SOME FIRST RESULTS ON THE CONSISTENCY OF SPATIAL REGRESSION WITH PARTIAL DIFFERENTIAL EQUATION REGULARIZATION

We study the consistency of the estimator in a spatial regression with partial differential equation (PDE) regularization. This new smoothing technique allows us to accurately estimate spatial fields over complex two-dimensional domains, starting from noisy observations. The regularizing term involv...

Full description

Saved in:
Bibliographic Details
Published in:Statistica Sinica 2022-01, Vol.32 (1), p.209-238
Main Authors: Arnone, Eleonora, Kneip, Alois, Nobile, Fabio, Sangalli, Laura M.
Format: Article
Language:English
Citations: Items that cite this one
Online Access:Get full text
Tags: Add Tag
No Tags, Be the first to tag this record!
cited_by cdi_FETCH-LOGICAL-c301t-f5679e09644b9fe9952304d58ff6ebe608180fc4055a177862c75904fe700f5b3
cites
container_end_page 238
container_issue 1
container_start_page 209
container_title Statistica Sinica
container_volume 32
creator Arnone, Eleonora
Kneip, Alois
Nobile, Fabio
Sangalli, Laura M.
description We study the consistency of the estimator in a spatial regression with partial differential equation (PDE) regularization. This new smoothing technique allows us to accurately estimate spatial fields over complex two-dimensional domains, starting from noisy observations. The regularizing term involves a PDE that formalizes problem-specific information about the phenomenon at hand. In contrast to classical smoothing methods, the solution to the infinite-dimensional estimation problem cannot be computed analytically. An approximation is obtained using the finite-element method, considering a suitable triangulation of the spatial domain. We first consider the consistency of the estimator in the infinite-dimensional setting. We then study the consistency of the finite-element estimator resulting from the approximated PDE. We study the bias and variance of the estimators with respect to the sample size and the value of the smoothing parameter. Lastly, simulation studies provide numerical evidence of the rates derived for the bias, variance, and mean square error.
doi_str_mv 10.5705/ss.202019.0346
format article
fullrecord <record><control><sourceid>jstor_cross</sourceid><recordid>TN_cdi_crossref_primary_10_5705_ss_202019_0346</recordid><sourceformat>XML</sourceformat><sourcesystem>PC</sourcesystem><jstor_id>27108521</jstor_id><sourcerecordid>27108521</sourcerecordid><originalsourceid>FETCH-LOGICAL-c301t-f5679e09644b9fe9952304d58ff6ebe608180fc4055a177862c75904fe700f5b3</originalsourceid><addsrcrecordid>eNo9kNFLwzAQxoMoOOZefRPyD7Re2qZpHktN10BtZ5Mi-lLWmsDEMWn24n9vtolPd9_x-467D6F7AiFlQB-dCyOIgPAQ4iS9QgvCeRpkFNi174GwABKgt2jl3G4E4EBJBvEC7VX7LHApO6VxJ1Rfa4XbButK4KJtlFRaNMUbbkusNrmWee2ptQeV9NSr1BXe5N15_iTLUnSiOQvx0nvaI57u67yT72d5h27s9suZ1V9dor4UuqiCul3LIq-DKQZyDCxNGTfA0yQZuTWc0yiG5INm1qZmNClk_ng7-YfoljCWpdHEKIfEGgZg6RgvUXjZO80H52Zjh-95t9_OPwOB4ZTX4NxwyWs45eUNDxfDpzse5n86YgQyGpH4F2UhXgE</addsrcrecordid><sourcetype>Aggregation Database</sourcetype><iscdi>true</iscdi><recordtype>article</recordtype></control><display><type>article</type><title>SOME FIRST RESULTS ON THE CONSISTENCY OF SPATIAL REGRESSION WITH PARTIAL DIFFERENTIAL EQUATION REGULARIZATION</title><source>JSTOR Archival Journals and Primary Sources Collection</source><creator>Arnone, Eleonora ; Kneip, Alois ; Nobile, Fabio ; Sangalli, Laura M.</creator><creatorcontrib>Arnone, Eleonora ; Kneip, Alois ; Nobile, Fabio ; Sangalli, Laura M.</creatorcontrib><description>We study the consistency of the estimator in a spatial regression with partial differential equation (PDE) regularization. This new smoothing technique allows us to accurately estimate spatial fields over complex two-dimensional domains, starting from noisy observations. The regularizing term involves a PDE that formalizes problem-specific information about the phenomenon at hand. In contrast to classical smoothing methods, the solution to the infinite-dimensional estimation problem cannot be computed analytically. An approximation is obtained using the finite-element method, considering a suitable triangulation of the spatial domain. We first consider the consistency of the estimator in the infinite-dimensional setting. We then study the consistency of the finite-element estimator resulting from the approximated PDE. We study the bias and variance of the estimators with respect to the sample size and the value of the smoothing parameter. Lastly, simulation studies provide numerical evidence of the rates derived for the bias, variance, and mean square error.</description><identifier>ISSN: 1017-0405</identifier><identifier>EISSN: 1996-8507</identifier><identifier>DOI: 10.5705/ss.202019.0346</identifier><language>eng</language><publisher>Institute of Statistical Science, Academia Sinica</publisher><ispartof>Statistica Sinica, 2022-01, Vol.32 (1), p.209-238</ispartof><lds50>peer_reviewed</lds50><oa>free_for_read</oa><woscitedreferencessubscribed>false</woscitedreferencessubscribed><citedby>FETCH-LOGICAL-c301t-f5679e09644b9fe9952304d58ff6ebe608180fc4055a177862c75904fe700f5b3</citedby></display><links><openurl>$$Topenurl_article</openurl><openurlfulltext>$$Topenurlfull_article</openurlfulltext><thumbnail>$$Tsyndetics_thumb_exl</thumbnail><linktopdf>$$Uhttps://www.jstor.org/stable/pdf/27108521$$EPDF$$P50$$Gjstor$$H</linktopdf><linktohtml>$$Uhttps://www.jstor.org/stable/27108521$$EHTML$$P50$$Gjstor$$H</linktohtml><link.rule.ids>314,780,784,4024,27923,27924,27925,58238,58471</link.rule.ids></links><search><creatorcontrib>Arnone, Eleonora</creatorcontrib><creatorcontrib>Kneip, Alois</creatorcontrib><creatorcontrib>Nobile, Fabio</creatorcontrib><creatorcontrib>Sangalli, Laura M.</creatorcontrib><title>SOME FIRST RESULTS ON THE CONSISTENCY OF SPATIAL REGRESSION WITH PARTIAL DIFFERENTIAL EQUATION REGULARIZATION</title><title>Statistica Sinica</title><description>We study the consistency of the estimator in a spatial regression with partial differential equation (PDE) regularization. This new smoothing technique allows us to accurately estimate spatial fields over complex two-dimensional domains, starting from noisy observations. The regularizing term involves a PDE that formalizes problem-specific information about the phenomenon at hand. In contrast to classical smoothing methods, the solution to the infinite-dimensional estimation problem cannot be computed analytically. An approximation is obtained using the finite-element method, considering a suitable triangulation of the spatial domain. We first consider the consistency of the estimator in the infinite-dimensional setting. We then study the consistency of the finite-element estimator resulting from the approximated PDE. We study the bias and variance of the estimators with respect to the sample size and the value of the smoothing parameter. Lastly, simulation studies provide numerical evidence of the rates derived for the bias, variance, and mean square error.</description><issn>1017-0405</issn><issn>1996-8507</issn><fulltext>true</fulltext><rsrctype>article</rsrctype><creationdate>2022</creationdate><recordtype>article</recordtype><recordid>eNo9kNFLwzAQxoMoOOZefRPyD7Re2qZpHktN10BtZ5Mi-lLWmsDEMWn24n9vtolPd9_x-467D6F7AiFlQB-dCyOIgPAQ4iS9QgvCeRpkFNi174GwABKgt2jl3G4E4EBJBvEC7VX7LHApO6VxJ1Rfa4XbButK4KJtlFRaNMUbbkusNrmWee2ptQeV9NSr1BXe5N15_iTLUnSiOQvx0nvaI57u67yT72d5h27s9suZ1V9dor4UuqiCul3LIq-DKQZyDCxNGTfA0yQZuTWc0yiG5INm1qZmNClk_ng7-YfoljCWpdHEKIfEGgZg6RgvUXjZO80H52Zjh-95t9_OPwOB4ZTX4NxwyWs45eUNDxfDpzse5n86YgQyGpH4F2UhXgE</recordid><startdate>20220101</startdate><enddate>20220101</enddate><creator>Arnone, Eleonora</creator><creator>Kneip, Alois</creator><creator>Nobile, Fabio</creator><creator>Sangalli, Laura M.</creator><general>Institute of Statistical Science, Academia Sinica</general><scope>AAYXX</scope><scope>CITATION</scope></search><sort><creationdate>20220101</creationdate><title>SOME FIRST RESULTS ON THE CONSISTENCY OF SPATIAL REGRESSION WITH PARTIAL DIFFERENTIAL EQUATION REGULARIZATION</title><author>Arnone, Eleonora ; Kneip, Alois ; Nobile, Fabio ; Sangalli, Laura M.</author></sort><facets><frbrtype>5</frbrtype><frbrgroupid>cdi_FETCH-LOGICAL-c301t-f5679e09644b9fe9952304d58ff6ebe608180fc4055a177862c75904fe700f5b3</frbrgroupid><rsrctype>articles</rsrctype><prefilter>articles</prefilter><language>eng</language><creationdate>2022</creationdate><toplevel>peer_reviewed</toplevel><toplevel>online_resources</toplevel><creatorcontrib>Arnone, Eleonora</creatorcontrib><creatorcontrib>Kneip, Alois</creatorcontrib><creatorcontrib>Nobile, Fabio</creatorcontrib><creatorcontrib>Sangalli, Laura M.</creatorcontrib><collection>CrossRef</collection><jtitle>Statistica Sinica</jtitle></facets><delivery><delcategory>Remote Search Resource</delcategory><fulltext>fulltext</fulltext></delivery><addata><au>Arnone, Eleonora</au><au>Kneip, Alois</au><au>Nobile, Fabio</au><au>Sangalli, Laura M.</au><format>journal</format><genre>article</genre><ristype>JOUR</ristype><atitle>SOME FIRST RESULTS ON THE CONSISTENCY OF SPATIAL REGRESSION WITH PARTIAL DIFFERENTIAL EQUATION REGULARIZATION</atitle><jtitle>Statistica Sinica</jtitle><date>2022-01-01</date><risdate>2022</risdate><volume>32</volume><issue>1</issue><spage>209</spage><epage>238</epage><pages>209-238</pages><issn>1017-0405</issn><eissn>1996-8507</eissn><abstract>We study the consistency of the estimator in a spatial regression with partial differential equation (PDE) regularization. This new smoothing technique allows us to accurately estimate spatial fields over complex two-dimensional domains, starting from noisy observations. The regularizing term involves a PDE that formalizes problem-specific information about the phenomenon at hand. In contrast to classical smoothing methods, the solution to the infinite-dimensional estimation problem cannot be computed analytically. An approximation is obtained using the finite-element method, considering a suitable triangulation of the spatial domain. We first consider the consistency of the estimator in the infinite-dimensional setting. We then study the consistency of the finite-element estimator resulting from the approximated PDE. We study the bias and variance of the estimators with respect to the sample size and the value of the smoothing parameter. Lastly, simulation studies provide numerical evidence of the rates derived for the bias, variance, and mean square error.</abstract><pub>Institute of Statistical Science, Academia Sinica</pub><doi>10.5705/ss.202019.0346</doi><tpages>30</tpages><oa>free_for_read</oa></addata></record>
fulltext fulltext
identifier ISSN: 1017-0405
ispartof Statistica Sinica, 2022-01, Vol.32 (1), p.209-238
issn 1017-0405
1996-8507
language eng
recordid cdi_crossref_primary_10_5705_ss_202019_0346
source JSTOR Archival Journals and Primary Sources Collection
title SOME FIRST RESULTS ON THE CONSISTENCY OF SPATIAL REGRESSION WITH PARTIAL DIFFERENTIAL EQUATION REGULARIZATION
url http://sfxeu10.hosted.exlibrisgroup.com/loughborough?ctx_ver=Z39.88-2004&ctx_enc=info:ofi/enc:UTF-8&ctx_tim=2024-12-30T20%3A40%3A49IST&url_ver=Z39.88-2004&url_ctx_fmt=infofi/fmt:kev:mtx:ctx&rfr_id=info:sid/primo.exlibrisgroup.com:primo3-Article-jstor_cross&rft_val_fmt=info:ofi/fmt:kev:mtx:journal&rft.genre=article&rft.atitle=SOME%20FIRST%20RESULTS%20ON%20THE%20CONSISTENCY%20OF%20SPATIAL%20REGRESSION%20WITH%20PARTIAL%20DIFFERENTIAL%20EQUATION%20REGULARIZATION&rft.jtitle=Statistica%20Sinica&rft.au=Arnone,%20Eleonora&rft.date=2022-01-01&rft.volume=32&rft.issue=1&rft.spage=209&rft.epage=238&rft.pages=209-238&rft.issn=1017-0405&rft.eissn=1996-8507&rft_id=info:doi/10.5705/ss.202019.0346&rft_dat=%3Cjstor_cross%3E27108521%3C/jstor_cross%3E%3Cgrp_id%3Ecdi_FETCH-LOGICAL-c301t-f5679e09644b9fe9952304d58ff6ebe608180fc4055a177862c75904fe700f5b3%3C/grp_id%3E%3Coa%3E%3C/oa%3E%3Curl%3E%3C/url%3E&rft_id=info:oai/&rft_id=info:pmid/&rft_jstor_id=27108521&rfr_iscdi=true