 2 Biostatistics, School of Public Health, University of Minnesota, Minneapolis, Minnesota, U.S.A. 2 Department of Forestry & Department
More informationIntroduction to Spatial Data and Models
Introduction to Spatial Data and Models 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
More informationBayesian Linear Models
Bayesian Linear Models Sudipto Banerjee 1 and Andrew O. Finley 2 1 Department of Forestry & Department of Geography, Michigan State University, Lansing Michigan, U.S.A. 2 Biostatistics, School of Public
More informationHierarchical Modeling for non-gaussian Spatial Data
Hierarchical Modeling for non-gaussian Spatial 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
More informationBayesian Linear Regression
Bayesian Linear Regression Sudipto Banerjee 1 Biostatistics, School of Public Health, University of Minnesota, Minneapolis, Minnesota, U.S.A. September 15, 2010 1 Linear regression models: a Bayesian perspective
More informationHierarchical Modeling for Univariate Spatial Data
Hierarchical Modeling for Univariate Spatial Data Geography 890, Hierarchical Bayesian Models for Environmental Spatial Data Analysis February 15, 2011 1 Spatial Domain 2 Geography 890 Spatial Domain This
More informationHierarchical Modelling for Multivariate Spatial Data
Hierarchical Modelling for Multivariate Spatial Data Geography 890, Hierarchical Bayesian Models for Environmental Spatial Data Analysis February 15, 2011 1 Point-referenced spatial data often come as
More informationGaussian predictive process models for large spatial data sets.
Gaussian predictive process models for large spatial data sets. Sudipto Banerjee, Alan E. Gelfand, Andrew O. Finley, and Huiyan Sang Presenters: Halley Brantley and Chris Krut September 28, 2015 Overview
More informationspbayes: An R Package for Univariate and Multivariate Hierarchical Point-referenced Spatial Models
spbayes: An R Package for Univariate and Multivariate Hierarchical Point-referenced Spatial Models Andrew O. Finley 1, Sudipto Banerjee 2, and Bradley P. Carlin 2 1 Michigan State University, Departments
More informationSpatial statistics, addition to Part I. Parameter estimation and kriging for Gaussian random fields
Spatial statistics, addition to Part I. Parameter estimation and kriging for Gaussian random fields 1 Introduction Jo Eidsvik Department of Mathematical Sciences, NTNU, Norway. (joeid@math.ntnu.no) February
More informationIntroduction to Spatial Data and Models
Introduction to Spatial Data and Models Sudipto Banerjee 1 and Andrew O. Finley 2 1 Department of Forestry & Department of Geography, Michigan State University, Lansing Michigan, U.S.A. 2 Biostatistics,
More informationLecture 23. Spatio-temporal Models. Colin Rundel 04/17/2017
Lecture 23 Spatio-temporal Models Colin Rundel 04/17/2017 1 Spatial Models with AR time dependence 2 Example - Weather station data Based on Andrew Finley and Sudipto Banerjee s notes from National Ecological
More informationHierarchical Modelling for non-gaussian Spatial Data
Hierarchical Modelling for non-gaussian Spatial Data Sudipto Banerjee 1 and Andrew O. Finley 2 1 Department of Forestry & Department of Geography, Michigan State University, Lansing Michigan, U.S.A. 2
More informationHierarchical Modeling for Univariate Spatial Data
Univariate spatial models Spatial Domain Hierarchical Modeling for Univariate Spatial Data Sudipto Banerjee and Andrew O. Finley 2 Biostatistics, School of Public Health, University of Minnesota, Minneapolis,
More informationAn Additive Gaussian Process Approximation for Large Spatio-Temporal Data
An Additive Gaussian Process Approximation for Large Spatio-Temporal Data arxiv:1801.00319v2 [stat.me] 31 Oct 2018 Pulong Ma Statistical and Applied Mathematical Sciences Institute and Duke University
More informationParameter Estimation in the Spatio-Temporal Mixed Effects Model Analysis of Massive Spatio-Temporal Data Sets
Parameter Estimation in the Spatio-Temporal Mixed Effects Model Analysis of Massive Spatio-Temporal Data Sets Matthias Katzfuß Advisor: Dr. Noel Cressie Department of Statistics The Ohio State University
More informationPrinciples of Bayesian Inference
Principles of Bayesian Inference 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
More informationBayesian hierarchical space time model applied to highresolution hindcast data of significant wave height
Bayesian hierarchical space time model applied to highresolution hindcast data of Presented at Wave workshop 2013, Banff, Canada Erik Vanem Introduction The Bayesian hierarchical space-time models were
More informationSpatio-temporal modelling of daily air temperature in Catalonia
Spatio-temporal modelling of daily air temperature in Catalonia M. Saez 1,, M.A. Barceló 1,, A. Tobias 3, D. Varga 1,4 and R. Ocaña-Riola 5 1 Research Group on Statistics, Applied Economics and Health
More informationA full scale, non stationary approach for the kriging of large spatio(-temporal) datasets
A full scale, non stationary approach for the kriging of large spatio(-temporal) datasets Thomas Romary, Nicolas Desassis & Francky Fouedjio Mines ParisTech Centre de Géosciences, Equipe Géostatistique
More informationData Integration Model for Air Quality: A Hierarchical Approach to the Global Estimation of Exposures to Ambient Air Pollution
Data Integration Model for Air Quality: A Hierarchical Approach to the Global Estimation of Exposures to Ambient Air Pollution Matthew Thomas 9 th January 07 / 0 OUTLINE Introduction Previous methods for
More informationGaussian processes for spatial modelling in environmental health: parameterizing for flexibility vs. computational efficiency
Gaussian processes for spatial modelling in environmental health: parameterizing for flexibility vs. computational efficiency Chris Paciorek March 11, 2005 Department of Biostatistics Harvard School of
More informationA Note on the comparison of Nearest Neighbor Gaussian Process (NNGP) based models
A Note on the comparison of Nearest Neighbor Gaussian Process (NNGP) based models arxiv:1811.03735v1 [math.st] 9 Nov 2018 Lu Zhang UCLA Department of Biostatistics Lu.Zhang@ucla.edu Sudipto Banerjee UCLA
More informationClimatography of the United States No
Climate Division: AK 5 NWS Call Sign: ANC Month (1) Min (2) Month(1) Extremes Lowest (2) Temperature ( F) Lowest Month(1) Degree s (1) Base Temp 65 Heating Cooling 90 Number of s (3) Jan 22.2 9.3 15.8
More informationarxiv: v4 [stat.me] 14 Sep 2015
Does non-stationary spatial data always require non-stationary random fields? Geir-Arne Fuglstad 1, Daniel Simpson 1, Finn Lindgren 2, and Håvard Rue 1 1 Department of Mathematical Sciences, NTNU, Norway
More informationModels for spatial data (cont d) Types of spatial data. Types of spatial data (cont d) Hierarchical models for spatial data
Hierarchical models for spatial data Based on the book by Banerjee, Carlin and Gelfand Hierarchical Modeling and Analysis for Spatial Data, 2004. We focus on Chapters 1, 2 and 5. Geo-referenced data arise
More informationGeostatistical Modeling for Large Data Sets: Low-rank methods
Geostatistical Modeling for Large Data Sets: Low-rank methods Whitney Huang, Kelly-Ann Dixon Hamil, and Zizhuang Wu Department of Statistics Purdue University February 22, 2016 Outline Motivation Low-rank
More informationWrapped Gaussian processes: a short review and some new results
Wrapped Gaussian processes: a short review and some new results Giovanna Jona Lasinio 1, Gianluca Mastrantonio 2 and Alan Gelfand 3 1-Università Sapienza di Roma 2- Università RomaTRE 3- Duke University
More informationNRCSE. Misalignment and use of deterministic models
NRCSE Misalignment and use of deterministic models Work with Veronica Berrocal Peter Craigmile Wendy Meiring Paul Sampson Gregory Nikulin The choice of spatial scale some questions 1. Which spatial scale
More informationSpatial smoothing using Gaussian processes
Spatial smoothing using Gaussian processes Chris Paciorek paciorek@hsph.harvard.edu August 5, 2004 1 OUTLINE Spatial smoothing and Gaussian processes Covariance modelling Nonstationary covariance modelling
More informationBayesian Hierarchical Modelling: Incorporating spatial information in water resources assessment and accounting
Bayesian Hierarchical Modelling: Incorporating spatial information in water resources assessment and accounting Grace Chiu & Eric Lehmann (CSIRO Mathematics, Informatics and Statistics) A water information
More informationBayesian SAE using Complex Survey Data Lecture 4A: Hierarchical Spatial Bayes Modeling
Bayesian SAE using Complex Survey Data Lecture 4A: Hierarchical Spatial Bayes Modeling Jon Wakefield Departments of Statistics and Biostatistics University of Washington 1 / 37 Lecture Content Motivation
More informationBayesian Linear Models
Bayesian Linear Models Sudipto Banerjee September 03 05, 2017 Department of Biostatistics, Fielding School of Public Health, University of California, Los Angeles Linear Regression Linear regression is,
More informationA short introduction to INLA and R-INLA
A short introduction to INLA and R-INLA Integrated Nested Laplace Approximation Thomas Opitz, BioSP, INRA Avignon Workshop: Theory and practice of INLA and SPDE November 7, 2018 2/21 Plan for this talk
More informationHierarchical Modeling for Spatial Data
Bayesian Spatial Modelling Spatial model specifications: P(y X, θ). Prior specifications: P(θ). Posterior inference of model parameters: P(θ y). Predictions at new locations: P(y 0 y). Model comparisons.
More informationSpatio-temporal precipitation modeling based on time-varying regressions
Spatio-temporal precipitation modeling based on time-varying regressions Oleg Makhnin Department of Mathematics New Mexico Tech Socorro, NM 87801 January 19, 2007 1 Abstract: A time-varying regression
More informationSpace-time downscaling under error in computer model output
Space-time downscaling under error in computer model output University of Michigan Department of Biostatistics joint work with Alan E. Gelfand, David M. Holland, Peter Guttorp and Peter Craigmile Introduction
More informationBayesian hierarchical modelling for data assimilation of past observations and numerical model forecasts
Bayesian hierarchical modelling for data assimilation of past observations and numerical model forecasts Stan Yip Exeter Climate Systems, University of Exeter c.y.yip@ex.ac.uk Joint work with Sujit Sahu
More informationSome notes on efficient computing and setting up high performance computing environments
Some notes on efficient computing and setting up high performance computing environments Andrew O. Finley Department of Forestry, Michigan State University, Lansing, Michigan. April 17, 2017 1 Efficient
More informationA Complete Spatial Downscaler
A Complete Spatial Downscaler Yen-Ning Huang, Brian J Reich, Montserrat Fuentes 1 Sankar Arumugam 2 1 Department of Statistics, NC State University 2 Department of Civil, Construction, and Environmental
More informationOn Some Computational, Modeling and Design Issues in Bayesian Analysis of Spatial Data
On Some Computational, Modeling and Design Issues in Bayesian Analysis of Spatial Data A DISSERTATION SUBMITTED TO THE FACULTY OF THE GRADUATE SCHOOL OF THE UNIVERSITY OF MINNESOTA BY Qian Ren IN PARTIAL
More informationNonstationary spatial process modeling Part II Paul D. Sampson --- Catherine Calder Univ of Washington --- Ohio State University
Nonstationary spatial process modeling Part II Paul D. Sampson --- Catherine Calder Univ of Washington --- Ohio State University this presentation derived from that presented at the Pan-American Advanced
More informationIntroduction to Spatial Data and Models
Introduction to Spatial Data and Models Researchers in diverse areas such as climatology, ecology, environmental health, and real estate marketing are increasingly faced with the task of analyzing data
More informationBayesian data analysis in practice: Three simple examples
Bayesian data analysis in practice: Three simple examples Martin P. Tingley Introduction These notes cover three examples I presented at Climatea on 5 October 0. Matlab code is available by request to
More informationA Divide-and-Conquer Bayesian Approach to Large-Scale Kriging
A Divide-and-Conquer Bayesian Approach to Large-Scale Kriging Cheng Li DSAP, National University of Singapore Joint work with Rajarshi Guhaniyogi (UC Santa Cruz), Terrance D. Savitsky (US Bureau of Labor
More informationClimatography of the United States No
Climate Division: CA 5 NWS Call Sign: Elevation: 6 Feet Lat: 37 Month (1) Min (2) Month(1) Extremes Lowest (2) Temperature ( F) Lowest Month(1) Degree s (1) Base Temp 65 Heating Cooling 1 Number of s (3)
More informationClimatography of the United States No
Climate Division: CA 4 NWS Call Sign: Elevation: 2 Feet Lat: 37 Month (1) Min (2) Month(1) Extremes Lowest (2) Temperature ( F) Lowest Month(1) Degree s (1) Base Temp 65 Heating Cooling 1 Number of s (3)
More informationClimatography of the United States No
Climate Division: CA 5 NWS Call Sign: Elevation: 1,14 Feet Lat: 36 Month (1) Min (2) Month(1) Extremes Lowest (2) Temperature ( F) Lowest Month(1) Degree s (1) Base Temp 65 Heating Cooling 1 Number of
More informationClimatography of the United States No
Climate Division: CA 4 NWS Call Sign: Elevation: 13 Feet Lat: 36 Month (1) Min (2) Month(1) Extremes Lowest (2) Temperature ( F) Lowest Month(1) Degree s (1) Base Temp 65 Heating Cooling 1 Number of s
More informationStatistics for extreme & sparse data
Statistics for extreme & sparse data University of Bath December 6, 2018 Plan 1 2 3 4 5 6 The Problem Climate Change = Bad! 4 key problems Volcanic eruptions/catastrophic event prediction. Windstorms
More informationEfficient Posterior Inference and Prediction of Space-Time Processes Using Dynamic Process Convolutions
Efficient Posterior Inference and Prediction of Space-Time Processes Using Dynamic Process Convolutions Catherine A. Calder Department of Statistics The Ohio State University 1958 Neil Avenue Columbus,
More informationClimatography of the United States No
Climate Division: CA 6 NWS Call Sign: LAX Elevation: 1 Feet Lat: 33 Month (1) Min (2) Month(1) Extremes Lowest (2) Temperature ( F) Lowest Month(1) Degree s (1) Base Temp 65 Heating Cooling 1 Number of
More informationClimatography of the United States No
Climate Division: CA 6 NWS Call Sign: TOA Elevation: 11 Feet Lat: 33 2W Temperature ( F) Month (1) Min (2) Month(1) Extremes Lowest (2) Lowest Month(1) Degree s (1) Base Temp 65 Heating Cooling 1 Number
More informationStatistical integration of disparate information for spatially-resolved PM exposure estimation
Statistical integration of disparate information for spatially-resolved PM exposure estimation Chris Paciorek Department of Biostatistics May 4, 2006 www.biostat.harvard.edu/~paciorek L Y X - FoilT E X
More informationBruno Sansó. Department of Applied Mathematics and Statistics University of California Santa Cruz bruno
Bruno Sansó Department of Applied Mathematics and Statistics University of California Santa Cruz http://www.ams.ucsc.edu/ bruno Climate Models Climate Models use the equations of motion to simulate changes
More informationHierarchical Modelling for Univariate and Multivariate Spatial Data
Hierarchical Modelling for Univariate and Multivariate Spatial Data p. 1/4 Hierarchical Modelling for Univariate and Multivariate Spatial Data Sudipto Banerjee sudiptob@biostat.umn.edu University of Minnesota
More informationBayesian spatial hierarchical modeling for temperature extremes
Bayesian spatial hierarchical modeling for temperature extremes Indriati Bisono Dr. Andrew Robinson Dr. Aloke Phatak Mathematics and Statistics Department The University of Melbourne Maths, Informatics
More informationBayesian Hierarchical Models
Bayesian Hierarchical Models Gavin Shaddick, Millie Green, Matthew Thomas University of Bath 6 th - 9 th December 2016 1/ 34 APPLICATIONS OF BAYESIAN HIERARCHICAL MODELS 2/ 34 OUTLINE Spatial epidemiology
More informationClimatography of the United States No
Climate Division: CA 2 NWS Call Sign: Month (1) Min (2) Month(1) Extremes Lowest (2) Temperature ( F) Lowest Month(1) Degree s (1) Base Temp 65 Heating Cooling 100 Number of s (3) Jan 55.6 38.8 47.2 81
More informationClimatography of the United States No
Climate Division: CA 2 NWS Call Sign: Month (1) Min (2) Month(1) Extremes Lowest (2) Temperature ( F) Lowest Month(1) Degree s (1) Base Temp 65 Heating Cooling 100 Number of s (3) Jan 53.5 37.6 45.6 78
More informationClimatography of the United States No
Climate Division: CA 1 NWS Call Sign: Month (1) Min (2) Month(1) Extremes Lowest (2) Temperature ( F) Lowest Month(1) Degree s (1) Base Temp 65 Heating Cooling 1 Number of s (3) Jan 56.2 4.7 48.5 79 1962
More informationClimatography of the United States No
Climate Division: CA 4 NWS Call Sign: Month (1) Min (2) Month(1) Extremes Lowest (2) Temperature ( F) Lowest Month(1) Degree s (1) Base Temp 65 Heating Cooling 100 Number of s (3) Jan 61.9 42.0 52.0 89
More informationClimatography of the United States No
Climate Division: CA 4 NWS Call Sign: Month (1) Min (2) Month(1) Extremes Lowest (2) Temperature ( F) Lowest Month(1) Degree s (1) Base Temp 65 Heating Cooling 100 Number of s (3) Jan 61.4 33.1 47.3 82+
More informationClimatography of the United States No
Climate Division: CA 1 NWS Call Sign: Month (1) Min (2) Month(1) Extremes Lowest (2) Temperature ( F) Lowest Month(1) Degree s (1) Base Temp 65 Heating Cooling 100 Number of s (3) Jan 50.2 31.2 40.7 65+
More informationClimatography of the United States No
Climate Division: CA 6 NWS Call Sign: Month (1) Min (2) Month(1) Extremes Lowest (2) Temperature ( F) Lowest Month(1) Degree s (1) Base Temp 65 Heating Cooling 100 Number of s (3) Jan 66.1 38.3 52.2 91
More informationClimatography of the United States No
Climate Division: CA 1 NWS Call Sign: Month (1) Min (2) Month(1) Extremes Lowest (2) Temperature ( F) Lowest Month(1) Degree s (1) Base Temp 65 Heating Cooling 100 Number of s (3) Jan 52.4 35.4 43.9 69
More informationMultivariate modelling and efficient estimation of Gaussian random fields with application to roller data
Multivariate modelling and efficient estimation of Gaussian random fields with application to roller data Reinhard Furrer, UZH PASI, Búzios, 14-06-25 NZZ.ch Motivation Microarray data: construct alternative
More informationStatistícal Methods for Spatial Data Analysis
Texts in Statistícal Science Statistícal Methods for Spatial Data Analysis V- Oliver Schabenberger Carol A. Gotway PCT CHAPMAN & K Contents Preface xv 1 Introduction 1 1.1 The Need for Spatial Analysis
More informationClimatography of the United States No
Climate Division: CA 4 NWS Call Sign: Month (1) Min (2) Month(1) Extremes Lowest (2) Temperature ( F) Lowest Month(1) Degree s (1) Base Temp 65 Heating Cooling 100 Number of s (3) Jan 55.6 39.3 47.5 77
More informationClimatography of the United States No
Climate Division: CA 5 NWS Call Sign: Month (1) Min (2) Month(1) Extremes Lowest (2) Temperature ( F) Lowest Month(1) Degree s (1) Base Temp 65 Heating Cooling 100 Number of s (3) Jan 56.6 36.5 46.6 81
More informationClimatography of the United States No
Climate Division: CA 5 NWS Call Sign: Month (1) Min (2) Month(1) Extremes Lowest (2) Temperature ( F) Lowest Month(1) Degree s (1) Base Temp 65 Heating Cooling 100 Number of s (3) Jan 44.8 25.4 35.1 72
More informationClimatography of the United States No
Climate Division: CA 1 NWS Call Sign: Month (1) Min (2) Month(1) Extremes Lowest (2) Temperature ( F) Lowest Month(1) Degree s (1) Base Temp 65 Heating Cooling 100 Number of s (3) Jan 57.9 38.9 48.4 85
More informationClimatography of the United States No
Climate Division: CA 4 NWS Call Sign: Month (1) Min (2) Month(1) Extremes Lowest (2) Temperature ( F) Lowest Month(1) Degree s (1) Base Temp 65 Heating Cooling 100 Number of s (3) Jan 49.4 37.5 43.5 73
More informationClimatography of the United States No
Climate Division: CA 6 NWS Call Sign: Month (1) Min (2) Month(1) Extremes Lowest (2) Temperature ( F) Lowest Month(1) Degree s (1) Base Temp 65 Heating Cooling 100 Number of s (3) Jan 69.4 46.6 58.0 92
More informationClimatography of the United States No
Climate Division: CA 1 NWS Call Sign: Month (1) Min (2) Month(1) Extremes Lowest (2) Temperature ( F) Lowest Month(1) Degree s (1) Base Temp 65 Heating Cooling 1 Number of s (3) Jan 57.8 39.5 48.7 85 1962
More informationClimatography of the United States No
Climate Division: CA 4 NWS Call Sign: Month (1) Min (2) Month(1) Extremes Lowest (2) Temperature ( F) Lowest Month(1) Degree s (1) Base Temp 65 Heating Cooling 1 Number of s (3) Jan 58.5 38.8 48.7 79 1962
More informationClimatography of the United States No
Climate Division: CA 6 NWS Call Sign: Month (1) Min (2) Month(1) Extremes Lowest (2) Temperature ( F) Lowest Month(1) Degree s (1) Base Temp 65 Heating Cooling 1 Number of s (3) Jan 67.5 42. 54.8 92 1971
More informationClimatography of the United States No
No. 2 1971-2 Asheville, North Carolina 2881 COOP ID: 42713 Climate Division: CA 7 NWS Call Sign: Elevation: -3 Feet Lat: 32 Month (1) Min (2) Month(1) Extremes Lowest (2) Temperature ( F) Lowest Month(1)
More informationClimatography of the United States No
No. 2 1971-2 Asheville, North Carolina 2881 COOP ID: 46175 Climate Division: CA 6 NWS Call Sign: 3L3 Elevation: 1 Feet Lat: 33 Month (1) Min (2) Month(1) Extremes Lowest (2) Temperature ( F) Lowest Month(1)
More informationA STATISTICAL TECHNIQUE FOR MODELLING NON-STATIONARY SPATIAL PROCESSES
A STATISTICAL TECHNIQUE FOR MODELLING NON-STATIONARY SPATIAL PROCESSES JOHN STEPHENSON 1, CHRIS HOLMES, KERRY GALLAGHER 1 and ALEXANDRE PINTORE 1 Dept. Earth Science and Engineering, Imperial College,
More informationPractical Bayesian Modeling and Inference for Massive Spatial Datasets On Modest Computing Environments
Practical Bayesian Modeling and Inference for Massive Spatial Datasets On Modest Computing Environments Lu Zhang UCLA Department of Biostatistics arxiv:1802.00495v1 [stat.me] 1 Feb 2018 Lu.Zhang@ucla.edu
More information