Spectral density estimation

The spectral density of a stationary stochastic process allows deep insights into the structure of the process, especially when it comes to knowledge about periodicities . It is therefore important that given data, e.g. B. a concrete time series , the spectral density can be estimated well.

Most of the estimators are based on the periodogram , which goes back to Arthur Schuster in 1898.

Definitions

Spectral density

Let ( the set of whole numbers ) be a (possibly complex-valued ) stationary stochastic process with ${\displaystyle X_{t};t\in \mathbb {Z} }$ ${\displaystyle \mathbb {Z} }$

• Expected value ${\displaystyle \operatorname {E} X_{t}=0}$
• Covariance function . ${\displaystyle \operatorname {E} X_{t}X_{t+h}=\gamma (h)}$

If so, the spectral representation of : ${\displaystyle \textstyle \sum _{h=-\infty }^{\infty }|\gamma (h)|<\infty }$ ${\displaystyle \gamma (h)}$

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

The function is called spectral density . Its function value indicates the intensity of the frequency in the spectrum of . ${\displaystyle f}$ ${\displaystyle f(\lambda )}$ ${\displaystyle \lambda }$ ${\displaystyle X_{t}}$

Periodogram

Be realizations of a stationary stochastic process with . Then the expression is called ${\displaystyle x_{1},\dots ,x_{n}}$ ${\displaystyle X_{t}}$ ${\displaystyle \operatorname {E} X_{t}=0}$

${\displaystyle I_{n}(\lambda ):={\frac {1}{2\pi n}}\left|\sum _{k=1}^{n}x_{k}\,e^{-i\lambda k}\right|^{2}}$

Periodogram of the concrete time series . ${\displaystyle x_{1},\dots ,x_{n}}$

Spectral density estimates

Inconsistent estimates

The periodogram can be transformed into

${\displaystyle I_{n}(\lambda )={\frac {1}{2\pi }}\sum _{k=-(n-1)}^{n-1}e^{-i\lambda k}{\hat {\gamma }}(k);\ {\hat {\gamma }}(k)={\frac {1}{n}}\sum _{t=1}^{n-k}x_{t+k}x_{t}}$.

${\displaystyle I_{n}(\lambda )}$thus turns out to be the (empirical) Fourier transform of the empirical covariance function . Since the Fourier transform of is, one can heuristically expect that to be a suitable estimate for . In fact, the periodogram is an asymptotically unbiased estimate of the spectral density, but it is inconsistent ; H. in unmodified form only partially suitable for estimating the spectral density. ${\displaystyle {\hat {\gamma }}(k)}$ ${\displaystyle f(\lambda )}$ ${\displaystyle \gamma (h)}$ ${\displaystyle I_{n}(\lambda )}$ ${\displaystyle f(\lambda )}$

Consistent estimates

Expectant and consistent estimates for are generated by appropriate weighted averages of from a suitable environment of . A general representation for this is ${\displaystyle f(\lambda )}$ ${\displaystyle I_{n}(\lambda ^{\prime })}$ ${\displaystyle \lambda }$

${\displaystyle {\hat {f}}_{n}(\lambda )={\tfrac {1}{2\pi }}\int _{-\pi }^{\pi }W_{n}(\nu )I_{n}(\lambda -\nu )d\nu }$

with suitable spectral window . As a rule, the above is generated discretely as a sum, specifically for the so-called Fourier frequencies , where it is chosen that the following applies. Then you have the structure ${\displaystyle W_{n}}$ ${\displaystyle {\hat {f}}_{n}(\lambda )}$ ${\displaystyle \lambda _{j}={\frac {2\pi }{n}}j}$ ${\displaystyle j\in \mathbb {Z} }$ ${\displaystyle -\pi <\lambda _{j}<\pi }$

${\displaystyle {\hat {f}}_{n}(\lambda _{j})=\sum _{|k_{n}|\leq m(n)}a_{k_{n}}I_{n}(\lambda _{j+k_{n}})}$.

If the and have the following properties, you enforce consistency: ${\displaystyle a_{k_{n}}}$ ${\displaystyle m(n)}$

${\displaystyle a_{k_{n}}\geq 0;\ a_{k_{n}}=a_{-k_{n}};\ \sum _{|k_{n}|\leq m(n)}a_{k_{n}}=1;\ \lim _{n\to \infty }\sum _{|k_{n}|\leq m(n)}a_{k_{n}}^{2}=0;\ \lim _{n\to \infty }m(n)=\infty ;\ \lim _{n\to \infty }{\frac {m(n)}{n}}=0}$.

The weights are usually by symmetrical core functions with generated according to: ${\displaystyle a_{k_{n}}}$ ${\displaystyle K\colon [-1,1]\to [0,\infty )}$ ${\displaystyle \int _{-1}^{1}K^{2}(x)dx<\infty }$

${\displaystyle a_{k_{n}}={\frac {K({\frac {k_{n}}{m(n)}})}{\sum _{i=-m(n)}^{m(n)}K({\frac {i}{m(n)}})}};\ \ -m(n)\leq k_{n}\leq m(n)}$.

Examples

see e.g. B. Simplified, we write now and instead of and . ${\displaystyle k}$ ${\displaystyle m}$ ${\displaystyle k_{n}}$ ${\displaystyle m(n)}$

• Truncated periodogram , generated by the rectangular core . Here is the indicator function . So it's for and otherwise. ${\displaystyle K(x)=\chi _{[-1,1]}(x)}$ ${\displaystyle \chi _{[-1,1]}}$ ${\displaystyle a_{k}=1}$ ${\displaystyle k\leq m}$ ${\displaystyle a_{k}=0}$
• Bartlett's estimate , generated by the triangle kernel , it's for and otherwise. ${\displaystyle K(x)=\max(1-|x|,0)}$ ${\displaystyle a_{k}={\frac {m-|k|}{m^{2}}}}$ ${\displaystyle k\leq m}$ ${\displaystyle a_{k}=0}$
• Parzen estimate generated by a more complicated kernel that gives a favorable asymptotic variance:

${\displaystyle K(x)={\begin{cases}1-6|x|^{2}+6|x|^{3}&{\text{wenn}}\ |x|<{\frac {1}{2}}\\2(1-|x|)^{3}&{\text{wenn}}\ {\frac {1}{2}}\leq |x|\leq 1\\0&{\text{sonst}}\end{cases}}}$.

Individual evidence

1. ^ A. Schuster: On the investigation of hidden periodicities with application to a supposed 26 day period of meteorological phenomena. In: Terrestrial Magnetism and Atmospheric Electricity. 3, 1898, pp. 13-41, doi : 10.1029 / TM003i001p00013 .
2. ^ J. Anděl: Statistical analysis of time series. Akademie-Verlag, Berlin 1984.
3. ^ U. Grenander , M. Rosenblatt : Statistical Analysis of Stationary Time Series. Wiley 1957. (Reprint: American Mathematical Societey, 2008) full text
4. ^ E. Parzen: Mathematical Considerations in the Estimation of Spectra. In: Technometrics. Vol. 3, 1961, pp. 167-190, doi : 10.1080 / 00401706.1961.10489939 , JSTOR 1266111 .
5. ^ ^{A b} P. J. Brockwell and RA Davis: Time Series: Theory and Methods. Springer 1987 (most recent edition 2009)