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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
where and CYY(k) is the autocovariance function such that .

Consider the autocovariance generating function

where CYY(k) = CYY(-k) and z is a complex scalar. The spectral density function can be represented as
The stationary ARMA(p,q) process is denoted:
where and do not have common roots. Note that the autocovariance generating function of the linear process is given by
For the ARMA(p,q) process, .Therefore, the spectral density function of the stationary ARMA(p,q) process becomes
The spectral density function of a white noise is a constant.
The spectral density function of the AR(1) process is given by
The spectrum of the AR(1) process has its minimum at g=0 and its maximum at if , while the spectral density function attains its maximum at g=0 and its minimum at , if . When the series is positively autocorrelated, its spectral density function is dominated by low frequencies. It is interesting to observe that the spectrum approaches as .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 equals

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).

where and g denotes frequency. The autocovariance function can also be written as
Consider the following stationary AR(p) process:
where is a white noise with mean zero and constant variance .

The autocovariance function of white noise equals

where if k=0; otherwise, . Therefore, the power spectral density of the white noise is ,. Note that, with ,
Using the following autocovariance function of Yt,
the autocovariance function of the white noise is denoted as
On the other hand, another formula of the gives
Therefore,
Since ,the rational spectrum of Yt is
To compute the power spectrum, estimated values of white noise variance and AR coefficients are used. The order of the AR process can be determined by using the minimum AIC procedure.

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