翻訳と辞書 Words near each other ・ Morar railway station ・ Morar Railway Viaduct ・ Morar, Gwalior district ・ Moran Glacier ・ Moran House ・ Moran Independent School District ・ Moran Lake ・ Moran language ・ Moran Lavi ・ Moran Mazor ・ Moran Medal ・ Moran Municipal Generation Station ・ Moran Norris ・ Moran of the Lady Letty ・ Moran of the Marines ・ Moran process ・ Moran River ・ Moran Roth ・ Moran Sarkar ・ Moran Shipping Agencies ・ Moran State Park ・ Moran Station ・ Moran sternwheelers ・ Moran Town ・ Moran Township ・ Moran Township, Michigan ・ Moran Township, Todd County, Minnesota ・ Moran v Pyle National (Canada) Ltd ・ Moran v. Household International, Inc. ・ Moran's I Dictionary Lists mini英和辞書 mini和英辞書 Webster 1913 Latin-English FOLDOC Wikipedia English ウィキペディア

翻訳と辞書　辞書検索 [ 開発暫定版 ] スポンサード リンク Moran process ： ウィキペディア英語版 Moran process A Moran process or Moran model is a simple stochastic process used in biology to describe finite populations. The process is named after Patrick Moran, who first proposed the model in 1958. It can be used to model variety-increasing processes such as mutation as well as variety-reducing effects such as genetic drift and natural selection. The process can describe the probabilistic dynamics in a finite population of constant size ''N'' in which two alleles A and B are competing for dominance. The two alleles are considered to be true replicators (i.e. entities that make copies of themselves). In each time step a random individual (which is of either type A or B) is chosen for reproduction and a random individual is chosen for death; thus ensuring that the population size remains constant. To model selection, one type has to have a higher fitness and is thus more likely to be chosen for reproduction. The same individual can be chosen for death and for reproduction in the same step. ==Neutral drift== Neutral drift is the idea that a neutral mutation can spread throughout a population, so that eventually the original allele is lost. A neutral mutation does not bring any fitness advantage or disadvantage to its bearer. The simple case of the Moran process can describe this phenomenon. If the number of A individuals is given by ''i'' then the Moran process is defined on the state space . Since the number of A individuals can change at most by one at each time step, a transition exists only between state ''i'' and state and . Thus the transition matrix of the stochastic process is tri-diagonal in shape and the transition probabilities are :$\backslash begin$ P_&=1\\ P_ &= \frac \frac\\ P_ &= 1- P_ - P_\\ P_ &= \frac \frac\\ P_&=1. \end The entry $P\_$ denotes the probability to go from state ''i'' to state ''j''. To understand the formulas for the transition probabilities one has to look at the definition of the process which states that always one individual will be chosen for reproduction and one is chosen for death. Once the A individuals have died out, they will never be reintroduced into the population since the process does not model mutations (A cannot be reintroduced into the population once it has died out and ''vice versa'') and thus $P\_=1$. For the same reason the population of A individuals will always stay ''N'' once they have reached that number and taken over the population and thus $P\_=1$. The states 0 and ''N'' are called ''absorbing'' while the states are called ''transient''. The intermediate transition probabilities can be explained by considering the first term to be the probability to choose the individual whose abundance will increase by one and the second term the probability to choose the other type for death. Obviously, if the same type is chosen for reproduction and for death, then the abundance of one type does not change. Eventually the population will reach one of the absorbing states and then stay there forever. In the transient states, random fluctuations will occur but eventually the population of A will either go extinct or reach fixation. This is one of the most important differences to deterministic processes which cannot model random events. The expected value and the variance of the number of A individuals at timepoint ''t'' can be computed when an initial state is given: :$\backslash begin$ E(= i ) &= i \\ Var(X(t)|X(0) = i) &= \tfrac \left(1-\tfrac \right ) \frac \right )^t}} \end : $\backslash begin$ E (| X(t-1) = i ) &= (i-1)P_ + iP_ + (i+1)P_\\ &= 2ip(1-p) + i(p^2 + (1-p)^2) \\ &= i. \end Writing $Y\; =\; X(t)$ and $Z\; =\; X(t-1)$, and applying the law of total expectation, $E()\; =\; E.$ Applying the argument repeatedly gives $E()\; =\; E(),$ or $E(=\; i)\; =\; i.$ For the variance the calculation runs as follows. Writing $V\_t\; =\; Var(X(t)|X(0)\; =\; i),$ we have : $\backslash begin$ V_1 &= E \left(= i \right ) - E()^2 \\ &= (i-1)^2p(1-p) + i^2 \left (p^2+(1-p)^2 \right ) + (i+1)^2p(1-p) - i^2 \\ &= 2p(1-p) \end For all , $(X(t)|X(t-1)\; =\; i)$ and $(X(1)|X(0)\; =\; i)$ are identically distributed, so their variances are equal. Writing as before $Y\; =\; X(t)$ and $Z\; =\; X(t-1)$, and applying the law of total variance, : $\backslash begin$ Var(Y) &= E() + Var(E()) \\ &= E \left ((\frac \right) \left(1-\frac \right ) \right ) + Var(Z)\\ &= \left(\frac \right ) \left (1-\frac \right) + \left(1-\frac\right)Var(Z). \end If $X(0)\; =\; i$, we obtain :$V\_t\; =\; V\_1\; +\; \backslash left\; (1-\backslash frac\; \backslash right)V\_.$ Rewriting this equation as :$V\_t\; -\; \backslash frac\}\; =\; \backslash left\; (1-\backslash frac\; \backslash right\; )\backslash left(V\_-\backslash frac\}\backslash right)\; =\; \backslash left\; (1-\backslash frac\; \backslash right)^\; \backslash left(V\_1-\backslash frac\}\backslash right)$ yields : $V\_t\; =\; V\_1\; \backslash frac\; \backslash right)^t\}\}$ as desired. ---- }} The probability of A to reach fixation is called ''fixation probability''. For the simple Moran process this probability is Since all individuals have the same fitness, they also have the same chance of becoming the ancestor of the whole population; this probability is and thus the sum of all ''i'' probabilities (for all A individuals) is just The mean time to absorption starting in state ''i'' is given by : $k\_i\; =\; N\; \backslash left(\backslash sum\_^\; \backslash frac\; +\; \backslash sum\_^\; \backslash frac\; \backslash right)$ The variable $y\_i^\; =\; k\_^j-\; k\_^j$ is used and the equation becomes : $\backslash begin$ y_^ &= y_i^ -\frac \\ \\ \sum_^m y_i^ &= (k_^j- k_^j) + (k_^j- k_^j) + \cdots + (k_^j- k_^j) + (k_^j- k_^j) \\ &= k_^j - k_^j \\ \sum_^m y_i^ &= k_^j \\ \\ y_1^ &= (k_^j- k_^j) = k_^j \\ y_2^ &= y_1^ -\frac = k_1^ -\frac \\ y_3^ &= k_1^ -\frac -\frac \\ & \vdots \\ y_i^ &= k_1^ -\sum_^ \frac = \begin k_1^j & j \geq i\\ k_1^j - \frac & j \leq i \end \\ \\ k_i^j &= \sum_^i y_m^ = \begin i \cdot k_1^j & j \geq i\\ i \cdot k_1^j - \frac & j \leq i \end \end Knowing that $k\_N^j\; =\; 0$ and :$q\_j\; =\; P\_=\backslash frac\; \backslash frac$ we can calculate $k\_1^j$: : $\backslash begin$ k_N^j = \sum_^m y_i^ = N \cdot k_1^j &- \frac = 0 \\ k_1^j &= \frac \end Therefore : with $k\_j^j\; =\; N$. Now , the total time until fixation starting from state ''i'', can be calculated : $\backslash begin$ k_i = \sum_^k_i^j &= \sum_^k_i^j + \sum_^k_i^j \\ &= \sum_^N \frac + \sum_^N \frac \end ---- }} For large ''N'' the approximation : $\backslash lim\_\; k\_i\; \backslash approx\; -N^2\; \backslash left((1-x\_i)\; \backslash ln(1-x\_i)\; +\; x\_i\; \backslash ln(x\_i)\; \backslash right)$ holds. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Moran process」の詳細全文を読む スポンサード リンク 翻訳と辞書 : 翻訳のためのインターネットリソース Copyright(C) kotoba.ne.jp 1997-2016. All Rights Reserved.