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翻訳と辞書　辞書検索 [ 開発暫定版 ] スポンサード リンク Stochastic matrix ： ウィキペディア英語版 Stochastic matrix : ''For a matrix whose elements are stochastic, see Random matrix In mathematics, a stochastic matrix (also termed probability matrix, transition matrix, substitution matrix, or Markov matrix) is a matrix used to describe the transitions of a Markov chain. Each of its entries is a nonnegative real number representing a probability. It has found use in probability theory, statistics, mathematical finance and linear algebra, as well as computer science and population genetics. There are several different definitions and types of stochastic matrices: :A right stochastic matrix is a real square matrix, with each row summing to 1. :A left stochastic matrix is a real square matrix, with each column summing to 1. :A doubly stochastic matrix is a square matrix of nonnegative real numbers with each row and column summing to 1. In the same vein, one may define stochastic vector (also called probability vector) as a vector whose elements are nonnegative real numbers which sum to 1. Thus, each row of a right stochastic matrix (or column of a left stochastic matrix) is a stochastic vector. A common convention in English language mathematics literature is to use row vectors of probabilities and right stochastic matrices rather than column vectors of probabilities and left stochastic matrices; this article follows that convention. ==Definition and properties== A stochastic matrix describes a Markov chain $\backslash boldsymbol\_$ over a finite state space ''S''. If the probability of moving from $i$ to $j$ in one time step is $Pr(j|i)=P\_$, the stochastic matrix ''P'' is given by using $P\_$ as the $i^$ row and $j^$ column element, e.g., : p_&p_&\dots&p_&\dots\\ \vdots&\vdots&\ddots&\vdots&\ddots\\ p_&p_&\dots&p_&\dots\\ \vdots&\vdots&\ddots&\vdots&\ddots \end\right). Since the total of transition probability from a state $i$ to all other states must be 1, this matrix is a right stochastic matrix, so that :$\backslash sum\_j\; P\_=1.\backslash ,$ The product of two right stochastic matrices is also right stochastic. In particular, the $k$-th power $P^k$ of a right stochastic matrix $P$ is also right stochastic. The probability of transitioning from $i$ to $j$ in two steps is then given by the $(i,j)^$ element of the square of $P$: :$\backslash left(P\; ^\backslash right)\_.$ In general the probability transition of going from any state to another state in a finite Markov chain given by the matrix $P$ in ''k'' steps is given by $P^k$. An initial distribution is given as a row vector. A stationary probability vector $\backslash boldsymbol$ is defined as a distribution, written as a row vector, that does not change under application of the transition matrix; that is, it is defined as a probability distribution on the set $\backslash$ which is also a row eigenvector of the probability matrix, associated with eigenvalue 1: :$\backslash boldsymbolP=\backslash boldsymbol.$ The right spectral radius of every right stochastic matrix is clearly at most 1. Additionally, every right stochastic matrix has an obvious column eigenvector associated to the eigenvalue 1: The vector $\backslash boldsymbol$, whose coordinates are all equal to 1. As left and right eigenvalues of a square matrix are the same, every stochastic matrix has, at least, a row eigenvector associated to the eigenvalue 1 and the largest absolute value of all its eigenvalues is also 1. Finally, the Brouwer Fixed Point Theorem (applied to the compact convex set of all probability distributions of the finite set $\backslash$) implies that there is some left eigenvector which is also a stationary probability vector. On the other hand, the Perron–Frobenius theorem also ensures that every irreducible stochastic matrix has such a stationary vector, and that the largest absolute value of an eigenvalue is always 1. However, this theorem cannot be applied directly to such matrices because they need not be irreducible. In general, there may be several such vectors. However, for a matrix with strictly positive entries (or, more generally, for an irreducible aperiodic stochastic matrix), this vector is unique and can be computed by observing that for any $i$ we have the following limit, :$\backslash lim\_\backslash left(P^k\; \backslash right)\_=\backslash boldsymbol\_j,$ where $\backslash boldsymbol\_$ is the $j^$ element of the row vector $\backslash boldsymbol$. Among other things, this says that the long-term probability of being in a state $j$ is independent of the initial state $i$. That both of these computations give the same stationary vector is a form of an ergodic theorem, which is generally true in a wide variety of dissipative dynamical systems: the system evolves, over time, to a stationary state. Intuitively, a stochastic matrix represents a Markov chain; the application of the stochastic matrix to a probability distribution redistributes the probability mass of the original distribution while preserving its total mass. If this process is applied repeatedly, the distribution converges to a stationary distribution for the Markov chain. 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「Stochastic matrix」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.