2. Symbolic time series analysis

The concept of symbolization has its roots in dynamical systems theory, particularly in the study of nonlinear systems, which can exhibit bifurcation and chaos. In [21], it is asserted that symbolic dynamics is a method for studying nonlinear discrete-time systems by taking a previously codified trajectory using strings of symbols from a finite set, also called an alphabet. According to [22], symbolic dynamics and symbolic analysis are connected but are different concepts. In fact, the former is the practice of modeling a dynamical system by a discrete space. However, the latter is an empirical approach to characterize highly noisy data by considering a partition, discretizing the data, and obtaining a string representing the very dynamic of the process.

As asserted by [23], symbolization involves transformation of raw time series measurements into a series of discretized symbols that are processed to extract information about the generating process. In this way, we can search for nonrandom patterns and dependence by transforming a given time series {x1, x2,…, xT} into a symbolic string {s1, s2, …, sT}.

The STSA approach is easy to apply but the definition of the right partition is the most difficult thing to do. Generally, it applied an equiprobable partition implying to take the empirical distribution of a given time series {x1, x2,…, xT} and establishing two or more equally probable regions. For instance, for a Gaussian time series, we can define two equally probable regions considering as partition the mean equal to zero. After that, we can assign the symbol si = 0 for negative values and si = 1 for positive ones. In this way, we transform a continuously random series into a discrete string similar to the outcomes from flipping a coin.
