PICAR: An Efficient Extendable Approach for Fitting Hierarchical Spatial Models

Hierarchical spatial models are very flexible and popular for a vast array of applications in areas such as ecology, social science, public health, and atmospheric science. It is common to carry out Bayesian inference for these models via Markov chain Monte Carlo (MCMC). Each iteration of the MCMC a...

Ausführliche Beschreibung

Gespeichert in:
Bibliographische Detailangaben
Veröffentlicht in:Technometrics Jg. 64; H. 2; S. 187 - 198
Hauptverfasser: Lee, Ben Seiyon, Haran, Murali
Format: Journal Article
Sprache:Englisch
Veröffentlicht: Alexandria Taylor & Francis 03.04.2022
American Society for Quality
Schlagworte:
Algorithms
Arrays
Atmospheric models
Basis functions
Basis representation
Bayesian analysis
Cloud cover
Computational efficiency
Computing costs
Cross correlation
Fields (mathematics)
Finite element method
Gaussian random field
Iterative methods
Markov analysis
Markov chain Monte Carlo
Markov chains
Monte Carlo simulation
Non-Gaussian spatial data
Ordinal spatial data
Predictions
Programming languages
Public health
Spatial data
Spatially varying coefficients
Statistical inference
ISSN:0040-1706, 1537-2723
Online-Zugang:Volltext
Tags: Tag hinzufügen
Keine Tags, Fügen Sie den ersten Tag hinzu!
Abstract Hierarchical spatial models are very flexible and popular for a vast array of applications in areas such as ecology, social science, public health, and atmospheric science. It is common to carry out Bayesian inference for these models via Markov chain Monte Carlo (MCMC). Each iteration of the MCMC algorithm is computationally expensive due to costly matrix operations. In addition, the MCMC algorithm needs to be run for more iterations because the strong cross-correlations among the spatial latent variables result in slow mixing Markov chains. To address these computational challenges, we propose a projection-based intrinsic conditional autoregression (PICAR) approach, which is a discretized and dimension-reduced representation of the underlying spatial random field using empirical basis functions on a triangular mesh. Our approach exhibits fast mixing as well as a considerable reduction in computational cost per iteration. PICAR is computationally efficient and scales well to high dimensions. It is also automated and easy to implement for a wide array of user-specified hierarchical spatial models. We show, via simulation studies, that our approach performs well in terms of parameter inference and prediction. We provide several examples to illustrate the applicability of our method, including (i) a high-dimensional cloud cover dataset that showcases its computational efficiency, (ii) a spatially varying coefficient model that demonstrates the ease of implementation of PICAR in the probabilistic programming languages stan and nimble, and (iii) a watershed survey example that illustrates how PICAR applies to models that are not amenable to efficient inference via existing methods.
AbstractList Hierarchical spatial models are very flexible and popular for a vast array of applications in areas such as ecology, social science, public health, and atmospheric science. It is common to carry out Bayesian inference for these models via Markov chain Monte Carlo (MCMC). Each iteration of the MCMC algorithm is computationally expensive due to costly matrix operations. In addition, the MCMC algorithm needs to be run for more iterations because the strong cross-correlations among the spatial latent variables result in slow mixing Markov chains. To address these computational challenges, we propose a projection-based intrinsic conditional autoregression (PICAR) approach, which is a discretized and dimension-reduced representation of the underlying spatial random field using empirical basis functions on a triangular mesh. Our approach exhibits fast mixing as well as a considerable reduction in computational cost per iteration. PICAR is computationally efficient and scales well to high dimensions. It is also automated and easy to implement for a wide array of user-specified hierarchical spatial models. We show, via simulation studies, that our approach performs well in terms of parameter inference and prediction. We provide several examples to illustrate the applicability of our method, including (i) a high-dimensional cloud cover dataset that showcases its computational efficiency, (ii) a spatially varying coefficient model that demonstrates the ease of implementation of PICAR in the probabilistic programming languages stan and nimble, and (iii) a watershed survey example that illustrates how PICAR applies to models that are not amenable to efficient inference via existing methods.
Author Haran, Murali
Lee, Ben Seiyon
Author_xml – sequence: 1
  givenname: Ben Seiyon
  orcidid: 0000-0003-0658-7458
  surname: Lee
  fullname: Lee, Ben Seiyon
  organization: Department of Statistics, George Mason University
– sequence: 2
  givenname: Murali
  surname: Haran
  fullname: Haran, Murali
  organization: Department of Statistics, Pennsylvania State University
BookMark eNqFkMtKAzEUhoNUsFUfQRhwPTWXySSjG0tpVVAqXtbhmEk0Mk3GTIr27Z3SunGhq3_zX875RmjggzcInRA8JljiM4wLTAQuxxRTMiYVY7wq99CQcCZyKigboOHGk29MB2jUde8YE0alGKLF_c108nCeTXw2s9ZpZ3zKZl_J-BpeGpNN2jYG0G-ZDTGbu5Scf82unYkQ9ZvT0GSPLSTX612oTdMdoX0LTWeOd3qInuezp-l1fru46pduc82YTLkpLGGl5biwoibMEi7Fiyg5qa3EIACDtJJLbTgA5xWRmjKDBaFCSAJ1wQ7R6ba3P-9jZbqk3sMq-n5SUVlKWpVFUfYuvnXpGLouGqva6JYQ14pgtWGnftipDTu1Y9fnLn7ltEv9m8GnCK75N325TTvfU1vCZ4hNrRKsmxBtBK9dp9jfFd-eoIc9
CitedBy_id crossref_primary_10_1002_env_2703
crossref_primary_10_1088_2752_5295_adf56e
crossref_primary_10_1080_10618600_2024_2365728
crossref_primary_10_1016_j_ecosta_2024_11_001
crossref_primary_10_1007_s13253_024_00619_9
crossref_primary_10_1080_00401706_2022_2115558
crossref_primary_10_1007_s11222_023_10227_1
crossref_primary_10_1214_22_BA1357
crossref_primary_10_3390_rs16010069
crossref_primary_10_1214_25_BA1532
Cites_doi 10.1080/01621459.1993.10476321
10.1093/biomet/asaa008
10.1080/10618600.2014.914946
10.18637/jss.v076.i01
10.1198/016214503000170
10.1080/01621459.2019.1677471
10.1198/106186006X100470
10.1002/(SICI)1097-0258(19970415)16:7<741::AID-SIM501>3.0.CO;2-1
10.1111/anzs.12148
10.1016/j.spasta.2013.02.002
10.1007/s13253-018-00348-w
10.1023/A:1009662704779
10.1080/01621459.2015.1123632
10.1002/env.1131
10.1111/j.1467-9868.2011.00777.x
10.1007/978-3-642-17086-7_3
10.1007/978-3-540-24806-4_4
10.1016/S0167-9473(99)00103-6
10.1080/01621459.2015.1044091
10.1016/j.csda.2020.107081
10.1080/10618600.2016.1172487
10.1093/biomet/ass068
10.1111/j.1467-9868.2012.01041.x
10.1093/biomet/asr013
10.1111/j.1467-9868.2007.00633.x
10.1111/j.1467-9868.2008.00663.x
10.1111/1467-9876.00113
10.1111/rssc.12069
10.1023/A:1009666805688
10.1080/01621459.2017.1409123
10.1111/j.2517-6161.1996.tb02080.x
10.1016/j.spasta.2013.06.003
10.18637/jss.v063.i19
10.1002/env.2485
10.1080/10618600.2018.1425625
10.1198/1061860031833
10.1111/j.1467-9868.2008.00700.x
10.1137/1.9780898719628
10.1201/b17115
10.1007/BF00116466
10.1198/106186006X132178
10.1007/978-1-4612-1494-6
10.1080/10618600.2012.719844
10.1214/16-SS115
10.1002/(SICI)1097-0258(19980930)17:18<2025::AID-SIM865>3.0.CO;2-M
10.1093/biomet/37.1-2.17
10.1198/016214507000000888
10.1016/j.csda.2010.02.024
ContentType Journal Article
Copyright 2021 American Statistical Association and the American Society for Quality 2021
Copyright American Society for Quality May 2022
Copyright_xml – notice: 2021 American Statistical Association and the American Society for Quality 2021
– notice: Copyright American Society for Quality May 2022
DBID AAYXX
CITATION
DOI 10.1080/00401706.2021.1933596
DatabaseName CrossRef
DatabaseTitle CrossRef
DatabaseTitleList

DeliveryMethod fulltext_linktorsrc
Discipline Engineering
Statistics
Mathematics
Public Health
EISSN 1537-2723
EndPage 198
ExternalDocumentID 10_1080_00401706_2021_1933596
1933596
Genre Research Article
GroupedDBID -ET
-~X
..I
.7F
.DC
.QJ
07G
0BK
0R~
123
29Q
30N
4.4
5RE
7WY
85S
8FL
96U
AAAVZ
AAENE
AAGDL
AAHIA
AAJMT
AALDU
AAMIU
AAPUL
AAQRR
ABCCY
ABEHJ
ABFAN
ABFIM
ABJNI
ABLIJ
ABPAQ
ABPEM
ABPPZ
ABTAI
ABXUL
ABXYU
ABYWD
ACBEA
ACGFO
ACGFS
ACGOD
ACIWK
ACMTB
ACNCT
ACTIO
ACTMH
ADCVX
ADGTB
AEGXH
AEISY
AELLO
AENEX
AEOZL
AEPSL
AEYOC
AFAZI
AFRVT
AFVYC
AGDLA
AGMYJ
AHDZW
AIAGR
AIJEM
AIYEW
AKBRZ
AKBVH
AKOOK
ALMA_UNASSIGNED_HOLDINGS
ALQZU
AMVHM
AMXXU
AQRUH
AQTUD
AVBZW
AWYRJ
BCCOT
BLEHA
BPLKW
C06
CCCUG
CS3
DGEBU
DKSSO
DU5
DWIFK
EBS
E~A
E~B
F5P
GTTXZ
H13
HFX
HF~
HZ~
H~P
I-F
IHF
IPNFZ
J.P
JAA
K60
K6~
KYCEM
LJTGL
M4Z
MS~
MW2
NA5
NY~
O9-
P2P
RIG
RNANH
ROSJB
RTWRZ
RWL
S-T
SNACF
TAE
TAQ
TASJS
TBQAZ
TDBHL
TEJ
TFL
TFT
TFW
TN5
TOXWX
TTHFI
TUROJ
U5U
UB9
UT5
UU3
WH7
WZA
YNT
ZGOLN
~02
~S~
AAYXX
CITATION
ID FETCH-LOGICAL-c338t-e4f136f504f7d13f1587b7651df80a7a0a8f858ce5aa55918c23e07127781ad43
IEDL.DBID TFW
ISICitedReferencesCount 10
ISICitedReferencesURI http://www.webofscience.com/api/gateway?GWVersion=2&SrcApp=Summon&SrcAuth=ProQuest&DestLinkType=CitingArticles&DestApp=WOS_CPL&KeyUT=000681170200001&url=https%3A%2F%2Fcvtisr.summon.serialssolutions.com%2F%23%21%2Fsearch%3Fho%3Df%26include.ft.matches%3Dt%26l%3Dnull%26q%3D
ISSN 0040-1706
IngestDate Mon Nov 10 02:56:09 EST 2025
Sat Nov 29 03:42:51 EST 2025
Tue Nov 18 22:17:26 EST 2025
Mon Oct 20 23:47:07 EDT 2025
IsPeerReviewed true
IsScholarly true
Issue 2
Language English
LinkModel DirectLink
MergedId FETCHMERGED-LOGICAL-c338t-e4f136f504f7d13f1587b7651df80a7a0a8f858ce5aa55918c23e07127781ad43
Notes ObjectType-Article-1
SourceType-Scholarly Journals-1
ObjectType-Feature-2
content type line 14
ORCID 0000-0003-0658-7458
PQID 2868296446
PQPubID 24108
PageCount 12
ParticipantIDs informaworld_taylorfrancis_310_1080_00401706_2021_1933596
crossref_citationtrail_10_1080_00401706_2021_1933596
proquest_journals_2868296446
crossref_primary_10_1080_00401706_2021_1933596
PublicationCentury 2000
PublicationDate 2022-04-03
PublicationDateYYYYMMDD 2022-04-03
PublicationDate_xml – month: 04
  year: 2022
  text: 2022-04-03
  day: 03
PublicationDecade 2020
PublicationPlace Alexandria
PublicationPlace_xml – name: Alexandria
PublicationTitle Technometrics
PublicationYear 2022
Publisher Taylor & Francis
American Society for Quality
Publisher_xml – name: Taylor & Francis
– name: American Society for Quality
References CIT0030
CIT0032
Brenner S. (CIT0011) 2007; 15
CIT0034
Cressie N. (CIT0015) 2015
CIT0036
CIT0035
CIT0038
CIT0037
CIT0039
CIT0041
CIT0040
CIT0043
CIT0042
CIT0001
CIT0044
Stribling J. B. (CIT0053) 1998
Tibshirani R. (CIT0055) 1994; 58
CIT0003
Park J. (CIT0045) 2020
CIT0047
CIT0002
CIT0046
CIT0005
CIT0049
CIT0004
CIT0048
CIT0007
CIT0006
CIT0009
CIT0008
CIT0050
CIT0052
CIT0051
CIT0010
CIT0054
CIT0012
CIT0056
Williams C. (CIT0057) 2001; 13
CIT0014
CIT0058
CIT0013
Hjelle Ø. (CIT0031) 2006
CIT0016
CIT0059
CIT0018
CIT0017
CIT0019
CIT0021
CIT0020
CIT0023
CIT0022
CIT0025
CIT0024
CIT0027
CIT0026
CIT0029
CIT0028
Johnson V. E. (CIT0033) 2006
References_xml – ident: CIT0040
– volume: 15
  volume-title: The Mathematical Theory of Finite Element Methods
  year: 2007
  ident: CIT0011
– ident: CIT0001
  doi: 10.1080/01621459.1993.10476321
– ident: CIT0035
  doi: 10.1093/biomet/asaa008
– volume: 13
  start-page: 682
  volume-title: Advances in Neural Information Processing Systems
  year: 2001
  ident: CIT0057
– ident: CIT0044
  doi: 10.1080/10618600.2014.914946
– volume-title: Ordinal Data Modeling
  year: 2006
  ident: CIT0033
– ident: CIT0002
– ident: CIT0012
  doi: 10.18637/jss.v076.i01
– ident: CIT0021
  doi: 10.1198/016214503000170
– ident: CIT0010
  doi: 10.1080/01621459.2019.1677471
– ident: CIT0013
  doi: 10.1198/106186006X100470
– ident: CIT0006
  doi: 10.1002/(SICI)1097-0258(19970415)16:7<741::AID-SIM501>3.0.CO;2-1
– ident: CIT0049
  doi: 10.1111/anzs.12148
– ident: CIT0048
  doi: 10.1016/j.spasta.2013.02.002
– ident: CIT0027
  doi: 10.1007/s13253-018-00348-w
– ident: CIT0056
  doi: 10.1023/A:1009662704779
– ident: CIT0034
  doi: 10.1080/01621459.2015.1123632
– ident: CIT0025
  doi: 10.1002/env.1131
– ident: CIT0038
  doi: 10.1111/j.1467-9868.2011.00777.x
– ident: CIT0054
  doi: 10.1007/978-3-642-17086-7_3
– start-page: 1
  year: 2020
  ident: CIT0045
  publication-title: Journal of Computational and Graphical Statistics
– ident: CIT0022
  doi: 10.1007/978-3-540-24806-4_4
– volume-title: Development of a Benthic Index of Biotic Integrity for Maryland Streams (CBWP-MANTA EA-98-3
  year: 1998
  ident: CIT0053
– ident: CIT0017
  doi: 10.1016/S0167-9473(99)00103-6
– ident: CIT0016
  doi: 10.1080/01621459.2015.1044091
– ident: CIT0059
  doi: 10.1016/j.csda.2020.107081
– ident: CIT0018
  doi: 10.1080/10618600.2016.1172487
– ident: CIT0003
  doi: 10.1093/biomet/ass068
– ident: CIT0032
  doi: 10.1111/j.1467-9868.2012.01041.x
– ident: CIT0008
  doi: 10.1093/biomet/asr013
– ident: CIT0014
  doi: 10.1111/j.1467-9868.2007.00633.x
– ident: CIT0005
  doi: 10.1111/j.1467-9868.2008.00663.x
– ident: CIT0019
  doi: 10.1111/1467-9876.00113
– ident: CIT0043
  doi: 10.1111/rssc.12069
– ident: CIT0046
– ident: CIT0028
  doi: 10.1023/A:1009666805688
– ident: CIT0039
  doi: 10.1080/01621459.2017.1409123
– volume: 58
  start-page: 267
  year: 1994
  ident: CIT0055
  publication-title: Journal of the Royal Statistical Society, Series B
  doi: 10.1111/j.2517-6161.1996.tb02080.x
– volume-title: Triangulations and Applications, Springer Science & Business Media
  year: 2006
  ident: CIT0031
– ident: CIT0052
  doi: 10.1016/j.spasta.2013.06.003
– volume-title: Statistics for Spatio-Temporal Data
  year: 2015
  ident: CIT0015
– ident: CIT0037
  doi: 10.18637/jss.v063.i19
– ident: CIT0042
  doi: 10.1002/env.2485
– ident: CIT0023
  doi: 10.1080/10618600.2018.1425625
– ident: CIT0026
  doi: 10.1198/1061860031833
– ident: CIT0047
  doi: 10.1111/j.1467-9868.2008.00700.x
– ident: CIT0036
  doi: 10.1137/1.9780898719628
– ident: CIT0004
  doi: 10.1201/b17115
– ident: CIT0007
  doi: 10.1007/BF00116466
– ident: CIT0020
  doi: 10.1198/106186006X132178
– ident: CIT0050
  doi: 10.1007/978-1-4612-1494-6
– ident: CIT0024
– ident: CIT0051
  doi: 10.1080/10618600.2012.719844
– ident: CIT0009
  doi: 10.1214/16-SS115
– ident: CIT0058
  doi: 10.1002/(SICI)1097-0258(19980930)17:18<2025::AID-SIM865>3.0.CO;2-M
– ident: CIT0041
  doi: 10.1093/biomet/37.1-2.17
– ident: CIT0029
  doi: 10.1198/016214507000000888
– ident: CIT0030
  doi: 10.1016/j.csda.2010.02.024
SSID ssj0013287
Score 2.3989358
Snippet Hierarchical spatial models are very flexible and popular for a vast array of applications in areas such as ecology, social science, public health, and...
SourceID proquest
crossref
informaworld
SourceType Aggregation Database
Enrichment Source
Index Database
Publisher
StartPage 187
SubjectTerms Algorithms
Arrays
Atmospheric models
Basis functions
Basis representation
Bayesian analysis
Cloud cover
Computational efficiency
Computing costs
Cross correlation
Fields (mathematics)
Finite element method
Gaussian random field
Iterative methods
Markov analysis
Markov chain Monte Carlo
Markov chains
Monte Carlo simulation
Non-Gaussian spatial data
Ordinal spatial data
Predictions
Programming languages
Public health
Spatial data
Spatially varying coefficients
Statistical inference
Title PICAR: An Efficient Extendable Approach for Fitting Hierarchical Spatial Models
URI https://www.tandfonline.com/doi/abs/10.1080/00401706.2021.1933596
https://www.proquest.com/docview/2868296446
Volume 64
WOSCitedRecordID wos000681170200001&url=https%3A%2F%2Fcvtisr.summon.serialssolutions.com%2F%23%21%2Fsearch%3Fho%3Df%26include.ft.matches%3Dt%26l%3Dnull%26q%3D
hasFullText 1
inHoldings 1
isFullTextHit
isPrint
journalDatabaseRights – providerCode: PRVAWR
  databaseName: Taylor & Francis Journals Complete
  customDbUrl:
  eissn: 1537-2723
  dateEnd: 99991231
  omitProxy: false
  ssIdentifier: ssj0013287
  issn: 0040-1706
  databaseCode: TFW
  dateStart: 19590201
  isFulltext: true
  titleUrlDefault: https://www.tandfonline.com
  providerName: Taylor & Francis
link http://cvtisr.summon.serialssolutions.com/2.0.0/link/0/eLvHCXMwrV3NS8MwFA8yPMyDH1NxOiUHrx1N0jSptzJWJugcMnW3kmYNDMaUdfr3-5K2Y0NkBz328EL68j6T934PoVvw-VFmIubRUIReoI3xshCyFIg0DOdC89B1yL0-iOFQTibRqKomLKqySptDmxIowtlqq9wqK-qKOIvH41BfILujpAsRCOORBd0G129Vc5y8bbwjSFHXzVmSuofnt1W2vNMWdukPW-0cUHL0D1s_RodV9InjUlxO0F6-aKGDDUxC-HpcA7kWLdS0wWiJ5XyKnkb3vfj5DscL3HfIE-CwcN_dotsGLBxX-OQY_gcnM1dRjQcz2-LsJq7MsR2ADAKP7QS2eXGGXpL-uDfwqoEMnoZMduXlgSEsNNwPjJgSZgiXIhMhJ1MjfSWUr6SRXOqcKwWZCpGashxiGCqEJGoasHPUWLwv8guE80yBsaGKg1BATAhcYNTXJMqnVHPFRBsF9UGkukIrt0Mz5ilZg5qWrEwtK9OKlW3UXZN9lHAduwiizVNOV-6exJRDTVK2g7ZTi0RaaX6RUhlK-5QdhJd_WPoKNants7AlQqyDGqvlZ36N9vUXHPnyxsn4N92R8kE
linkProvider Taylor & Francis
linkToHtml http://cvtisr.summon.serialssolutions.com/2.0.0/link/0/eLvHCXMwrV1LSwMxEB58gXrwLb7NwetKk2we661IS8VaRerjFtJ0A4VSxVZ_v5PsbqmIeNDjHiZkJ5PMTDLzfQBn6POzns94wqSSSeq8T3oSsxSMNLwQygkZO-Qe26rT0c_P2WwvTCirDDm0L4Ai4lkdNne4jK5K4gIgT4R9wfSO0XMMQbjI5DwsRnY6tOlu82nmJUGrqnIuyFRdPD8N88U_fUEv_XZaRxfUXP-PyW_AWhmAknphMZswl4-2YHUGlhC_bqZYruMtWAnxaAHnvA23d1eX9fsLUh-RRgSfQJ9FGvEiPfRgkXoJUU7wh0hzEIuqSWsQupwj6cqQBA5ktHkSSNiG4x14aDa6l62k5GRIHCazkyRPPeXSi1rqVZ9yT4VWPSUF7Xtds8rWrPZaaJcLazFZodoxnmMYw5TS1PZTvgsLo5dRvgck71k8b5gVaBcYFqIWOKs5muV95oTlah_SaiWMKwHLA2_G0NAprmmhShNUaUpV7sP5VOy1QOz4TSCbXWYziVclvuA1MfwX2aPKJky5-ceGaanDa3YqD_4w9Ckst7o3bdO-6lwfwgoLbRehYogfwcLk7T0_hiX3gcv_dhIN_hNc0PZv
linkToPdf http://cvtisr.summon.serialssolutions.com/2.0.0/link/0/eLvHCXMwrV1LTxsxEB6VFCE4tLwq0lLwgetGsb1-LLeIZkXUkEZVeNwsx7uWkKIUkcDvZ-zdjYJQxaE97mEs73js-cae-QbgDH1-NvUZT5hUMkmd98lUYpSCSMMLoZyQsULuZqhGI313l43rbMJFnVYZYmhfEUXEszps7ofCNxlxgY8nsr5gdMdoBxEIF5ncgI8InWXYm5P8du0hQasmcS7INEU8fxvmlXt6RV765rCOHij__B_mvgufavhJepW97MGHcr4PO2ukhPh1tWJyXezDdkCjFZnzAfwaDy56v89Jb076kXoCPRbpx2v0UIFFejVBOcH_Ifl9TKkml_ehxjm2XJmR0AEZLZ6EFmyzxSFc5_3JxWVSd2RIHIayy6RMPeXSi27qVUG5p0KrqZKCFl53rbJdq70W2pXCWgxVqHaMlwhimFKa2iLlX6A1_zMvj4CUU4unDbMCrQJBIWqBs66jWVkwJyxXbUibhTCupisPXTNmhq5YTStVmqBKU6uyDZ2V2EPF1_GeQLa-ymYZL0p81dXE8HdkjxuTMPXWXximpQ5v2an8-g9Dn8LW-EduhoPRz2-wzULNRUgX4sfQWj4-ld9h0z3j6j-eRHN_AU6r9Rg
openUrl ctx_ver=Z39.88-2004&ctx_enc=info%3Aofi%2Fenc%3AUTF-8&rfr_id=info%3Asid%2Fsummon.serialssolutions.com&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.atitle=PICAR%3A+An+Efficient+Extendable+Approach+for+Fitting+Hierarchical+Spatial+Models&rft.jtitle=Technometrics&rft.au=Lee%2C+Ben+Seiyon&rft.au=Haran%2C+Murali&rft.date=2022-04-03&rft.issn=0040-1706&rft.eissn=1537-2723&rft.volume=64&rft.issue=2&rft.spage=187&rft.epage=198&rft_id=info:doi/10.1080%2F00401706.2021.1933596&rft.externalDBID=n%2Fa&rft.externalDocID=10_1080_00401706_2021_1933596
thumbnail_l http://covers-cdn.summon.serialssolutions.com/index.aspx?isbn=/lc.gif&issn=0040-1706&client=summon
thumbnail_m http://covers-cdn.summon.serialssolutions.com/index.aspx?isbn=/mc.gif&issn=0040-1706&client=summon
thumbnail_s http://covers-cdn.summon.serialssolutions.com/index.aspx?isbn=/sc.gif&issn=0040-1706&client=summon