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Class of stochastic process
statistics, a stationary process (also called a strict/strictly stationary process or strong/strongly stationary process) is a stochastic process whose statistical
Stationary_process
Stochastic process that exhibits both stationarity and ergodicity
stationary ergodic process is a stochastic process which exhibits both stationarity and ergodicity. In essence this implies that the random process will
Stationary_ergodic_process
Stochastic process in time series analysis
trend-stationary process is a stochastic process from which an underlying trend (function solely of time) can be removed, leaving a stationary process. The
Trend-stationary_process
Feature of some stochastic processes
time series to be non-stationary, yet have no unit root and be trend-stationary. In both unit root and trend-stationary processes, the mean can be growing
Unit_root
Topics referred to by the same term
supplies stationary process, in mathematics and statistics, a process whose probability distribution does not change over time. stationary set, in set
Stationary
Statistical model
stochastic process is strict-sense stationary. However, for a Gaussian stochastic process the two concepts are equivalent. Therefore, stationary Gaussian
Gaussian_process
Index of articles associated with the same name
distribution of a stationary process or stationary time series The set of joint probability distributions of a stationary process or stationary time series
Stationary_distribution
Formula in probability theory
theory, Rice's formula counts the average number of times an ergodic stationary process X(t) per unit time crosses a fixed level u. Adler and Taylor describe
Rice's_formula
Concept in probability and statistics
time. If { X t } {\displaystyle \left\{X_{t}\right\}} is a weakly stationary (WSS) process, then the following are true: μ t 1 = μ t 2 ≜ μ {\displaystyle
Autocovariance
Type of queue model in queueing theory
proportion of time which the server is occupied. The probability that the stationary process is in state i (contains i customers, including those in service) is
M/M/1_queue
states; every stationary process in N outcomes is a Bernoulli scheme, and vice versa. Bessel process Birth–death process Branching process Branching random
List of stochastic processes topics
List_of_stochastic_processes_topics
Representation of a type of random process
coefficients are allowed to change over time to model evolving or non-stationary processes. TVAR models are widely applied in cases where the underlying dynamics
Autoregressive_model
Correlation of a signal with a time-shifted copy of itself, as a function of shift
autocorrelation, such as unit root processes, trend-stationary processes, autoregressive processes, and moving average processes. In statistics, the autocorrelation
Autocorrelation
Topic in mathematics
[-\log c(n,n,X)]} exist and are equal for any stationary process including the stationary ergodic process X. Denote it as H. Argue that both c ( i , k
Asymptotic equipartition property
Asymptotic_equipartition_property
The Airy processes are a family of stationary stochastic processes that appear as limit processes in the theory of random growth models and random matrix
Airy_process
Trending periodic processes
Trend periodic non-stationary processes (or trend cyclostationary processes) are a type of cyclostationary process that exhibits both periodic behavior
Trend periodic nonstationary processes
Trend_periodic_nonstationary_processes
Square matrix of cyclically shifted rows
translation-invariant problems on periodic domains, and in the study of stationary stochastic processes. In orthogonal frequency-division multiplexing, the cyclic
Circulant_matrix
In probability theory, a stochastic process is said to have stationary increments if its change only depends on the time span of observation, but not on
Stationary_increments
Signal with properties that vary cyclically with time
interleaved stationary processes. For example, the maximum daily temperature in New York City can be modeled as a cyclostationary process: the maximum
Cyclostationary_process
Probability theory concept
n-fBm. n-fBm is a Gaussian, self-similar, non-stationary process whose increments of order n are stationary. For n = 1, n-fBm is classical fBm. Like the
Fractional_Brownian_motion
Summary statistic
X.} In other words, a process is integrated to order d if taking repeated differences d times yields a stationary process. In particular, if a series
Order_of_integration
Stochastic process modeling random walk with friction
The Ornstein–Uhlenbeck process is a stationary Gauss–Markov process, which means that it is a Gaussian process, a Markov process, and is temporally homogeneous
Ornstein–Uhlenbeck_process
Type of motion in mechanical engineering
Mathematically, random vibration is characterized as an ergodic and stationary process. The acceleration spectral density (ASD) or power spectral density
Random_vibration
Topics referred to by the same term
The term "stationary source" may refer to one of the following: A source of data produced by a stationary process, in the mathematical theory of probability
Stationary_source
Time series statistical test
time series to be non-stationary, have no unit root yet be trend-stationary. In both unit root and trend-stationary processes, the mean can be growing
KPSS_test
Mathematical decompositions of vector fields
discrete spaces. In particular, it applies to decompositions of stationary stochastic processes, and to edge-flows over graphs and simplicial complexes. It
Helmholtz–Hodge_decomposition
Time density of the average information in a stochastic process
\infty }H(X_{n}|X_{n-1},X_{n-2},\dots X_{1})} For strongly stationary stochastic processes, H ( X ) = H ′ ( X ) {\displaystyle H(X)=H'(X)} . The entropy
Entropy_rate
Set of laboratory techniques for separation of mixtures
hold the stationary phase in place. The separation process in CPC is governed solely by the partitioning of solutes between the stationary and mobile
Chromatography
Random process independent of past history
same stationary distribution as the forward process. A chain is said to be reversible if the reversed process is the same as the forward process (in distribution)
Markov_chain
Series of activities
system in a given state Lévy process, a stochastic process with independent, stationary increments Poisson process, a point process consisting of randomly located
Process
Power spectrum of a noise signal
stationary process. The other major difference between this and the previous method is that the differencing used to make the time series stationary (δ
Colors_of_noise
Random sequence whose joint probability distribution is invariant over time
probability theory – specifically in the theory of stochastic processes, a stationary sequence is a random sequence whose joint probability distribution
Stationary_sequence
Theorem relating stationary processes' autocorrelations and power spectra
power spectral density of a wide-sense-stationary random process is equal to the Fourier transform of that process's autocorrelation function. Norbert Wiener
Wiener–Khinchin_theorem
Relative importance of certain frequencies in a composite signal
an input signal. If the signal being analyzed can be considered a stationary process, the STFT is a good smoothed estimate of its power spectral density
Spectral_density
Theorem of stationary processes
t} , but it is a stochastic process and it is also covariance-stationary, it cannot be an arbitrary deterministic process that violates stationarity.
Wold's_theorem
Signal (re-)construction algorithm
generally converge if the sample sequence comes from sampling almost any stationary process, in which case the sample sequence is not square summable, and is
Whittaker–Shannon interpolation formula
Whittaker–Shannon_interpolation_formula
Term in proability theory
frequently are non-stationary. They are typically modelled as either trend-stationary or difference stationary. A trend stationary process {yt} evolves according
Stochastic_drift
Collection of random variables
distribution of the stationary stochastic process remains the same. A sequence of random variables forms a stationary stochastic process only if the random
Stochastic_process
Power law growth of entropy of language or a stochastic process
term "Hilberg's law" or "Hilberg's condition" often applies to any stationary process that satisfies a power-law scaling of H ( n ) − h n {\displaystyle
Hilberg's_hypothesis
Trading strategy
portfolio with a stationary spread series. Regardless of how the portfolio is constructed, if the spread series is a stationary processes, then it can be
Pairs_trade
Signal processing algorithm
linear time-invariant (LTI) filtering of an observed noisy process, assuming known stationary signal and noise spectra, and additive noise. The Wiener filter
Wiener_filter
Spatial statistics function
{s} _{1})-Z(\mathbf {s} _{2})\right)^{2}\right].} In the case of a stationary process, the variogram and semivariogram can be represented as a function
Variogram
Phenomenon in linguistics and data analysis
and short-range dependent stationary process is in terms of their autocovariance functions. For a short-range dependent process, the coupling between values
Long-range_dependence
Part of signal analysis and signal processing
time-varying spectrum for non-stationary processes is defined from the expected Wigner–Ville distribution. Locally stationary processes appear in many physical
Bilinear time–frequency distribution
Bilinear_time–frequency_distribution
Concept in statistics
of a process whereby ergodicity measure is applied. A wide-sense stationary process X ( t ) {\displaystyle X(t)} has constant mean μ X = E [ X ( t ) ]
Ergodic_process
Type of random mathematical object
the Poisson process located in some region of space. The resulting point process is called a homogeneous or stationary Poisson point process. In the second
Poisson_point_process
Property of measure-preserving dynamical systems
probability p {\displaystyle p} of heads on each toss. The resulting stationary process is ergodic: apart from events of probability zero, there is no further
Ergodicity
Statistical model used in time series analysis
generalizations of the autoregressive moving average (ARMA) model to non-stationary series and periodic variation, respectively. All these models are fitted
Autoregressive integrated moving average
Autoregressive_integrated_moving_average
Average uncertainty in variable's states
generated by a given stochastic process: this will always be equal to the entropy rate in the case of a stationary process.) Other quantities of information
Entropy_(information_theory)
Stochastic process generalizing Brownian motion
continuous-time stochastic process named after Norbert Wiener. It is one of the best known Lévy processes (càdlàg stochastic processes with stationary independent increments)
Wiener_process
Stochastic process in probability theory
probability theory, a Lévy process, named after the French mathematician Paul Lévy, is a stochastic process with independent, stationary increments: it represents
Lévy_process
Process to separate compounds by properties
mobile phase, carrying mixtures through a stationary phase, packed inside a column. The chromatographic process is harnessed to create a solid-liquid extractive
Solid-phase_extraction
Operates industrial machinery and equipment that provide energy in various forms
A stationary engineer (also called an operating engineer, power engineer or process operator) is a technically trained professional who operates, troubleshoots
Stationary_engineer
State in which variables of a system are unchanging in time
types of equilibrium Stability (disambiguation) State function Stationary process Stationary state Time invariance Transient state "AC analysis intro 1 (Video)"
Steady_state
Stochastic processes
Gaussian processes and Markov processes. A stationary Gauss–Markov process is unique[citation needed] up to rescaling; such a process is also known as an Ornstein–Uhlenbeck
Gauss–Markov_process
Mathematical model of financial markets
liquidity risk, which is difficult to hedge; the assumption of a stationary process, yielding volatility risk, which can be hedged with volatility hedging;
Black–Scholes_model
Theory of the origin of the state
The theory of the stationary bandit is a political economy theory explaining the origins of the state, primarily developed by the American economist Mancur
Stationary_bandit_theory
American mathematician
best known for his work in information theory, ergodic theory, and stationary process theory. Kieffer received his elementary and high school education
John_Kieffer
unemployment rate exhibits hysteresis, then it follows a statistically non-stationary process, because the expected value of the unemployment rate now and in the
Hysteresis_(economics)
Sequence of data points over time
methods. The parametric approaches assume that the underlying stationary stochastic process has a certain structure which can be described using a small
Time_series
Industrial process
sulfuric acid are first fed onto a stationary reaction plate where an initial reaction takes place. The stationary plate is up to 6 m (20 ft) in diameter
Mannheim_process
American-Israeli mathematician
group theory, number theory and combinatorics". Furstenberg, Harry, Stationary processes and prediction theory, Princeton, N.J., Princeton University Press
Hillel_Furstenberg
Mathematical model for sequential decision making under uncertainty
decision process (MDP) is a mathematical model for sequential decision making when outcomes are uncertain. It is a type of stochastic decision process, and
Markov_decision_process
These random processes are either stationary or non-stationary. The first moments of a stationary process are invariant with time. Cottis, R.A. (1996), "The
Electrochemical_noise
Theorem that tells the maximum rate at which information can be transmitted
Gaussian stationary process noise. This formula's way of introducing frequency-dependent noise cannot describe all continuous-time noise processes. For example
Shannon–Hartley_theorem
Estimate of the spectral density of a signal
average of the corresponding elements of all the periodograms. For stationary processes, this reduces the noise variance of each element by approximately
Periodogram
Hypothetical mechanism for extracting energy from rotating black holes
cannot keep up with the rotation of the black hole, as the trajectories of stationary (from the outside perspective) objects become space-like, rather than
Penrose_process
Russian mathematician
limit theorems, giving a definition of a stationary process and laying a foundation for the theory of such processes. Khinchin made significant contributions
Aleksandr_Khinchin
Equation relating transport coefficients to correlation functions
the heat flux is the gradient of 1 / T {\displaystyle 1/T} . For a stationary process, the mean squared increment of A can be written as ⟨ [ A ( t ) − A
Green–Kubo_relations
Statistical hypothesis test for forecasting
series is a stationary process, the test is performed using the level values of two (or more) variables. If the variables are non-stationary, then the test
Granger_causality
software Static analysis Stationary distribution Stationary ergodic process Stationary process Stationary sequence Stationary subspace analysis Statistic
List_of_statistics_articles
Physics demonstration device
released, it strikes the stationary spheres, compressing them and thereby transmitting a pressure wave through the stationary spheres, which creates a
Newton's_cradle
Higher-order frequency analysis
generalizations. In his original publications, Brillinger considers a stationary process (the signal) z ( t ) {\displaystyle z(t)} and its multi-time cumulant
Polyspectra
817–843. Ferrari, Pablo A.; Mari, Nevena "Quasi stationary distributions and Fleming Viot processes" Archived 2016-03-03 at the Wayback Machine, Lecture
Fleming–Viot_process
including those for which the rare-event rates are time-dependent (non-stationary process). To treat systems in which there is time dependence in the dynamics
Stochastic process rare event sampling
Stochastic_process_rare_event_sampling
British statistician
Rebecca Killick is a British statistician whose work concerns non-stationary processes and changepoint detection. They are a professor of statistics at
Rebecca_Killick
of process capability Time-to-time variability Appearance of a non-stationary process Examine process inputs or steps for evidence of shifts or drifts
Multi-vari_chart
Radar signal processing technique
reflected from the object will survive this process because of constructive interference. If all objects are stationary, the two samples will cancel out and
Moving_target_indication
∈ [ 1 / 2 , ∞ ) {\displaystyle H\in [1/2,\infty )} . A wide-sense stationary process ( X n ) n ≥ 0 {\displaystyle (X_{n})_{n\geq 0}} is called exactly
Self-similar_process
The stationary wavelet transform (SWT) is a wavelet transform algorithm designed to overcome the lack of translation-invariance of the discrete wavelet
Stationary_wavelet_transform
Aspect of probability theory
a stochastic process expressed in any formalism to have a product form stationary distribution (assuming that the process is stationary). The theorem
Reversed compound agent theorem
Reversed_compound_agent_theorem
Theorem in probability theory
states that for a stationary continuous-time Markov chain, a process defined as the time-reversed process has the same stationary distribution as the
Kelly's_lemma
Probability amplitude in quantum scattering theory
spherical wave relative to the incoming plane wave in a stationary-state scattering process. Scattering in quantum mechanics begins with a physical model
Scattering_amplitude
Estimation technique for serially correlated observations
{var} (\varepsilon _{t-1})+\mathrm {var} (e_{t})} Noting that for a stationary process, variance is constant over time, ( 1 − ρ 2 ) v a r ( ε t ) = v a r
Prais–Winsten_estimation
Region outside of a rotating black hole's event horizon
of this dragging effect, an object within the ergosphere cannot appear stationary with respect to an outside observer at a great distance unless that object
Ergosphere
Time series model
stochastic structure. Contrary to the AR model, the finite MA model is always stationary. The moving-average model should not be confused with the moving average
Moving-average_model
Automatic conversion of spoken language into text
stationary signal or a short-time stationary signal. In a short time scale (e.g. 10 milliseconds), speech can be approximated as a stationary process
Speech_recognition
Stochastic process for effort or wear
time. The gamma process has independent and stationary increments which follow the gamma distribution, hence the name. The gamma process is studied in mathematics
Gamma_process
Mathematical models of changing DNA
) Q . {\displaystyle \mathbf {p} '(t)=\mathbf {p} (t)Q\,.} For a stationary process, where Q {\displaystyle Q} does not depend on time t, this differential
Models_of_DNA_evolution
Type of signal processing statistic
measurement process, or by the spectral signal processing can contribute to or corrupt the coherence. If the signals are non-stationary, (and therefore
Coherence_(signal_processing)
Random walk Monte Carlo Renewal theory Skorokhod's embedding theorem Stationary process Stochastic calculus Itô calculus Malliavin calculus Stratonovich integral
List_of_probability_topics
Formulas to determine the energy balance of a nonlinear wave
conservation of energy. It requires that energy storage in the circuit is a stationary process that varies with time only due to the oscillations and not due to
Manley–Rowe_relations
Web page delivered to web browser as-is
A static web page, sometimes called a flat page or a stationary page, is a web page that is delivered to a web browser exactly as stored, in contrast to
Static_web_page
Probability concept
time-reversed process is defined to be X ^ t = X T − t {\displaystyle {\hat {X}}_{t}=X_{T-t}} . By Kelly's lemma this process has the same stationary distribution
Continuous-time_Markov_chain
network data. Suppose X {\displaystyle X} be a weakly stationary (2nd-order stationary) process with mean μ {\displaystyle \mu } , variance σ 2 {\displaystyle
Self-Similarity of Network Data Analysis
Self-Similarity_of_Network_Data_Analysis
Solution to a stochastic differential equation
statistics, diffusion processes are a class of continuous-time Markov process with almost surely continuous sample paths. Diffusion processes are stochastic
Diffusion_process
State of extreme dryness or process of thorough drying
joining (NHEJ) pathway are more sensitive to prolonged desiccation during stationary phase than wild-type strains. NHEJ appears to be the preferred pathway
Desiccation
Field of electrical engineering
Recurrence relations Transform theory Time-frequency analysis – for processing non-stationary signals Linear canonical transformation Spectral estimation –
Signal_processing
Large-format camera
other to facilitate focusing on close objects (macrophotography). The stationary process camera is used for copying nearly flat artwork that is held to a copyboard
View_camera
Denoising scheme in information theory
noiseless sequence x {\displaystyle \mathbf {x} } is a realization of a stationary process X {\displaystyle \mathbf {X} } , the DUDE asymptotically performs
Discrete_Universal_Denoiser
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