**3.1. Fractal fluctuations quantification**

The dynamics of a time series can be explored through its correlation properties, or in other words, the time ordering of the series. Fractal analysis is an appropriate method to characterize complex time series by focusing on the time-evolutionary properties on the data series and on their correlation properties. In this context, the *detrended fluctuation analysis* (DFA) method was developed specifically to distinguish between intrinsic fluctuations generated by complex systems and those caused by external or environmental stimuli acting on the system [20]. The DFA method can quantify the temporal organization of the fluctuations in a given non-stationary time series by a single scaling exponent *α*, a self-similarity parameter that represents the long-range power-law correlation properties of the signal. The scaling exponent *α* is obtained by computing the root-mean-square fluctuation *F*(*n*) of integrated and detrended time series at different observation windows of size *n* and plotting *F*(*n*) against *n* on a log-log scale. Fractal signals are characterized by a power law relation between the average magnitudes of the fluctuations *F*(*n*) and the number of points *n*, *F*(*n*) ∼ *nα*. The slope of the regression line relating log(*F*(*n*)) to log(*n*) determines the scaling exponent *α*.

#### **3.2. Chaos degree quantification**

The principle of chaos analysis is to transform the properties of a time series into the topological properties of a geometrical object (attractor) constructed out of a time series, which is embedded in a *state/phase space*. The concept of phase space reconstruction is central to the analysis of nonlinear dynamics. A valid phase space is any vector space in which the state of the dynamical system can be unequivocally defined at any point [21]. The most used way of reconstructing the full dynamics of the system from scalar time measurements is based on the embedding theorem [21], which justifies the transformation of a time series into a *m*-dimensional multivariate time series. This is done by associating to each *m* successive samples distant a certain number *τ* of samples, a point in the phase space.

10.5772/53407

185

http://dx.doi.org/10.5772/53407

entropy measures. Thus, a complexity measures widely used is the proposed by Lempel and Ziv [20], which will be referred to as *Lempel-Ziv complexity* (LZC). This metric provides a measure of complexity related to the number of distinct substrings and the rate of their occurrence along a given sequence, larger values of LZC corresponding to more complex series. Another metric habitually used is the *Shannon entropy* (ShEn) [21]. This index gives a number that characterize the probability that different words occur. Thus, counting the relative frequency of each word, the ShEn is estimated as the sum of the relative frequencies weighted by the logarithm of the inverse of the relative frequencies (i.e. when the frequency is low, the weight is high, and vice versa). For a very regular binary sequence, only a few distinct words occur. Hence, ShEn would be small because the probability for these patterns is high and only little information is contained in the whole sequence. For a random binary sequence, all possible words occur with the same probability and the ShEn is maximal.

The Contribution of Nonlinear Methods in the Understanding of Atrial Fibrillation

*Approximate entropy* (ApEn) provides a measure of the degree of irregularity or randomness within a series of data. ApEn assigns a non-negative number to a sequence or time series, with larger values corresponding to greater process randomness or serial irregularity, and smaller values corresponding to more instances of recognizable features or patterns in the data [21]. ApEn measures the logarithmic likelihood that runs of patterns that are close (within a tolerance window *r*) for length *m* continuous observations remain close (within the same tolerance *r*) on next incremental comparison. The input variables *m* and *r* must be fixed to calculate ApEn. The method can be applied to relatively short time series, but the amounts of data points has an influence on the value of ApEn. This is due to the fact that the algorithm counts each sequence as matching itself to avoid the occurrence of ln(0) in the calculations. The *sample entropy* (SampEn) algorithm excludes self-matches in the analysis

On the other hand, the *multiscale entropy* (MSE) has been developed as a more robust measure of regularity of physiological time series which typically exhibit structure over multiple time scales [23]. For its computation, the sample mean inside each non-overlapping window of the original time series is calculated, thus constituting this set of sample means a new time series. Repeating the process *N* times with a set of window lengths starting from 1 to a certain length *N*, this will give a set of *N* time series of sample means. The MSE is obtained by computing any entropy measure (SampEn is suggested) for each time series, and displaying it as a function of the number of data points *N* inside the window (i.e. of the

Another index that can be used to quantity the regularity of a time series is the *conditional entropy* (CE) [24]. This index computed for a time series measures the amount of information carried by its most recent sample which is not explained by the knowledge of a predetermined conditioning vector containing information about the past of the observed multivariate process. The CE computation can be expressed as the difference between the ShEn calculated for the time series divided both in *L* and *L* − 1 sample-length patterns. Thus, this index measures the amount of information obtained when the pattern length is augmented from *L* − 1 to *L*. If a process is periodic (i.e. perfectly predictable) and has been observed for a sufficient time, it will be possible to predict the next samples. Therefore, there will be no increase of information by increasing the pattern length and CE will go to zero

**3.4. Irregularity quantification**

scale).

and is less dependent of the length of data series [22].

Several methods and algorithms are currently available to characterize a reconstructed phase space. Thus, two features widely used to emphasize the geometrical properties of the attractor are the *correlation dimension* (CD) and the *correlation entropy* (CorEn). The CD is a measure of the dimensionality of the attractor, i.e., of the organization of points in the phase space. Although there are several algorithms for its estimation, the CD can be computed by first calculating the correlation sum of the time series, which is defined as the number of points in the phase space that are closer than a certain threshold *r* [21]. Then, the CD is defined as the slope of the line fitting the log-log plot of the correlation sum as a function of the threshold. On the other hand, the CorEn is a measure of how fast the distance between two initially nearby states in phase space grows in time. This can be envisaged by taking a point in the reconstructed phase space, which corresponds to a segment in the time series. Another point in phase space located closely to the first one refers to a different segment in the time series. Thus, the CorEn is a measure of how fast these time segments loose their resemblance when both the segments are lengthened.

*Lyapunov exponents* (LEs) are also found habitually in the literature to enhance the dynamics of trajectories in the phase space. Precisely, these exponents quantify the exponential divergence or convergence of initially close phase space trajectories. LEs quantify also the amount of instability or predictability of the process. An *m*-dimensional dynamical system has *m* exponents but in most applications it is sufficient to compute only the largest LE (LLE), which can be computed as follows. First, a starting point is selected in the reconstructed phase space and all the points which are closer to this point than a predetermined distance, *ǫ*, are found. Then the average value of the distances between the trajectory of the initial point and the trajectories of the neighboring points are calculated as the system evolves. The slope of the line obtained by plotting the logarithms of these average values versus time gives the LLE. To remove the dependence of calculated values on the starting point, the procedure is repeated for different starting points and the LLE is taking as the average.

#### **3.3. Information content quantification**

Symbolic time series analysis involves the transformation of the original time series into a series of discrete symbols that are processed to extract useful information about the state of the system generating the process [20]. The first step of symbolic time series analysis is, hence, the transformation of the time series into a symbolic/binary sequence using a context-dependent symbolization procedure. After symbolization, the next step is the construction of words from the symbol series by collecting groups of symbols together in temporal order. This process typically involves definition of a finite word-length template that can be moved along the symbol series one step at a time, each step revealing a new sequence.

Quantitative measures of word sequence frequencies include statistics of words (word frequency or transition probabilities between words) and information theoretic based on entropy measures. Thus, a complexity measures widely used is the proposed by Lempel and Ziv [20], which will be referred to as *Lempel-Ziv complexity* (LZC). This metric provides a measure of complexity related to the number of distinct substrings and the rate of their occurrence along a given sequence, larger values of LZC corresponding to more complex series. Another metric habitually used is the *Shannon entropy* (ShEn) [21]. This index gives a number that characterize the probability that different words occur. Thus, counting the relative frequency of each word, the ShEn is estimated as the sum of the relative frequencies weighted by the logarithm of the inverse of the relative frequencies (i.e. when the frequency is low, the weight is high, and vice versa). For a very regular binary sequence, only a few distinct words occur. Hence, ShEn would be small because the probability for these patterns is high and only little information is contained in the whole sequence. For a random binary sequence, all possible words occur with the same probability and the ShEn is maximal.
