An important aspect of the analysis of time series is the possibility of uncover the character of the process underlying the series. It is especially interesant to discriminate if one time series represents a nonlinear process, a linear process or if it the series does not give more information than random noise would do. One useful tool to address such a question is the method of surrogate time series [97,109]. Surrogate time series are series derived from the original series, which are designed to test against a null hypothesis whether the original series belongs to a definite class of processes, such as those described above.
To construct a set of surrogate series it is required to find some parameters of the original series, which are associated to a definite class of processes.
For example, the mutual independence of the data in a symbolic sequence is expressed in statistics which are independent of the relative positions of the symbols. Hence, by maintaining the probabilities of finding each symbol from the original series by making random shuffles of their positions, one constructs a surrogate set of sequences without temporal correlations. Consequently, if the dynamical entropies of ordern ≥1of the original series are not significantly different from those of the surrogates, this can be regarded as evidence for that the original series (and probably the underlying process) describes independent random variables, with no temporal correlations up to the order n.
Nevertheless, it is not a trivial task to find a set of parameters which cor-respond to a definite class of processes. Only two cases are relatively evident:
the one-symbol probabilities, which are conserved in processes of independent variables as above; and the amplitudes of the Fourier transform, which are con-served in processes of linear correlations. In the latter case, the construction of surrogates is possible when the amplitudes of the periodogram estimator of the power spectrum are multiplied by random phases and then transformed back to the time domain:
where0≤αk ≤2π are independent random numbers.
Given a class of parameters which is the desired to represent the process of the null hypothesis, the construction of the surrogates is often not an easy matter as well. One could think the construction oftypical realizations can be helpful, that is, constructing surrogate series using the specific values of the parameters which characterize the original series. This approach is not very useful, since it requires often fitting procedures, which introduces systematic errors, needs the specification of fitting parameters and leads to series which fluctuate around the data of the original series. On top of that, it is not possible to recover the true underlying process by any fitting. Hence, the most appealing method to construct surrogates is to create constrained realizations [110]. For this means, it is useful to consider the measurable properties of the time series rather tan its underlying model equations. This is possible constructing surrogates with the same second order properties as the measured data, but which are otherwise random, as shown above with the frequencies of the data for the sequence of independent random variables, or the Fourier amplitudes for the linear (possibly stochastic) processes. Schreiber [96, 97]
describes recent developments in the constrained randomization approach.
To discriminate the character of a process with some high significance, surrogate series are used in hypotheses tests. Rigorous tests are designed constructing M =K/α−1surrogates, so that, with the original series, there are K/αseries. If the series of the measured data belongs to theK smallest or largest values, the null hyphotesis can be rejected with a significance of α. In two-sided tests,M = 2K/α−1, resulting in a probabilityαthat the data gives either one of the K smallest or largest values. In the present work surrogate series will not be used to test an hypothesis about the character of the studied series, but to look for the relative difference of the entropies with those of a process of independent random variables. In this way surrogates will serve the purpose of selecting the most important blocks of a series, acting as an additional criterium facing the unknown variances of higher-order entropies.
Hence, the standard definition of significance in terms of the deviation with respect to the variance will be used, without attempting to interprete the results assuming Gaussian-distributed surrogates (section 3.4.3).
The Southern Oscillation:
Interannual variability of the tropical Pacific
The most important mode of variability in the tropical Pacific is known as El Niño and the Southern Oscillation [115]. In this light, it is hard to imagine that El Niño was discovered by Peruvian sailors as a warm ocean current from the north, which became its name (the Child Jesus) since it appeared shortly after Christmas. As this term is now commonly related to devastating droughts over the western tropical Pacific, torrential floods over the eastern tropical Pacific, and unusual weather patterns over various parts of the world, the view of it has become pejorative. Nevertheless El Niño is a phase of an extremely interesting natural oscillation known as the Southern Oscillation [84]. Episodes of El Niño repeat irregularly after being separated by periods where oceanic and atmospheric conditions are benign and opposite to those of El Niño. The term La Niña (the girl) is used for these opposite phases of the Southern Oscillation, when sea surface temperature in the central
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and eastern tropical Pacific are unusually low and when the trade winds are very intense. The physics and the general problematic of El Niño and the Southern Oscillation will be addressed in some detail in this chapter.