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Comparison of Two Learning Networks for Time Series Prediction
Published in Takushi Tanaka, Setsuo Ohsuga, Moonis Ali, Industrial and Engineering Applications of Artificial Intelligence and Expert Systems, 2022
Daniel Nikovski, Mehdi Zargham
The problem of predicting the continuation of a given time series is fundamentaily a machine learning problem, since such a continuation requires the extraction by the prediction method of the rules that govern the dynamics of the time series. This extraction has to be done from examples of past values of the time series. The use of past values of the time series for reconstructing the dynamics of the underlying dynamic system is justified by Takens’ theorem [WG94].
The Emergence of Chaos in Time
Published in Pier Luigi Gentili, Untangling Complex Systems, 2018
A feature of a chaotic time series is its unpredictability in the long term. It has been demonstrated (Sugihara and May 1990) that the accuracy of a nonlinear forecasting method falls off when it tries to predict chaotic time series, and the prediction-time interval is increased. On the other hand, the nonlinear forecasting method is roughly independent of the prediction time interval when it tries to predict time series that are uncorrelated noise.10 For the discernment of a chaotic from a white noise time series, we first need to build its phase space by exploiting the Takens’ theorem. Then, we plot the time series in its phase space. For each point A¯i of the trajectory, it is possible to find its neighbors, which are A¯j (with j = 1, …, k). For the prediction of the value A¯i+τΔt, which lies τΔt ahead in its phase space (see Figure 10.20), the nonlinear predictor exploits the corresponding A¯j+τΔt (with j = 1, …, k) and the following algorithm: [] A¯i+τΔt=∑j=1kW(A¯j,A¯i)*A¯j+τΔt
Attractor dynamics of elite performance: the high bar longswing
Published in Sports Biomechanics, 2021
Sophie Burton, Domenico Vicinanza, Timothy Exell, Karl M. Newell, Gareth Irwin, Genevieve K. R. Williams
Poincaré plots: Poincaré plots (Kantz & Schreiber, 2004) were used to denote the CM trajectory in the phase space as an initial step to examine the presence of attractor dynamics (Vicinanza et al., 2018). Takens (1981) theorem states that the Takens’ vector enables the reconstruction of an equivalent dynamical system to the original system that is produced by the observed time series. Embedding the time series in an n-dimensional space generates the set of Takens’ vectors. The n-th Takens’ vector is defined as:
Modeling interaction as a complex system
Published in Human–Computer Interaction, 2021
Niels van Berkel, Simon Dennis, Michael Zyphur, Jinjing Li, Andrew Heathcote, Vassilis Kostakos
Furthermore, DeAngelis and Yurek (DeAngelis & Yurek, 2015) point out the central role of equations in modern science, stating that “mathematics has not had the “unreasonable effectiveness” in ecology that it has had in physics” (DeAngelis & Yurek, 2015). This stems from the fact that it is near impossible to parameterize all aspects of an ecological system in a single model. As such, rather than formulating equations to construct a model, the authors state that the collected data should directly determine the model (DeAngelis & Yurek, 2015). This notion forms the basis of (Ye et al., 2015) equation-free ecosystem forecasting using empirical dynamic modeling. Equation-based modeling in HCI faces the same problems as identified in ecological modeling. Capturing and measuring all aspects of the interaction between a user and an artifact, including a complete overview of the user’s context, is near impossible regardless of the care a researcher takes in controlling a study. Takens’ theorem describes how the future state of a complex dynamic system can be predicted using time series data of only a single variable of that system (Takens, 1981). This is an important property for the analysis of observational, in-the-wild studies. Given the nature of in-the-wild studies, researchers are unable to control for all confounding variables which may potentially affect the variable of interest. Takens’ theorem suggests that these latent variables nevertheless leave an imprint on the variables captured by the researcher. Returning to the example of wolves and sheep introduced at the onset of this paper, it is easy to imagine that the availability of grass affects the sheep population. Even though the variable ‘grass’ may not be measured by the researcher, changes in the availability of grass are reflected in changes in the sheep population. As such, the analysis can determine whether there is a relationship between wolves and sheep without necessarily measuring the amount of grass, rain, or other potentially confounding variables.