When Daily Sunspot Births Become Positively Correlated

Abstract

We study the first differences $w(t)$ of the International Sunspot Number (ISSN) daily series for the time span 1850 – 2013. The one-day correlations $\rho_{1}$ between $w(t)$ and $w(t+1)$ are computed within four-year sliding windows and are found to shift from negative to positive values near the end of Cycle 17 (${\sim}\,1945$). They remain positive during the last Grand Maximum and until ${\sim}\,2009$, when they fall to zero. We also identify a prominent regime change in ${\sim}\,1915$, strengthening previous evidence of major anomalies in solar activity at this date. We test an autoregressive process of order 1 (AR(1)) as a model that can reproduce the high-frequency component of ISSN: we compute $\rho_{1}$ for this AR(1) process and find that it is negative. Positive values of $\rho_{1}$ are found only if the process involves positive correlation: this leads us to suggest that the births of successive spots are positively correlated during the last Grand Maximum.

Appendix: Analytical computation of correlations

For the common AR(1) process defined by Equation (2) without modulation, when the random variables $\eta_{k}$ have a finite variance $\mathbf{Var}(\eta_{k})=\sigma_{k}^{2}$, the auto-correlation can be calculated analytically. We set $g(t)=G(t+1)-G(t)$. The covariance is

$$\begin{aligned} \mathbf{Cov} \bigl(g(1),g(0) \bigr) =& \lim_{l\to\infty} \mathbf{Cov} \Biggl(-(1-a)a^{l+1}G(-l) \\ &{}-(1-a)\sum_{k=-1}^{l} a^{k+1} \eta_{-k} +\eta_{2}, -(1-a)a^{l}G(-l) \\ &{}-(1-a)\sum_{k=0}^{l}a^{k} \eta_{-k}+\eta_{1} \Biggr). \end{aligned}$$

(5)

If the random variables $\eta$ are independent, the terms $\mathbf{Cov}(\eta_{i},\eta_{j})$ disappear for $i\ne j$, and Equation (5) is simplified into

$$ \mathbf{Cov} \bigl(g(1),g(0) \bigr)= (1-a)^{2}\sum _{k=0}^{\infty}\sigma_{-k}^{2}a^{2k+1}-(1-a) \sigma_{1}^{2}. $$

(6)

If $\sigma_{k}=\sigma$ for all $k$, then

$$ \mathbf{Cov} \bigl(g(1),g(0) \bigr)= -\frac{(1-a)}{1+a} \sigma^{2}. $$

(7)

Since the variance of the one-step differences is equal to

$$\mathbf{Var}\bigl(g(1)\bigr)=\mathbf{Var}\bigl(g(0)\bigr)= (1-a)^{2} \mathbf{Cov} \Biggl(\sum_{k=0}^{\infty}a^{k} \eta_{-k},\sum_{k=0}^{\infty}a^{k} \eta_{-k} \Biggr)+ \mathbf{Var}(\eta_{1})=\frac{2}{1+a} \sigma^{2}, $$

the correlation $\rho_{1}$ between the daily differences is

$$ \rho_{1}=\rho \bigl(g(1),g(0) \bigr)=- \frac{1-a}{2}. $$

(8)

From Equation (8) it follows that the one-day auto-correlation of the AR(1)-process is negative and goes to zero as the auto-correlation factor $a$ tends to 1.