In
statistics
Statistics (from German language, German: ''wikt:Statistik#German, Statistik'', "description of a State (polity), state, a country") is the discipline that concerns the collection, organization, analysis, interpretation, and presentation of ...
, Wold's decomposition or the Wold representation theorem (not to be confused with the Wold theorem that is the discrete-time analog of the
Wiener–Khinchin theorem
In applied mathematics, the Wiener–Khinchin theorem or Wiener–Khintchine theorem, also known as the Wiener–Khinchin–Einstein theorem or the Khinchin–Kolmogorov theorem, states that the autocorrelation function of a wide-sense-stationary r ...
), named after
Herman Wold, says that every
covariance-stationary time series
In mathematics, a time series is a series of data points indexed (or listed or graphed) in time order. Most commonly, a time series is a sequence taken at successive equally spaced points in time. Thus it is a sequence of discrete-time data. Exa ...
can be written as the sum of two time series, one ''deterministic'' and one ''stochastic''.
Formally
:
where:
:*
is the
time series
In mathematics, a time series is a series of data points indexed (or listed or graphed) in time order. Most commonly, a time series is a sequence taken at successive equally spaced points in time. Thus it is a sequence of discrete-time data. Exa ...
being considered,
:*
is an uncorrelated sequence which is the
innovation process to the process
– that is, a white noise process that is input to the
linear filter
Linear filters process time-varying input signals to produce output signals, subject to the constraint of linearity. In most cases these linear filters are also time invariant (or shift invariant) in which case they can be analyzed exactly using ...
.
:*
is the ''possibly'' infinite vector of moving average weights (coefficients or parameters)
:*
is a deterministic time series, such as one represented by a sine wave.
The moving average coefficients have these properties:
# Stable, that is square summable
<
# Causal (i.e. there are no terms with ''j'' < 0)
# Minimum delay
# Constant (
independent of ''t'')
# It is conventional to define
This theorem can be considered as an existence theorem: any stationary process has this seemingly special representation. Not only is the existence of such a simple linear and exact representation remarkable, but even more so is the special nature of the moving average model. Imagine creating a process that is a moving average but not satisfying these properties 1–4. For example, the coefficients
could define an acausal and model. Nevertheless the theorem assures the existence of a causal that exactly represents this process. How this all works for the case of causality and the minimum delay property is discussed in Scargle (1981), where an extension of the Wold Decomposition is discussed.
The usefulness of the Wold Theorem is that it allows the
dynamic
Dynamics (from Greek δυναμικός ''dynamikos'' "powerful", from δύναμις ''dynamis'' "power") or dynamic may refer to:
Physics and engineering
* Dynamics (mechanics)
** Aerodynamics, the study of the motion of air
** Analytical dynam ...
evolution of a variable
to be approximated by a
linear model
In statistics, the term linear model is used in different ways according to the context. The most common occurrence is in connection with regression models and the term is often taken as synonymous with linear regression model. However, the term ...
. If the innovations
are
independent
Independent or Independents may refer to:
Arts, entertainment, and media Artist groups
* Independents (artist group), a group of modernist painters based in the New Hope, Pennsylvania, area of the United States during the early 1930s
* Independ ...
, then the linear model is the only possible representation relating the observed value of
to its past evolution. However, when
is merely an
uncorrelated
In probability theory and statistics, two real-valued random variables, X, Y, are said to be uncorrelated if their covariance, \operatorname ,Y= \operatorname Y- \operatorname \operatorname /math>, is zero. If two variables are uncorrelated, there ...
but not independent sequence, then the linear model exists but it is not the only representation of the dynamic dependence of the series. In this latter case, it is possible that the linear model may not be very useful, and there would be a nonlinear model relating the observed value of
to its past evolution. However, in practical
time series analysis
In mathematics, a time series is a series of data points indexed (or listed or graphed) in time order. Most commonly, a time series is a sequence taken at successive equally spaced points in time. Thus it is a sequence of discrete-time data. Exa ...
, it is often the case that only linear predictors are considered, partly on the grounds of simplicity, in which case the Wold decomposition is directly relevant.
The Wold representation depends on an infinite number of parameters, although in practice they usually decay rapidly. The
autoregressive model
In statistics, econometrics and signal processing, an autoregressive (AR) model is a representation of a type of random process; as such, it is used to describe certain time-varying processes in nature, economics, etc. The autoregressive model spe ...
is an alternative that may have only a few coefficients if the corresponding moving average has many. These two models can be combined into an
autoregressive-moving average (ARMA) model, or an autoregressive-integrated-moving average (ARIMA) model if non-stationarity is involved. See and references there; in addition this paper gives an extension of the Wold Theorem that allows more generality for the moving average (not necessarily stable, causal, or minimum delay) accompanied by a sharper characterization of the innovation (identically and independently distributed, not just uncorrelated). This extension allows the possibility of models that are more faithful to physical or astrophysical processes, and in particular can sense ″the
arrow of time
The arrow of time, also called time's arrow, is the concept positing the "one-way direction" or " asymmetry" of time. It was developed in 1927 by the British astrophysicist Arthur Eddington, and is an unsolved general physics question. This ...
.″
References
*
*
*
*
Wold, H. (1954) ''A Study in the Analysis of Stationary Time Series'', Second revised edition, with an Appendix on "Recent Developments in Time Series Analysis" by
Peter Whittle. Almqvist and Wiksell Book Co., Uppsala.
{{Statistics, analysis
Theorems in statistics
Time series