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## Example: avoiding periodicity artefacts

> Figure: Progressive stages of the simulated annealing scheme. The data used in Fig. 6 is used to generate an annealed surrogate that minimises over all permutations of the data. From top to bottom, the values for E are: 0 (original data), 1.01 (random scramble), 0.51, 0.12, 0.015, and 0.00013.

Let us illustrate the use of the annealing method in the case of the standard null hypothesis of a rescaled linear process. We will show how the periodicity artefact discussed in Sec. 4.5 can be avoided by using a more suitable cost function. We prepare a surrogate for the data shown in Fig. 6 (almost unstable AR(2) process) without truncating its length. We minimise the cost function given by Eq.(23), involving all lags up to . Also, we excluded the first and last points from permutations as a cheap way of imposing the long range correlation. In Fig. 11 we show progressive stages of the annealing procedure, starting from a random scramble. The temperature T is decreased by 0.1% after either permutations have been tried or have been successful. The final surrogate neither has spuriously matching ends nor the additional high frequency components we saw in Fig. 6. The price we had to pay was that the generation of one single surrogate took 6 h of CPU time on a Pentium II PC at 350 MHz. If we had taken care of the long range correlation by leaving the end points loose but taking , convergence would have been prohibitively slow. Note that for a proper test, we would need at least 19 surrogates. We should stress that this example with its very slow decay of correlations is particularly nasty -- but still feasible. Obviously, sacrificing 10% of the points to get rid of the end point mismatch is preferable here to spending several days of CPU time on the annealing scheme. In other cases, however, we may not have such a choice.

Next: Combinatorial minimisation and accuracy Up: General constrained randomisation Previous: Computational issues of simulated