solifc.blogg.se - Galton watson process

Notably, new names can be created, existing names can be changed over a person's lifetime, and people historically have often assumed names of unrelated persons, particularly nobility. If in the non-trivial case the averaged reproduction mean per couple stays bounded over all generations and will not exceed 1 for a sufficiently large population size, then the probability of final extinction is always 1.Ĭiting historical examples of Galton–Watson process is complicated due to the history of family names often deviating significantly from the theoretical model. However, excluding the non-trivial case, the concept of the averaged reproduction mean (Bruss (1984)) allows for a general sufficient condition for final extinction, treated in the next section.

Since the total reproduction within a generation depends now strongly on the mating function, there exists in general no simple necessary and sufficient condition for final extinction as is the case in the classical Galton–Watson process. Now the analogue of the trivial case corresponds to the case of each male and female reproducing in exactly one couple, having one male and one female descendant, and that the mating function takes the value of the minimum of the number of males and females (which are then the same from the next generation onwards). As before, reproduction of different couples are considered to be independent of each other. ( Bisexual in this context refers to the number of sexes involved, not sexual orientation.) In this process, each child is supposed as male or female, independently of each other, with a specified probability, and a so-called "mating function" determines how many couples will form in a given generation. (Likewise, if mitochondrial transmission is analyzed, only women need to be considered, since only females transmit their mitochondria to descendants.)Ī model more closely following actual sexual reproduction is the so-called "bisexual Galton–Watson process", where only couples reproduce.

This effectively means that reproduction can be modeled as asexual. In the classical family surname Galton–Watson process described above, only men need to be considered, since only males transmit their family name to descendants. Suppose the number of a man's sons to be a random variable distributed on the set For a detailed history see Kendall (19).Īssume, for the sake of the model, that surnames are passed on to all male children by their father. Galton and Watson appear to have derived their process independently of the earlier work by I. Together, they then wrote an 1874 paper titled "On the probability of the extinction of families" in the Journal of the Anthropological Institute of Great Britain and Ireland (now the Journal of the Royal Anthropological Institute). Galton originally posed a mathematical question regarding the distribution of surnames in an idealized population in an 1873 issue of The Educational Times, and the Reverend Henry William Watson replied with a solution. There was concern amongst the Victorians that aristocratic surnames were becoming extinct.

4 Extinction criterion for Galton–Watson process.

The formula is of limited usefulness in understanding actual family name distributions, since in practice family names change for many other reasons, and dying out of name line is only one factor. Likewise, since mitochondria are inherited only on the maternal line, the same mathematical formulation describes transmission of mitochondria. This is an accurate description of Y chromosome transmission in genetics, and the model is thus useful for understanding human Y-chromosome DNA haplogroups. The process models family names as patrilineal (passed from father to son), while offspring are randomly either male or female, and names become extinct if the family name line dies out (holders of the family name die without male descendants). The Galton–Watson process is a branching stochastic process arising from Francis Galton's statistical investigation of the extinction of family names.

But the probability of survival of a new type may be quite low even if λ > 1 and the population as a whole is experiencing quite strong exponential increase. For λ ≤ 1, eventual extinction will occur with probability 1. Galton–Watson survival probabilities for different exponential rates of population growth, if the number of children of each parent node can be assumed to follow a Poisson distribution.