Analysis of the complexity

The Kolmogorov complexity was introduced by Kolmogorov (1903-1987). This

is a method of analysis for binary combination that allows to determine the degree of

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information needs to describe a particular object (Charpentier et al. 2007, Durand &

Zvonkin 2007).

This measure was the basis in developing an algorithm to calculate the index of

complexity c (n) which is an approximation of Kolmogorov complexity that estimates the

degree of randomness in a time series (Lempel & Ziv 1976). This measure allows the

characterization of spatial-temporal patterns in non-linear systems (Kilby et al. 2014).

Similar patterns in a time series are detected through this algorithm. The information can

be compressed and as result a shorter time series is obtained when repeated patterns are

found.

The complexity index c (n) measures the number of different patterns in a given

time series. According to the Lempel-Ziv (LZC) algorithm, the complexity of a time

series {Xi}, i = 1, 2, 3, 4... n, is calculated as follows:

1. In a first step, the original series is encoded.

2. The complexity index c (n) is a function of the length of the sequence N. The values of

c (n), approaching a maximum value b (n) when N approaches infinity. For instance:

, (18) 3. The measurement of standardized information is calculated and defined by:

(19)

A high value of C (n) indicates increased randomness and a lower level of predictability

(Kilby et al. 2014).

Binary series 0 and 1, are often used to facilitate the understanding of this method.

The corresponding complexity measure c (n) is obtained by normalization.

A detail description of the Kolmogorov complexity can be found in Durand & Zvonkin

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4.1.2.4.1 Permutation Entropy (PE)

This concept was introduced by Bandt & Pompe (2002) as a complexity measure

for time series. Entropy can be approximately defined as the degree of disorder or

uncertainty in a system, thus it is an indicator of its state. Statistical equations for entropy

have been derived by Boltzmann in physics who links entropy to energy and by Shannon

in the field of information theory. Both parameters have an inverse relationship. That is,

adding information to the system leads to the more efficient use of energy, and thus

lowering entropy (Bailey 2001). Initially, only monotonous self-maps of one-dimensional

intervals were considered and the permutation entropy (PE) formulation was made under

the framework of dynamical systems. However, this concept has been generalized by the

use of ordinal symbols related to arbitrary finite partitions in a dynamical system where

the symbols are the labels of the partition sets. These labels are ordinal patterns.

Therefore, this theory is related to the measure of the amount of information based on the

presence of a pattern which is defined by a natural encoding of the time series into a

sequence of symbols (Amigó & Keller 2013).

An ordinal pattern of length L is a vector displaying the rank order of consecutive

entries in a random time series. The permutation entropy of order L is defined as the

Shannon entropy of the ordinal L-patterns. This is an average measure of uncertainty and

related to the average amount of information contained in a random variable. The

permutation of the values of a time series is determined by this method and the

characterization of the structure of the local order in a time series is a measure of the

complexity in dynamic systems.

According to Riedl et al. (2013), the calculation of the PE of a given time series

{x} of length N is made by the following method: The permutation order is defined as m

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The time series {xi} index is i = 1, ... n where n is the counter for each pattern.

The dynamic representation is presented in Fig. 15 for, m = 3 (Riedl et al. 2013).

Fig.15. Representation of the permutations for m = 3 and its frequencies in a signal. (Riedl et al. 2013).

The rank of values is calculated following a certain sequence and the resultant

values are indexes in ascending order. The natural encoding reflects the rank order of

successive components of the time series (xi) in sequences of length n and the permutation

entropy is defined by:

!

(20)

Where represents the permutation or a relative frequency of the possible patterns

detected in the sequence of symbols. The permutation per symbol is given by:

1/ 1

!

(21)

This calculation is necessary for a possible comparison entropy permutation with

different values of m. In this work m=4. The highest value of de Pen is 1, which indicates

that all permutations have the same probability of occurrence. However, if PE is zero,

indicates that the time series is very regular. The calculation of PE depends on the choice

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The principle of maximum entropy, interpreted as maximum uncertainty, has been

used in the analysis of rainfall time series (Tapiador 2007, Lumley et al. 2014). In

particular to explain the origin of clustering and persistence of the rainfall occurrence

process, (Koutsoyiannis 2006) or to explore temporal changes in the dynamics of El

Niño/Southern Oscillation (ENSO) (Saco et al. 2010).This concept has been also applied

to the solar wind time series (Suyal et al. 2012).

4.2. Statistical tests