Galton–Watson process

Galton–Watson process

The Galton–Watson process is a stochastic process arising from Francis Galton's statistical investigation of the extinction of surnames.

History

There was concern amongst the Victorians that aristocratic surnames were becoming extinct. Galton originally posed the question regarding the probability of such an event in the Educational Times of 1873, and the Reverend Henry William Watson replied with a solution. Together, they then wrote an 1874 paper entitled "On the probability of extinction of families". Galton and Watson appear to have derived their process independently of the earlier work by I. J. Bienaymé; see Heyde and Seneta 1977. For a detailed history see Kendall (1966 and 1975).

Concepts

Assume, as was taken for granted in Galton's time, that surnames are passed on to all male children by their father. Suppose the number of a man's sons to be a random variable distributed on the set { 0, 1, 2, 3, ...}. Further suppose the numbers of different men's sons to be independent random variables, all having the same distribution.

Then the simplest substantial mathematical conclusion is that if the average number of a man's sons is 1 or less, then their surname will surely die out, and if it is more than 1, then there is more than zero probability that it will survive forever.

Modern applications include the survival probabilities for a new mutant gene, or the initiation of a nuclear chain reaction, or the dynamics of disease outbreaks in their first generations of spread, or the chances of extinction of small population of organisms; as well as explaining (perhaps closest to Galton's original interest) why only a handful of males in the deep past of humanity now have "any" surviving male-line descendants, reflected in a rather small number of distinctive human Y-chromosome DNA haplogroups.

A corollary of high extinction probabilities is that if a lineage "has" survived, it is likely to have experienced, purely by chance, an unusually high growth rate in its early generations at least when compared to the rest of the population.

Mathematical definition

A Galton-Watson process is a stochastic process {"X""n"} which evolves according to the recurrence formula "X"0 = 1 and

:X_{n+1} = sum_{j=1}^{X_n} xi_j^{(n+1)}

where for each "n", xi_j^{(n)} is a sequence of IID natural number-valued random variables. The extinction probability is given by

:lim_{n o infty} Pr(X_n = 0)

and is equal to one if "E"{"ξ1"} ≤ 1 and strictly less than one if "E"{"ξ1"} > 1.

The process can be treated analytically using the method of probability generating functions.

If the number of children "ξ j" at each node follows a Poisson distribution, a particularly simple recurrence can be found for the total extinction probability "xn" for a process starting with a single individual at time "n" = 0:

:x_{n+1} = e^{lambda (x_n - 1)}

giving the curves plotted above.

Bisexual Galton–Watson process

In the (classical) Galton–Watson process defined above, only men count, that is, the reproduction can be understoodas being asexual. The more natural corresponding version for (bi)sexual reproduction is the so-called 'Bisexual Galton–Watson process',where only couples can reproduce.In this process, each child is supposed to be 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. As before, reproduction of different couples are considered to be independent of each other. Since the total reproduction within a generation depends now also on the mating function,there exists in general no simple necessary and sufficient for final extinction as it is the case in the classical Galton–Watson process. However, the concept of the 'averaged reproduction mean' (Bruss (1984)) allows for a general and simplesufficient condition for final extinction: If the averaged reproduction mean per couple stays bounded and will not exceed 1for a sufficiently large population size, then the probability of final extinction is always one.

Example

Countries that have used family names for many generations exhibit the Galton–Watson process in their low number of surviving family names:
* Korean names are the most striking example, with 250 family names, and 45% of the population sharing 3 family names
* Chinese names are similar, with 22% of the population sharing 3 family names (numbering close to 300 million people), and the top 200 names covering 96% of the population.

By contrast:
* Dutch names have only included a family name since the Napoleonic Wars in the early 19th century, and there are over 68,000 Dutch family names.
* Thai names have only included a family name since 1920, and only a single family can use a given family name, hence there are a great number of Thai names. Further, Thai people change their family names with some frequency, complicating the analysis.

ee also

*Branching process

References

* F T Bruss (1984). "A Note on Extinction Criteria for Bisexual Galton–Watson Processes". "Journal of Applied Probability" 21: 915–919.
* C C Heyde and E Seneta (1977). "I.J. Bienayme: Statistical Theory Anticipated". Berlin, Germany.
* D G Kendall (1966). "Journal of the London Mathematical Society" 41: 385–406
* D G Kendall (1975). "Bulletin of the London Mathematical Society" 7: 225–253
* H W Watson and Francis Galton, "On the Probability of the Extinction of Families", "Journal of the Anthropological Institute of Great Britain", volume 4, pages 138–144, 1875.

External links

* [http://galton.org/essays/1870-1879/galton-1874-jaigi-family-extinction.pdf The original Galton–Watson paper: On the Probability of the Extinction of Families]
* [http://web.archive.org/web/20040401131411/http://www-users.york.ac.uk/~pml1/stats/gwproc.ps "Survival of a Single Mutant" by Peter M. Lee of the University of York]


Wikimedia Foundation. 2010.

Игры ⚽ Поможем сделать НИР

Look at other dictionaries:

  • Processus de Galton-Watson — Le processus de Galton Watson est un processus stochastique permettant de décrire des dynamiques de populations. Sommaire 1 Historique 2 Formulation générale 3 Paramètre critique et classification des processus de Galton Watson …   Wikipédia en Français

  • Galton (disambiguation) — Galton may refer to:People with the given name Galton:* Charles Galton Darwin (1887 ndash;1962), English physicist * Galton Blackiston (21st century), English chefPeople with the surname Galton:* Douglas Strutt Galton (1822 ndash;1899), British… …   Wikipedia

  • Branching process — In probability theory, a branching process is a Markov process that models a population in which each individual in generation n produces some random number of individuals in generation n + 1, according to a fixed probability distribution that… …   Wikipedia

  • Henry William Watson — Rev. Henry William Watson (25 February 1827 ndash; 11 January 1903) was a noted mathematician and author of a number of mathematics books.Watson was educated at King s College London, winning the first mathematical scholarship which had been set… …   Wikipedia

  • List of statistics topics — Please add any Wikipedia articles related to statistics that are not already on this list.The Related changes link in the margin of this page (below search) leads to a list of the most recent changes to the articles listed below. To see the most… …   Wikipedia

  • List of mathematics articles (G) — NOTOC G G₂ G delta space G networks Gδ set G structure G test G127 G2 manifold G2 structure Gabor atom Gabor filter Gabor transform Gabor Wigner transform Gabow s algorithm Gabriel graph Gabriel s Horn Gain graph Gain group Galerkin method… …   Wikipedia

  • List of probability topics — This is a list of probability topics, by Wikipedia page. It overlaps with the (alphabetical) list of statistical topics. There are also the list of probabilists and list of statisticians.General aspects*Probability *Randomness, Pseudorandomness,… …   Wikipedia

  • Outline of probability — Probability is the likelihood or chance that something is the case or will happen. Probability theory is used extensively in statistics, mathematics, science and philosophy to draw conclusions about the likelihood of potential events and the… …   Wikipedia

  • Chinese surname — Chinese family names have been historically used by Han Chinese and Sinicized Chinese ethnic groups in mainland China, Taiwan, Hong Kong, and among overseas Chinese communities. In ancient times two types of surnames, family names (Chinese: 姓;… …   Wikipedia

  • Yuan (surname) — Yuan (, Audio|zh yuan2.ogg|pronunciation) is a Chinese surname ranked 37th in China by population.Is Yuen in Canton and Hong Kong, Cantonese Phonetic is Yuen . [Chinese surnames pronounced yuán which still exist include: , , , , , and ; surnames… …   Wikipedia

Share the article and excerpts

Direct link
Do a right-click on the link above
and select “Copy Link”