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Time Series Analysis and Control Examples

Spectral Analysis

The autocovariance function of the random variable Yt is defined as
CYY(k) = E(Yt+k Yt)
where EYt = 0. When the real valued process Yt is stationary and its autocovariance is absolutely summable, the population spectral density function is obtained using the Fourier transform of the autocovariance function
f(g) = \frac{1}{2 \pi} 
 \sum_{k = -\infty}^{\infty} C_{YY}(k) \exp(-igk)
 \hspace*{0.15in} -\pi \leq g \leq \pi
where i=\sqrt{-1} and CYY(k) is the autocovariance function such that \sum_{k = -\infty}^{\infty} | C_{YY}(k)| \lt \infty.

Consider the autocovariance generating function

\gamma(z) = \sum_{k = -\infty}^{\infty} C_{YY}(k) z^k
where CYY(k) = CYY(-k) and z is a complex scalar. The spectral density function can be represented as
f(g) = \frac{1}{2 \pi} \gamma(\exp(-ig))
The stationary ARMA(p,q) process is denoted:
\phi(B) y_t = \theta(B) \epsilon_t \hspace*{0.15in} 
 \epsilon_t \sim(0,\sigma^2)
where \phi(B) and \theta(B) do not have common roots. Note that the autocovariance generating function of the linear process y_t = \psi(B)\epsilon_t is given by
\gamma(B) = \sigma^2\psi(B)\psi(B^{-1})
For the ARMA(p,q) process, \psi(B) = \frac{\theta(B)}{\phi(B)}.Therefore, the spectral density function of the stationary ARMA(p,q) process becomes
f(g) = \frac{\sigma^2}{2\pi}|\frac{\theta(\exp(-ig))
The spectral density function of a white noise is a constant.
f(g) = \frac{\sigma^2}{2\pi}
The spectral density function of the AR(1) process (\phi(B) = 1-\phi_1 B) is given by
f(g) = \frac{\sigma^2}{2\pi(1-\phi_1 \cos(g) + \phi_1^2)}
The spectrum of the AR(1) process has its minimum at g=0 and its maximum at g=+-\pi if \phi_1 \lt 0, while the spectral density function attains its maximum at g=0 and its minimum at g=+-\pi, if \phi_1\gt. When the series is positively autocorrelated, its spectral density function is dominated by low frequencies. It is interesting to observe that the spectrum approaches \frac{\sigma^2}{4\pi}\frac{1}{1-\cos(g)} as \phi_1arrow 1.This relationship shows that the series is difference-stationary if its spectral density function has a remarkable peak near 0.

The spectrum of AR(2) process (\phi(B)=1-\phi_1 B-\phi_2 B^2) equals

f(g) = \frac{\sigma^2}{2\pi}\frac{1}
 {4\phi_2}]^2 + \frac{(1+\phi_2)^2(4\phi_2 + \phi_1^2)}

Refer to Anderson (1971) for details of the characteristics of this spectral density function of the AR(2) process.

In practice, the population spectral density function cannot be computed. There are many ways of computing the sample spectral density function. The TSBAYSEA and TSMLOCAR calls compute the power spectrum using AR coefficients and the white noise variance.

The power spectral density function of Yt is derived using the Fourier transformation of CYY(k).

f_{YY}(g) = \sum_{k=-\infty}^\infty \exp(-2\pi igk)C_{YY}(k),
 \hspace*{.25in} -\frac{1}2\leq g \leq\frac{1}2
where i=\sqrt{-1} and g denotes frequency. The autocovariance function can also be written as
C_{YY}(k) = \int_{-1/2}^{1/2} \exp(2\pi igk)f_{YY}(g)dg
Consider the following stationary AR(p) process:
y_t - \sum_{i=1}^p \phi_i y_{t-i} = \epsilon_t
where \epsilon_t is a white noise with mean zero and constant variance \sigma^2.

The autocovariance function of white noise \epsilon_t equals

C_{\epsilon\epsilon}(k) = \delta_{k0}\sigma^2
where \delta_{k0}=1 if k=0; otherwise, \delta_{k0}=0. Therefore, the power spectral density of the white noise is f_{\epsilon\epsilon}(g) = \sigma^2,-\frac{1}2\leq g \leq\frac{1}2. Note that, with \phi_0 = -1,
C_{\epsilon\epsilon}(k) = \sum_{m=0}^p \sum_{n=0}^p \phi_m\phi_n
Using the following autocovariance function of Yt,
C_{YY}(k) = \int_{-1/2}^{1/2} \exp(2\pi igk)f_{YY}(g)dg
the autocovariance function of the white noise is denoted as
C_{\epsilon\epsilon}(k) & = & \sum_{m=0}^p \sum_{n=0}^p \phi_m\phi_n
 ...2}^{1/2} \exp(2\pi igk)
 \,| 1-\sum_{m=1}^p \phi_m\exp(-2\pi igm)|^2 f_{YY}(g)dg
On the other hand, another formula of the C_{\epsilon\epsilon}(k)gives
C_{\epsilon\epsilon}(k) = \int_{-1/2}^{1/2} \exp(2\pi igk)
f_{\epsilon\epsilon}(g) = | 1-\sum_{m=1}^p \phi_m\exp(-2\pi igm)
 |^2 f_{YY}(g)
Since f_{\epsilon\epsilon}(g) = \sigma^2,the rational spectrum of Yt is
f_{YY}(g) = \frac{\sigma^2}
 {| 1-\sum_{m=1}^p \phi_m\exp(-2\pi igm)|^2}
To compute the power spectrum, estimated values of white noise variance \hat{\sigma}^2 and AR coefficients \hat{\phi}_mare used. The order of the AR process can be determined by using the minimum AIC procedure.

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Copyright © 1999 by SAS Institute Inc., Cary, NC, USA. All rights reserved.