## Rate of change and trend

When sampling is repeated within a plot each new state will differ from the previous as a consequence of limited precision in measuring or change taking place in the system. If change occurs, it can be blurred by noise and a trend may only emerge when sufficiently strong. Consequently, distinguishing randomness from trend is a mandatory prerequisite for any further step when looking at time series, and a method has to be found to investigate the nature and possible causes of change.

In the simplest case a time series consists of subsequent states separated by even time steps (Figure 9.2, left side). The five states documented

Figure 9.2 Measuring rate of change in time series of multistate systems. The matrix depicts distance (dissimilarity).

by five releves are then compared by calculating similarities or distances (see Section 4.2), which yields a 5 by 5 distance matrix (Figure 9.2). In the diagonal the self-comparisons are found; these are all zero (identity). All comparisons of states separated by one time step can be found in the first off-diagonal vector. I call these rate of change of order 1. The number of comparisons, c, is:

c(o) = n — o where n is sample size and o is the order of change. For order 1 there are four possible comparisons, for order 2 only three and so on. The distance matrix can now be interpreted. Simple reasoning leads to the following considerations:

1 A change of any order does not ultimately allow identification of the existence of a trend, as it can arise from any type of error.

2 Small increases in distance from short to long time steps may have been caused by an increase of measurement error. Errors of this kind can usually be avoided by quality control.

3 If distances continuously increase with the number of time steps, the rate of change is likely to be constant in time. A trend emerges.

In most systems, whether disturbed or undisturbed, the rate of change will vary. In succession one can expect phases in which change is fast and others in which it is slow. The same holds for short-time fluctuation as well. Probably the best way to recognize multivariate trends is to display the states in phase space; that is, to ordinate the releves. If the data originate from one plot only, trends manifest in a single sequence of points in which the order of points accords with time. If this forms a smooth line then there is temporal dependence occurring, since each state evolves from the previous, causing minor changes to be likely to happen. To decide whether this is a local phenomenon or a spatially relevant, a spatial sample is needed, as shown in Figure 9.3. This is a subset of the data used by Wildi & Schutz (2000) in which releves document succession in 8 (out of 59) plots located in the Swiss National Park. The time steps are all adjusted to five-year interval and the investigation period ranges from 1917 until about 1996. The individual series are overlapping, forming a long, horseshoe-shaped temporal gradient. From the symbols one can even see that the whole successional gradient has a range of a great many time steps (approximately 80), corresponding to about 400 years. Different types of succession can be distinguished. Plot Tr6 shows a perfect trend in one main direction: the rate of change is almost constant. Plot Tr5 also fits into the series, but only after the first four time steps; before that there is a different process underway. Pin4, finally, fits perfectly into the temporal gradient but almost no change can be found.

Once we know that plots exhibit a common trend, we can inspect the distance matrices, as explained in Figure 9.2. Taking plot Tr6, there are 16 states available from 1921 until 1994. The resulting distance matrix is shown in Figure 9.4, left side. The distances increase monotonely as the order of rate of change increases - when moving away from the diagonal.

Figure 9.3 Ordination of data from eight plots in the Swiss National Park (WiLdi & Schutz 2000). Succession proceeds from right to left.
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Figure 9.4 Rate of change in plots Tr6 (Left, 16 time steps) and Pin4 (right, 12 time steps) in the Swiss National Park (see Figure 9.3). Explanations are given in Figure 9.2.

Figure 9.4 Rate of change in plots Tr6 (Left, 16 time steps) and Pin4 (right, 12 time steps) in the Swiss National Park (see Figure 9.3). Explanations are given in Figure 9.2.

The pattern confirms that the trend occurs at all temporal scales, at short and also at long time intervals. The right-hand side shows the distance matrix of the plot named Pin4. The series consists of 12 time steps starting in 1940 and ending in 1996. The distance matrix suggests that there is a faint trend in the first nine time steps, but then the average distances start shrinking again. When inspecting the respective plot in Figure 9.3 it can be seen that the trend still perfectly fits the overall successional sequence, but it does not evolve any further and may have reached some equilibrium stage.