A Model of Sunspot Number with a Modified Logistic Function

Abstract

Solar cycles are studied with the Version 2 monthly smoothed international sunspot number, the variations of which are found to be well represented by the modified logistic differential equation with four parameters: maximum cumulative sunspot number or total sunspot number x m , initial cumulative sunspot number x 0 , maximum emergence rate r 0 , and asymmetry α. A two-parameter function is obtained by taking α and r 0 as fixed value. In addition, it is found that x m and x 0 can be well determined at the start of a cycle. Therefore, a prediction model of sunspot number is established based on the two-parameter function. The prediction for cycles 4 − 23 shows that the solar maximum can be predicted with average relative error being 8.8% and maximum relative error being 22% in cycle 15 at the start of solar cycles if solar minima are already known. The quasi-online method for determining solar minimum moment shows that we can obtain the solar minimum 14 months after the start of a cycle. Besides, our model can predict the cycle length with the average relative error being 9.5% and maximum relative error being 22% in cycle 4. Furthermore, we predict the sunspot number Corresponding author: G. Qin qingang@hit.edu.cn 2 Qin and Wu variations of cycle 24 with the relative errors of the solar maximum and ascent time being 1.4% and 12%, respectively, and the predicted cycle length is 11.0 (95% confidence interval is 8.3−12.9) years. The comparison to the observation of cycle 24 shows that our prediction model has good effectiveness.(1) with four parameters, namely, amplitude a, starting time t 0 , rise time b, and asymmetry c. They found that the asymmetry c can be taken as a fixed value and the rise time b is relevant with the amplitude a, so that the function could be reduced to a two-parameter form and the sunspot number could still be fitted well.They found that the amplitude can be estimated at the start of a solar cycle by using the correlation between Sunspot Number Model

The Sunspot Number Forecasting Using a Hybridization Model of EMD, LSTM and Attention Mechanism

“…The sunspot number is decomposed into eight sub-sequences by EMD, and every sub-sequence is fed into the prediction module for prediction value. The differences between the observed value and prediction value, namely prediction error, are divided into 9 intervals and such as [0,0.2], (0.2, 0.5], (0.5,1], (1,1.5], (1.5,2],(2,3], (3,4], (4,5], (5, +∞) and compute the proportion of prediction error in different intervals as shown in Table III. The difference proportion was accumulated in [0,1] and [0,2], respectively.…”

“…The sequence of sunspot number, which is a classical time series with the feature of non‐stationary and complex, is constructed by historical data. Therefore, the prediction of sunspot number (SSN) has a significant meaning [3].…”

Prediction of Yearly Mean Sunspot Number using Machine Learning Methods

“…Various approaches have been proposed for predicting the number of sunspots: statistical learning models include autoregressive modeling (AR) (Werner, 2012), a modified logistic function (Qin and Wu, 2018), and warped Gaussian process regression ( Ítalo G. Gonçalves et al, 2020). Physical models for the solar cycle involve solar dynamo theories (Jones et al, 2010;Karak et al, 2018).…”

Relationship of Transport Coefficients with Statistical Quantities of Charged Particles

“…In previous investigations, scientists have found that the spatial transport coefficients of charged energetic particles, including the parallel and perpendicular diffusion coefficients, are the key parameters describing the modulation of galactic cosmic rays (Parker 1965;Burger & Hattingh 1998;Qin 2007;Moraal 2013;Potgieter 2013;Qin & Zhang 2014;Qin & Shen 2017;Qin & Wu 2018;Oughton & Engelbrecht 2021), transport of solar energetic particles (Reames et al 1996(Reames et al , 1997Droege 2000;Zank et al 2000;Qin et al 2006Qin et al , 2013, diffusive acceleration of charged particles by shocks (Zank et al 2000(Zank et al , 2006Li et al 2003Li et al , 2005Li et al , 2012Dosch & Shalchi 2010;Hu et al 2017), etc. Thus, the spatial transport coefficient formulas and corresponding spatial transport equations have to be obtained (Schlickeiser 2002;Schlickeiser & Shalchi 2008;Shalchi 2009Shalchi , 2021bWang & Qin 2018, 2019.…”