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Matrix used to describe the transitions of a Markov chain
It is also called a probability matrix, transition matrix, substitution matrix, or Markov matrix. The stochastic matrix was first developed by Andrey Markov
Stochastic_matrix
Type of square matrix
probability and combinatorics, a doubly stochastic matrix (also called bistochastic matrix) is a square matrix X = ( x i j ) {\displaystyle X=(x_{ij})}
Doubly_stochastic_matrix
Stochastic matrix representing links between entities
A Google matrix is a particular stochastic matrix that is used by Google's PageRank algorithm. The matrix represents a graph with edges representing links
Google_matrix
Random process independent of past history
identity matrix of size n, and 0n,n is the zero matrix of size n×n. Multiplying together stochastic matrices always yields another stochastic matrix, so Q
Markov_chain
Matrix describing continuous-time Markov chains
Stochastic matrix Suhov & Kelbert 2008, Definition 2.1.1. Asmussen, S. R. (2003). "Markov Jump Processes". Applied Probability and Queues. Stochastic
Transition-rate_matrix
Randomly determined process
word stochastic is used to describe other terms and objects in mathematics. Examples include a stochastic matrix, which describes a stochastic process
Stochastic
Matrix with no negative elements
of non-negative matrices, e.g. stochastic matrix; doubly stochastic matrix; symmetric non-negative matrix. Metzler matrix Berman, Abraham; Plemmons, Robert
Nonnegative_matrix
Mathematical inequality
An n × n matrix P is doubly stochastic precisely if both P and its transpose PT are stochastic matrices. A stochastic matrix is a square matrix of nonnegative
Muirhead's_inequality
Matrix representation of a graph
theory, the Laplacian matrix, also called the graph Laplacian, admittance matrix, Kirchhoff matrix, or discrete Laplacian, is a matrix representation of a
Laplacian_matrix
Topics referred to by the same term
Transition matrix may refer to: Change-of-basis matrix, associated with a change of basis for a vector space. Stochastic matrix, a square matrix used to
Transition_matrix
Array of numbers
and sum up to one. Stochastic matrices are used to define Markov chains with finitely many states. A row of the stochastic matrix gives the probability
Matrix_(mathematics)
Probability concept
move to a different state as specified by the probabilities of a stochastic matrix. An equivalent formulation describes the process as changing state
Continuous-time_Markov_chain
Concept in probability theory
as a stochastic kernel or probability kernel) is a map that in the general theory of Markov processes plays the role that the transition matrix does in
Markov_kernel
orthogonal matrix Precision matrix — a symmetric n×n matrix, formed by inverting the covariance matrix. Also called the information matrix. Stochastic matrix —
List_of_named_matrices
stochastic vector, since the product of any two stochastic matrices is a stochastic matrix, and the product of a stochastic vector and a stochastic matrix
Probabilistic_automaton
In mathematics, a unistochastic matrix (also called unitary-stochastic) is a doubly stochastic matrix whose entries are the squares of the absolute values
Unistochastic_matrix
Matrix representing the frequency of evolution of a protein or nucleotide sequence
dissimilarity between compared sequences. It is an application of a stochastic matrix. Substitution matrices are usually seen in the context of amino acid
Substitution_matrix
denoted A and B is a doubly stochastic matrix D such that DA = BD. If the doubly stochastic matrix is a permutation matrix, then it constitutes a graph
Fractional_graph_isomorphism
Topics referred to by the same term
probability distributions Regular stochastic matrix, a stochastic matrix such that all the entries of some power of the matrix are positive Free regular set
Regular
Probability concept
be described by a stochastic matrix, which lists the probabilities of moving to each state from any individual state. From this matrix, the probability
Discrete-time_Markov_chain
Probabilistic optimal control
time t realization of the stochastic n × n state transition matrix, Bt is the time t realization of the stochastic n × k matrix of control multipliers,
Stochastic_control
Collection of random variables
In probability theory and related fields a stochastic (/stəˈkæstɪk/) or random process is a mathematical object usually defined as a family of random variables
Stochastic_process
Topics referred to by the same term
Regular matrix may refer to: Regular stochastic matrix, a stochastic matrix such that all the entries of some power of the matrix are positive The opposite
Regular_matrix
Doubly stochastic matrix
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
Orthostochastic_matrix
Concept in network science
communities exactly. The community sizes and probability matrix may be known or unknown. Stochastic block models exhibit a sharp threshold effect reminiscent
Stochastic_block_model
Square matrix whose off-diagonal entries are nonnegative
differential equation M-matrix P-matrix Q-matrix, a specific kind of Metzler matrix Z-matrix Hurwitz-stable matrix Stochastic matrix Positive systems Berman
Metzler_matrix
Measure in graph theory
{x} )} is the diagonal matrix of vector x {\displaystyle \mathbf {x} } . N {\displaystyle \mathbf {N} } is a row-stochastic matrix. The normalized eigenvector
Eigenvector_centrality
Table in statistics
x_{2}=b_{j})=T_{kj},} with k and j ranging over K values, create a K×K matrix. This matrix is a stochastic matrix since the columns sum to 1; i.e. ∑ k T k j = 1 {\displaystyle
Conditional_probability_table
Optimization algorithm
Stochastic gradient descent (often abbreviated SGD) is an iterative method for optimizing an objective function with suitable smoothness properties (e
Stochastic_gradient_descent
Square of numbers with equal row, column and diagonal totals
will yield a doubly stochastic matrix, whose row sums and column sums equal to unity. However, unlike the doubly stochastic matrix, the diagonal sums of
Magic_square
Theorem in linear algebra
possible that none of these will be positive. A row (column) stochastic matrix is a square matrix each of whose rows (columns) consists of non-negative real
Perron–Frobenius_theorem
Computing technique in probability theory
complicated version of the matrix geometric method and is the classical solution method for M/G/1 chains. An M/G/1-type stochastic matrix is one of the form P
Matrix_analytic_method
Russian mathematician (1856–1922)
source Markov network Markov number Markov property Stochastic matrix (also known as Markov matrix) Subjunctive possibility (Russian: Андре́й Андре́евич
Andrey_Markov
Award for advancements in discrete mathematics
Waerden's conjecture that the matrix with all entries equal has the smallest permanent of any doubly stochastic matrix. 1985: Jozsef Beck for tight bounds
Fulkerson_Prize
Tool for working with matrices
lottery on deterministic allocations. A bistochastic matrix (also called: doubly-stochastic) is a matrix in which all elements are greater than or equal to
Birkhoff_algorithm
transition rate from state i to state j. Similarly, for a stochastic matrix P, P is a lumpable matrix on a partition T if and only if, for any subsets ti and
Lumpability
Measure of covariance of components of a random vector
covariance matrix (also known as auto-covariance matrix, dispersion matrix, variance matrix, or variance–covariance matrix) is a square matrix giving the
Covariance_matrix
Every square matrix with positive entries can be written in a certain standard form
elements such that D1AD2 is doubly stochastic. The matrices D1 and D2 are unique up to multiplying the first matrix by a positive number and dividing the
Sinkhorn's_theorem
Algorithm used by Google Search to rank web pages
p_{j})=1} , i.e. the elements of each column sum up to 1, so the matrix is a stochastic matrix (for more details see the computation section below). Thus this
PageRank
Topics referred to by the same term
Doubly stochastic may refer to: Doubly stochastic model Doubly stochastic matrix This disambiguation page lists articles associated with the title Doubly
Doubly_stochastic
Triangular matrix Tridiagonal matrix Block matrix Sparse matrix Hessenberg matrix Hessian matrix Vandermonde matrix Stochastic matrix Toeplitz matrix Circulant
Outline_of_linear_algebra
Standard example in game theory
as a stochastic process and M is a stochastic matrix, allowing all of the theory of stochastic processes to be applied. One result of stochastic theory
Prisoner's_dilemma
Measure of joint variability in statistics
In probability and statistics, given two stochastic processes { X t } {\displaystyle \left\{X_{t}\right\}} and { Y t } {\displaystyle \left\{Y_{t}\right\}}
Cross-covariance
Overview of and topical guide to machine learning
Stephen Wolfram Stochastic block model Stochastic cellular automaton Stochastic diffusion search Stochastic grammar Stochastic matrix Stochastic universal sampling
Outline_of_machine_learning
Matrix equal to its conjugate-transpose
In mathematics, a Hermitian matrix (or self-adjoint matrix) is a square matrix with complex-valued entries that is equal to its own conjugate transpose
Hermitian_matrix
Matrix with exactly one 1 per row and column
In mathematics, particularly in matrix theory, a permutation matrix is a square binary matrix that has exactly one entry of 1 in each row and each column
Permutation_matrix
Vector with non-negative entries that add up to one
In mathematics and statistics, a probability vector or stochastic vector is a vector with non-negative entries that add up to one. Underlying every probability
Probability_vector
Correlation of a signal with a time-shifted copy of itself, as a function of shift
interchangeably. The definition of the autocorrelation coefficient of a stochastic process is ρ X X ( t 1 , t 2 ) = K X X ( t 1 , t 2 ) σ t 1 σ t 2 = E
Autocorrelation
Polytope
polytope are the permutation matrices, and therefore that any doubly stochastic matrix may be represented as a convex combination of permutation matrices;
Birkhoff_polytope
Cellular automaton with probabilistic rules
A stochastic cellular automaton (SCA), also known as a probabilistic cellular automaton (PCA), is a type of computational model. It consists of a grid
Stochastic_cellular_automaton
Calculus on stochastic processes
Stochastic calculus is a branch of mathematics that operates on stochastic processes. It allows a consistent theory of integration to be defined for integrals
Stochastic_calculus
Examples of the probabilistic construct
type j. Notice that the rows of P sum to 1: this is because P is a stochastic matrix. The weather on day 0 (today) is known to be sunny. This is represented
Examples_of_Markov_chains
Type of biased random walk on a graph
using this edge after visiting i {\displaystyle i} . Formally, find a stochastic matrix S {\displaystyle S} (containing the transition probabilities of a
Maximal_entropy_random_walk
Specialized notation for multivariable calculus
the derivative as approximating linear mapping. Matrix calculus is used for deriving optimal stochastic estimators, often involving the use of Lagrange
Matrix_calculus
Linear combination whose coefficients sum to 1
plane, and the trivial cases, a point or the whole space). When a stochastic matrix, A, acts on a column vector, b→, the result is a column vector whose
Affine_combination
Sum of elements on the main diagonal
sophisticated stochastic estimators of trace have been developed. If a 2 x 2 real matrix has zero trace, its square is a diagonal matrix. The trace of
Trace_(linear_algebra)
Degree of connectedness within a graph
A can be real numbers representing connection strengths, as in a stochastic matrix. Katz centrality is a generalization of degree centrality. Degree
Centrality
Deep learning method
K trans {\displaystyle K_{\text{trans}}} can be represented as a stochastic matrix: [ K trans ] = [ ( 1 − 3 p ) p p p p ( 1 − 3 p ) p p p p ( 1 − 3 p
Generative adversarial network
Generative_adversarial_network
Product-form solution / spr Quantum Markov chain / phs Semi-Markov process Stochastic matrix / anl Telegraph process / (U:B) Variable-order Markov model Wiener
Catalog of articles in probability theory
Catalog_of_articles_in_probability_theory
Stochastic equicontinuity Stochastic gradient descent Stochastic grammar Stochastic investment model Stochastic kernel estimation Stochastic matrix Stochastic
List_of_statistics_articles
Study of matrices and their algebraic properties
Orthogonal matrix, unitary matrix Symmetric matrix, antisymmetric matrix Stochastic matrix Matrix polynomial Matrix exponential Some authors, e.g. Horn and
Matrix_analysis
Evolutionary algorithm
matrix adaptation evolution strategy (CMA-ES) is a particular kind of strategy for numerical optimization. Evolution strategies (ES) are stochastic,
CMA-ES
Pore-forming membrane protein
combination with the stochastic matrix to determine the stable distribution matrix by solving the equation PX=X, where P is the stochastic matrix and X is the
Ion_channel
Random variable with multiple component dimensions
types of aggregate random variables, e.g. a random matrix, random tree, random sequence, stochastic process, etc. Formally, a multivariate random variable
Multivariate_random_variable
Vector satisfying some of the criteria of an eigenvector
algebra, a generalized eigenvector of an n × n {\displaystyle n\times n} matrix A {\displaystyle A} is a vector which satisfies certain criteria which are
Generalized_eigenvector
Category whose objects are natural numbers and whose morphisms are matrices
{\displaystyle \mathbb {F} } . A stochastic matrix is a real matrix of nonnegative entries, such that the sum of each column is one. Stochastic matrices include the
Category_of_matrices
Polynomial of the elements of a matrix
that the minimum permanent among all n × n doubly stochastic matrices is n!/nn, achieved by the matrix for which all entries are equal to 1/n. Proofs of
Permanent_(mathematics)
V {\displaystyle A\subset V} . Let P {\displaystyle P} denote the stochastic matrix of the graph, and let λ 2 {\displaystyle \lambda _{2}} be the second
Expander_walk_sampling
Matrix of partial derivatives of a vector-valued function
vector calculus, the Jacobian matrix (/dʒəˈkoʊbiən/, /dʒɪ-, jɪ-/) of a vector-valued function of several variables is the matrix of all its first-order partial
Jacobian matrix and determinant
Jacobian_matrix_and_determinant
Bound on eigenvalues
Doubly stochastic matrix Hurwitz-stable matrix – Matrix whose eigenvalues have negative real part Joel Lee Brenner Metzler matrix – Square matrix whose
Gershgorin_circle_theorem
Process of finding a spatial transformation that aligns two point clouds
result due to Sinkhorn, which states that a doubly stochastic matrix is obtained from any square matrix with all positive entries by the iterative process
Point-set_registration
Mathematical concept
;x_{0})} to the stochastic differential equation { d X = f ( X ) d t + ε d W ( t ) ; X ( 0 ) = x 0 ; {\displaystyle \left\{{\begin{matrix}\mathrm {d} X=f(X)\
Random_dynamical_system
Russian-British mathematician
represent every n × n {\displaystyle n\times n} doubly stochastic matrix, and that some doubly stochastic matrices need this many permutation matrices. In modern
Leon_Mirsky
Family of iterative methods
Stochastic approximation methods are a family of iterative methods typically used for root-finding problems or for optimization problems. The recursive
Stochastic_approximation
Preorder on vectors of real numbers
{\displaystyle \mathbf {x} =\mathbf {D} \mathbf {y} } for some doubly stochastic matrix D {\displaystyle \mathbf {D} } . In particular, x {\displaystyle \mathbf
Majorization
Matrix of second derivatives
In mathematics, the Hessian matrix, Hessian or (less commonly) Hesse matrix is a square matrix of second-order partial derivatives of a scalar-valued function
Hessian_matrix
Matrix-valued random variable
probability theory and mathematical physics, a random matrix is a matrix-valued random variable—that is, a matrix in which some or all of its entries are sampled
Random_matrix
Measure of distinguishability between two quantum states
_{j}(\log q_{j})P_{ij}),} where Pi j = |vi*wj|2. Since the matrix (Pi j)i j is a doubly stochastic matrix and -log is a convex function, the above expression
Quantum_relative_entropy
Statistical model to calculate the value of multiple quantities as they change over time
the vectors in order to write a VAR(p) as a stochastic matrix difference equation, with a concise matrix notation: Y = B Z + U {\displaystyle Y=BZ+U\
Vector_autoregression
perfect fractional matching, then the matrix representation of M {\displaystyle M} is a doubly stochastic matrix – the sum of elements in each row and
Fractional_matching
In mathematics, invariant of square matrices
theory of stochastic dynamics and stochastic differential equations. Determinants as treated above admit several variants: the permanent of a matrix is defined
Determinant
Grammar model in linguistics
A stochastic grammar (statistical grammar) is a grammar framework with a probabilistic notion of grammaticality: Stochastic context-free grammar Statistical
Stochastic_grammar
Quantum analog of probabilistic automata
Another generalization that should be immediately apparent is to use a stochastic matrix for the transition matrices, and a probability vector for the state;
Quantum_finite_automaton
problem states that given a càdlàg process {X(t), t ≥ 0} and an M-matrix R, then stochastic processes {W(t), t ≥ 0} and {Z(t), t ≥ 0} are said to solve the
Skorokhod_problem
Algorithms for matrix decomposition
Distributed Nonnegative Matrix Factorization (DNMF), Scalable Nonnegative Matrix Factorization (ScalableNMF), Distributed Stochastic Singular Value Decomposition
Non-negative matrix factorization
Non-negative_matrix_factorization
Russian mathematician (1938–2023)
Waerden's conjecture that the matrix with all entries equal has the smallest permanent of any doubly stochastic matrix. Egorychev was a professor in the
Georgy_Egorychev
Result of repeatedly applying a mathematical function
systematic. If the function is linear and can be described by a stochastic matrix, that is, a matrix whose rows or columns sum to one, then the iterated system
Iterated_function
Interpretation of quantum mechanics
Stochastic quantum mechanics is a framework for describing the dynamics of particles that are subjected to intrinsic random processes as well as various
Stochastic_quantum_mechanics
Representation of a matrix as a product
algebra, a matrix decomposition or matrix factorization is a factorization of a matrix into a product of matrices. There are many different matrix decompositions;
Matrix_decomposition
Measure of connection disorder in a network
context, network entropy is the entropy of a stochastic matrix associated with the graph adjacency matrix ( A i j ) {\displaystyle (A_{ij})} and the random
Network_entropy
Topics referred to by the same term
kernel, the stochastic discount factor used in mathematical finance Positive-definite kernel, a generalization of a positive-definite matrix Kernel trick
Kernel
Computer-based method for summarizing a text
of similarity. Once the graph is constructed, it is used to form a stochastic matrix, combined with a damping factor (as in the random surfing model),
Automatic_summarization
Method of machine learning
maximize ad revenue, portfolio optimization, shortest path prediction (with stochastic weights, e.g. traffic on roads for a maps application), spam filtering
Online_machine_learning
Topics referred to by the same term
generator (stochastic processes), of a stochastic process infinitesimal generator matrix, of a continuous time Markov chain, a class of stochastic processes
Infinitesimal_generator
Stochastic differential equation
In mathematics — specifically, in stochastic analysis — the infinitesimal generator of a Feller process (i.e. a continuous-time Markov process satisfying
Infinitesimal generator (stochastic processes)
Infinitesimal_generator_(stochastic_processes)
American mathematician
Newman, Morris (1959). "On the minimum of the permanent of a doubly stochastic matrix". Duke Mathematical Journal. 26. doi:10.1215/S0012-7094-59-02606-7
Marvin_Marcus
Discipline within mathematical theory
{\displaystyle U_{n}=X_{A_{n}-}} . This is a discrete-time Markov chain with stochastic matrix: P = ( 1 − a 0 a 0 0 0 0 ⋯ 1 − ( a 0 + a 1 ) a 1 a 0 0 0 ⋯ 1 − ( a
G/M/1_queue
Square matrix that is a generalization of the Hadamard matrix
DNA-RNA Genetic Code Analysis Using Information Theory of Double Stochastic Matrix,” IntechOpen, Book Chapter, April 17, 2022. [Available in Online:
Jacket_matrix
Optimization algorithm
perturbation stochastic approximation (SPSA) is an algorithmic method for optimizing systems with multiple unknown parameters. It is a type of stochastic approximation
Simultaneous perturbation stochastic approximation
Simultaneous_perturbation_stochastic_approximation
Study of mathematical algorithms for optimization problems
Dynamic programming is the approach to solve the stochastic optimization problem with stochastic, randomness, and unknown model parameters. It studies
Mathematical_optimization
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