Spectral representation of stationary stochastic processes

The spectral representation of stationary stochastic processes is in a certain sense an analogue to the Fourier series expansion of a function. Every limited continuous function can be represented as an additive superposition of harmonic oscillations . A stationary stochastic process can also be represented as an additive superposition of harmonic oscillations, but with a random amplitude . The spectral representation of a stationary process usually offers deeper insights into the structure of the process, especially if it is a mixture of different periodic components.

Mathematical description

Be the set of integers and a discrete-time stationary stochastic process with expectation and covariance function that depends only on the difference of the times because of the stationarity, so only the function of a variable is . ${\displaystyle \mathbb {Z} }$ ${\displaystyle X_{t};\ t\in \mathbb {Z} }$ ${\displaystyle \operatorname {E} X_{t}=0}$ ${\displaystyle \operatorname {Cov} (X_{t},X_{t+h})=\Gamma (t,t+h)}$ ${\displaystyle \Gamma (t,t+h)=:\gamma (h)}$

Spectral representation of ${\displaystyle X_{t}}$

Every stationary process with has the so-called spectral representation ${\displaystyle X_{t}}$ ${\displaystyle \operatorname {E} X_{t}=0}$

${\displaystyle X_{t}=\int _{-\pi }^{\pi }e^{it\lambda }dZ(\lambda )}$.

This is a stochastic integral with respect to a process with uncorrelated increases, i. H. for are the increases and are uncorrelated. ${\displaystyle Z(\lambda )}$ ${\displaystyle \lambda _{1}<\lambda _{2}<\lambda _{3}<\lambda _{4}}$ ${\displaystyle Y(\lambda _{2})-Y(\lambda _{1})}$ ${\displaystyle Y(\lambda _{4})-Y(\lambda _{3})}$

If only has a finite number of gains, e.g. B. increases in , then the above integral can be written as a sum: ${\displaystyle Z(\lambda )}$ ${\displaystyle n}$ ${\displaystyle A(\lambda _{1}),\dots ,A(\lambda _{n})}$ ${\displaystyle -\pi <\lambda _{1}<\dots <\lambda _{n}<\pi }$

${\displaystyle X_{t}=\sum _{k=1}^{n}A(\lambda _{k})e^{it\lambda _{k}}}$.

Each summand is a harmonic oscillation with frequency and the random amplitude . ${\displaystyle \lambda _{k}}$ ${\displaystyle A(\lambda _{k})}$

Spectral representation of the covariance function

The covariance function is a symmetrical and positive semidefinite function and thus has the representation according to Bochner's theorem (in a discrete variant called Herglotz's theorem) ${\displaystyle \gamma (h)}$

${\displaystyle \gamma (h)=\int _{-\pi }^{\pi }e^{ih\lambda }dF(\lambda )}$.

It is called the spectral distribution function . It doesn't fall on monotonous and it applies . The relationship ${\displaystyle F}$ ${\displaystyle [-\pi ,\pi ]}$ ${\displaystyle F(-\pi )=0;\ F(\pi )=\gamma (0)=\operatorname {Var} X_{t}}$

${\displaystyle \operatorname {E} |Z(\lambda _{2}]-Z(\lambda _{1})|^{2}=F(\lambda _{2})-F(\lambda _{1});\ \lambda _{2}\geq \lambda _{1}}$

represents the connection between the spectral representation of and the spectral representation of. ${\displaystyle X_{t}}$ ${\displaystyle \gamma (h)}$

Spectral density

If so , then the spectral representation of can be written as a Riemann integral : ${\displaystyle \sum _{h=-\infty }^{\infty }|\gamma (h)|<\infty }$ ${\displaystyle \gamma (h)}$

${\displaystyle \gamma (h)=\int _{-\pi }^{\pi }f(\lambda )e^{ih\lambda }d\lambda }$.

The function is called the spectral density of . In clear terms, indicates the intensity with which the frequency occurs in the spectrum of . The spectral density itself has the representation ${\displaystyle f}$ ${\displaystyle X_{t}}$ ${\displaystyle f(\lambda )}$ ${\displaystyle \lambda }$ ${\displaystyle X_{t}}$

${\displaystyle f(\lambda )={\frac {1}{2\pi }}\sum _{h=-\infty }^{\infty }e^{-ih\lambda }\gamma (h)}$.

${\displaystyle f}$is the Fourier transform of , or is the inverse Fourier transform of . For is special ${\displaystyle \gamma }$ ${\displaystyle \gamma }$ ${\displaystyle f}$ ${\displaystyle h=0}$

${\displaystyle \gamma (0)=\operatorname {Var} X_{t}=\int _{-\pi }^{\pi }f(\lambda )d\lambda }$.

This can be interpreted as a dispersion breakdown (signaling power distribution ) on the different frequencies . ${\displaystyle \lambda }$

Constant fall

Now be a stationary process with a real value . Then the above formulas are modified to: ${\displaystyle X_{t}}$ ${\displaystyle t\in \mathbb {R} }$

${\displaystyle X_{t}=\int _{-\infty }^{\infty }e^{i\lambda t}dZ(\lambda );\ \ \gamma (h)=\int _{-\infty }^{\infty }e^{i\lambda h}dF(\lambda )}$.

This is again a stochastic process with uncorrelated growth. If so , then the spectral distribution function has a spectral density and we have: ${\displaystyle Z(\lambda )}$ ${\displaystyle \int _{-\infty }^{\infty }|\gamma (h)|dh<\infty }$ ${\displaystyle F}$ ${\displaystyle f}$

${\displaystyle \gamma (h)=\int _{-\infty }^{\infty }e^{i\lambda h}f(\lambda )d\lambda ;\ \ f(\lambda )={\frac {1}{2\pi }}\int _{-\infty }^{\infty }\gamma (h)e^{-ih\lambda }dh}$.

Examples

• A stationary process with the frequently used covariance function has the spectral density . ${\displaystyle X_{t}}$ ${\displaystyle \gamma (h)=\sigma ^{2}e^{-\alpha |h|}}$ ${\displaystyle f(\lambda )={\frac {\sigma ^{2}\alpha }{\pi (\lambda ^{2}+\alpha ^{2})}}}$
• White noise has the covariance function and the spectral density . ${\displaystyle \gamma (h)={\begin{cases}\sigma ^{2}&{\text{wenn }}h=0\\0&{\text{sonst }}\end{cases}}\ }$ ${\displaystyle f(\lambda )={\frac {1}{2\pi }}\sigma ^{2}}$

The spectral density is therefore constant. All frequencies are equally represented in the spectrum (analogy to white light).

Applications

Spectral representations are required in time series analysis , in signal processing (see e.g. also spectral power density ), in the construction of suitable filters ( e.g. low pass , high pass or band pass ).

Suitable methods for spectral density estimation are particularly important in the applications .

Individual evidence

1. ↑ ^{a b} J.L. Doob: Stochastic Processes , Wiley 1953
2. ↑ ^{a b} A.M. Jaglom, Introduction to the theory of stationary random functions , Berlin Akademieverlag 1959 (English: An introduction to the theory of stationary random functions , Prentice Hall 1962, Dover 2004)
3. ^ Teubner-Taschenbuch der Mathematik (editor E. Zeidler), Teubner 1996, p. 1083