Geostatistical Modeling for Large Data Sets: Low-rank methods
|
|
- Terence Howard
- 5 years ago
- Views:
Transcription
1 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
2 Outline Motivation Low-rank methods
3 Gaussian process (GP) geostatistics Model: where Y (s) = µ(s) + η(s) + ε(s), s S R d µ(s) = E [Y (s)] = X(s) T β {η(s)} s S GP (0, C (, )) ε(s) iid N(0, τ 2 ) s S
4 Likelihood When the process Y (s) is observed at n locations, say, s 1, s n ie, Y = (Y (s 1 ),, Y (s n )) T Log-likelihood: where l n 1 2 log Σ Y 1 2 (Y XT β) T [Σ Y ] 1 (Y Xβ) Σ Y = [Cov {Y (s i ), Y (s j )}] i,j=1,,n [ ] = C (s i, s j ) + τ 2 1 {si =s j } i,j=1,,n
5 Big n Problem in geostatistics Modern environmental instruments have produced a wealth of space time data n is big Evaluation of the likelihood function involves factorizing large covariance matrices that generally requires O(n 3 ) operations O(n 2 ) memory Modeling strategies are needed to deal with large spatial data set parameter estimation MLE, Bayesian spatial interpolation Kriging multivariate spatial data (np np), spatio-temporal data (nt nt)
6 Big n Problem in geostatistics Modern environmental instruments have produced a wealth of space time data n is big Evaluation of the likelihood function involves factorizing large covariance matrices that generally requires O(n 3 ) operations O(n 2 ) memory Modeling strategies are needed to deal with large spatial data set parameter estimation MLE, Bayesian spatial interpolation Kriging multivariate spatial data (np np), spatio-temporal data (nt nt)
7 Big n Problem in geostatistics Modern environmental instruments have produced a wealth of space time data n is big Evaluation of the likelihood function involves factorizing large covariance matrices that generally requires O(n 3 ) operations O(n 2 ) memory Modeling strategies are needed to deal with large spatial data set parameter estimation MLE, Bayesian spatial interpolation Kriging multivariate spatial data (np np), spatio-temporal data (nt nt)
8 Outline Motivation Low-rank methods
9 Low rank approximation Goal: want to approximate Y = X T β + η + ε into a lower-dimensional structure Hierarchical Representation (assume zero mean) Y = η + ε, ε N n (0, Σ ε ) η = AZ + ξ, ξ N n (0, Σ ξ ) Z N r (0, Σ Z ) Z = (Z 1,, Z r ) T, r n A is the (n r) expansion matrix that maps Z to η Σ ε and Σ ξ are (usually assumed to be) diagonal η 0 = A 0 Z + ξ 0 η(s 0 ) = r j=1 a(s 0) T Z j + ξ(s 0 )
10 Low rank approximation Goal: want to approximate Y = X T β + η + ε into a lower-dimensional structure Hierarchical Representation (assume zero mean) Y = η + ε, ε N n (0, Σ ε ) η = AZ + ξ, ξ N n (0, Σ ξ ) Z N r (0, Σ Z ) Z = (Z 1,, Z r ) T, r n A is the (n r) expansion matrix that maps Z to η Σ ε and Σ ξ are (usually assumed to be) diagonal η 0 = A 0 Z + ξ 0 η(s 0 ) = r j=1 a(s 0) T Z j + ξ(s 0 )
11 Low rank approximation Goal: want to approximate Y = X T β + η + ε into a lower-dimensional structure Hierarchical Representation (assume zero mean) Y = η + ε, ε N n (0, Σ ε ) η = AZ + ξ, ξ N n (0, Σ ξ ) Z N r (0, Σ Z ) Z = (Z 1,, Z r ) T, r n A is the (n r) expansion matrix that maps Z to η Σ ε and Σ ξ are (usually assumed to be) diagonal η 0 = A 0 Z + ξ 0 η(s 0 ) = r j=1 a(s 0) T Z j + ξ(s 0 )
12 Low rank approximation Goal: want to approximate Y = X T β + η + ε into a lower-dimensional structure Hierarchical Representation (assume zero mean) Y = η + ε, ε N n (0, Σ ε ) η = AZ + ξ, ξ N n (0, Σ ξ ) Z N r (0, Σ Z ) Z = (Z 1,, Z r ) T, r n A is the (n r) expansion matrix that maps Z to η Σ ε and Σ ξ are (usually assumed to be) diagonal η 0 = A 0 Z + ξ 0 η(s 0 ) = r j=1 a(s 0) T Z j + ξ(s 0 )
13 Computational savings using low rank approximation To carry out parameter estimation and spatial interpolation one need to compute ( AΣZ A T + V ) 1 where V = Σ ε + Σ ξ Sherman Morrison Woodbury formula (A + BCD) 1 = A 1 A 1 B ( C 1 + DA 1 B ) 1 DA 1 In low rank approximation, we have ( AΣZ A T + V ) 1 = V 1 V 1 A ( Σ 1 Z + AT V 1 A ) 1 A T V 1
14 Low rank approximation: expansion matrix A η(s 1 ) η(s 2 ) η(s n ) }{{} η A 11 A 12 A 1r A 21 A 22 A 2r A n1 A n2 A nr }{{} A Z 1 Z 2 Z r }{{} Z Question How to choose A? How to determine Z = (Z 1,, Z r ) T? choose r knots s 1,, s r S and let Z = (Z (s 1 ),, Z (s r )) T
15 Low rank approximation: expansion matrix A η(s 1 ) η(s 2 ) η(s n ) }{{} η A 11 A 12 A 1r A 21 A 22 A 2r A n1 A n2 A nr }{{} A Z 1 Z 2 Z r }{{} Z Question How to choose A? How to determine Z = (Z 1,, Z r ) T? choose r knots s 1,, s r S and let Z = (Z (s 1 ),, Z (s r )) T
16 Choice of A Orthogonal basis functions: eg, Fourier, orthogonal polynomial, etc Karhune Loéve expansion: Cov(η(s 1 ), η(s 2 )) = j=1 λ jφ j (s 1 )φ j (s 2 ) where Cov(η(s 1 ), η(s 2 ))φ j (s 1 ) ds 1 = λ j φ j (s 2 ) S Empirical orthogonal functions (EOFs) Nonorthogonal Basis Functions: represent the spatial process as combination of kernel functions η(s) = r a(s, s j )Z j + ξ(s) j=1
17 Fixed Rank Kriging (Cressie & Johannesson 08) Fixed-rank kriging outlines how kriging (ie, spatial best linear unbiased predictor) can be applied in the low-rank parameterization of a spatial process A: (multiresolutional) non orthogonal basis functions ( ) a(s) T = {1 ( s s1(l) /ρ 1(l)) 2 } 2 +,, {1 ( s sr(l) /ρ r(l)) 2 } 2 + Z = (Z (s 1 ),, Z (s r )) T, Σ Z to be estimated from data The fixed rank kriging is equivalent to the following low rank model Y (s) = X (s) T β + r a(s s j(l) )Z j + ε(s) + ξ(s) j=1
18 Multiresolutional basis functions Figure: courtesy of Cressie & Johannesson 08
19 Nonstationary covariance function Figure: courtesy of Cressie & Johannesson 08
20 Gaussian Predictive Process (Banerjee et al 08) Y (s) = X (s) T β + η(s) + ε(s), η GP(0, C(, ; θ) Reasoning: Given Z, we seek a linear combination a(s) T Z such that [ 2 E η (s) a (s) Z] T ds S is minimized In predictive process one let Z = η = {η(s i )}p i=1 N p(0, C (θ)) [ ] p where C (θ) = C(si, s j ; θ) and i,j=1 ] j=1,,p A = [C(s i, s j ; θ) [C ] 1 i=1,,n
21 Covaraince function approximation Figure: courtesy of Banerjee et al 08
22 Covaraince function approximation cont d Figure: courtesy of Banerjee et al 08
23 Discussion
24 References S Banerjee, A E Gelfand, A O Finley, and H Sang Gaussian Predictive Process Models for Large Spatial Data Sets JRSSB, 70: , 2008 N A C Cressie, and G Johannesson Fixed Rank Kriging for Very Large Spatial Data Sets JRSSB, 70: , 2008 C K Wikle Low-rank representations for spatial processes, Chapter 8 of Handbook of Spatial Statistics Chapman & Hall/CRC, 2010
Low-rank methods and predictive processes for spatial models
Low-rank methods and predictive processes for spatial models Sam Bussman, Linchao Chen, John Lewis, Mark Risser with Sebastian Kurtek, Vince Vu, Ying Sun February 27, 2014 Outline Introduction and general
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 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 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 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 informationNonparametric Bayesian Methods
Nonparametric Bayesian Methods Debdeep Pati Florida State University October 2, 2014 Large spatial datasets (Problem of big n) Large observational and computer-generated datasets: Often have spatial and
More informationKarhunen-Loeve Expansion and Optimal Low-Rank Model for Spatial Processes
TTU, October 26, 2012 p. 1/3 Karhunen-Loeve Expansion and Optimal Low-Rank Model for Spatial Processes Hao Zhang Department of Statistics Department of Forestry and Natural Resources Purdue University
More informationChapter 4 - Fundamentals of spatial processes Lecture notes
TK4150 - Intro 1 Chapter 4 - Fundamentals of spatial processes Lecture notes Odd Kolbjørnsen and Geir Storvik January 30, 2017 STK4150 - Intro 2 Spatial processes Typically correlation between nearby sites
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 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 informationSpace-time data. Simple space-time analyses. PM10 in space. PM10 in time
Space-time data Observations taken over space and over time Z(s, t): indexed by space, s, and time, t Here, consider geostatistical/time data Z(s, t) exists for all locations and all times May consider
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 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 informationSTATISTICAL INTERPOLATION METHODS RICHARD SMITH AND NOEL CRESSIE Statistical methods of interpolation are all based on assuming that the process
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 STATISTICAL INTERPOLATION METHODS
More informationAn Introduction to Spatial Statistics. Chunfeng Huang Department of Statistics, Indiana University
An Introduction to Spatial Statistics Chunfeng Huang Department of Statistics, Indiana University Microwave Sounding Unit (MSU) Anomalies (Monthly): 1979-2006. Iron Ore (Cressie, 1986) Raw percent data
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 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 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 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 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 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 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 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 informationGroundwater permeability
Groundwater permeability Easy to solve the forward problem: flow of groundwater given permeability of aquifer Inverse problem: determine permeability from flow (usually of tracers With some models enough
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 informationLatent Gaussian Processes and Stochastic Partial Differential Equations
Latent Gaussian Processes and Stochastic Partial Differential Equations Johan Lindström 1 1 Centre for Mathematical Sciences Lund University Lund 2016-02-04 Johan Lindström - johanl@maths.lth.se Gaussian
More informationRegionalization of Multiscale Spatial Processes Using a Criterion for Spatial Aggregation Error. Christopher K. Wikle
Regionalization of Multiscale Spatial Processes Using a Criterion for Spatial Aggregation Error Christopher K. Wikle University of Missouri TAMU Spatial Workshop, January 2015 Collaborators: Jonathan Bradley
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 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 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 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 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 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 informationAsymptotic Multivariate Kriging Using Estimated Parameters with Bayesian Prediction Methods for Non-linear Predictands
Asymptotic Multivariate Kriging Using Estimated Parameters with Bayesian Prediction Methods for Non-linear Predictands Elizabeth C. Mannshardt-Shamseldin Advisor: Richard L. Smith Duke University Department
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 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 informationHierarchical Low Rank Approximation of Likelihoods for Large Spatial Datasets
Hierarchical Low Rank Approximation of Likelihoods for Large Spatial Datasets Huang Huang and Ying Sun CEMSE Division, King Abdullah University of Science and Technology, Thuwal, 23955, Saudi Arabia. July
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 informationA Bayesian Spatio-Temporal Geostatistical Model with an Auxiliary Lattice for Large Datasets
Statistica Sinica (2013): Preprint 1 A Bayesian Spatio-Temporal Geostatistical Model with an Auxiliary Lattice for Large Datasets Ganggang Xu 1, Faming Liang 1 and Marc G. Genton 2 1 Texas A&M University
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 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 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 informationSpatial Backfitting of Roller Measurement Values from a Florida Test Bed
Spatial Backfitting of Roller Measurement Values from a Florida Test Bed Daniel K. Heersink 1, Reinhard Furrer 1, and Mike A. Mooney 2 1 Institute of Mathematics, University of Zurich, CH-8057 Zurich 2
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 informationCOVARIANCE APPROXIMATION FOR LARGE MULTIVARIATE SPATIAL DATA SETS WITH AN APPLICATION TO MULTIPLE CLIMATE MODEL ERRORS 1
The Annals of Applied Statistics 2011, Vol. 5, No. 4, 2519 2548 DOI: 10.1214/11-AOAS478 Institute of Mathematical Statistics, 2011 COVARIANCE APPROXIMATION FOR LARGE MULTIVARIATE SPATIAL DATA SETS WITH
More informationModel Selection for Geostatistical Models
Model Selection for Geostatistical Models Richard A. Davis Colorado State University http://www.stat.colostate.edu/~rdavis/lectures Joint work with: Jennifer A. Hoeting, Colorado State University Andrew
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 informationLearning gradients: prescriptive models
Department of Statistical Science Institute for Genome Sciences & Policy Department of Computer Science Duke University May 11, 2007 Relevant papers Learning Coordinate Covariances via Gradients. Sayan
More informationA full-scale approximation of covariance functions for large spatial data sets
A full-scale approximation of covariance functions for large spatial data sets Huiyan Sang Department of Statistics, Texas A&M University, College Station, USA. Jianhua Z. Huang Department of Statistics,
More informationStatistical modeling of MODIS cloud data using the spatial random effects model
University of Wollongong Research Online Centre for Statistical & Survey Methodology Working Paper Series Faculty of Engineering and Information Sciences 2013 Statistical modeling of MODIS cloud data using
More informationA Multi-resolution Gaussian process model for the analysis of large spatial data sets.
1 2 3 4 A Multi-resolution Gaussian process model for the analysis of large spatial data sets. Douglas Nychka, Soutir Bandyopadhyay, Dorit Hammerling, Finn Lindgren, and Stephan Sain August 13, 2013 5
More informationESTIMATING THE MEAN LEVEL OF FINE PARTICULATE MATTER: AN APPLICATION OF SPATIAL STATISTICS
ESTIMATING THE MEAN LEVEL OF FINE PARTICULATE MATTER: AN APPLICATION OF SPATIAL STATISTICS Richard L. Smith Department of Statistics and Operations Research University of North Carolina Chapel Hill, N.C.,
More informationMA 575 Linear Models: Cedric E. Ginestet, Boston University Mixed Effects Estimation, Residuals Diagnostics Week 11, Lecture 1
MA 575 Linear Models: Cedric E Ginestet, Boston University Mixed Effects Estimation, Residuals Diagnostics Week 11, Lecture 1 1 Within-group Correlation Let us recall the simple two-level hierarchical
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 informationLinear Regression. In this problem sheet, we consider the problem of linear regression with p predictors and one intercept,
Linear Regression In this problem sheet, we consider the problem of linear regression with p predictors and one intercept, y = Xβ + ɛ, where y t = (y 1,..., y n ) is the column vector of target values,
More informationSpatial Extremes in Atmospheric Problems
Spatial Extremes in Atmospheric Problems Eric Gilleland Research Applications Laboratory (RAL) National Center for Atmospheric Research (NCAR), Boulder, Colorado, U.S.A. http://www.ral.ucar.edu/staff/ericg
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 informationShort Questions (Do two out of three) 15 points each
Econometrics Short Questions Do two out of three) 5 points each ) Let y = Xβ + u and Z be a set of instruments for X When we estimate β with OLS we project y onto the space spanned by X along a path orthogonal
More informationFast Direct Methods for Gaussian Processes
Fast Direct Methods for Gaussian Processes Mike O Neil Departments of Mathematics New York University oneil@cims.nyu.edu December 12, 2015 1 Collaborators This is joint work with: Siva Ambikasaran Dan
More informationHierarchical Matrices. Jon Cockayne April 18, 2017
Hierarchical Matrices Jon Cockayne April 18, 2017 1 Sources Introduction to Hierarchical Matrices with Applications [Börm et al., 2003] 2 Sources Introduction to Hierarchical Matrices with Applications
More informationGaussian Multiscale Spatio-temporal Models for Areal Data
Gaussian Multiscale Spatio-temporal Models for Areal Data (University of Missouri) Scott H. Holan (University of Missouri) Adelmo I. Bertolde (UFES) Outline Motivation Multiscale factorization The multiscale
More informationTAMS39 Lecture 2 Multivariate normal distribution
TAMS39 Lecture 2 Multivariate normal distribution Martin Singull Department of Mathematics Mathematical Statistics Linköping University, Sweden Content Lecture Random vectors Multivariate normal distribution
More informationA BAYESIAN SPATIO-TEMPORAL GEOSTATISTICAL MODEL WITH AN AUXILIARY LATTICE FOR LARGE DATASETS
Statistica Sinica 25 (2015), 61-79 doi:http://dx.doi.org/10.5705/ss.2013.085w A BAYESIAN SPATIO-TEMPORAL GEOSTATISTICAL MODEL WITH AN AUXILIARY LATTICE FOR LARGE DATASETS Ganggang Xu 1, Faming Liang 1
More informationA Sequential Split-Conquer-Combine Approach for Analysis of Big Spatial Data
A Sequential Split-Conquer-Combine Approach for Analysis of Big Spatial Data Min-ge Xie Department of Statistics & Biostatistics Rutgers, The State University of New Jersey In collaboration with Xuying
More informationGaussian Process Regression Model in Spatial Logistic Regression
Journal of Physics: Conference Series PAPER OPEN ACCESS Gaussian Process Regression Model in Spatial Logistic Regression To cite this article: A Sofro and A Oktaviarina 018 J. Phys.: Conf. Ser. 947 01005
More informationModified Linear Projection for Large Spatial Data Sets
Modified Linear Projection for Large Spatial Data Sets Toshihiro Hirano July 15, 2014 arxiv:1402.5847v2 [stat.me] 13 Jul 2014 Abstract Recent developments in engineering techniques for spatial data collection
More informationSPATIO-TEMPORAL METHODS IN CLIMATOLOGY. Christopher K. Wikle Department of Statistics, University of Missouri Columbia, USA
SPATIO-TEMPORAL METHODS IN CLIMATOLOGY Christopher K. Wikle Department of Statistics, University of Missouri Columbia, USA Keywords: Data assimilation, Empirical orthogonal function, Principal oscillation
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 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 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 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 informationEcon 2148, fall 2017 Gaussian process priors, reproducing kernel Hilbert spaces, and Splines
Econ 2148, fall 2017 Gaussian process priors, reproducing kernel Hilbert spaces, and Splines Maximilian Kasy Department of Economics, Harvard University 1 / 37 Agenda 6 equivalent representations of the
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 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 information*Department Statistics and Operations Research (UPC) ** Department of Economics and Economic History (UAB)
Wind power: Exploratory space-time analysis with M. P. Muñoz*, J. A. Sànchez*, M. Gasulla*, M. D. Márquez** *Department Statistics and Operations Research (UPC) ** Department of Economics and Economic
More informationA Generalized Convolution Model for Multivariate Nonstationary Spatial Processes
A Generalized Convolution Model for Multivariate Nonstationary Spatial Processes Anandamayee Majumdar, Debashis Paul and Dianne Bautista Department of Mathematics and Statistics, Arizona State University,
More informationSummary STK 4150/9150
STK4150 - Intro 1 Summary STK 4150/9150 Odd Kolbjørnsen May 22 2017 Scope You are expected to know and be able to use basic concepts introduced in the book. You knowledge is expected to be larger than
More informationBasics of Point-Referenced Data Models
Basics of Point-Referenced Data Models Basic tool is a spatial process, {Y (s), s D}, where D R r Chapter 2: Basics of Point-Referenced Data Models p. 1/45 Basics of Point-Referenced Data Models Basic
More informationOn fixed effects estimation in spline-based semiparametric regression for spatial data
Libraries Conference on Applied Statistics in Agriculture 015-7th Annual Conference Proceedings On fixed effects estimation in spline-based semiparametric regression for spatial data Guilherme Ludwig University
More informationTutorial on Fixed Rank Kriging (FRK) of CO 2 data. M. Katzfuss, The Ohio State University N. Cressie, The Ohio State University
Tutorial on Fixed Rank Kriging (FRK) of CO 2 data M. Katzfuss, The Ohio State University N. Cressie, The Ohio State University Technical Report No. 858 July, 20 Department of Statistics The Ohio State
More informationSpatial Statistics with Image Analysis. Lecture L02. Computer exercise 0 Daily Temperature. Lecture 2. Johan Lindström.
C Stochastic fields Covariance Spatial Statistics with Image Analysis Lecture 2 Johan Lindström November 4, 26 Lecture L2 Johan Lindström - johanl@maths.lth.se FMSN2/MASM2 L /2 C Stochastic fields Covariance
More informationLecture 14 Bayesian Models for Spatio-Temporal Data
Lecture 14 Bayesian Models for Spatio-Temporal Data Dennis Sun Stats 253 August 13, 2014 Outline of Lecture 1 Recap of Bayesian Models 2 Empirical Bayes 3 Case 1: Long-Lead Forecasting of Sea Surface Temperatures
More informationHandbook of Spatial Statistics Chapter 2: Continuous Parameter Stochastic Process Theory by Gneiting and Guttorp
Handbook of Spatial Statistics Chapter 2: Continuous Parameter Stochastic Process Theory by Gneiting and Guttorp Marcela Alfaro Córdoba August 25, 2016 NCSU Department of Statistics Continuous Parameter
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 informationCross-covariance Functions for Tangent Vector Fields on the Sphere
Cross-covariance Functions for Tangent Vector Fields on the Sphere Minjie Fan 1 Tomoko Matsuo 2 1 Department of Statistics University of California, Davis 2 Cooperative Institute for Research in Environmental
More informationA general science-based framework for dynamical spatio-temporal models
Test (2010) 19: 417 451 DOI 10.1007/s11749-010-0209-z INVITED PAPER A general science-based framework for dynamical spatio-temporal models Christopher K. Wikle Mevin B. Hooten Received: 2 October 2009
More informationRegularized Estimation of High Dimensional Covariance Matrices. Peter Bickel. January, 2008
Regularized Estimation of High Dimensional Covariance Matrices Peter Bickel Cambridge January, 2008 With Thanks to E. Levina (Joint collaboration, slides) I. M. Johnstone (Slides) Choongsoon Bae (Slides)
More informationKriging Luc Anselin, All Rights Reserved
Kriging Luc Anselin Spatial Analysis Laboratory Dept. Agricultural and Consumer Economics University of Illinois, Urbana-Champaign http://sal.agecon.uiuc.edu Outline Principles Kriging Models Spatial Interpolation
More informationA Probability Review
A Probability Review Outline: A probability review Shorthand notation: RV stands for random variable EE 527, Detection and Estimation Theory, # 0b 1 A Probability Review Reading: Go over handouts 2 5 in
More informationClimate Change: the Uncertainty of Certainty
Climate Change: the Uncertainty of Certainty Reinhard Furrer, UZH JSS, Geneva Oct. 30, 2009 Collaboration with: Stephan Sain - NCAR Reto Knutti - ETHZ Claudia Tebaldi - Climate Central Ryan Ford, Doug
More informationFixed rank filtering for spatio-temporal data
University of Wollongong Research Online Faculty of Informatics - Papers (Archive) Faculty of Engineering and Information Sciences 2010 Fixed rank filtering for spatio-temporal data Noel Cressie University
More informationSparse Covariance Selection using Semidefinite Programming
Sparse Covariance Selection using Semidefinite Programming A. d Aspremont ORFE, Princeton University Joint work with O. Banerjee, L. El Ghaoui & G. Natsoulis, U.C. Berkeley & Iconix Pharmaceuticals Support
More informationA HIERARCHICAL MAX-STABLE SPATIAL MODEL FOR EXTREME PRECIPITATION
Submitted to the Annals of Applied Statistics arxiv: arxiv:0000.0000 A HIERARCHICAL MAX-STABLE SPATIAL MODEL FOR EXTREME PRECIPITATION By Brian J. Reich and Benjamin A. Shaby North Carolina State University
More informationA multi-resolution Gaussian process model for the analysis of large spatial data sets.
National Science Foundation A multi-resolution Gaussian process model for the analysis of large spatial data sets. Doug Nychka Soutir Bandyopadhyay Dorit Hammerling Finn Lindgren Stephen Sain NCAR/TN-504+STR
More informationAnalysing geoadditive regression data: a mixed model approach
Analysing geoadditive regression data: a mixed model approach Institut für Statistik, Ludwig-Maximilians-Universität München Joint work with Ludwig Fahrmeir & Stefan Lang 25.11.2005 Spatio-temporal regression
More informationDouglas Nychka, Soutir Bandyopadhyay, Dorit Hammerling, Finn Lindgren, and Stephan Sain. October 10, 2012
A multi-resolution Gaussian process model for the analysis of large spatial data sets. Douglas Nychka, Soutir Bandyopadhyay, Dorit Hammerling, Finn Lindgren, and Stephan Sain October 10, 2012 Abstract
More informationKarhunen-Loève Approximation of Random Fields Using Hierarchical Matrix Techniques
Institut für Numerische Mathematik und Optimierung Karhunen-Loève Approximation of Random Fields Using Hierarchical Matrix Techniques Oliver Ernst Computational Methods with Applications Harrachov, CR,
More informationExperiences with Model Reduction and Interpolation
Experiences with Model Reduction and Interpolation Paul Constantine Stanford University, Sandia National Laboratories Qiqi Wang (MIT) David Gleich (Purdue) Emory University March 7, 2012 Joe Ruthruff (SNL)
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 informationMIT Spring 2015
Regression Analysis MIT 18.472 Dr. Kempthorne Spring 2015 1 Outline Regression Analysis 1 Regression Analysis 2 Multiple Linear Regression: Setup Data Set n cases i = 1, 2,..., n 1 Response (dependent)
More information