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<front>
<journal-meta>
<journal-id journal-id-type="publisher-id">JDS</journal-id>
<journal-title-group><journal-title>Journal of Data Science</journal-title></journal-title-group>
<issn pub-type="epub">1683-8602</issn><issn pub-type="ppub">1680-743X</issn><issn-l>1680-743X</issn-l>
<publisher>
<publisher-name>School of Statistics, Renmin University of China</publisher-name>
</publisher>
</journal-meta>
<article-meta>
<article-id pub-id-type="publisher-id">JDS1085</article-id>
<article-id pub-id-type="doi">10.6339/23-JDS1085</article-id>
<article-categories><subj-group subj-group-type="heading">
<subject>Statistical Data Science</subject></subj-group></article-categories>
<title-group>
<article-title>Central Posterior Envelopes for Bayesian Functional Principal Component Analysis</article-title>
</title-group>
<contrib-group>
<contrib contrib-type="author">
<name><surname>Boland</surname><given-names>Joanna</given-names></name><xref ref-type="aff" rid="j_jds1085_aff_001">1</xref>
</contrib>
<contrib contrib-type="author">
<name><surname>Telesca</surname><given-names>Donatello</given-names></name><xref ref-type="aff" rid="j_jds1085_aff_001">1</xref>
</contrib>
<contrib contrib-type="author">
<name><surname>Sugar</surname><given-names>Catherine</given-names></name><xref ref-type="aff" rid="j_jds1085_aff_001">1</xref><xref ref-type="aff" rid="j_jds1085_aff_002">2</xref><xref ref-type="aff" rid="j_jds1085_aff_003">3</xref>
</contrib>
<contrib contrib-type="author">
<name><surname>Guindani</surname><given-names>Michele</given-names></name><xref ref-type="aff" rid="j_jds1085_aff_001">1</xref>
</contrib>
<contrib contrib-type="author">
<name><surname>Jeste</surname><given-names>Shafali</given-names></name><xref ref-type="aff" rid="j_jds1085_aff_004">4</xref>
</contrib>
<contrib contrib-type="author">
<name><surname>Dickinson</surname><given-names>Abigail</given-names></name><xref ref-type="aff" rid="j_jds1085_aff_003">3</xref>
</contrib>
<contrib contrib-type="author">
<name><surname>DiStefano</surname><given-names>Charlotte</given-names></name><xref ref-type="aff" rid="j_jds1085_aff_004">4</xref>
</contrib>
<contrib contrib-type="author">
<name><surname>Şentürk</surname><given-names>Damla</given-names></name><email xlink:href="mailto:dsenturk@ucla.edu">dsenturk@ucla.edu</email><xref ref-type="aff" rid="j_jds1085_aff_001">1</xref><xref ref-type="aff" rid="j_jds1085_aff_002">2</xref><xref ref-type="corresp" rid="cor1">∗</xref>
</contrib>
<aff id="j_jds1085_aff_001"><label>1</label>Department of Biostatistics, <institution>University of California</institution>, Los Angeles, Los Angeles, CA 90025, <country>USA</country></aff>
<aff id="j_jds1085_aff_002"><label>2</label>Department of Statistics, <institution>University of California</institution>, Los Angeles, Los Angeles, CA 90025, <country>USA</country></aff>
<aff id="j_jds1085_aff_003"><label>3</label>Department of Psychiatry and Biobehavioral Sciences, <institution>University of California</institution>, Los Angeles, Los Angeles, CA 90025, <country>USA</country></aff>
<aff id="j_jds1085_aff_004"><label>4</label>Division of Neurology, <institution>Children’s Hospital Los Angeles</institution>, Los Angeles, CA 90027, <country>USA</country></aff>
</contrib-group>
<author-notes>
<corresp id="cor1"><label>∗</label>Corresponding author. Email: <ext-link ext-link-type="uri" xlink:href="mailto:dsenturk@ucla.edu">dsenturk@ucla.edu</ext-link>.</corresp>
</author-notes>
<pub-date pub-type="ppub"><year>2023</year></pub-date><pub-date pub-type="epub"><day>19</day><month>1</month><year>2023</year></pub-date><volume>21</volume><issue>4</issue><fpage>715</fpage><lpage>734</lpage><supplementary-material id="S1" content-type="document" xlink:href="jds1085_s001.pdf" mimetype="application" mime-subtype="pdf">
<caption>
<title>Supplementary Material</title>
<p>Supplementary material online includes: derived posterior distributions for model estimation; algorithm for alignment of posterior eigenfunction estimates; details on data generation for simulation studies and additional simulation results; pre-processing of the EEG data featured in Section 5; and plots illustrating band depth, and estimates from simulation studies and data analysis. The R code for the proposed methodology is made publicly available on the Github page <uri>https://github.com/dsenturk/FDpostSumms_BFPCA</uri>, along with a tutorial for step-by-step implementation using simulated data.</p>
</caption>
</supplementary-material><history><date date-type="received"><day>5</day><month>7</month><year>2022</year></date><date date-type="accepted"><day>13</day><month>1</month><year>2023</year></date></history>
<permissions><copyright-statement>2023 The Author(s). Published by the School of Statistics and the Center for Applied Statistics, Renmin University of China.</copyright-statement><copyright-year>2023</copyright-year>
<license license-type="open-access" xlink:href="https://creativecommons.org/licenses/by/4.0/">
<license-p>Open access article under the <ext-link ext-link-type="uri" xlink:href="https://creativecommons.org/licenses/by/4.0/">CC BY</ext-link> license.</license-p></license></permissions>
<abstract>
<p>Bayesian methods provide direct uncertainty quantification in functional data analysis applications without reliance on bootstrap techniques. A major tool in functional data applications is the functional principal component analysis which decomposes the data around a common mean function and identifies leading directions of variation. Bayesian functional principal components analysis (BFPCA) provides uncertainty quantification on the estimated functional model components via the posterior samples obtained. We propose central posterior envelopes (CPEs) for BFPCA based on functional depth as a descriptive visualization tool to summarize variation in the posterior samples of the estimated functional model components, contributing to uncertainty quantification in BFPCA. The proposed BFPCA relies on a latent factor model and targets model parameters within a hierarchical modeling framework using modified multiplicative gamma process shrinkage priors on the variance components. Functional depth provides a center-outward order to a sample of functions. We utilize modified band depth and modified volume depth for ordering of a sample of functions and surfaces, respectively, to derive at CPEs of the mean and eigenfunctions within the BFPCA framework. The proposed CPEs are showcased in extensive simulations. Finally, the proposed CPEs are applied to the analysis of a sample of power spectral densities from resting state electroencephalography where they lead to novel insights on diagnostic group differences among children diagnosed with autism spectrum disorder and their typically developing peers across age.</p>
</abstract>
<kwd-group>
<label>Keywords</label>
<kwd>electroencephalography</kwd>
<kwd>functional data analysis</kwd>
<kwd>modified band depth</kwd>
<kwd>modified volume depth</kwd>
<kwd>uncertainty quantification</kwd>
</kwd-group>
<funding-group><award-group><funding-source xlink:href="https://doi.org/10.13039/100000025">National Institute of Mental Health</funding-source><award-id>R01 MH122428</award-id></award-group><funding-statement>This research was supported by National Institute of Mental Health [R01 MH122428 (DS, DT, CS, SJ)]. </funding-statement></funding-group>
</article-meta>
</front>
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