8.13 The Fast Fourier  8. Spectral Analysis  8.11 Removing the trend

8.12 Some statistical characteristics of estimators

We must always work with estimators for the power spectrum and autocovariance function, so it is important to consider the uncertainties associated with those estimators. If the observed values in the series are $\{f_k\}$, observed at N points with sampling interval $\Delta T$ and sample duration T, the estimators are

$$\hat \Gamma(\nu) = {{\Delta T}\over{N}} \Bigl\{\bigl(\sum_{k=......igr)^2+\bigl(\sum_k f_k\sin(2\pi\nu k\Delta T)\bigr)^2\Bigr\}$$ (8.62)

 

and^8.1

$$\hat V_{XX}(\tau=j\Delta T) = {{\Delta T}\over{T}} \sum_{k=1}^{N-j}x_kx_{k+j}$$ (8.63)

 

valid at the analyzed frequencies $\nu=j/\Delta T$ where j varies from -N/2 to N/2 and so the frequency interval is $-\nu_N\le\nu\le\nu_N$. For a finite sample interval from 0 to T, these estimators are related via

$$\hat \Gamma(\nu) = \int_{-T}^T \hat V_{XX}(\tau) e^{-i2\pi\nu\tau}d\tau,~~~-\infty\le\nu\le\infty$$ (8.64)

 

$$\hat V_{xx}(\tau)=\int_{-\infty}^\infty \hat \Gamma(\nu)e^{i2\pi\nu\tau}d\nu,~~~-T\le\tau\le T.$$ (8.65)

 

Extended to discrete samples, these formulas become

$$\hat \Gamma(\nu)=\Delta T\sum_{k=-N/2}^{(N/2)-1} \hat V_{xx}(k\Delta T)e^{-i2\pi\nu k\Delta T}$$ (8.66)

 

$$\hat V_{xx}(\tau)=\Delta\nu\sum_{j=-N/2}^{(N/2)-1}\hat \Gamma(j\Delta\nu)e^{i2\pi\tauj\Delta\nu}d\nu$$ (8.67)

 

where $\Delta\nu=1/T$.

For a given lag, the number of points contributing to the estimator for the autocovariance in (8.67) increases linearly with the sample duration T, so the variance reduces in proportion to 1/T as expected for a mean value. However, the same is not true of the estimator for the variance spectrum $\hat \Gamma(\nu)$. As Tincreases, the range in frequencies affected by a particular value of $V_{XX}(\tau)$ decreases as 1/T, so the number of measurements contributing to a given value of $\hat \Gamma(\nu)$ does not increase as T increases and thus the variance in $\hat \Gamma(\nu)$ does not decrease. Instead, the variance spectrum can be estimated at more frequencies and those frequencies are closer spaced, but the variance in any one of the estimates remains constant.

It can be demonstrated that the standard deviation in any single estimate of the variance spectrum is equal to the value of the variance spectrum. This 100% uncertainty makes variance spectra of little use unless averaging is used to reduce the variance. It is always important to consider uncertainty limits when evaluating the significance of features in the variance spectra. Perhaps the simplest way to reduce the variance while preserving resolution is to divide the frequency range into logarithmically spaced intervals, then average all spectral estimates that lie in the same interval. For particularly long series, the computation can be more efficient if the spectrum is calculated for segments of the complete time series and these results are then averaged to reduce the variance in the estimate of the spectrum.

   
 8.13 The Fast Fourier  8. Spectral Analysis  8.11 Removing the trend