Background
The genealogies of a population reflect population events of the past. Lineages, at the organismal level, are lines of descent carrying genetic information through a population's genealogies. These lineages are a link between genetic evidence and population events of the past. In contrast to estimates of purely genetic quantities, estimates of quantities about lineages can have closer correspondence to nongenetic evidence, such as archaeological, geological, and cultural evidence.
Definition of lineal admixture time
We define lineal admixture time per lineage. This measure in time duration depends on two reference time points:
a past time horizon when all ancestors are members of separate nonadmixed subpopulations, and
an observation time, such as the present.
Given a lineage, the lineal admixture time is defined as:
the amount of time since fertilization of the first admixed individual in the lineage (or zero if the lineage has no such individual).
Stated more compactly, lineal admixture time is time since a lineage's first admixed fertilization. Admixed individuals are all individual whose parents are not from the same nonadmixed subpopulations.
The average lineal admixture time of an individual is an average over all lineages passing through that individual. At every merging of lineages through offspring, equal weighting is given to the respective sets of maternal and paternal lineages. The average lineal admixture time of a group of individuals is the average lineal admixture time across all those individuals.
It is worth noting that nonzero admixture times do not remain constant if the observation time changes. It follows that nonzero average admixture times of individuals are not fixed. They will increase if the observation time increases with age.
An example with Mendel's peas
We consider an example of lineal admixture times with Mendel's
famous peas
[1].
We imagine one of his classic experiments starting in 1860 and ending
in 1862. Because the pea plant is an annual plant, each generation is
separated by one year. During the lifetime of one generation,
fertilization of the next generation occurs, in the form of seeds
(illustrated with a .
symbol).
1860 A.Y B.Z
 
1861 M.N

1862 Q
In this example we will treat the nonadmixed ancestral
subpopulations to be truebreeding pea plant varieties in 1860. Plants
A
and B
in the diagram
are nonadmixed plants with round peas. Plants
Y
and Z
are nonadmixed
plants with wrinkled peas. These two subpopulations grow and are
crossed in 1860 producing seeds that same year. Because they were
crossed, the offspring M
and
N
are both admixed (hybrid). This first admixed
(hybrid) generation will produce seeds in 1861 for the second admixed
generation, plant Q
.
Plant Q
has four lineages:
QMA
QMY
QNB
QNZ
In these lineages the first admixed individuals are
M
and N
, both of which
were fertilized in 1860. The first admixed fertilizations occur in
1860. Thus the lineal admixture time for all lineages of
Q
, observed in 1862, is 2
years.
Hybrid generation numbers
A related definition measures generation number rather than time. Given a lineage, the lineal hybrid generation number is defined as:
the number of admixed individual in the lineage.
Because peas are annual, the lineal admixture times in years coincide exactly with the lineal hybrid generation numbers. Today, the first and second hybrid generations are often referred to as $F_1$ and $F_2$ [2]. Lineal average hybird generation numbers are a generalization of indexes given to hybrid generations $F_1$, $F_2$, etc...
An example with humans
These definitions are inspired by the use of generation numbers to infer admixture timing in the study of admixture in Greenland [3].
We give a hypothetical example in Greenland at an observation time
of 1600 CE. We categorize the ancestors at a time horizon of 1000 CE
into separate noninterbreeding Inuit and European subpopulations. We
denote the fertilization and death of an individual with symbols
.
and x
, respectively,
and write names at the time an individual is a 3monthold baby.
1500 Erik
1510  Aaju
1520   Natar Atuat
1530 .  
1540  Tagak  .
1550   x x Mikak 
1560 x   
1570 . x
1580 x Kiviaq 
1590  x
1600 
We imagine Erik as only having European ancestors and Aaju, Natar and Atuat only having Intuit ancestors. As of 1600 CE, the lineages are:
KiviaqTagakErik
KiviaqTagakAaju
KiviaqMikakNatar
KiviaqMikakAtuat
For the Tagak lineages, the first admixed fertilization is in 1530. In contrast, the Mikak lineages have a first admixed fertilization in 1570. Thus the lineal admixture times are equal parts 70 years and 30 years, for an average lineal admixture time of 50 years. The respective lineal hybrid generation numbers are 1 and 2 resulting in an average lineal hybrid generation number of 1.5.
Advantages of lineal admixture time
Lineal admixture time has a number of advantages as a quantity for research.
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. But 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.
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.
Thirdly, we conjecture that distributions of lineal admixture times across individuals, and groups of individuals will prove to be a useful mathematical tool in the timing of admixture. This conjecture is based on notyetdocumented mathematical work by ECE. Lineal admixture time is conveniently representable as a random variable from a stochastic process in which lineages are random objects.
The same populations events that affect admixture also affect hybridization, introgression, and gene flow. We conjecture that statistical tools for estimating distributions of lineal admixture time will also be useful to these related topics in addition to admixture timing.
The Greenland example illustrates how average lineal admixture time is easily testable against nongenetic lines of evidence. If a genetic model estimates the average lineal admixture time of present day Greenlanders to be 500 years, we can consider historical and archaeological evidence to falsify that estimate.
End
This document presents a definition of lineal admixture time. 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:
Is lineal admixture time easily understood and interpretable without fluency in probability theory?
Does lineal admixture time benefit from being comparable to nongenetic lines of evidence (i.e archaeological, geological, linguistic, historical, cultural)?
What are existing terms or literature for this specific definition of admixture time?
Are the following terms:
lineage
genealogy
lineal admixture time
lineal hybrid generation number
potentially confusing in the way they are used in this document?