Telegraph process

Last updated June 15, 2023

In probability theory, the telegraph process is a memoryless continuous-time stochastic process that shows two distinct values. It models burst noise (also called popcorn noise or random telegraph signal). If the two possible values that a random variable can take are $c_{1}$ and $c_{2}$ , then the process can be described by the following master equations:

where $\lambda _{1}$ is the transition rate for going from state $c_{1}$ to state $c_{2}$ and $\lambda _{2}$ is the transition rate for going from going from state $c_{2}$ to state $c_{1}$ . The process is also known under the names Kac process (after mathematician Mark Kac),^[1] and dichotomous random process.^[2]

Solution

The master equation is compactly written in a matrix form by introducing a vector $\mathbf {P} =[P(c_{1},t|x,t_{0}),P(c_{2},t|x,t_{0})]$ ,

{\frac {d\mathbf {P} }{dt}}=W\mathbf {P}

where

W={\begin{pmatrix}-\lambda _{1}&\lambda _{2}\\\lambda _{1}&-\lambda _{2}\end{pmatrix}}

is the transition rate matrix. The formal solution is constructed from the initial condition $\mathbf {P} (0)$ (that defines that at $t=t_{0}$ , the state is $x$ ) by

\mathbf {P} (t)=e^{Wt}\mathbf {P} (0)

.

It can be shown that^[3]

e^{Wt}=I+W{\frac {(1-e^{-2\lambda t})}{2\lambda }}

where $I$ is the identity matrix and $\lambda =(\lambda _{1}+\lambda _{2})/2$ is the average transition rate. As $t\rightarrow \infty$ , the solution approaches a stationary distribution $\mathbf {P} (t\rightarrow \infty )=\mathbf {P} _{s}$ given by

\mathbf {P} _{s}={\frac {1}{2\lambda }}{\begin{pmatrix}\lambda _{2}\\\lambda _{1}\end{pmatrix}}

Properties

Knowledge of an initial state decays exponentially. Therefore, for a time $t\gg (2\lambda )^{-1}$ , the process will reach the following stationary values, denoted by subscript s:

Mean:

\langle X\rangle _{s}={\frac {c_{1}\lambda _{2}+c_{2}\lambda _{1}}{\lambda _{1}+\lambda _{2}}}.

Variance:

\operatorname {var} \{X\}_{s}={\frac {(c_{1}-c_{2})^{2}\lambda _{1}\lambda _{2}}{(\lambda _{1}+\lambda _{2})^{2}}}.

One can also calculate a correlation function:

\langle X(t),X(u)\rangle _{s}=e^{-2\lambda |t-u|}\operatorname {var} \{X\}_{s}.

Application

This random process finds wide application in model building:

In physics, spin systems and fluorescence intermittency show dichotomous properties. But especially in single molecule experiments probability distributions featuring algebraic tails are used instead of the exponential distribution implied in all formulas above.
In finance for describing stock prices^[1]
In biology for describing transcription factor binding and unbinding.

Related Research Articles

In mathematics, a Hermitian matrix is a complex square matrix that is equal to its own conjugate transpose—that is, the element in the $i$ -th row and $j$ -th column is equal to the complex conjugate of the element in the $j$ -th row and $i$ -th column, for all indices $i$ and $j$ :

In linear algebra, a QR decomposition, also known as a QR factorization or QU factorization, is a decomposition of a matrix A into a product A = QR of an orthonormal matrix Q and an upper triangular matrix R. QR decomposition is often used to solve the linear least squares problem and is the basis for a particular eigenvalue algorithm, the QR algorithm.

In mathematics, the Rayleigh quotient for a given complex Hermitian matrix $and nonzero vector is defined as:$

In mathematics, the Hodge star operator or Hodge star is a linear map defined on the exterior algebra of a finite-dimensional oriented vector space endowed with a nondegenerate symmetric bilinear form. Applying the operator to an element of the algebra produces the Hodge dual of the element. This map was introduced by W. V. D. Hodge.

In physics, the Rabi cycle is the cyclic behaviour of a two-level quantum system in the presence of an oscillatory driving field. A great variety of physical processes belonging to the areas of quantum computing, condensed matter, atomic and molecular physics, and nuclear and particle physics can be conveniently studied in terms of two-level quantum mechanical systems, and exhibit Rabi flopping when coupled to an optical driving field. The effect is important in quantum optics, magnetic resonance and quantum computing, and is named after Isidor Isaac Rabi.

In physics, the S-matrix or scattering matrix relates the initial state and the final state of a physical system undergoing a scattering process. It is used in quantum mechanics, scattering theory and quantum field theory (QFT).

In physics, mathematics and statistics, scale invariance is a feature of objects or laws that do not change if scales of length, energy, or other variables, are multiplied by a common factor, and thus represent a universality.

In the theory of stochastic processes, the Karhunen–Loève theorem, also known as the Kosambi–Karhunen–Loève theorem states that a stochastic process can be represented as an infinite linear combination of orthogonal functions, analogous to a Fourier series representation of a function on a bounded interval. The transformation is also known as Hotelling transform and eigenvector transform, and is closely related to principal component analysis (PCA) technique widely used in image processing and in data analysis in many fields.

In geometry, the line element or length element can be informally thought of as a line segment associated with an infinitesimal displacement vector in a metric space. The length of the line element, which may be thought of as a differential arc length, is a function of the metric tensor and is denoted by .

In quantum mechanics, a two-state system is a quantum system that can exist in any quantum superposition of two independent quantum states. The Hilbert space describing such a system is two-dimensional. Therefore, a complete basis spanning the space will consist of two independent states. Any two-state system can also be seen as a qubit.

In quantum mechanics, the Hellmann–Feynman theorem relates the derivative of the total energy with respect to a parameter to the expectation value of the derivative of the Hamiltonian with respect to that same parameter. According to the theorem, once the spatial distribution of the electrons has been determined by solving the Schrödinger equation, all the forces in the system can be calculated using classical electrostatics.

The name paravector is used for the combination of a scalar and a vector in any Clifford algebra, known as geometric algebra among physicists.

In physics, the algebra of physical space (APS) is the use of the Clifford or geometric algebra Cl_3,0(R) of the three-dimensional Euclidean space as a model for (3+1)-dimensional spacetime, representing a point in spacetime via a paravector.

The Jaynes–Cummings model is a theoretical model in quantum optics. It describes the system of a two-level atom interacting with a quantized mode of an optical cavity, with or without the presence of light. It was originally developed to study the interaction of atoms with the quantized electromagnetic field in order to investigate the phenomena of spontaneous emission and absorption of photons in a cavity.

The theoretical and experimental justification for the Schrödinger equation motivates the discovery of the Schrödinger equation, the equation that describes the dynamics of nonrelativistic particles. The motivation uses photons, which are relativistic particles with dynamics described by Maxwell's equations, as an analogue for all types of particles.

In mathematics, the Schur orthogonality relations, which were proven by Issai Schur through Schur's lemma, express a central fact about representations of finite groups. They admit a generalization to the case of compact groups in general, and in particular compact Lie groups, such as the rotation group SO(3).

A Werner state is a × -dimensional bipartite quantum state density matrix that is invariant under all unitary operators of the form $. That is, it is a bipartite quantum state that satisfies$

Within bayesian statistics for machine learning, kernel methods arise from the assumption of an inner product space or similarity structure on inputs. For some such methods, such as support vector machines (SVMs), the original formulation and its regularization were not Bayesian in nature. It is helpful to understand them from a Bayesian perspective. Because the kernels are not necessarily positive semidefinite, the underlying structure may not be inner product spaces, but instead more general reproducing kernel Hilbert spaces. In Bayesian probability kernel methods are a key component of Gaussian processes, where the kernel function is known as the covariance function. Kernel methods have traditionally been used in supervised learning problems where the input space is usually a space of vectors while the output space is a space of scalars. More recently these methods have been extended to problems that deal with multiple outputs such as in multi-task learning.

In machine learning, the kernel embedding of distributions comprises a class of nonparametric methods in which a probability distribution is represented as an element of a reproducing kernel Hilbert space (RKHS). A generalization of the individual data-point feature mapping done in classical kernel methods, the embedding of distributions into infinite-dimensional feature spaces can preserve all of the statistical features of arbitrary distributions, while allowing one to compare and manipulate distributions using Hilbert space operations such as inner products, distances, projections, linear transformations, and spectral analysis. This learning framework is very general and can be applied to distributions over any space $on which a sensible kernel function may be defined. For example, various kernels have been proposed for learning from data which are: vectors in, discrete classes/categories, strings, graphs/networks, images, time series, manifolds, dynamical systems, and other structured objects. The theory behind kernel embeddings of distributions has been primarily developed by Alex Smola, Le Song, Arthur Gretton, and Bernhard Schölkopf. A review of recent works on kernel embedding of distributions can be found in.$

Low-rank matrix approximations are essential tools in the application of kernel methods to large-scale learning problems.

References

1 2 Bondarenko, YV (2000). "Probabilistic Model for Description of Evolution of Financial Indices". Cybernetics and Systems Analysis. 36 (5): 738–742. doi:10.1023/A:1009437108439. S2CID 115293176.
↑ Margolin, G; Barkai, E (2006). "Nonergodicity of a Time Series Obeying Lévy Statistics". Journal of Statistical Physics. 122 (1): 137–167. arXiv: cond-mat/0504454 . Bibcode:2006JSP...122..137M. doi:10.1007/s10955-005-8076-9. S2CID 53625405.
↑ Balakrishnan, V. (2020). Mathematical Physics: Applications and Problems. Springer International Publishing. pp. 474

This page is based on this Wikipedia article
Text is available under the CC BY-SA 4.0 license; additional terms may apply.
Images, videos and audio are available under their respective licenses.

[Kac-1] 1 2 Bondarenko, YV (2000). "Probabilistic Model for Description of Evolution of Financial Indices". Cybernetics and Systems Analysis. 36 (5): 738–742. doi:10.1023/A:1009437108439. S2CID 115293176.

[2] Margolin, G; Barkai, E (2006). "Nonergodicity of a Time Series Obeying Lévy Statistics". Journal of Statistical Physics. 122 (1): 137–167. arXiv: cond-mat/0504454 . Bibcode:2006JSP...122..137M. doi:10.1007/s10955-005-8076-9. S2CID 53625405.

[3] Balakrishnan, V. (2020). Mathematical Physics: Applications and Problems. Springer International Publishing. pp. 474

[1]

[2]

[3]

v t e Stochastic processes
Discrete time	Bernoulli process Branching process Chinese restaurant process Galton–Watson process Independent and identically distributed random variables Markov chain Moran process Random walk Loop-erased Self-avoiding Biased Maximal entropy
Continuous time	Additive process Bessel process Birth–death process pure birth Brownian motion Bridge Excursion Fractional Geometric Meander Cauchy process Contact process Continuous-time random walk Cox process Diffusion process Empirical process Feller process Fleming–Viot process Gamma process Geometric process Hawkes process Hunt process Interacting particle systems Itô diffusion Itô process Jump diffusion Jump process Lévy process Local time Markov additive process McKean–Vlasov process Ornstein–Uhlenbeck process Poisson process Compound Non-homogeneous Schramm–Loewner evolution Semimartingale Sigma-martingale Stable process Superprocess Telegraph process Variance gamma process Wiener process Wiener sausage
Both	Branching process Galves–Löcherbach model Gaussian process Hidden Markov model (HMM) Markov process Martingale Differences Local Sub- Super- Random dynamical system Regenerative process Renewal process Stochastic chains with memory of variable length White noise
Fields and other	Dirichlet process Gaussian random field Gibbs measure Hopfield model Ising model Potts model Boolean network Markov random field Percolation Pitman–Yor process Point process Cox Poisson Random field Random graph
Time series models	Autoregressive conditional heteroskedasticity (ARCH) model Autoregressive integrated moving average (ARIMA) model Autoregressive (AR) model Autoregressive–moving-average (ARMA) model Generalized autoregressive conditional heteroskedasticity (GARCH) model Moving-average (MA) model
Financial models	Binomial options pricing model Black–Derman–Toy Black–Karasinski Black–Scholes Chan–Karolyi–Longstaff–Sanders (CKLS) Chen Constant elasticity of variance (CEV) Cox–Ingersoll–Ross (CIR) Garman–Kohlhagen Heath–Jarrow–Morton (HJM) Heston Ho–Lee Hull–White LIBOR market Rendleman–Bartter SABR volatility Vašíček Wilkie
Actuarial models	Bühlmann Cramér–Lundberg Risk process Sparre–Anderson
Queueing models	Bulk Fluid Generalized queueing network M/G/1 M/M/1 M/M/c
Properties	Càdlàg paths Continuous Continuous paths Ergodic Exchangeable Feller-continuous Gauss–Markov Markov Mixing Piecewise deterministic Predictable Progressively measurable Self-similar Stationary Time-reversible
Limit theorems	Central limit theorem Donsker's theorem Doob's martingale convergence theorems Ergodic theorem Fisher–Tippett–Gnedenko theorem Large deviation principle Law of large numbers (weak/strong) Law of the iterated logarithm Maximal ergodic theorem Sanov's theorem Zero–one laws (Blumenthal, Borel–Cantelli, Engelbert–Schmidt, Hewitt–Savage, Kolmogorov, Lévy)
Inequalities	Burkholder–Davis–Gundy Doob's martingale Doob's upcrossing Kunita–Watanabe Marcinkiewicz–Zygmund
Tools	Cameron–Martin formula Convergence of random variables Doléans-Dade exponential Doob decomposition theorem Doob–Meyer decomposition theorem Doob's optional stopping theorem Dynkin's formula Feynman–Kac formula Filtration Girsanov theorem Infinitesimal generator Itô integral Itô's lemma Karhunen–Loève theorem Kolmogorov continuity theorem Kolmogorov extension theorem Lévy–Prokhorov metric Malliavin calculus Martingale representation theorem Optional stopping theorem Prokhorov's theorem Quadratic variation Reflection principle Skorokhod integral Skorokhod's representation theorem Skorokhod space Snell envelope Stochastic differential equation Tanaka Stopping time Stratonovich integral Uniform integrability Usual hypotheses Wiener space Classical Abstract
Disciplines	Actuarial mathematics Control theory Econometrics Ergodic theory Extreme value theory (EVT) Large deviations theory Mathematical finance Mathematical statistics Probability theory Queueing theory Renewal theory Ruin theory Signal processing Statistics Stochastic analysis Time series analysis Machine learning
List of topics Category