**1. Introduction**

Presently, time series forecasting is applied to many areas such as weather, finance, ecology, health, electrochemical reactions, computer networks, and so on [1]. Among the most popular and effective methods stand the classical time series models such as the Simple Exponential Smoothing (SES) and the Autoregressive Integrated Moving Average (ARIMA). Also, forecasting methods of machine learning such as Neural Networks have gained popularity after the results of the Smyl winning method of the M4 Competition, and the benchmark forecasting methods of Theta, ARIMA, and ETS [2]. In the forecasting area, researchers agree that it is too difficult to identify a suitable forecasting method for a particular time series beforehand, even knowing its specific statistical characteristics [3]. For instance, time series (TS) complexity [3] is a widely debated measure, which it is supposed to quantify the *intricacy* of the time series, allowing choice of the forecasting methods to be applied. Shannon's entropy has been used to measure the complexity of discrete systems [4]. Although the entropy formula was conceived in the thermodynamic area, the entropy concept has spread to different disciplines adapting

its meaning in regard to the applied area and making tools for many applications [5–7]. For example, in [8] a package with functions to measure emergence, self-organization, and complexity applied to discrete and continuous data is presented as a framework; the present study is based on them. However, to the best of the authors' knowledge, these formulae have not been applied to assess the *forecastability* of time series. Furthermore, this framework is extended with other measures. We present four complexity measures based on entropy and a methodology for determining the relationships between these complexity measures and the forecasting error of the Smyl [9], Theta [10], ARIMA [11,12], and ETS [13] methods; all of them were participants of the M4 Competition [14]. This study was made for a dataset with some synthetic time series [15] and more than 20,000 time series taken from M4 Competition [16], which is a reference point used by many researchers. We obtain the prediction error with the forecasting values of each one of the four selected methods, and we determine four complexity measures based on the relationship between Entropy and Mean Absolute Scaled Error (MASE) error [17], but for functionality we use the logarithm values of MASE error *log*(*MASE*). We present a complexity *log*(*MASE*) analysis, and we apply a visualization method [18] for the time series of the dataset. Finally, the experimentation shows that the permutation and 2-regimen complexities are the measures that identify patterns of the distribution of TS on the two-dimensional space; also we found a relationship between the permutation complexity and the *log*(*MASE*) values and finally we make a comparison between the four forecasting methods reinforcing the known *No-free lunch* theorem.

This paper is organized as follows. Section 2 presents the materials used in this research; Section 3 describes the methods, parameters settings, methodology and the dataset used in the experimentation. In Section 4, we provide the results of the experimentation. Finally, Section 5 presents the conclusion for this work.
