## 5 Tidal Effects on the Local Magnetometers
### Statistical Analysis  
A statistical analysis is performed to confirm or deny the proposed hypothesis that the correlation between local magnetic fields (with the eliminated diel cycle) measured at different locations around the world depends on the absolute difference between the meridian coordinates of those locations (Hypothesis 1) and that the SRCI correlates with the TWCI (Hypothesis 2). We define the following statistical variables: \(X_{i,1}\)--SRCI of the \(i\)th region in the 0-1 Hz range, \(X_{i,2}\)--SRCI of the \(i\)th region in the 0-3.5 Hz range, \(X_{i,3}\)--SRCI of the \(i\)th region in the 3.5-36 Hz range, \(X_{i,4}\)--TWCI of the \(i\)th region. [PERSON]'s linear correlation and [PERSON]'s rank correlation coefficients are commonly used in the study of the correlation relationship in statistics. [PERSON]'s coefficient is applied when the data are normal. If the condition of normality of the variables is not satisfied or the data are sparse, [PERSON]'s rank coefficient is applied. We check whether the normality condition is satisfied. Applying the chi-square test with a significance level of \(\alpha=0.05\), we test the following hypotheses:  
\[ \begin{split} H_{0}:X_{i,j}&\sim N(\hat{\mu}_{i,j},\hat{\sigma}_{i,j}),\\ H_{a}:X_{i,j}&\approx N(\hat{\mu}_{i,j},\hat{ \sigma}_{i,j})\end{split}  \tag{6}\]  
here \(\hat{\mu}_{i,j},\hat{\sigma}_{i,j}\) (\(i\in\{1,2,3,4,5\},j\in\{1,2,3,4\}\)) --parameter estimates found by the maximum likelihood method. Since in all cases the \(p\)-value is lower than the selected level of significance (Table 9), the hypotheses of the normality of variables \(X_{i,1},X_{i,2},X_{i,3},X_{i,4},i\in\{1,2,3,4,5\}\) are rejected. As the normality condition is not satisfied, we will apply [PERSON]'s rank correlation coefficient. Suppose we have pairs of variables \((X_{i,m},X_{i,5}),i\in\{1,2,3,4,5\},m,s\in\{1,2,3,4\}\) and observations \((X^{1}_{i,m^{\prime}},X^{1}_{i,s}),(X^{2}_{i,m^{\prime}},X^{2}_{i,s}),\dots,( X^{m}_{i,m^{\prime}},X^{n}_{i,s})\). We arrange the observations \(X^{1}_{i,m^{\prime}},X^{2}_{i,m^{\prime}},\dots,X^{n}_{i,m}\) and \(X^{1}_{i,s^{\prime}},X^{2}_{i,s^{\prime}},\dots,X^{n}_{i,s}\) separately. Let \(R(X^{k}_{i,l})\) be the rank of observation \(X^{k}_{i,l}\). Then, the [PERSON]'s rank correlation coefficient is given by  
\[r^{i}_{m,s}=\frac{\sum\limits_{k=1}^{n}(R(X^{k}_{i,m})-\frac{n+1}{2})(R(X^{k}_ {i,s})-\frac{n+1}{2})}{\sqrt{\sum\limits_{k=1}^{n}(R(X^{k}_{i,m})-\frac{n+1}{2 })^{2}}\sqrt{\sum\limits_{k=1}^{n}(R(X^{k}_{i,s})-\frac{n+1}{2})^{2}}} \tag{7}\]  
We calculate the [PERSON]'s correlation coefficient between the variables in each region (Table 10 represents the [PERSON]'s correlation coefficients in California, Table 11 the coefficients in Canada, Table 12 those in Lithuania, Table 13 those in Saudi Arabia, and Table 14 the coefficients in New Zealand). The results obtained suggest that the correlation between pairs of variables \(X_{i,1},X_{i,2}\), \(X_{i,3},i\in\{1,2,3,4,5\}\), in each region is strong or very strong, with the exception of the Canadian region, where the pair \((X_{2,1},X_{2,3})\) exhibits a moderate strength of connection. However, it should be noted that in each region between variables \(X_{i,1},X_{i,2},X_{i,3}\) and variable \(X_{i,4},i\in\{1,2,3,4,5\}\) there is a very weak or non-existent relationship. \begin{table}
\begin{tabular}{c c c c c} \hline \hline \(r_{m,s}^{i}\) & \(X_{i,1}\) & \(X_{i,2}\) & \(X_{i,3}\) & \(X_{i,4}\) \\ \hline \(X_{i,1}\) & \(1\) & \(0.95\) & \(0.72\) & \(0.22\) \\ \hline \(X_{i,2}\) & \(0.95\) & \(1\) & \(0.82\) & \(0.18\) \\ \hline \(X_{i,3}\) & \(0.72\) & \(0.82\) & \(1\) & \(0.25\) \\ \hline \(X_{i,4}\) & \(0.22\) & \(0.18\) & \(0.25\) & \(1\) \\ \hline \end{tabular}
\end{table} 
Table 10: [PERSON]’s correlation coefficient values (\(i=1\), region—California).