**Time Series Complexities and Their Relationship to Forecasting Performance**

## **Mirna Ponce-Flores 1,\*,†, Juan Frausto-Solís 1,\*,†, Guillermo Santamaría-Bonfil 2,\*,†, Joaquín Pérez-Ortega 3 and Juan J. González-Barbosa 1**


Received: 17 December 2019; Accepted: 7 January 2020; Published: 10 January 2020

**Abstract:** Entropy is a key concept in the characterization of uncertainty for any given signal, and its extensions such as Spectral Entropy and Permutation Entropy. They have been used to measure the complexity of time series. However, these measures are subject to the discretization employed to study the states of the system, and identifying the relationship between complexity measures and the expected performance of the four selected forecasting methods that participate in the M4 Competition. This relationship allows the decision, in advance, of which algorithm is adequate. Therefore, in this paper, we found the relationships between entropy-based complexity framework and the forecasting error of four selected methods (Smyl, Theta, ARIMA, and ETS). Moreover, we present a framework extension based on the Emergence, Self-Organization, and Complexity paradigm. The experimentation with both synthetic and M4 Competition time series show that the feature space induced by complexities, visually constrains the forecasting method performance to specific regions; where the logarithm of its metric error is poorer, the Complexity based on the emergence and self-organization is maximal.

**Keywords:** classical forecasting methods; complexity; entropy; error measures; symbolic analysis; M4 competition
