Spectral Analysis

Spectral Density AR(p) process

Given that an autoregressive process $X$ of p'th order is defined in a regressive fashion as

\[X_t = a_1 X_{t-1} + a_2 X_{t-2} + ... + a_p X_{t-p} + \epsilon_t\]

the two functions arSpectrum compute an estimate of the spectral density of the above defined AR(p) process.

Either the already estimated or predetermined parameters with coefficents ($a_1, a_2, ..., a_p$) and variance of the error terms $\sigma_{\epsilon}$ can be used or for a given time series $y_t$ the parameters can be estimated in a first step with one of the selected estimators (Burg, OLS, Yule-Walker).

For both methods the spectral density function is calculated as a function $h$ of the frequency component $\omega$ as

\[ h(\omega) = \frac{\sigma_{\epsilon}}{2\pi} \frac{1}{|1 - a_1 exp(-i\omega) - ... - a_p exp(-ip\omega)|^2}\]

For an early treatment of this procedure see Akaike (1969)^[1]

arSpectrum(Φ :: Vector, σ::Float64; T = 200)

arSpectrum(y :: Vector; p::Int = 3, method=:burg)

Periodogram

For a time series $X(t)$ the periodogram is defined as the modulus-squared of the finite Fourier transform in accordance with the definition of Schulz (1898)^[2]:

\[ I(\omega) = \frac{1}{2 \pi T} | \sum_{t=0}^{T-1} exp(-i\omega t) X(t)|^2,\]

for $-\pi \leq \omega \leq \pi$ and $\omega_j = \frac{2 \pi j}{T}$, ($j = 1,...,m$ where $m = \lceil (T-1) / 2 \rceil$)

With regard to the autocovariance function $\gamma (p)$ one can write the periodogram also as

\[ I(\omega) = \frac{1}{ 2\pi} \sum_{j=-T-1}^{T-1} \gamma(j) cos(j \omega).\]

Using the factor $\frac{1}{2 \pi}$ here lets the periodogram also serve as a direct estimate of the spectral density or power spectrum of a covariance stationary process (see Brillinger 1975^[3]).

periodogram(Y :: Vector; trunc::Int=-1)

Band-Pass Filter

Baxter-King

Baxter-King (1999)^[4] provide an approximate band-pass filter, denoted by the authors as $BK_{K}(pl, pu)$, where K states the numbers of lags (or leads) to include and the bandwidth is stated in terms of the periodicity of cycles ($2\pi / \omega$) to include (pl denotes the lowest and pu the highest period included).

baxterKing(x :: Vector, pl::Real, pu::Real, K::Int)