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<front>
<journal-meta>
<journal-id/>
<journal-title-group>
</journal-title-group>
<issn/>
<publisher>
<publisher-name/>
</publisher>
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<article-meta>
<title-group>
<article-title>Lineal admixture time</article-title>
</title-group>
<contrib-group>
<contrib contrib-type="author">
<contrib-id contrib-id-type="orcid">0000-0002-5014-4809</contrib-id>
<name>
<surname>Ellerman</surname>
<given-names>E. Castedo</given-names>
</name>
<email>castedo@castedo.com</email>
</contrib>
</contrib-group>
<pub-date date-type="eprint" publication-format="electronic" iso-8601-date="2022-03-31">
<day>31</day>
<month>3</month>
<year>2022</year>
</pub-date>
<permissions>
<copyright-statement>© 2022, Ellerman et al</copyright-statement>
<copyright-year>2022</copyright-year>
<copyright-holder>Ellerman et al</copyright-holder>
<license license-type="open-access">
<ali:license_ref xmlns:ali="http://www.niso.org/schemas/ali/1.0/">https://creativecommons.org/licenses/by/4.0/</ali:license_ref>
<license-p>This document is distributed under a Creative Commons
Attribution 4.0 International license.</license-p>
</license>
</permissions>
<abstract>
<p>This discussion document defines lineal admixture time, a key
construct under mathematical study by ECE. Desired future outcomes are
mathematical and computation tools, based on lineal admixture time, that
become useful to downstream empirical research, specifically the
estimation of the timing and extent of interbreeding between previously
separated populations. Feedback is greatly appreciated, especially
regarding the relevance to downstream empirical research.</p>
</abstract>
</article-meta>
</front>
<body>
<disp-quote>
<p>
<bold>WORKING DRAFT</bold>
</p>
</disp-quote>
<sec id="introduction">
<title>Introduction</title>
<p>Genealogies are a link between genetic evidence and population
events of the past. By genealogies we mean the parent-offspring
relations of a group of individuals and their ancestors. In contrast
to estimates of purely genetic quantities, estimates of quantities
about genealogies can have closer correspondence to non-genetic lines
of evidence, such as archaeological, geological, and cultural
evidence.</p>
</sec>
<sec id="definition-of-lineal-admixture-time">
<title>Definition of lineal admixture time</title>
<p>We define <italic>lineal admixture time</italic> per lineage. By
(organismal) lineage we mean a linear chain of parent-offspring pairs
descending through a genealogy. The following definition assumes a
past point in time when all individuals are classified into separate
(non-admixed) subpopulations. Lineages do not have a lineal admixture
time <italic>point</italic> if they consist of individuals from only
one non-admixed subpopulation. For all other lineages, the lineal
admixture time <italic>point</italic> is the fertilization time of the
first individual with parents from two distinct non-admixed
subpopulations. From an observation time at the end of a lineage, the
lineal admixture time <italic>duration</italic> is the time since the
linear admixture time <italic>point</italic>. If this time point does
not exist, then the time duration is zero.</p>
<p>The <italic>average lineal admixture time</italic> of an individual
is averaged across all lineages ending in that individual. At every
merging of lineages, through the mother and father of offspring, equal
weighting is given to the two respective sets of lineages. The
<italic>average lineal admixture time</italic> of a group of
individuals at an observation time point, is the average lineal
admixture time across all those individuals.</p>
</sec>
<sec id="advantages-of-lineal-admixture-time">
<title>Advantages of lineal admixture time</title>
<p>Lineal admixture time has a number of advantages as a quantity for
research.</p>
<p>Firstly, it is independent of any particular model of an admixture
process. With enough genealogical information one could calculate
lineal admixture times of a real group of individuals. There is one
true set of lineal admixture times for a real population, even though
realistically, we can only hope to estimate those numbers based on
evidence and models. But the definition exists independent of any
particular model used for estimation.</p>
<p>Secondly, the interpretation of lineal admixture time does not
require fluency in probability theory. Researchers with interests in
empirical evidence and not mathematics can make use of estimates of
lineal admixture times.</p>
<p>Thirdly, we conjecture that <italic>distributions</italic> of
lineal admixture times across individuals, groups of individuals, and
chromosomes will prove to be a useful mathematical tool in the timing
of admixture. This conjecture is based on not-yet-documented
mathematical work by ECE. Lineal admixture time is conveniently
representable as a random variable from a stochastic process in which
lineages are random objects.</p>
</sec>
<sec id="mendels-hybrid-generation-numbers">
<title>Mendel's hybrid generation numbers</title>
<p>Lineal admixture time can be interpreted as a generalization of
Mendel's hybrid generation numbers. In his famous experiments with
peas, Mendel
<xref alt="1" rid="ref-abbott_experiments_2016" ref-type="bibr">1</xref>
assigns ordinal numbers (first, second, third, ...) to successive
generations of hybrids. Today, the first and second hybrid generations
are often referred to as <inline-formula><alternatives><tex-math><![CDATA[F_1]]></tex-math><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:msub><mml:mi>F</mml:mi><mml:mn>1</mml:mn></mml:msub></mml:math></alternatives></inline-formula>
and <inline-formula><alternatives><tex-math><![CDATA[F_2]]></tex-math><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:msub><mml:mi>F</mml:mi><mml:mn>2</mml:mn></mml:msub></mml:math></alternatives></inline-formula>
<xref alt="2" rid="ref-hartl_essential_2006" ref-type="bibr">2</xref>.
An example of using these generation numbers to infer admixture timing
is in the study of admixture in Greenland
<xref alt="3" rid="ref-waples_genetic_2021" ref-type="bibr">3</xref>.</p>
<p>Mendel's hybrid generation numbers correspond to a simple special
case with the following assumptions:</p>
<list list-type="bullet">
<list-item>
<p>generation numbers are assigned to mature individuals at
non-negative integer time points</p>
</list-item>
<list-item>
<p>after time zero, all mature individuals are admixed</p>
</list-item>
<list-item>
<p>parents are mature and reproducing for only one time point</p>
</list-item>
<list-item>
<p>the fertilization time of offspring is when the parents are
reproducing</p>
</list-item>
<list-item>
<p>maturation time follows exactly one time unit after
fertilization</p>
</list-item>
</list>
<p>With these assumptions, lineal admixture time durations observed at
positive integer time points are exactly the hybrid generation
numbers.</p>
</sec>
<sec id="fin">
<title>Fin</title>
<p>This document presents a definition of <italic>lineal admixture
time</italic>. This quantity is the basis for current mathematical
studies of ECE. Feedback and input is greatly appreciated, especially
regarding the relevance to downstream empirical research. In
particular, the following questions are of particular interest:</p>
<list list-type="order">
<list-item>
<p>Is <italic>lineal admixture time</italic> easily understood and
interpretable without fluency in probability theory?</p>
</list-item>
<list-item>
<p>Does <italic>lineal admixture time</italic> benefit from being
comparable to non-genetic lines of evidence (i.e archaeological,
geological, linguistic, historical, cultural)?</p>
</list-item>
<list-item>
<p>What are existing terms or literature for this specific
definition of admixture time?</p>
</list-item>
<list-item>
<p>Are there reasons for confusion with these names chosen in this
document</p>
<list list-type="bullet">
<list-item>
<p>lineal admixture time</p>
</list-item>
<list-item>
<p>lineage</p>
</list-item>
<list-item>
<p>genealogy</p>
</list-item>
</list>
<p>or how they are used?</p>
</list-item>
</list>
</sec>
<sec id="references">
<title>References</title>
</sec>
</body>
<back>
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</article>
