The goodness of global fitting in the reconstruction scheme

In the reconstruction scheme, the results for predictions from chaotic time series are accurate. This work introduces the chi-square test for goodness-of-fit and the statistic R-squared in this scenario. These new features facilitate the choice between different predictors and improve the predictive capacity of the LinMapTS package. Program Title: LinMapTS Program Files doi: http://dx.doi.org/10.17632/pnhy9zymrp.2 Licensing provisions: GPLv3 Programming language: Maple 17 Journal reference of previous version: P R.L. Alves, L.G.S. Duarte, L.A.C.P. da Mota, Comput. Phys. Commun. 215 (2017) 265–268 Does the new version supersede the previous version?: Yes Reasons for the new version: The global fitting captures the underlying dynamics from a time series. It is quite convenient to test if a map is proper to describe the time evolutions of an observable or not. The introduction of the chi-square test for goodness-of-fit in the reconstruction scheme attends this demand [1]. On the other hand, a satisfactory global map may be more accurate than another. The statistic R-squared is a powerful aid to one’s choosing the best predictor. [2]. Summary of revisions: If the command LinGfiTS has the optional argument Analysis=1, then the output now presents both results of the test and the statistic R-squared for the global fitting. To illustrate the computational implementation of these new features, we revisit some predictions of chaotic time series with the LinMapTS package [3,4]. The commands below are sufficient to reconstruct the state vectors and to forecast of the dynamical variable of order 723 (XP) if one analyzes the time series – for a Lorenz System – stored in the file ts37.txt. In the next step, this communication presents the commands for the analysis of a time series of the real-world – a chaotic circuit [5,3,6] – and outputs of the new features and results for different polynomial predictors. Download : Download high-res image (143KB)Download : Download full-size image Download : Download high-res image (147KB)Download : Download full-size image Fig. 1 presents well-succeed global fittings for the Lorenz System (see Fig. 1(a)) and the chaotic voltage in a four-dimensional reconstruct phase space (see Fig. 1(b)). In both study cases, the most accurate forecasts correspond to the largest R-squared between all polynomial predictors (see Table 1, Table 2). So these applications suggest that the new features presented in this communication may improve the forecast capability significantly. Nature of problem: Time series analysis and improving forecast capability. Solution method: The method of solution is published in [3]. Additional comments including restrictions and unusual features: Depending on the data inputted, the chi-square test may not run properly. In this case, one must introduce the optional argument ChiSquare=0 to perform the graphical analysis and compute the statistic R-squared. Declaration of competing interest The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper. Acknowledgments L.G.S. Duarte and L.A.C.P. da Mota wish to thank Fundação de Amparo à Pesquisa do Estado do Rio de Janeiro (FAPERJ) – Registro n.o 8182/UERJ/2013 and Deliberacão n.o 25/2013 – for the Research Grant. References: [1] D. Sheskin, Handbook of parametric and nonparametric statistical procedures, Chapman & Hall/CRC, Boca Raton, 2011 (2011). [2] N. Draper, Applied regression analysis, Wiley, New York, 1998 (1998). [3] P. Alves, L. Duarte, L. da Mota, Improvement in global forecast for chaotic time series, Computer Physics Communications 207 (2016) 325 – 340 (2016). https://doi.org/10.1016/j.cpc.2016.05.011. URL http://www.sciencedirect.com/science/article/pii/S0010465516301278 [4] P. Alves, L. Duarte, L. da Mota, Alternative predictors in chaotic time series, Computer Physics Communications 215 (2017) 265 – 268 (2017). https://doi.org/10.1016/j.cpc.2017.02.013. URL http://www.sciencedirect.com/science/article/pii/S0010465517300656 [5] P. Alves, L. Duarte, L. da Mota, A new characterization of chaos from a time series, Chaos, Solitons and Fractals 104 (2017) 323 – 326 (2017). https://doi.org/10.1016/j.chaos.2017.08.033. URL http://www.sciencedirect.com/science/article/pii/S096007791730365X [6] P. McSharry, Nonlinear dynamics and chaos workshop ((accessed 09.08.2014)). URL http://people.maths.ox.ac.uk/mcsharry/lectures/ndc/ndcworkshop.shtml