Derivation of stochastic differential equations for sunspot activity

Allen, E.J.; Huff, Cierra

doi:10.1051/0004-6361/200913978

Cited by 8 publications

(6 citation statements)

References 16 publications

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“…Dataset E-IV: Three time-series of the sunspots numbers for the period of 1976–2013 38 , the daily sunspots numbers depicts a noisy “pseudo-sinusoidal” behavior. It is accepted that magnetic cycles in the Sun are generated by a solar dynamo produced through nonlinear interactions between solar plasmas and magnetic fields 54 , 55 . However, the fluctuations in the period in the cycles is still difficult to understand 56 .…”

Section: Datasetsmentioning

confidence: 99%

Discriminating chaotic and stochastic time series using permutation entropy and artificial neural networks

Boaretto

Budzinski

Rossi

et al. 2021

Sci Rep

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Extracting relevant properties of empirical signals generated by nonlinear, stochastic, and high-dimensional systems is a challenge of complex systems research. Open questions are how to differentiate chaotic signals from stochastic ones, and how to quantify nonlinear and/or high-order temporal correlations. Here we propose a new technique to reliably address both problems. Our approach follows two steps: first, we train an artificial neural network (ANN) with flicker (colored) noise to predict the value of the parameter, $$\alpha$$ α , that determines the strength of the correlation of the noise. To predict $$\alpha$$ α the ANN input features are a set of probabilities that are extracted from the time series by using symbolic ordinal analysis. Then, we input to the trained ANN the probabilities extracted from the time series of interest, and analyze the ANN output. We find that the $$\alpha$$ α value returned by the ANN is informative of the temporal correlations present in the time series. To distinguish between stochastic and chaotic signals, we exploit the fact that the difference between the permutation entropy (PE) of a given time series and the PE of flicker noise with the same $$\alpha$$ α parameter is small when the time series is stochastic, but it is large when the time series is chaotic. We validate our technique by analysing synthetic and empirical time series whose nature is well established. We also demonstrate the robustness of our approach with respect to the length of the time series and to the level of noise. We expect that our algorithm, which is freely available, will be very useful to the community.

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Section: Datasetsmentioning

confidence: 99%

Discriminating chaotic and stochastic time series using permutation entropy and artificial neural networks

Boaretto

Budzinski

Rossi

et al. 2021

Sci Rep

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show abstract

“…To describe the sunspot numbers the underlying Brownian motion must remain non-negative, but there is a significant probability of observing a zero sunspot number. For this reason it is difficult to extend the stochastic differential equation formulation of Allen & Huff (2010) to account for both solar maximum and minimum without using an ad-hoc treatment of the stochastic process at zero. In the Fokker-Planck approach the non-negativity constraint on s(t) means that probability in s > 0 cannot move into the region s < 0 and the appropriate boundary condition at s = 0 is the zero probability flux condition…”

Section: Derivation Of a Fokker-planck Equationmentioning

confidence: 99%

“…In this paper, we introduce a formal statistical framework for modeling day-to-day fluctuations in sunspot number. Our approach is similar to a recent paper by Allen & Huff (2010) in that it treats the sunspot numbers as a diffusion process, but it has a number of specific advantages over this earlier method. First, our model provides a framework for combining deterministic (including chaotic) models for secular variation in solar cycles with statistical analysis of the sunspot number time series.…”

Section: Introductionmentioning

confidence: 99%

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Modeling the Sunspot Number Distribution With a Fokker-Planck Equation

Noble

Wheatland

2011

ApJ

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Sunspot numbers exhibit large short-timescale (daily-monthly) variation in addition to longer-timescale variation due to solar cycles. A formal statistical framework is presented for estimating and forecasting randomness in sunspot numbers on top of deterministic (including chaotic) models for solar cycles. The Fokker-Planck approach formulated assumes a specified long-term or secular variation in sunspot number over an underlying solar cycle via a driver function. The model then describes the observed randomness in sunspot number on top of this driver function. We consider a simple harmonic choice for the driver function, but the approach is general and can easily be extended to include other drivers which account for underlying physical processes and/or empirical features of the sunspot numbers. The framework is consistent during both solar maximum and minimum, and requires no parameter restrictions to ensure non-negative sunspot numbers. Model parameters are estimated using statistically optimal techniques. The model agrees both qualitatively and quantitatively with monthly sunspot data even with the simplistic representation of the periodic solar cycle. This framework should be particularly useful for solar cycle forecasters and is complementary to existing modeling techniques. An analytic approximation for the Fokker-Planck equation is presented, which is analogous to the Euler approximation, which allows for efficient maximum likelihood estimation of large data sets and/or when using difficult to evaluate driver functions.

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“…Aguirre, Letellier, and Maquet, 2008;Hanslmeier and Brajša, 2010;and Letellier et al, 2006), statistical techniques (e.g. Akaike, 1978;Yule, 1927;Allen and Huff, 2010), and neural networks (e.g. Conway, 1998).…”

Section: Introductionmentioning

confidence: 99%