QSD to those of the normalised full simulation and those drawn from the compressed repre- sentations. To test the correspondence between two vectors, we needed to select a distance measure. A number of measures have been suggested for comparing probability distributions (Cha, 2007); however, we could find no study in the literature that considered whether these measures were comparable between vectors of different lengths, an important consideration since we wished to understand the performance of the different algorithms for different num- bers of patches. We therefore tested a range of distance metrics (see Appendix D.1) for bias in measuring the distance between uniform random vectors of different lengths. The only measure among those tested for which the mean distance was not sensitive to the length of the vector was the Pearson correlation coefficient r (or more properly, the Pearson distance 1− r), and this was used for the majority of the comparisons in the remainder of this study. Post-processing to obtain distance statistics between vectors was conducted in Java. Figures were made in the statistical programming language R (R Core Team, 2012), using ggplot2 (Wickham, 2009).
5.3
Accuracy of the full simulation
In Experiment 1, we tested the correspondence between the full simulated QSD and the ex- act QSD derived from the eigenvector calculations. Our findings show that for the SRLM, the algorithm proposed by de Oliveira and Dickman (2005) does produce an accurate ap- proximation of the full QSD in reasonable time (i.e. not solely for an order parameter of the system, as considered by these authors). However, refinements were required in the case of systems with low persistence in which the exploration of the state space is highly dependent on initial conditions.
Some of the Figures in this section serve as illustrations. For these, a sample landscape with 5 patches was used, and this is shown in Figure 5.3.
5.3.1
Algorithm refinement for least persistent systems
Informal testing prior to conducting the full experiment showed that the algorithm proposed in de Oliveira and Dickman (2005) appeared to be effective for simulating not just the marginal distribution of the number of occupied patches, but also the full QSD. However, it sometimes failed when the system had low persistence because the full state space was not explored sufficiently thoroughly. Figure 5.4 illustrates two runs (a and b) of the algorithm for the landscape shown in Figure 5.3 and parameters leading to low persistence, compared with the exact QSD. The problem in the run shown in (b) appears to be to not visiting certain states at all in the simulator.
5.3. Accuracy of the full simulation 103 0 1 2 3 4 5 6 0 1 2 3 4 5 6
Figure 5.3: Sample landscape with five patches and log normally distributed patch areas.
In order to encourage better exploration of the state space, we increased the number of times that chains were restarted in a state drawn uniform randomly from all states. The number of random jumps required depended on the persistence of the system described by e and the number of patches n, and in all experiments described below, the number of times a uniform random state was chosen upon extinction was set to be 20× n × e.
a b 0.00 0.25 0.50 0.75 0 10 20 30 0 10 20 30 States Probability Form Exact Simulation
Figure 5.4: Two runs of the baseline algorithm for the landscape shown in Figure 5.3 and parameters giving low persistence.
5.3. Accuracy of the full simulation 104
5.3.2
Trend as n increases
We begin by providing an illustration of the accuracy of the algorithm (all discussion from here on uses the modified version of the algorithm described in Subsection 5.3.1 above). Figure 5.5 shows a histogram and scatterplot comparing the corresponding exact QSD and the full simulation method for a high persistence system for the landscape in Figure 5.3. For this example, the Pearson correlation coefficient between the two vectors is r = 0.999898, showing excellent correspondence. We choose a system with 5 patches for this illustration because it becomes difficult to interpret the histogram for the QSD with more patches. Al- though the particular parameter values illustrated give a better accuracy than the average for this number of patches (lower accuracy is achieved for systems with lower persistence across all patch size distributions), if we increase the number of patches to ten, this correla- tion is fairly typical, and thus it is representative of the accuracy for the larger systems which constitute our main interest in this work.
0.0 0.1 0.2 0.3 0.4 0 10 20 30 States P roba bi li ty Algorithm Exact Full sim. 0.0 0.1 0.2 0.3 0.4 0.0 0.1 0.2 0.3 0.4 Probabability (exact) P roba bi li ty (ful l s im ul at ion)
Figure 5.5: Left panel: State probabilities of exact QSD and full simulation (refined algo- rithm). Right panel: Scatter of state probabilities of full simulation against exact QSD. Both for the landscape shown in Figure 5.3 (e = 0.1, α = 0.5).
Figure 5.6 shows the median Pearson correlation coefficient between the numerical QSD and the full simulation for a range of values of n and e. At least as far as n = 10, the full sim- ulation improves in accuracy as n increases. The convergence of the full simulated QSD to the numerical QSD depends on the total number of events and the number of random jumps. Greater accuracy could therefore be achieved by simulating for longer, and particularly for low persistence, by including a higher number of random jumps. Among the 8000 simulation runs used to produce Figure 5.6, the lowest correlation coefficient was for a landscape with