Wold's decomposition

The Wold decomposition describes a special decomposition in time series analysis , a sub-area of ​​mathematical statistics.

Bulk

The decomposition is named after Herman Wold , who showed in 1938 that the random variables of a time-discrete covariance stationary , non-deterministic stochastic process can be decomposed into two parts: ${\displaystyle x_{t}}$

• into a deterministic part and ${\displaystyle \mu _{t}}$
• into a purely non-deterministic part, which is created by smoothing random variables . All in all: ${\displaystyle u_{t}}$

${\displaystyle x_{t}=\mu _{t}+\sum _{j=0}^{\infty }\psi _{j}u_{t-j}}$

The random variables have the expected value zero and a constant variance and are uncorrelated in pairs : ${\displaystyle u_{t}}$

${\displaystyle \operatorname {E} (u_{t})=0}$ and ${\displaystyle \operatorname {E} (u_{t}u_{s})={\begin{cases}\sigma ^{2}&\mathrm {f{\ddot {u}}r} \quad t=s\\0&{\text{sonst}}\end{cases}}}$

The smoothing sequence is ${\displaystyle \psi _{j}}$

• possibly infinitely long (but can also be finite)
• square sum label: ${\displaystyle \sum _{j=0}^{\infty }|\psi _{j}|^{2}<\infty }$
• "Causal" (there are no terms ) ${\displaystyle j<0}$
• they are constant (i.e. independent of time ) ${\displaystyle \psi _{j}}$ ${\displaystyle t}$

Usually the following is set:

${\displaystyle \psi _{0}=1}$

The purely non-deterministic part is also called white noise . A linear deterministic component such as can be perfectly predicted based on its own past values ​​(but can also contain random elements). The deterministic part can have a mean value that is constant over time, but also includes, for example, periodic, polynomial or exponential sequences in the points in time . ${\displaystyle (u_{t})}$ ${\displaystyle \mu _{t}}$ ${\displaystyle t}$

The required quadratic convergence of the series of guarantees the existence of the second moments of the process. For the validity of this decomposition no distribution assumptions have to be made and does not have to be independent; uncorrelatedness suffices . ${\displaystyle \psi _{j}}$ ${\displaystyle u_{t}}$

The expected value is obtained ${\displaystyle \operatorname {E} (x_{t}-\mu _{t})=\operatorname {E} \left(\sum _{j=0}^{\infty }\psi _{j}u_{t-j}\right)=\sum _{j=0}^{\infty }\psi _{j}\operatorname {E} (u_{t-j})=0}$

that is, the following applies:

${\displaystyle \operatorname {E} (x_{t})=\mu _{t}}$

The variance is calculated as follows:

${\displaystyle \operatorname {V} (x_{t})=\operatorname {E} [(x_{t}-\mu _{t})^{2}]=\operatorname {E} [(u_{t}+\psi _{1}u_{t-1}+\psi _{2}u_{t-2}+\cdots )^{2}]}$

Because of for this expression simplifies to ${\displaystyle \operatorname {E} (u_{t}u_{t-j})=0}$ ${\displaystyle j\neq 0}$

${\displaystyle \operatorname {V} (x_{t})=\operatorname {E} (u_{t}^{2})+\psi _{1}^{2}\operatorname {E} (u_{t-1}^{2})+\psi _{2}^{2}\operatorname {E} (u_{t-2}^{2})+\cdots =\sigma ^{2}\sum _{j=0}^{\infty }\psi _{j}^{2}}$

The variance is therefore finite and independent of time. According obtained with the autocovariances ${\displaystyle \tau >0}$

${\displaystyle {\begin{aligned}&{}\qquad \operatorname {Cov} (x_{t},x_{t+\tau })=\operatorname {E} [(x_{t}-\mu _{t})(x_{t+\tau }-\mu _{t+\tau })]\\&=\operatorname {E} \left[(u_{t}+\psi _{1}u_{t-1}+\cdots +\psi _{\tau }u_{t-\tau }+\psi _{\tau +1}u_{t-\tau -1}+\cdots )\right.\\&{}\qquad \qquad \left.(u_{t+\tau }+\psi _{1}u_{t+\tau -1}+\cdots +\psi _{\tau }u_{t}+\psi _{\tau +1}u_{t-1}+\cdots )\right]\\&=\sigma ^{2}(\psi _{\tau }+\psi _{1}\psi _{\tau +1}+\psi _{2}\psi _{\tau +2}+\cdots )\\&=\sigma ^{2}\sum _{j=0}^{\infty }\psi _{j}\psi _{\tau +j}<\infty \end{aligned}}}$

with . It can be seen that the autocovariances are only a function of the time difference . Thus all conditions for the covariance stationarity are fulfilled. The autocorrelation function can be written as follows: ${\displaystyle \psi _{0}=1}$ ${\displaystyle \tau }$

${\displaystyle \rho (\tau )={\frac {\sum _{j=0}^{\infty }\psi _{j}\psi _{\tau +j}}{\sum _{j=0}^{\infty }\psi _{j}^{2}}}{\text{ mit }}\tau =1,2,3,\dots }$

For example, ARMA models can be brought into Wold's representation. This representation is more of theoretical interest, because in practical applications models with an infinite number of parameters are useless.

Wold decomposition in functional analysis

There is also a Wold decomposition in functional analysis, see shift operator .

literature

• Herman Wold A Study in the Analysis of Stationary Time Series , Stockholm: Almquist and Wicksell 1938
• Gerhard Kirchgässner, Jürgen Wolters: Introduction to modern time series analysis , 1st edition, Munich: Vahlen, 2006, ISBN 978-3-800-63268-8 , p. 19f

See also

• ARMA model

References and comments

1. ↑ Used here in the sense that the covariance Cov ( ) only depends on the time difference (ts). Kirchgässner, Wolters, Hassner, Introduction to modern time series analysis, Springer 2013, p. 14 (weakly stationary is defined there as covariance and mean value stationary) ${\displaystyle x_{t},x_{s}}$