Return Time

The return time, RT, is the time it takes for a trajectory to return to its mean value after being disturbed. For example, in the cone beetle time series the mean of the series is plotted as a horizontal broken line through the data. Notice that the trajectory crosses the mean six times. The first crossing occurs in less than one time step so the return time must be less than one (i.e., RT(1) < 1). The actual time taken for this first crossing is

where N(0) is the density of beetles at time zero, N(1) is the density of beetles at time one, and is the mean of the series. The next 3 segments also take less than 1 time period to return to the mean, but the fifth takes more than 2; i.e.,

Thus, the return time consists of the sum of the two components; (1) the number of complete time steps prior to crossing the mean (CTS), and (2) the fraction of the final time step to reach the mean (FTS). After calculating the return times for all segments of the trajectory, we compute the mean return time

where I is the number of measurements that are added together. The variance of the return time is given by

These values are presented in the statistics window of the PAS screen (see figure). Notice that, in the case of the cone beetle data, the variance is smaller than the mean. This indicates that the period of fluctuation is relatively constant over time. The mean and variance of the return time provide useful clues to the length of the time lag, and whether the data show trends or discontinuities:

1. MRT < 2 implies a time lag of one, or d = 1 (first order dynamics),
2. MRT > 2 suggests that d > 1, (higher order dynamics) and
3. VRT >> MRT implies a trend or discontinuity in the data: i.e., a non-stationary series.

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