When time series are not stationary, as in the gypsy moth time series, they must be subjected to some kind of manipulation before their endogenous dynamics can be analyzed.
When a time series is found to contain discontinuities, it can be split into two or more segments or sequences, and then each sequence analyzed separately. For example, we could divide the gypsy moth time series into two sequences at about year 11. If the pattern of fluctuation was the same in all sequences, indicating that similar mechanisms were operating, we could possibly splice them back together again after adjusting them to a common mean. Obviously this condition is not met with the gypsy moth series because the patterns of fluctuations are quite different in each sequence. Hence, the two sequences must be analyzed separately.
When trends are observed in the time series, it can be detrended by rotating the data around the mean. In other words, the difference between the expected value computed from a linear regression through the series and the data point is added to the mean of the series, or to some other value supplied by the user; i.e.,
where X'(t) is the detrended data point at time t, X(t) is the original data point, 
is the mean of the series (or another value supplied by the user), a is the regression intercept, and b is the regression slope. Data can be detrended in arithmetic or logarithmic model (click here to see the detrended cone beetle time series).
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©1997 Alan A. Berryman