Modelling Persistent Cycles in Solar Activity

Abstract

Solar activity at decadal time scales is characterised by persistent periodic patterns with global effects on the Earth’s climate. This article deals with the analysis and prediction of the revised monthly sunspot numbers, adopting a recently proposed time-series model for long-range dependent cycles. The methodology is based on the maximum-likelihood estimate of the model parameters and provides optimal signal extraction filters for cycle estimation and prediction. The analysis suggests the presence of stationary cyclical long memory in the sunspot-generating process. Moreover, our formulation provides a reliable method for solar-cycle predictions, yielding forecasts of the future Cycle 25. In particular, we claim a main peak will occur in early 2024 with an amplitude of 114 and an end of the cycle in early 2030.

Appendix: Testing for Nonstationary Cycles

This appendix illustrates how the test for a nonstationary cycle at a known frequency proposed by Robinson (1994) is carried out in our case. Consider the linear regression model with fSW errors

$$ \textstyle\begin{array}{l@{\quad }l@{\quad }l} y_{t} & = & \mathbf{x}_{t}'\boldsymbol {\beta }+ \psi _{t} \\ \psi _{t} & = & \alpha _{t} \cos (\lambda t) + \alpha ^{*}_{t} \sin ( \lambda t) \\ (1-L)^{d} \alpha _{t} & = & \eta _{t} \\ (1-L)^{d} \alpha ^{*}_{t} & = & \eta ^{*}_{t}, \\ \end{array} $$

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where $\mathbf{x}_{t}$ is a vector of exogenous variables and $[\eta _{t}, \eta ^{*}_{t}] \sim N(0,\sigma ^{2}_{\eta }I_{2})$.

The Lagrange Multiplier (LM) test by Robinson (1994) of the null hypothesis $H_{0}: d_{0} = 0.5$ against $H_{1}: d_{0} < 0.5$, is implemented as follows. Considering the complex fractional noise representation of the fSW process (see Proietti and Maddanu, 2021), we regress $\Phi (L)y_{t}$ on $\boldsymbol {w}_{t}=\Phi (L) \mathbf{x}_{t}$ where $\Phi (L) = (1 - 2 \cos (\lambda )L + L^{2})^{d_{0}}$. The resulting ordinary least-square (OLS) estimator $\hat{\boldsymbol {\beta }}= (\sum _{t=1}^{n} \boldsymbol {w}_{t} \boldsymbol {w}_{t}')^{-1}\sum _{t=1}^{n} \boldsymbol {w}_{t}\Phi (L)y_{t}$ is then used to compute the residuals $\hat{u}_{t} = \Phi (L)y_{t} - \hat{\boldsymbol {\beta }}'\boldsymbol {w}_{t}$ and the corresponding periodogram $I_{\hat{u}}(\omega _{j})=\frac{1}{2 \pi n} |\sum _{t=1}^{n} \hat{u}_{t} e^{-\imath \omega _{j} t} |^{2}$.

The LM statistic is

$$ LM(\lambda ,d_{0})=\frac{n}{\hat{\sigma }^{4}_{\eta }} \hat{a} \hat{A}^{-1} \hat{a}, $$

(21)

where

$$ \hat{a}= \frac{2 \pi }{n} \sum _{j=1}^{n-1} \Psi (\omega _{j}) \frac{I_{\hat{u}}(\omega _{j})}{g(\omega _{j})}, $$

and

$$ \hat{A}= 2 \mbox{Var}( \Psi (\omega _{j}) ) $$

with

$$ \Psi (\omega _{j})=\log (1 - 2 \cos (\omega _{j})e^{-\imath \omega _{j}} + e^{-2 \imath \omega _{j}}) $$

$$ g(\omega _{j}) = \frac{1}{4 \pi } \biggr\lbrace \biggr| \sin ( \frac{\omega _{j} - \lambda }{2}) \biggr|^{2 d_{0}} + \biggr| \sin ( \frac{\omega _{j} + \lambda }{2}) \biggr|^{2 d_{0}} \biggr\rbrace , $$

while the $\sigma ^{2}_{\eta }$ parameter is estimated as

$$ \hat{\sigma }^{2}_{\eta }= \exp (\hat{b}_{0}), $$

(22)

where $b_{0}$ is the intercept estimated via the linear regression $\ln (2\pi ) I_{\hat{u}}(\omega _{j}) + c= b_{0}+ 2 \sum _{i=1}^{K} b_{i} \cos (i \omega _{j}) + e_{t}$ ($c=0.57721$ is Euler’s constant), see Bloomfield (1973), where $K$ is chosen according to the Akaike information criterion.

Robinson (1994) shows that the LM statistic has a standard chi-square distribution with one degree of freedom.