Spectral analysis

A time series $x_{t}$ with stationary covariance, mean $μ$ and $k^{th}$ autocovariance $E\left( \left( x_{t} - \mu \right)\left( x_{t - k} - \mu \right) \right) = \gamma(k)$ can be described as a weighted sum of periodic trigonometric functions: sin$(\omega t)$ and cos$(\omega t)$, where $\omega$ denotes frequency. Spectral analysis investigates this frequency domain representation of $x_{t}$ to determine how important cycles of different frequencies are in accounting for the behaviour of $x_{t}$.

Assuming that the autocovariances $\gamma(k)$ are absolutely summable ($\sum_{k = - \infty}^{\infty}\left| \gamma(k) \right| < \infty$), the autocovariance generating function, which summarises these autocovariances through a scalar valued function, is given by equation [1]¹.

$acgf(z) = \sum_{k = - \infty}^{\infty}{z^{k}\gamma(k)}$, [1]

where $z$ denotes complex scalar.

Once the equation [1] is divided by $\pi$ and evaluated at some $z{= e}^{- i\omega} = cos\omega - isin\omega$, where $i = \sqrt{- 1}$ and $\omega$ is a real scalar,$\ - \infty < \ \omega < \infty$, the result of this transformation is called a population spectrum $f\left( \omega \right)\ $for $\ x_{t}$, given in equation [2]¹.

$f\left( \omega \right) = \frac{1}{\pi}\sum_{k = - \infty}^{\infty}{e^{- ik\omega}\gamma(k)}$ [2]

Therefore, the analysis of the population spectrum in the frequency domain is equivalent to the examination of the autocovariance function in the time domain analysis; however it provides an alternative way of inspecting the process. Because $f\left( \omega \right)\text{dω}$ is interpreted as a contribution to the variance of components with frequencies in the range $(\omega,\ \omega + d\omega)$, a peak in the spectrum indicates an important contribution to the variance at frequencies near the value that corresponds to this peak.

As $e^{- i\omega} = cos\omega - isin\omega,\ $the spectrum can be also expressed as in equation [3].

$f\left( \omega \right) = \frac{1}{\pi}\sum_{k = - \infty}^{\infty}{(cos\omega k - isin\omega k)\gamma(k)}$ [3]

Since $\gamma(k) = \gamma( - k)$ (i.e. $\gamma(k)\ $is an even function of $k$) and $\sin{( - x)}\ = \operatorname{-sin}x$, [3] can be presented as equation [4]².

$f\left( \omega \right) = \frac{1}{\pi}\left\lbrack \ \gamma(0) + 2\sum_{k = 1}^{\infty}{\ \gamma(k)}cos\text{ωk} \right\rbrack$, [4]

This implies that if autocovariances are absolutely summable the population spectrum exists and is a continuous, real-valued function of $\omega$. Due to the properties of trigonometric functions $\left( \cos\left( - \omega k \right) = \cos\left( \text{ωk} \right) \right.\ \ $and $\left. \ \cos\left( \omega + 2\pi j)k = cos(\omega k \right) \right)\ $the spectrum is a periodic, even function of $\omega$, symmetric around $\omega = 0$. Therefore, the analysis of the spectrum can be reduced to the interval $( - \pi,\pi).$ The spectrum is nonnegative for all $\omega \in ( - \pi,\pi)$.

The shortest cycle that can be distinguished in a time series lasts two periods. The frequency which corresponds to this cycle is $\omega = \pi$ and is called the Nyquist frequency. The frequency of the longest cycles that can be observed in the time series with $n$ observations is $\omega = \frac{2\pi}{n}$ and is called the fundamental (Fourier) frequency.

Note that if $x_{t}$ is a white noise process with zero mean and variance $\sigma^{2}$, then for all $\left| k \right| > 0$ $\gamma\left(k\right)=0$ and the spectrum of $x_{t}$ is constant ($f\left(\omega\right)= \frac{\sigma^{2}}{\pi}$) since each frequency in the specrum contributes equally to the variance of the process³.

The aim of spectral analysis is to determine how important cycles of different frequencies are in accounting for the behaviour of a time series¹. Since spectral analysis can be used to detect the presence of periodic components, it is a natural diagnostic tool for detecting trading day effects as well as seasonal effects⁴. Among the tools used for spectral analysis are the autoregressive spectrum and the periodogram.

The explanations given in the subsections of this node derive mainly from DE ANTONIO, D., and PALATE, J. (2015) and BROCKWELL, P.J., and DAVIS, R.A. (2006).