Analysis for the reconstruction of attractors by reconstruction of phase spaces using delay times

Being able to estimate the behavior of a system from observable data is one of the great difficulties that any system presents. This problem presents a challenge for researchers who perform scenario estimation and forecasting. In most problems it is proposed to perform data analysis, but in this article, we propose to perform synthesis in such a way that a dipheomorphic attractor is constructed. that models the system. In the treatment of the analysis, we start from the inputs and assume some equations that describe the system, in the case of synthesis the most important thing is the data produced by the system, since these are real with some associated noise, so from those data and using Takens’ theorem, we can build an attractor that models the system we model in a more real way.

Data resampling: An approach for improving characterization of complex dynamics from noisy interspike intervals

Extracting dynamics from point processes produced by different models describing spiking phenomena depends on several factors affecting the quality of reconstruction of nonuniformly sampled dynamical systems. Although its ability is verified by embedding theorems analogous to the Takens theorem for uniformly sampled time series, a limited amount of samples, a low firing rate and the presence of noise can provide significant computational errors and incorrect characterization of the analyzed oscillatory regimes. Here, we discuss how to improve the accuracy of the quantitative evaluation of complex oscillations from point processes using data resampling. This approach provides a more stable estimation of Lyapunov exponents for noisy datasets. The advantages of resampling-based reconstruction are confirmed by the analysis of various spiking mechanisms, including the generation of single firing events and chaotic bursts.

Economic Complexity: Correlations between Gross Domestic Product and Fitness

In this paper we study the causal relation between country Economic Fitness F c and its Gross Domestic Product per capita ( G D P ). Using the Takens’ theorem, as first suggested in (Sugihara, G. et al. 2012), we show that there exists a reasonable evidence of causal correlation between G D P and F c for relatively rich countries. This is not the case for relatively poor countries where F c and G D P do not show any significant causal relation. We also present some preliminary results to understand whether G D P or F c are driving factor for economic growth.

Use of False Nearest Neighbours for Selecting Variables and Embedding Parameters for State Space Reconstruction

If data are generated by a system with a d-dimensional attractor, then Takens’ theorem guarantees that reconstruction that is diffeomorphic to the original attractor can be built from the single time series in 2d+1-dimensional phase space. However, under certain conditions, reconstruction is possible even in a space of smaller dimension. This topic is very important because the size of the reconstruction space relates to the effectiveness of the whole subsequent analysis. In this paper, the false nearest neighbour (FNN) methods are revisited to estimate the optimum embedding parameters and the most appropriate observables for state space reconstruction. A modification of the false nearest neighbour method is introduced. The findings contribute to evidence that the length of the embedding time window (TW) is more important than the reconstruction delay time and the embedding dimension (ED) separately. Moreover, if several time series of the same system are observed, the choice of the one that is used for the reconstruction could also be critical. The results are demonstrated on two chaotic benchmark systems.

Research on the Magnitude Time Series Prediction Based on Wavelet Neural Network

Based on the fundamental principles of the wavelet analysis combining with BP neural network, the paper can obtain the minimum embedding dimension and delay time. According to the chaos theory, the phase space of the magnitude time series can be reconstructed by Takens theorem. The paper uses wavelet neural network to train and test the nonlinear magnitude time series in the reconstructed phase space. The simulation results show that the predictive effect of the magnitude time series is remarkable and the predictive performance of single-step prediction is superior to that of multi-step prediction.

Increasing Translation Invariant Morphological Forecasting Models for Stock Market Prediction

Statistical models have been widely used for the purpose of forecasting. However, it has some limitations regarding its performance, which prevents an automatic forecasting system development. In order to overcome such limitations, Artificial Neural Networks (ANNs), Evolutionary Algorithms (EAs) and Fuzzy Systems (FSs) based approaches have been proposed for nonlinear time series modeling. However, a dilemma arises from all these models regarding financial time series, which follow a Random Walk (RW) model, where the forecast of such time series exhibits a characteristic one step shift regarding original data. In this way, this work presents a new approach, referred to as Increasing Translation Invariant Morphological Forecasting (ITIMF) model, to overcome the RW dilemma for financial time series forecasting, which performs an evolutionary search for the minimum dimension to determining the characteristic phase space that generates the financial time series phenomenon. It is inspired on Takens Theorem and consists of an intelligent hybrid model composed of a Modular Morphological Neural Network (MMNN) combined with a Modified Genetic Algorithm (MGA), which searches for the particular time lags capable of a fine tuned characterization of the time series and estimates the initial (sub-optimal) parameters (weights, architecture and number of modules) of the MMNN. Each individual of the MGA population is trained by the Back Propagation (BP) algorithm to further improve the MMNN parameters supplied by the MGA. After adjusting the model, it performs a behavioral statistical test and a phase fix procedure to adjust time phase distortions observed in financial time series. Furthermore, an experimental analysis is conducted with the proposed model using ten real world financial time series. Five well-known performance metrics and an evaluation function are used to assess the performance of the proposed model and the obtained results are compared to classical models presented in literature.