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A doubly stochastic matrix is a square matrix of nonnegative real numbers with each row and column summing to 1. A substochastic matrix is a real square matrix whose row sums are all ; In the same vein, one may define a probability vector as a vector whose elements are nonnegative real numbers which sum to 1. Thus, each row of a right ...
A Markov chain can be described by a stochastic matrix, which lists the probabilities of moving to each state from any individual state. From this matrix, the probability of being in a particular state n steps in the future can be calculated. A Markov chain's state space can be partitioned into communicating classes that describe which states ...
Explain: The original matrix equation is equivalent to a system of n×n linear equations in n×n variables. And there are n more linear equations from the fact that Q is a right stochastic matrix whose each row sums to 1. So it needs any n×n independent linear equations of the (n×n+n) equations to solve for the n×n variables.
The term stochastic process first appeared in English in a 1934 paper by Joseph L. Doob. [1] For the term and a specific mathematical definition, Doob cited another 1934 paper, where the term stochastischer Prozeß was used in German by Aleksandr Khinchin, [22] [23] though the German term had been used earlier in 1931 by Andrey Kolmogorov. [24]
A continuous-time Markov chain (CTMC) is a continuous stochastic process in which, for each state, the process will change state according to an exponential random variable and then move to a different state as specified by the probabilities of a stochastic matrix. An equivalent formulation describes the process as changing state according to ...
Doubly stochastic matrix — a non-negative matrix such that each row and each column sums to 1 (thus the matrix is both left stochastic and right stochastic) Fisher information matrix — a matrix representing the variance of the partial derivative, with respect to a parameter, of the log of the likelihood function of a random variable.
The class of doubly stochastic matrices is a convex polytope known as the Birkhoff polytope.Using the matrix entries as Cartesian coordinates, it lies in an ()-dimensional affine subspace of -dimensional Euclidean space defined by independent linear constraints specifying that the row and column sums all equal 1.
In mathematics, an orthostochastic matrix is a doubly stochastic matrix whose entries are the squares of the absolute values of the entries of some orthogonal matrix. The detailed definition is as follows. A square matrix B of size n is doubly stochastic (or bistochastic) if all its rows and columns sum to 1 and all its entries are nonnegative ...