## Spacefortime substitution 941 Principle and method

In long-term investigations species eventually exhibit a characteristic pattern of change: constancy, increase, decrease, random fluctuation, periodicity and so on (Huismann et al. 1993). If the observation time is sufficiently long, many of these patterns turn out to be fragments of a bell-shaped response curve. In space-for-time substitution, one assumes that several different fragments of the same response curve can be found, but occurring in different plots. If these fragments overlap, the entire response curve can be restored. This principle is sketched schematically in Figure 9.8. The heavy lines show hypothetical response curves of the same species over eight time steps. At first glance they seem to be different in nature - sometimes with a tendency

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 time steps plot 1 plot 2 plot 3

■—•—plot 2, shifted ■■■ ■ piot 3, shifted synthetic time series plot 1 plot 2 plot 3

Figure 9.8 The principle of space-for-time substitution in the univariate case (Ghosh & Wildi 2007).

to increase and sometimes decreasing - however, when the curves of plots 2 and 3 are properly shifted (light lines), a single response evolves covering 14 time steps. This is of course just an interpretation, because such curves never fit perfectly. It is therefore essential that they overlap sufficiently, as is the case in Figure 9.8. In real situations, there are many species involved, and an overlap should yield a meaningful result for all of them.

The fact that a successional trend ends within a plot but continues in another plot has been known for long time. It has added much to the mon-oclimax theory strongly debated in the first half of the twentieth century (Clements 1916, see also Maarel 2005). Although there has never been a doubt that such phenomena exist, the handling and interpretation has led to controversy. Much of this can be read in the review by Pickett (1989): he warns against the uncritical use of such data. Although I give an example of a really successful application below, some of the most frequently mentioned shortcomings and pitfalls are listed first:

• Superimposing time series data is always hampered by statistical noise and disturbance. The resulting synthetic series therefore suffers from some uncertainty.

• Vegetation change within two different plots will never be identical as the plots will most likely differ in site condition as well as in the species pool.

• Succession may not proceed at the same speed in different plots, prohibiting a perfect fit of series.

• The overlay of series mimics monoclimax. Alternative paths as in poly-climax can hardly be identified by this approach alone.

There are so far no experiments known to me that would test the potentials and failures of space-for-time substitution, simply because these would last too long. All applications known are from surveys. Often, it is not even envisaged that long-term temporal patterns will be sought, just the data suggested in the course of the analysis. This was the case in the succession data from the Swiss National Park shown below. The method has been developed specifically to screen the more than a hundred time series available for a general temporal trend (Wildi & Schutz 2000). The aim of the method is to find an unequivocal solution to the superposition of several (i.e. more than two!) multivariate time series rather than having to search for an iterative result by trial and error.

The problem with finding the best solution when many time series are available is in identifying the most suitable pairs of response curves for fusion. The best candidates are those where the overlapping observations are the most similar compared to all other time series. In the following the similarity of time series is defined as the similarity of the two most similar observations in any two time series. This is shown in Figure 9.9, an example from two plots in the Swiss National Park. The species set is reduced to three for simplicity. The 'real' plots, AC1 and AC9, were surveyed in 'real' years: AC1 from 1930 until 1994 and AC9 from 1917 until 1982. When comparing the species composition (using Euclidean distance) the most similar observations are those from 1990 in AC1 and those from 1959 in AC9. The series are now shifted until these two observations are located in the same column. The new series AC1/9 covers 80 years: now just in the sense of age and without any specific dates. It can also be seen that the method requires time steps of the same length. In the present case minor deviations from identical time steps were corrected by interpolations. The steps leading to an unequivocal solution when fusing three or more time series are the following:

• Compute a resemblance matrix of time series. Resemblance (distance) is defined as shown in Figure 9.9.

• Derive the minimum spanning tree of time series (Gower & Ross 1969). This is a graph showing the nearest neighbours of all time series in the form of a tree.

 PlOt AC1 'Year 19.: 30 35 42 47 S3 57 60 65 63 74 81 35 90 94 'Aconitum' 72 71 70 56 44 26 48 40 34 31 24 23 22 22 'Deschampsia' 18 19 21 27 38 46 40 42 44 46 51 54 58 60 Trisetum' 8 7 6 9 7 14 7 9 11 13 12 11 8 8 Plot AC9 ■Year 19.: 17 22 25 32 35 40 47 50 53 59 65 68 74 82 'Aconitum' 65 85 86 65 72 57 39 33 25 23 16 29 24 16 'Oeschampsia' 12 10 12 14 22 32 40 43 46 55 57 47 49 46 'Trisetum' 3 3 1 0 2 5 7 9 10 8 11 11 11 15 Plot AC1i9 Age m 0 5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 'Aconitum' 72 0 71 0 70.0 70.5 65.0 56 0 66.5 56.0 45.5 35.0 28 5 24.0 22.5 19 0 29.0 24.0 16.0 'Oeschampsia' 18.0 16.0 21.0 19.5 24.0 29.0 270 32 0 38.0 43 0 47.0 50 0 56.5 53.5 47.0 49.0 46 0 Trisetum' 8.0 7 0 6 0 6.0 5.0 7.5 3 5 5.5 8 0 10.0 10.5 10.5 8 5 9 5 110 11.0 15.0

Figure 9.9 The similarity of time series. Plots AC1 and AC9 are Located in the Swiss National Park.

Figure 9.9 The similarity of time series. Plots AC1 and AC9 are Located in the Swiss National Park.

• Position the observations by overlapping the time series according to the order given in the minimum spanning tree (Figure 9.11). This yields the relative age of each series (Figure 9.12).

• Compute the average composition of the new synthetic time series by averaging all scores pertaining to the same time step.

The minimum spanning tree yields a unique solution to the problem. This is not necessarily the 'true' one, but it is the one delivering the shortest possible series based on the data used.

9.4.2 The Swiss National Park succession (example)

The results shown below are from Wildi & Schutz (2000). The original time series are of unusual length and the first observations date back to the year 1917, when J. Braun-Blanquet established the first permanent plots in the Park with the aim of documenting reforestation of pastures (Figure 9.10).

It took as much as 80 years to detect that the data could be interpreted according to the idea of space-for-time substitution. The plots do not constitute a statistical sampling design but they are dispersed all over the previous pastures in the park. The data set used below includes 59 of them, consisting of 751 releves (data set 'Snpser59', Appendix B). The species are summarized into six groups, carrying the names of 'dominats'.

Figure 9.10 Pinus mugo on a former pasture in the Swiss National Park, just escaping the reach of browsing red deer.
Figure 9.11 Minimum spanning tree of 59 time series from the Swiss National Park.
Figure 9.12 Ordering of 59 time series from the Swiss National Park.

Model year

Figure 9.13 Succession in abandoned pastures of the Swiss National Park, derived by space-for-time substitution. Every fourth time step is shown.

ooooooooooooooooooooo

Model year

Figure 9.13 Succession in abandoned pastures of the Swiss National Park, derived by space-for-time substitution. Every fourth time step is shown.

Data handling is explained in more detail in Wildi & Schtitz (2000). The steps involved in the analysis are the same as those shown in Section 9.4.1. In Figure 9.11 the minimum spanning tree for the 59 time series is shown. This is not just a single line as an ordering principle but a more complex tree. Processing this by fusing time series pairwise yields the arrangement in Figure 9.12. The resulting synthetic time steps (81) have to be multiplied by step length of 5 years, yielding a model time span of 405 years (Figure 9.13).

The overall trend can be interpreted as follows: an initial Aconitum phase, resulting from livestock grazing and fertilization, dominates for about 50 years after the cessation of grazing by livestock. A Deschampsia phase then emerges and is dominant for about 15 years. A later transition to a grassland dominated by Festuca rubra is most likely caused by grazing activity by red deer (Achermann et al. 2000). This is followed by a Carex sempervirens phase that may last 150 years. Finally, Pinus montana seedlings begin to establish, initiating the reforestation phase.

The pattern revealed in this example must be strongly nonlinear, as it has been shown in Section 9.3 that a linear Markov model fails to explain it. It has been shown by Wildi & Schutz (2007) that computed process length, a result of analysis, varies to some extent depending on the transformation chosen for the species scores. Unlike in Section 7.2.3 no reference measurement exists to tell us which of the estimations is best, leaving us with some uncertainty about succession velocity.