8.6 Relationship of autocovariance  8. Spectral Analysis  8.4 The variance spectrum

8.5 The autocorrelation and autocovariance functions

In the time series shown in the top or bottom panel of Fig. 8.1, consecutive measurements are correlated. An alternative to representing such a series by the variance spectrum is to represent these correlations by the autocovariance function, which may be thought of as representing correlations between observations as a function of the intervening time $\tau$:

$$V_{ff}(\tau) = {\rm Cov}\bigl(f(t),f(t+\tau)\bigr)$$ (8.21)

  

$$~~~~~ = \bigl\langle\bigl(f(t)-\overline{f(t)}\bigr)\bigl(f(t+\tau)-\overline{f(t)}\bigr)\bigr\rangle .$$ (8.22)

 

Similarly, the autocorrelation function is defined as

$$\rho(\tau) = {{V_{ff}(\tau)}\over{V_{ff}(0)}}$$ (8.23)

 

and so has the properties that $-1\le\rho(\tau)\le 1$, $\rho(0)=1$, and (for stationary processes) $\rho(\tau)=\rho(-\tau)$.

The autocovariance and autocorrelation functions can be estimated for a finite series from

$$\hat V_{ff}(k\Delta T) = {{1}\over{N-k}} \sum_{j=1}^{j=N-k}(f_j-\overline{f})(f_{j+k}-\overline{f})$$ (8.24)

  

$$\hat \rho(k\Delta T) = {{\hat V_{ff}(k\Delta T)}\over{\hat V_{ff}(0)}} \ .$$ (8.25)

  

   
 8.6 Relationship of autocovariance  8. Spectral Analysis  8.4 The variance spectrum