**5. Conclusions**

In this work, we proposed four possible characterizations of the state of a dynamic system based on Shannon entropy: a frequentist binning approach (distribution), the spectral probability density of the TS (spectral), and symbolic transformations (permutation and 2-regimes) defining the alphabet by ordinal rank patterns, and sequences of the first derivative sign. These characterizations are the measures of complexity, and they are bounded between zero (i.e., minimal Entropy/Complexity) and one (i.e., maximal Entropy/Complexity). One important feature for these measures is that Entropy is maximal when TS states are equiprobable. In contrast, Complexity is maximal when the system tends to high Self-Organization or high Emergence (i.e., discernible patterns with some noise or high noise with some discernible patterns). From those measures, we determined the principal components, and through its loadings, we found that *Cperm* and *<sup>C</sup>*2*reg* are those measures that represent patterns that identify TS groups with similar features. Also, by plotting the TS by its *log*(*MASE*) in different quartiles, we observed that the TS with low *log*(*MASE*) are concentrating along with the first principal component. Moreover, comparing the four forecasting methods, the behavior is very similar between them; it is important to emphasize that for every TS, the *log*(*MASE*) values displayed in this space are very close among each other. Thus, these plots only corroborate the supposition that the winning method is the best for the quantity of TS where the winning is individually. Another important result is that we found that from the four forecasting methods identified as the winner of each TS dispersed over the complete TS, we see that the two principal components are consistent with the *No-free lunch* theorem. Finally, we determine that the TS with complexity measures closer to zero correspond to a low *log*(*MASE*) error, whereas when complexities measures are high, the *log*(*MASE*) tends to be high.

**Author Contributions:** In this paper, M.P.-F., J.F.-S., and G.S.-B. made the conceptualization for obtaining Entropy, Complexity, and prediction error measure; M.P.-F., J.F.-S., and G.S.-B. made the formal analysis; G.S.-B. conceptualized the ESC entropy extensions and the CFS; M.P.-F. and G.S.-B. performed the experiments; M.P.-F., J.F.-S., G.S.-B., J.P.-O. and J.J.G.-B. validated the experiments and wrote the paper. All authors have read and agreed to the published version of the manuscript.

**Funding:** This research received no external funding.

**Acknowledgments:** The authors acknowledge CONACYT, Cátedras CONACYT program, and TecNM/ITCM for the use of its installations.

**Conflicts of Interest:** The authors declare no conflict of interest. The dataset collection, analyses, and results interpretation were completely made by the authors, as the writing of the manuscript as well. The decision to publish the results prepared for this paper was completely taken by the authors. The are no funders that had a role in the design of the study, in the collection, analyses, or interpretation of data; in the writing of the manuscript, or in the decision to publish the results.
