spbayes: An R Package for Univariate and Multivariate Hierarchical Point-referenced Spatial Models
|
|
- Baldric Scott
- 6 years ago
- Views:
Transcription
1 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 of Forestry and Geography 2 University of Minnesota, Division of Biostatistics July 29, spbayes
2 Overview Outline: Need for new software tools spbayes overview Spatial models and Bayesian inference Illustration Future direction Summary 2 spbayes
3 Need for new software tools Motivation Spatial data is used in an array of disciplines economics, genetics, natural sciences, sociology,... 3 spbayes
4 Need for new software tools Motivation Spatial data is used in an array of disciplines economics, genetics, natural sciences, sociology,... Trends in data availability and analytical needs: Increasing volume of spatial data Addressing interrelated variables of interest Summarizing uncertainty at many levels (Hierarchical modeling) Delivering spatial analysis products to end users 3 spbayes
5 Need for new software tools Scope Scope of interest: Point-referenced or geostatistical data vs. areally referenced data Point locations defined on R 2 (e.g., latitude-longitude or Easting-Northing) Multivariate generalized linear model Simple spatial dependence structures stationary and isotropic spatial processes 4 spbayes
6 Need for new software tools Software options Current software options: BUGS (Bayesian Inference Using Gibbs Sampling) Flexible coding environment Too slow (e.g., matrix operations) 5 spbayes
7 Need for new software tools Software options Current software options: BUGS (Bayesian Inference Using Gibbs Sampling) Flexible coding environment Too slow (e.g., matrix operations) geor and georglm Univariate only Limited prior specifications Too canned and cumbersome 5 spbayes
8 Need for new software tools Software options Current software options: BUGS (Bayesian Inference Using Gibbs Sampling) Flexible coding environment Too slow (e.g., matrix operations) geor and georglm Univariate only Limited prior specifications Too canned and cumbersome Model specific coding Efficient code C, C++, FORTRAN Optimized libs BLAS (Basic Linear Algebra Subprogs.), LAPACK (Linear Algebra Package) Too time consuming 5 spbayes
9 spbayes overview spbayes features Models: Univariate and multivariate spatial regression Choice of spatial and non-spatial covariance matrices Fully Bayesian specification Diagnostics: Model choice DIC (Deviance Information Criterion) CODA (Convergence Diagnosis and Output Analysis) ready output Interface and underlying code: R package Written in C++ and BLAS, LAPACK, and SPARSKIT R s foreign language interface permits OS portability and processor optimization 6 spbayes
10 spbayes overview spbayes features Models: Univariate and multivariate spatial regression Choice of spatial and non-spatial covariance matrices Fully Bayesian specification Diagnostics: Model choice DIC (Deviance Information Criterion) CODA (Convergence Diagnosis and Output Analysis) ready output Interface and underlying code: R package Written in C++ and BLAS, LAPACK, and SPARSKIT R s foreign language interface permits OS portability and processor optimization 6 spbayes
11 spbayes overview spbayes features Models: Univariate and multivariate spatial regression Choice of spatial and non-spatial covariance matrices Fully Bayesian specification Diagnostics: Model choice DIC (Deviance Information Criterion) CODA (Convergence Diagnosis and Output Analysis) ready output Interface and underlying code: R package Written in C++ and BLAS, LAPACK, and SPARSKIT R s foreign language interface permits OS portability and processor optimization 6 spbayes
12 Spatial models Point referenced spatial models and Bayesian inference (in a nutshell) 7 spbayes
13 Spatial models Univariate model Simple linear model + random spatial effects Y (s) = µ(s) + w(s) + ɛ(s), Response: Y (s) at some site Mean: µ = X(s) T β Spatial random effects: w(s) GP (0, σ 2 ρ(θ; s s )) Non-spatial variance: ɛ(s) iid N(0, τ 2 ) D Y (s), X(s) 8 spbayes
14 Spatial models Gaussian process Spatial Gaussian process (GP ): Say w(s) GP (0, σ 2 ρ( )) and Cov(w(s), w(s )) = σ 2 ρ ( θ; s s ) D w(s i) w(s j) 9 spbayes
15 Spatial models Gaussian process Spatial Gaussian process (GP ): Say w(s) GP (0, σ 2 ρ( )) and Cov(w(s), w(s )) = σ 2 ρ ( θ; s s ) Let w = [w(s i )] n i=1, then w MV N(0, σ 2 H(θ)), where H(θ) = [ρ(θ; s i s j )] n i,j=1 D w(s i) w(s j) 9 spbayes
16 Spatial models Gaussian process Realization of a Gaussian process: Changing θ and holding σ 2 = 1: w MV N(0, σ 2 H(θ)), where H(θ) = [ρ(θ; s i s j )] n i,j=1 10 spbayes
17 Spatial models Gaussian process Realization of a Gaussian process: Changing θ and holding σ 2 = 1: w MV N(0, σ 2 H(θ)), where H(θ) = [ρ(θ; s i s j )] n i,j=1 Correlation model for H(θ): e.g., exponential decay ρ(θ; t) = exp( θt) if t > spbayes
18 Spatial models Gaussian process Realization of a Gaussian process: Changing θ and holding σ 2 = 1: w MV N(0, σ 2 H(θ)), where H(θ) = [ρ(θ; s i s j )] n i,j=1 Correlation model for H(θ): e.g., exponential decay ρ(θ; t) = exp( θt) if t > 0. Other valid models e.g., Gaussian, Spherical, Matérn, etc. 10 spbayes
19 Spatial models Multivariate model Multivariate spatial model Y(s) = µ(s) + w(s) + ɛ(s), Response: Y(s) = [Y i (s)] q i=1 Mean: µ = X T (s)β, where X T (s) = [x T i (s)]q i=1 Spatial random effects: w(s) MV GP (0, K(s, s ; θ)) Non-spatial variance: ɛ(s) MV N(0, Ψ) 11 spbayes
20 Spatial models Multivariate model Multivariate spatial model Y(s) = µ(s) + w(s) + ɛ(s), Response: Y(s) = [Y i (s)] q i=1 Mean: µ = X T (s)β, where X T (s) = [x T i (s)]q i=1 Spatial random effects: w(s) MV GP (0, K(s, s ; θ)) Non-spatial variance: ɛ(s) MV N(0, Ψ) Care is needed in choosing K(s, s ; θ) insure symmetric and positive definite qn qn covariance matrix. 11 spbayes
21 Spatial models Multivariate model Multivariate spatial model Y(s) = µ(s) + w(s) + ɛ(s), Response: Y(s) = [Y i (s)] q i=1 Mean: µ = X T (s)β, where X T (s) = [x T i (s)]q i=1 Spatial random effects: w(s) MV GP (0, K(s, s ; θ)) Non-spatial variance: ɛ(s) MV N(0, Ψ) Care is needed in choosing K(s, s ; θ) insure symmetric and positive definite qn qn covariance matrix. K(s, s ; θ) = A[ q k=1 ρ k(s, s ; θ k )]A T 11 spbayes
22 Bayesian inference Hierarchical modeling Model parameterization 1 Unmarginalized likelihood: First stage: Y β, w, Ψ MV N(X T β + w, I n Ψ) 12 spbayes
23 Bayesian inference Hierarchical modeling Model parameterization 1 Unmarginalized likelihood: First stage: Y β, w, Ψ MV N(X T β + w, I n Ψ) Second stage: w A, θ MV N(0, Σ w ), where Σ w = [K(s i, s j ; θ)] n i,j=1 12 spbayes
24 Bayesian inference Hierarchical modeling Model parameterization 1 Unmarginalized likelihood: First stage: Y β, w, Ψ MV N(X T β + w, I n Ψ) Second stage: w A, θ MV N(0, Σ w ), where Σ w = [K(s i, s j ; θ)] n i,j=1 2 Marginalized likelihood (used in spbayes): Y Ω MV N(X T β, Σ w + I n Ψ), where Ω = {β, A, θ, Ψ}. 12 spbayes
25 Bayesian inference Hierarchical modeling Sampling and quantities of interest spbayes MCMC sampling scheme for Ω: β Gibbs (default) or Metropolis-Hastings (MH) A, Ψ, θ MH but soon slice sampling Recover w given Ω samples: P (w Data) P (w Ω, Data)P (Ω Data)dΩ. Prediction given Ω samples and new X(s 0 )... X(s m ): P (w Data) P (w w, Ω, Data)P (w Ω, Data)P (Ω Data)dΩdw, P (Y Data) P (Y Ω, Data)P (Ω Data)dΩ. 13 spbayes
26 Bayesian inference Hierarchical modeling Sampling and quantities of interest spbayes MCMC sampling scheme for Ω: β Gibbs (default) or Metropolis-Hastings (MH) A, Ψ, θ MH but soon slice sampling Recover w given Ω samples: P (w Data) P (w Ω, Data)P (Ω Data)dΩ. Prediction given Ω samples and new X(s 0 )... X(s m ): P (w Data) P (w w, Ω, Data)P (w Ω, Data)P (Ω Data)dΩdw, P (Y Data) P (Y Ω, Data)P (Ω Data)dΩ. 13 spbayes
27 Bayesian inference Hierarchical modeling Sampling and quantities of interest spbayes MCMC sampling scheme for Ω: β Gibbs (default) or Metropolis-Hastings (MH) A, Ψ, θ MH but soon slice sampling Recover w given Ω samples: P (w Data) P (w Ω, Data)P (Ω Data)dΩ. Prediction given Ω samples and new X(s 0 )... X(s m ): P (w Data) P (w w, Ω, Data)P (w Ω, Data)P (Ω Data)dΩdw, P (Y Data) P (Y Ω, Data)P (Ω Data)dΩ. 13 spbayes
28 Illustration Synthetic illustration Synthetic data Given two response variables (i.e., q = 2), exponential spatial correlation function, and β = ( 1 1 ) ( 1 2, K(0; θ) = 2 8 ) ( 9 0, Ψ = 0 2 ) ( 0.6, θ = 0.1 ). 14 spbayes
29 Illustration Synthetic illustration Synthetic data (cont d) Y 1 Y 2 15 spbayes
30 Illustration Synthetic illustration Potential candidate models fit with ggt.sp in spbayes Model 1: w = 0 (non-spatial) Model 2: A = σi q and Ψ = 0 Model 3: A = σi q and Ψ = τ 2 I q Model 4: A = diag[σ i ] q i=1 and Ψ = diag[τ i 2]q i=1 Model 5: A and Ψ = diag[τi 2]q i=1 Model 6: A, Ψ = diag[τi 2]q i=1, and θ = {φ k} q k=1 Model 7: A, Ψ, and θ = {φ k } q k=1 16 spbayes
31 Illustration Synthetic illustration Specifying a model using ggt.sp Recall Model 6: A, Ψ = diag[τ 2 i ]q i=1, and θ = {φ k} q k=1 Step 1: Define parameters priors, hyperpriors K.prior <- prior(dist="iwish", df=2, S=diag(c(3,6))) Psi.prior.1 <- prior(dist="ig", shape=2, scale=7) Psi.prior.2 <- prior(dist="ig", shape=2, scale=5) phi.prior <- prior(dist="unif", a=0.06, b=3) 17 spbayes
32 Illustration Synthetic illustration Step 2: Define parameters sampling info starting value, MH tuning, etc. var.update.control <- list( "K"=list(sample.order=0, starting=diag(1, 2), tuning=diag(c(0.1, 0.5, 0.1)), prior=k.prior), "Psi"=list(sample.order=1, starting=1, tuning=0.3, prior=list(psi.prior.1, Psi.prior.2)) "phi"=list(sample.order=2, starting=0.5, tuning=0.5, prior=list(phi.prior, phi.prior)) ) beta.control <- list(update="gibbs", prior=prior(dist="flat")) 18 spbayes
33 Illustration Synthetic illustration Step 3: Run control number of samples, etc. run.control <- list("n.samples"=5000, "sp.effects"=true) Step 4: Call ggt.sp ggt.model.6 <- ggt.sp( formula=list(y.1~1, Y.2~1), run.control=run.control, coords=coords, var.update.control=var.update.control, beta.update.control=beta.control, cov.model="exponential") Step 5: Prediction if desired sp.pred <-sp.predict(ggt.model.6, pred.coords=coords, pred.covars=covars) 19 spbayes
34 Illustration Synthetic illustration Step 3: Run control number of samples, etc. run.control <- list("n.samples"=5000, "sp.effects"=true) Step 4: Call ggt.sp ggt.model.6 <- ggt.sp( formula=list(y.1~1, Y.2~1), run.control=run.control, coords=coords, var.update.control=var.update.control, beta.update.control=beta.control, cov.model="exponential") Step 5: Prediction if desired sp.pred <-sp.predict(ggt.model.6, pred.coords=coords, pred.covars=covars) 19 spbayes
35 Illustration Synthetic illustration Step 3: Run control number of samples, etc. run.control <- list("n.samples"=5000, "sp.effects"=true) Step 4: Call ggt.sp ggt.model.6 <- ggt.sp( formula=list(y.1~1, Y.2~1), run.control=run.control, coords=coords, var.update.control=var.update.control, beta.update.control=beta.control, cov.model="exponential") Step 5: Prediction if desired sp.pred <-sp.predict(ggt.model.6, pred.coords=coords, pred.covars=covars) 19 spbayes
36 Illustration Synthetic illustration CODA ready posterior samples held in ggt.sp object. For example, summary of ggt.model.6$p.samples: Parameter Estimates: 50% (2.5%, 97.5%) β 1, (0.555, 1.628) β 2, (-1.619, 1.157) K 1, (0.542, 6.949) K 2, (-3.604, ) K 2, (4.645, ) Ψ 1, (3.020, ) Ψ 2, (0.832, 5.063) φ (0.243, 2.805) φ (0.073, 0.437) 20 spbayes
37 Illustration Synthetic illustration Finally, given the model object and new X(s 0 )... X(s m ) Y1 Y2 21 spbayes
38 Future direction Future of spbayes Extend to other GLMs More priors and sampling routines (e.g., slice sampling) Include general model specification (e.g., MCMC package s metrop) Address BIG-N challenges 22 spbayes
39 Future direction s New function sp.lm Fully Bayesian lm with spatial effects Minimal arguments for fitting and prediciton BIG-N through Predictive Process Knot-based dimension reduction Fit models with n>10,000 with common desktop computer See Banerjee et al. (2007) w w 23 spbayes
40 Summary Summary spbayes begins to meet a statistical computing need: Flexible model specification Efficient MCMC computation Useful parameter and predictive inference Portable and scalable code base 24 spbayes
41 Acknowledgements and References This work was supported by: NASA Headquarters under the Earth System Science Fellowship Grant NGT5 NSF Grant USDA Forest Service Forest Inventory and Analysis program University of Minnesota, School of Statistics References: Finley, A.O., Banerjee, S., and Carlin. B.P. (2007). spbayes: An R package for Univariate and Multivariate Hierarchical Point-referenced Spatial Models Journal of Statistical Software. 19(4). Finley, A.O., Banerjee, S., Ek, A.R. and McRoberts, R.E. (In press). Bayesian multivariate process modeling for prediction of forest attributes. Journal of Agricultural, Biological, and Environmental Statistics. Banerjee, S., and Finley, A.O. (In press). Bayesian multiresolution modeling of spatially replicated data. Journal of Statistical Planning and Inference. Finley, A.O., Banerjee, S., and McRoberts, R.E. (In press). A Bayesian approach to quantifying uncertainty in multi-source forest area estimates. Environmental and Ecological Statistics. 25 spbayes
42 Questions Questions 26 spbayes
Hierarchical 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 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 informationJournal of Statistical Software
JSS Journal of Statistical Software April 2007, Volume 19, Issue 4. http://www.jstatsoft.org/ spbayes: An R Package for Univariate and Multivariate Hierarchical Point-referenced Spatial Models Andrew O.
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 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 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 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, Sudipto Banerjee, and Bradley P. Carlin 1 Department Correspondence of Forest Resources,
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 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 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 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 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 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 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 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 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 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 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 informationHierarchical Modelling for non-gaussian Spatial Data
Hierarchical Modelling for non-gaussian Spatial Data Geography 890, Hierarchical Bayesian Models for Environmental Spatial Data Analysis February 15, 2011 1 Spatial Generalized Linear Models Often data
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 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 informationFrailty Modeling for Spatially Correlated Survival Data, with Application to Infant Mortality in Minnesota By: Sudipto Banerjee, Mela. P.
Frailty Modeling for Spatially Correlated Survival Data, with Application to Infant Mortality in Minnesota By: Sudipto Banerjee, Melanie M. Wall, Bradley P. Carlin November 24, 2014 Outlines of the talk
More informationMCMC algorithms for fitting Bayesian models
MCMC algorithms for fitting Bayesian models p. 1/1 MCMC algorithms for fitting Bayesian models Sudipto Banerjee sudiptob@biostat.umn.edu University of Minnesota MCMC algorithms for fitting Bayesian models
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 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 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 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 Sudipto Banerjee 1 and Andrew O. Finley 2 1 Biostatistics, School of Public Health, University of Minnesota,
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 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 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 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 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 informationIntroduction to Spatial Data and Models
Introduction to Spatial Data and Models Sudipto Banerjee and Andrew O. Finley 2 Biostatistics, School of Public Health, University of Minnesota, Minneapolis, Minnesota, U.S.A. 2 Department of Forestry
More informationAnalysis of Marked Point Patterns with Spatial and Non-spatial Covariate Information
Analysis of Marked Point Patterns with Spatial and Non-spatial Covariate Information p. 1/27 Analysis of Marked Point Patterns with Spatial and Non-spatial Covariate Information Shengde Liang, Bradley
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 Department of Forestry & Department of Geography, Michigan State University, Lansing
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 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 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 Framework for Daily Spatio-Temporal Stochastic Weather Simulation
A Framework for Daily Spatio-Temporal Stochastic Weather Simulation, Rick Katz, Balaji Rajagopalan Geophysical Statistics Project Institute for Mathematics Applied to Geosciences National Center for Atmospheric
More informationCBMS Lecture 1. Alan E. Gelfand Duke University
CBMS Lecture 1 Alan E. Gelfand Duke University Introduction to spatial data and models Researchers in diverse areas such as climatology, ecology, environmental exposure, public health, and real estate
More informationThe Wishart distribution Scaled Wishart. Wishart Priors. Patrick Breheny. March 28. Patrick Breheny BST 701: Bayesian Modeling in Biostatistics 1/11
Wishart Priors Patrick Breheny March 28 Patrick Breheny BST 701: Bayesian Modeling in Biostatistics 1/11 Introduction When more than two coefficients vary, it becomes difficult to directly model each element
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 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 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 informationComparing Non-informative Priors for Estimation and Prediction in Spatial Models
Environmentrics 00, 1 12 DOI: 10.1002/env.XXXX Comparing Non-informative Priors for Estimation and Prediction in Spatial Models Regina Wu a and Cari G. Kaufman a Summary: Fitting a Bayesian model to spatial
More informationMetropolis-Hastings Algorithm
Strength of the Gibbs sampler Metropolis-Hastings Algorithm Easy algorithm to think about. Exploits the factorization properties of the joint probability distribution. No difficult choices to be made to
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 informationST 740: Markov Chain Monte Carlo
ST 740: Markov Chain Monte Carlo Alyson Wilson Department of Statistics North Carolina State University October 14, 2012 A. Wilson (NCSU Stsatistics) MCMC October 14, 2012 1 / 20 Convergence Diagnostics:
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 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 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 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 informationRonald Christensen. University of New Mexico. Albuquerque, New Mexico. Wesley Johnson. University of California, Irvine. Irvine, California
Texts in Statistical Science Bayesian Ideas and Data Analysis An Introduction for Scientists and Statisticians Ronald Christensen University of New Mexico Albuquerque, New Mexico Wesley Johnson University
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 informationAdvanced Statistical Modelling
Markov chain Monte Carlo (MCMC) Methods and Their Applications in Bayesian Statistics School of Technology and Business Studies/Statistics Dalarna University Borlänge, Sweden. Feb. 05, 2014. Outlines 1
More informationPractical Bayesian Optimization of Machine Learning. Learning Algorithms
Practical Bayesian Optimization of Machine Learning Algorithms CS 294 University of California, Berkeley Tuesday, April 20, 2016 Motivation Machine Learning Algorithms (MLA s) have hyperparameters that
More informationTechnical Vignette 5: Understanding intrinsic Gaussian Markov random field spatial models, including intrinsic conditional autoregressive models
Technical Vignette 5: Understanding intrinsic Gaussian Markov random field spatial models, including intrinsic conditional autoregressive models Christopher Paciorek, Department of Statistics, University
More informationPrinciples of Bayesian Inference
Principles of Bayesian Inference Sudipto Banerjee University of Minnesota July 20th, 2008 1 Bayesian Principles Classical statistics: model parameters are fixed and unknown. A Bayesian thinks of parameters
More informationComputational statistics
Computational statistics Markov Chain Monte Carlo methods Thierry Denœux March 2017 Thierry Denœux Computational statistics March 2017 1 / 71 Contents of this chapter When a target density f can be evaluated
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 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 informationEmpirical Bayes methods for the transformed Gaussian random field model with additive measurement errors
1 Empirical Bayes methods for the transformed Gaussian random field model with additive measurement errors Vivekananda Roy Evangelos Evangelou Zhengyuan Zhu CONTENTS 1.1 Introduction......................................................
More informationMarkov Chain Monte Carlo
Markov Chain Monte Carlo Recall: To compute the expectation E ( h(y ) ) we use the approximation E(h(Y )) 1 n n h(y ) t=1 with Y (1),..., Y (n) h(y). Thus our aim is to sample Y (1),..., Y (n) from f(y).
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 informationBayesian non-parametric model to longitudinally predict churn
Bayesian non-parametric model to longitudinally predict churn Bruno Scarpa Università di Padova Conference of European Statistics Stakeholders Methodologists, Producers and Users of European Statistics
More informationCPSC 540: Machine Learning
CPSC 540: Machine Learning MCMC and Non-Parametric Bayes Mark Schmidt University of British Columbia Winter 2016 Admin I went through project proposals: Some of you got a message on Piazza. No news is
More informationMCMC 2: Lecture 2 Coding and output. Phil O Neill Theo Kypraios School of Mathematical Sciences University of Nottingham
MCMC 2: Lecture 2 Coding and output Phil O Neill Theo Kypraios School of Mathematical Sciences University of Nottingham Contents 1. General (Markov) epidemic model 2. Non-Markov epidemic model 3. Debugging
More informationBAYESIAN MODEL FOR SPATIAL DEPENDANCE AND PREDICTION OF TUBERCULOSIS
BAYESIAN MODEL FOR SPATIAL DEPENDANCE AND PREDICTION OF TUBERCULOSIS Srinivasan R and Venkatesan P Dept. of Statistics, National Institute for Research Tuberculosis, (Indian Council of Medical Research),
More informationOverview of Spatial Statistics with Applications to fmri
with Applications to fmri School of Mathematics & Statistics Newcastle University April 8 th, 2016 Outline Why spatial statistics? Basic results Nonstationary models Inference for large data sets An example
More informationBayesian Estimation of Input Output Tables for Russia
Bayesian Estimation of Input Output Tables for Russia Oleg Lugovoy (EDF, RANE) Andrey Polbin (RANE) Vladimir Potashnikov (RANE) WIOD Conference April 24, 2012 Groningen Outline Motivation Objectives Bayesian
More informationRegression with correlation for the Sales Data
Regression with correlation for the Sales Data Scatter with Loess Curve Time Series Plot Sales 30 35 40 45 Sales 30 35 40 45 0 10 20 30 40 50 Week 0 10 20 30 40 50 Week Sales Data What is our goal with
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 informationDefault Priors and Effcient Posterior Computation in Bayesian
Default Priors and Effcient Posterior Computation in Bayesian Factor Analysis January 16, 2010 Presented by Eric Wang, Duke University Background and Motivation A Brief Review of Parameter Expansion Literature
More informationApproaches for Multiple Disease Mapping: MCAR and SANOVA
Approaches for Multiple Disease Mapping: MCAR and SANOVA Dipankar Bandyopadhyay Division of Biostatistics, University of Minnesota SPH April 22, 2015 1 Adapted from Sudipto Banerjee s notes SANOVA vs MCAR
More informationIntroduction to Machine Learning CMU-10701
Introduction to Machine Learning CMU-10701 Markov Chain Monte Carlo Methods Barnabás Póczos & Aarti Singh Contents Markov Chain Monte Carlo Methods Goal & Motivation Sampling Rejection Importance Markov
More informationChapter 4 - Fundamentals of spatial processes Lecture notes
Chapter 4 - Fundamentals of spatial processes Lecture notes Geir Storvik January 21, 2013 STK4150 - Intro 2 Spatial processes Typically correlation between nearby sites Mostly positive correlation Negative
More informationOr How to select variables Using Bayesian LASSO
Or How to select variables Using Bayesian LASSO x 1 x 2 x 3 x 4 Or How to select variables Using Bayesian LASSO x 1 x 2 x 3 x 4 Or How to select variables Using Bayesian LASSO On Bayesian Variable Selection
More informationEffective Sample Size in Spatial Modeling
Int. Statistical Inst.: Proc. 58th World Statistical Congress, 2011, Dublin (Session CPS029) p.4526 Effective Sample Size in Spatial Modeling Vallejos, Ronny Universidad Técnica Federico Santa María, Department
More informationA general mixed model approach for spatio-temporal regression data
A general mixed model approach for spatio-temporal regression data Thomas Kneib, Ludwig Fahrmeir & Stefan Lang Department of Statistics, Ludwig-Maximilians-University Munich 1. Spatio-temporal regression
More informationFlexible Regression Modeling using Bayesian Nonparametric Mixtures
Flexible Regression Modeling using Bayesian Nonparametric Mixtures Athanasios Kottas Department of Applied Mathematics and Statistics University of California, Santa Cruz Department of Statistics Brigham
More informationPackage spatial.gev.bma
Type Package Package spatial.gev.bma February 20, 2015 Title Hierarchical spatial generalized extreme value (GEV) modeling with Bayesian Model Averaging (BMA) Version 1.0 Date 2014-03-11 Author Alex Lenkoski
More informationIntroduction to Bayes and non-bayes spatial statistics
Introduction to Bayes and non-bayes spatial statistics Gabriel Huerta Department of Mathematics and Statistics University of New Mexico http://www.stat.unm.edu/ ghuerta/georcode.txt General Concepts Spatial
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 informationA Review of Pseudo-Marginal Markov Chain Monte Carlo
A Review of Pseudo-Marginal Markov Chain Monte Carlo Discussed by: Yizhe Zhang October 21, 2016 Outline 1 Overview 2 Paper review 3 experiment 4 conclusion Motivation & overview Notation: θ denotes the
More informationStat 451 Lecture Notes Markov Chain Monte Carlo. Ryan Martin UIC
Stat 451 Lecture Notes 07 12 Markov Chain Monte Carlo Ryan Martin UIC www.math.uic.edu/~rgmartin 1 Based on Chapters 8 9 in Givens & Hoeting, Chapters 25 27 in Lange 2 Updated: April 4, 2016 1 / 42 Outline
More informationSTA 4273H: Statistical Machine Learning
STA 4273H: Statistical Machine Learning Russ Salakhutdinov Department of Computer Science! Department of Statistical Sciences! rsalakhu@cs.toronto.edu! h0p://www.cs.utoronto.ca/~rsalakhu/ Lecture 7 Approximate
More informationOverall Objective Priors
Overall Objective Priors Jim Berger, Jose Bernardo and Dongchu Sun Duke University, University of Valencia and University of Missouri Recent advances in statistical inference: theory and case studies University
More informationA Process over all Stationary Covariance Kernels
A Process over all Stationary Covariance Kernels Andrew Gordon Wilson June 9, 0 Abstract I define a process over all stationary covariance kernels. I show how one might be able to perform inference that
More informationMaking rating curves - the Bayesian approach
Making rating curves - the Bayesian approach Rating curves what is wanted? A best estimate of the relationship between stage and discharge at a given place in a river. The relationship should be on the
More informationBayesian Multivariate Logistic Regression
Bayesian Multivariate Logistic Regression Sean M. O Brien and David B. Dunson Biostatistics Branch National Institute of Environmental Health Sciences Research Triangle Park, NC 1 Goals Brief review of
More informationTAKEHOME FINAL EXAM e iω e 2iω e iω e 2iω
ECO 513 Spring 2015 TAKEHOME FINAL EXAM (1) Suppose the univariate stochastic process y is ARMA(2,2) of the following form: y t = 1.6974y t 1.9604y t 2 + ε t 1.6628ε t 1 +.9216ε t 2, (1) where ε is i.i.d.
More informationLecture 19. Spatial GLM + Point Reference Spatial Data. Colin Rundel 04/03/2017
Lecture 19 Spatial GLM + Point Reference Spatial Data Colin Rundel 04/03/2017 1 Spatial GLM Models 2 Scottish Lip Cancer Data Observed Expected 60 N 59 N 58 N 57 N 56 N value 80 60 40 20 0 55 N 8 W 6 W
More informationMultivariate Gaussian Random Fields with SPDEs
Multivariate Gaussian Random Fields with SPDEs Xiangping Hu Daniel Simpson, Finn Lindgren and Håvard Rue Department of Mathematics, University of Oslo PASI, 214 Outline The Matérn covariance function and
More informationGaussian Processes. Le Song. Machine Learning II: Advanced Topics CSE 8803ML, Spring 2012
Gaussian Processes Le Song Machine Learning II: Advanced Topics CSE 8803ML, Spring 01 Pictorial view of embedding distribution Transform the entire distribution to expected features Feature space Feature
More informationSpatial Models: A Quick Overview Astrostatistics Summer School, 2017
Spatial Models: A Quick Overview Astrostatistics Summer School, 2017 Murali Haran Department of Statistics Penn State University April 10, 2017 1 What this tutorial will cover I will explain why spatial
More informationMultivariate spatial modeling
Multivariate spatial modeling Point-referenced spatial data often come as multivariate measurements at each location Chapter 7: Multivariate Spatial Modeling p. 1/21 Multivariate spatial modeling Point-referenced
More informationMCMC Methods: Gibbs and Metropolis
MCMC Methods: Gibbs and Metropolis Patrick Breheny February 28 Patrick Breheny BST 701: Bayesian Modeling in Biostatistics 1/30 Introduction As we have seen, the ability to sample from the posterior distribution
More informationBayesian Methods for Machine Learning
Bayesian Methods for Machine Learning CS 584: Big Data Analytics Material adapted from Radford Neal s tutorial (http://ftp.cs.utoronto.ca/pub/radford/bayes-tut.pdf), Zoubin Ghahramni (http://hunch.net/~coms-4771/zoubin_ghahramani_bayesian_learning.pdf),
More informationBayesian hierarchical models for spatially misaligned data in R
Methods in Ecology and Evolution 24, 5, 54 523 doi:./24-2x.29 APPLICATION Bayesian hierarchical models for spatially misaligned data in R Andrew O. Finley *, Sudipto Banerjee 2 and Bruce D. Cook 3 Department
More informationLog Gaussian Cox Processes. Chi Group Meeting February 23, 2016
Log Gaussian Cox Processes Chi Group Meeting February 23, 2016 Outline Typical motivating application Introduction to LGCP model Brief overview of inference Applications in my work just getting started
More information