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10.21105.joss.00141.jats
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<?xml version="1.0" encoding="utf-8" ?>
<!DOCTYPE article PUBLIC "-//NLM//DTD JATS (Z39.96) Journal Publishing DTD v1.2 20190208//EN"
"JATS-publishing1.dtd">
<article xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink" dtd-version="1.2" article-type="other">
<front>
<journal-meta>
<journal-id></journal-id>
<journal-title-group>
<journal-title>Journal of Open Source Software</journal-title>
<abbrev-journal-title>JOSS</abbrev-journal-title>
</journal-title-group>
<issn publication-format="electronic">2475-9066</issn>
<publisher>
<publisher-name>Open Journals</publisher-name>
</publisher>
</journal-meta>
<article-meta>
<article-id pub-id-type="publisher-id">141</article-id>
<article-id pub-id-type="doi">10.21105/joss.00141</article-id>
<title-group>
<article-title>vbvs.concurrent: Fitting Methods for the Functional
Linear Concurrent Model</article-title>
</title-group>
<contrib-group>
<contrib contrib-type="author">
<contrib-id contrib-id-type="orcid">0000-0002-6150-8997</contrib-id>
<string-name>Jeff Goldsmith</string-name>
<xref ref-type="aff" rid="aff-1"/>
</contrib>
<aff id="aff-1">
<institution-wrap>
<institution>Columbia University</institution>
</institution-wrap>
</aff>
</contrib-group>
<pub-date date-type="pub" publication-format="electronic" iso-8601-date="2016-12-16">
<day>16</day>
<month>12</month>
<year>2016</year>
</pub-date>
<volume>1</volume>
<issue>8</issue>
<fpage>141</fpage>
<permissions>
<copyright-statement>Authors of papers retain copyright and release the
work under a Creative Commons Attribution 4.0 International License (CC
BY 4.0)</copyright-statement>
<copyright-year>2021</copyright-year>
<copyright-holder>The article authors</copyright-holder>
<license license-type="open-access" xlink:href="https://creativecommons.org/licenses/by/4.0/">
<license-p>Authors of papers retain copyright and release the work under
a Creative Commons Attribution 4.0 International License (CC BY
4.0)</license-p>
</license>
</permissions>
<kwd-group kwd-group-type="author">
<kwd>statistical analysis</kwd>
</kwd-group>
</article-meta>
</front>
<body>
<sec id="summary">
<title>Summary</title>
<p>Functional data analysis is concerned with understanding
measurements made over time, space, frequencies, and other domains for
multiple subjects. Given the ubiquity of wearable devices, it is
common to obtain several data streams monitoring blood pressure,
physical activity, heart rate, location, and other quantities on study
participants in parallel. Each of these data streams can be thought of
as functional data, and the functional linear concurrent model is
useful for relating predictor data to an outcome. This model can be
written <disp-formula><alternatives>
<tex-math><![CDATA[
Y_i(t) = \beta_0(t) + \sum_{k = 1}^{p}X_{ik}(t)\beta_k(t) + \delta_i(t)
]]></tex-math>
<mml:math display="block" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:msub><mml:mi>Y</mml:mi><mml:mi>i</mml:mi></mml:msub><mml:mrow><mml:mo stretchy="true" form="prefix">(</mml:mo><mml:mi>t</mml:mi><mml:mo stretchy="true" form="postfix">)</mml:mo></mml:mrow><mml:mo>=</mml:mo><mml:msub><mml:mi>β</mml:mi><mml:mn>0</mml:mn></mml:msub><mml:mrow><mml:mo stretchy="true" form="prefix">(</mml:mo><mml:mi>t</mml:mi><mml:mo stretchy="true" form="postfix">)</mml:mo></mml:mrow><mml:mo>+</mml:mo><mml:munderover><mml:mo>∑</mml:mo><mml:mrow><mml:mi>k</mml:mi><mml:mo>=</mml:mo><mml:mn>1</mml:mn></mml:mrow><mml:mi>p</mml:mi></mml:munderover><mml:msub><mml:mi>X</mml:mi><mml:mrow><mml:mi>i</mml:mi><mml:mi>k</mml:mi></mml:mrow></mml:msub><mml:mrow><mml:mo stretchy="true" form="prefix">(</mml:mo><mml:mi>t</mml:mi><mml:mo stretchy="true" form="postfix">)</mml:mo></mml:mrow><mml:msub><mml:mi>β</mml:mi><mml:mi>k</mml:mi></mml:msub><mml:mrow><mml:mo stretchy="true" form="prefix">(</mml:mo><mml:mi>t</mml:mi><mml:mo stretchy="true" form="postfix">)</mml:mo></mml:mrow><mml:mo>+</mml:mo><mml:msub><mml:mi>δ</mml:mi><mml:mi>i</mml:mi></mml:msub><mml:mrow><mml:mo stretchy="true" form="prefix">(</mml:mo><mml:mi>t</mml:mi><mml:mo stretchy="true" form="postfix">)</mml:mo></mml:mrow></mml:mrow></mml:math></alternatives></disp-formula>
where <inline-formula><alternatives>
<tex-math><![CDATA[Y_i(t)]]></tex-math>
<mml:math display="inline" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:msub><mml:mi>Y</mml:mi><mml:mi>i</mml:mi></mml:msub><mml:mrow><mml:mo stretchy="true" form="prefix">(</mml:mo><mml:mi>t</mml:mi><mml:mo stretchy="true" form="postfix">)</mml:mo></mml:mrow></mml:mrow></mml:math></alternatives></inline-formula>
is the functional response for subject <inline-formula><alternatives>
<tex-math><![CDATA[i]]></tex-math>
<mml:math display="inline" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>i</mml:mi></mml:math></alternatives></inline-formula>,
the <inline-formula><alternatives>
<tex-math><![CDATA[X_{ik}(t)]]></tex-math>
<mml:math display="inline" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:msub><mml:mi>X</mml:mi><mml:mrow><mml:mi>i</mml:mi><mml:mi>k</mml:mi></mml:mrow></mml:msub><mml:mrow><mml:mo stretchy="true" form="prefix">(</mml:mo><mml:mi>t</mml:mi><mml:mo stretchy="true" form="postfix">)</mml:mo></mml:mrow></mml:mrow></mml:math></alternatives></inline-formula>
are functional predictors, the <inline-formula><alternatives>
<tex-math><![CDATA[\beta_k(t)]]></tex-math>
<mml:math display="inline" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:msub><mml:mi>β</mml:mi><mml:mi>k</mml:mi></mml:msub><mml:mrow><mml:mo stretchy="true" form="prefix">(</mml:mo><mml:mi>t</mml:mi><mml:mo stretchy="true" form="postfix">)</mml:mo></mml:mrow></mml:mrow></mml:math></alternatives></inline-formula>
are functional coefficients of interest, and the
<inline-formula><alternatives>
<tex-math><![CDATA[\delta_i(t)]]></tex-math>
<mml:math display="inline" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:msub><mml:mi>δ</mml:mi><mml:mi>i</mml:mi></mml:msub><mml:mrow><mml:mo stretchy="true" form="prefix">(</mml:mo><mml:mi>t</mml:mi><mml:mo stretchy="true" form="postfix">)</mml:mo></mml:mrow></mml:mrow></mml:math></alternatives></inline-formula>
are possibly correlated errors. This package implements two
statistical methods (with and without variable selection) for
estimating the parameters in the functional linear concurrent model;
these methods are described in detail
(<xref alt="Goldsmith & Schwartz, 2017" rid="ref-goldsmith2017" ref-type="bibr">Goldsmith
& Schwartz, 2017</xref>).</p>
<p>Given tidy datasets containing functional responses and predictors
for all subjects, <monospace>vb_concurrent</monospace> and
<monospace>vbvs_concurrent</monospace> fit the functional linear
concurrent model using variational Bayes and variational Bayes with
variable selection, respectively. These functions produce objects of
class <monospace>flcm</monospace> and have the same structure.
Coefficients and predictions can be extracted or computed from
<monospace>flmc</monospace> objects using <monospace>coef</monospace>
and <monospace>predict</monospace>. Interactive visualizations of
<monospace>flmc</monospace> objects are supported through the
<monospace>refund.shiny</monospace> package Wrobel et al.
(<xref alt="2016" rid="ref-wrobel2016" ref-type="bibr">2016</xref>).</p>
</sec>
</body>
<back>
<ref-list>
<ref-list>
<ref id="ref-goldsmith2017">
<element-citation publication-type="article-journal">
<person-group person-group-type="author">
<name><surname>Goldsmith</surname><given-names>J</given-names></name>
<name><surname>Schwartz</surname><given-names>J E</given-names></name>
</person-group>
<article-title>Variable selection in the functional linear concurrent model</article-title>
<source>Under Review</source>
<year iso-8601-date="2017">2017</year>
</element-citation>
</ref>
<ref id="ref-refund.shiny">
<element-citation publication-type="book">
<person-group person-group-type="author">
<name><surname>Wrobel</surname><given-names>J</given-names></name>
<name><surname>Goldsmith</surname><given-names>J</given-names></name>
</person-group>
<source>Refund.shiny: Interactive plotting for functional data analyses</source>
<year iso-8601-date="2015">2015</year>
</element-citation>
</ref>
<ref id="ref-wrobel2016">
<element-citation publication-type="article-journal">
<person-group person-group-type="author">
<name><surname>Wrobel</surname><given-names>J</given-names></name>
<name><surname>Park</surname><given-names>S-Y</given-names></name>
<name><surname>Staicu</surname><given-names>A-M</given-names></name>
<name><surname>Goldsmith</surname><given-names>J</given-names></name>
</person-group>
<article-title>Interactive graphics for functional data analyses</article-title>
<source>Stat</source>
<year iso-8601-date="2016">2016</year>
<volume>5</volume>
<pub-id pub-id-type="doi">10.1002/sta4.109</pub-id>
</element-citation>
</ref>
</ref-list>
</ref-list>
</back>
</article>