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Analysis and Data-Based Reconstruction of Complex Nonlinear Dynamical Systems pp 207–213Cite as

Distinguishing Diffusive and Jumpy Behaviors in Real-World Time Series

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Part of the book series: Understanding Complex Systems ((UCS))

Abstract

Jumps are discontinuous variations in time series and with large amplitude can be considered as an extreme event. We expect the higher the jump activity to cause higher uncertainty in the stochastic behaviour of measured time series. Therefore, building statistical evidence to detect real jump seems of primary importance. In addition jump events can participate in the observed non-Gaussian feature of the increments’ (ramp up and ramp down) statistics of many time series [1]. This is the reason that most of the jump detection techniques are based on threshold values for differential of time series. There is not, however, a robust method for detection and characterisation of such discontinuous events that is able to estimate time-dependence of the “jump rate” and their amplitudes, etc.

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Notes

  1. 1.

    We note in variance gamma model for jump amplitude introduced in Chap. 12, one finds \({K_{j}^{(6)}(x,\tau ) }/{5 K_{j}^{(4)}(x,\tau )} \approx \sigma _{\xi } ^2 (x,\hat{ b}) (1+ 2 \hat{b})\), where \(\hat{b}\) is given by Eq. (12.15).

  2. 2.

    We note that for \(\alpha =2\), one can start from x(0) and \(x(\Delta )\), then find two coarse samples of time series which are not statistically independent, of course. We can average the estimated KM conditional moments from two series. For \(\alpha =n\), we have n possible coarse samples.

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Authors and Affiliations

  1. Department of Physics, Sharif University of Technology, Tehran, Iran

    M. Reza Rahimi Tabar

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  1. M. Reza Rahimi Tabar
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Correspondence to M. Reza Rahimi Tabar .

Problems

Problems

19.1

Jump Detection Measure Q(x)

Reproduce the plots in Fig. 19.1. The details of linear, nonlinear and jump-diffusion are given in the caption of Figs. 18.1 and 18.2. Choose \(\tau \in \{10^{-6}, 2\times 10^{-6}, 4\times 10^{-6}, 6\times 10^{-6},\ldots ,10^{-3}\}\), in integration of corresponding stochastic equations in Euler–Maruyama scheme.

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Tabar, M.R.R. (2019). Distinguishing Diffusive and Jumpy Behaviors in Real-World Time Series. In: Analysis and Data-Based Reconstruction of Complex Nonlinear Dynamical Systems. Understanding Complex Systems. Springer, Cham. https://doi.org/10.1007/978-3-030-18472-8_19

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