Autocorrelation in Dynamic Systems

When experimental data are collected over time or distance there is always a chance of having autocorrelated residuals. Box et al. (1994) provide an extensive treatment of correlated disturbances in discrete time models. The structure of the disturbance term is often moving average or autoregressive models. Detection of autocorrelation in the residuals can be established either from a time series plot of the residuals versus time (or experiment number) or from a lag plot. If we can see a pattern in the residuals over time, it probably means that there is correlation between the disturbances. 

If we have data collected at a constant sampling interval, any autocorrelation in the residuals can be readily detected by plotting the residual autocorrelation function versus lag number. The latter is defined as, 

The 95% confidence intervals of the autocorrelation function beyond lag 2 is simply given by 2/ 4N. If the estimated autocorrelation function has values at 

If we assume that the residuals satisfy Equation 8.69, we must transform the data so that the new residuals are not correlated. Substituting the definition of the residual (for the single-response case) into Equation 8.69 we obtain upon rearrangement, 

The above model equation now satisfies all the criteria for least squares estimation. As an initial guess for k we can use our estimated parameter values when it was assumed that no correlation was present. Of course, in the second step we have to include p in the list of the parameters to be determined. 

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Use of Direct Search Methods Autocorrelation in Dynamic Systems... 

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