## 5 Tidal Effects on the Local Magnetometers
### Statistical Analysis
\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 \(X_{i,1}\) & \(1\) & \(0.89\) & \(0.45\) & \(0.45\) \\ \hline \(X_{i,2}\) & \(0.89\) & \(1\) & \(0.70\) & \(0.22\) \\ \hline \(X_{i,3}\) & \(0.45\) & \(0.70\) & \(1\) & \(-0.23\) \\ \hline \(X_{i,4}\) & \(0.45\) & \(0.22\) & \(-0.23\) & \(1\) \\ \hline \end{tabular}
\end{table} 
Table 11: Spearman’s correlation coefficient values (\(i=2\), region—Canada). \begin{table}
\begin{tabular}{c c c c c c} \hline \hline \multirow{2}{*}{**Region**} & \(i\) & **Distribution Parameter Estimation** & \(p\)**-Value \(\leq\)** \\ \cline{2-6}  & & **j** & \(\hat{\mu}_{i,j}\) & \(\hat{\sigma}_{i,j}\) & \\ \hline \multirow{4}{*}{California} & \multirow{4}{*}{\(1\)} & \(1\) & \(74.36\) & \(10.54\) & \(1.8\cdot 10^{-8}\) \\  & & \(2\) & \(63.77\) & \(14.77\) & \(1.1\cdot 10^{-16}\) \\  & & \(3\) & \(48.39\) & \(20\) & \(0\) \\  & & \(4\) & \(12.23\) & \(1.55\) & \(4.6\cdot 10^{-4}\) \\ \hline \multirow{4}{*}{Canada} & \multirow{4}{*}{\(2\)} & \(1\) & \(58.95\) & \(14.34\) & \(1.4\cdot 10^{-7}\) \\  & & \(2\) & \(53.63\) & \(14.79\) & \(6.7\cdot 10^{-4}\) \\  & & \(3\) & \(59.48\) & \(15.81\) & \(0\) \\  & & \(4\) & \(11.82\) & \(3.04\) & \(2.4\cdot 10^{-14}\) \\ \hline \multirow{4}{*}{Lithuania} & \multirow{4}{*}{\(3\)} & \(1\) & \(94.13\) & \(2.95\) & \(4.2\cdot 10^{-3}\) \\  & & \(2\) & \(79.36\) & \(6.89\) & \(6.3\cdot 10^{-11}\) \\  & & \(3\) & \(76.9\) & \(10.31\) & \(3.7\cdot 10^{-4}\) \\  & & \(4\) & \(12.28\) & \(1.44\) & \(0\) \\ \hline \multirow{4}{*}{Saudi Arabia} & \multirow{4}{*}{\(4\)} & \(1\) & \(73.96\) & \(6.97\) & \(0\) \\  & & \(2\) & \(78.51\) & \(7.4\) & \(4\cdot 10^{-9}\) \\  & & \(3\) & \(72.82\) & \(9.68\) & \(9\cdot 10^{-4}\) \\  & & \(4\) & \(12.21\) & \(1.34\) & \(0\) \\ \hline \multirow{4}{To determine the statistical significance of the values of the correlation coefficients obtained, we subject the results to hypothesis testing. We test the following hypotheses:  
\[H_{0}:r^{i}_{m,s}=0, \tag{8}\] \[H_{a}:r^{i}_{m,s}\
eq 0\]  
where \(i\in\{1,2,3,4,5\},m\
eq s,(m,s)\in\{1,2,3,4\}\). The following statistical measure is used to test the hypothesis:  
\[T^{i}_{m,s}=r^{i}_{m,s}\sqrt{\frac{n-2}{1-(r^{i}_{m,s})^{2}}} \tag{9}\]  
When the null hypothesis \(H_{0}\) is true, the distribution of the statistic \(T^{i}_{m,s}\) follows a Student's t-distribution with \((n-2)\) degrees of freedom. Therefore, the \(p\)-value for the applied criterion is calculated as follows:  
\[P^{i}_{m,s}=P(|T|\geq|T^{i}_{m,s}|) \tag{10}\]  
where \(T\) is a random variable distributed according to the Student's t-distribution with \((n-2)\) degrees of freedom. We select the significance level \(\alpha=0.05\) to evaluate each hypothesis. The \(p\)-value for each hypothesis is then calculated. If the \(p\)-value is less than or equal to \(\alpha=0.05\) we reject the null hypothesis, indicating a statistically significant result. Conversely, if the \(p\)-value is greater than \(\alpha=0.05\), we fail to reject the null hypothesis, suggesting a lack of statistical significance. Table 15 indicates that, for the regions of the USA, Canada, Lithuania, and Saudi Arabia, the Spearman's rank correlation coefficients are significantly different from zero. This implies that the null hypotheses \(H_{0}\), which assert that the quantities are not correlated, are rejected in these cases. In the case of New Zealand, it should be noted that the hypotheses of the non-correlation between the quantities \(X_{5,1}\) and \(X_{5,A}\), \(X_{5,2}\) and \(X_{5,4}\) cannot be rejected. This outcome is likely due to the fact that the values of the Spearman's rank correlation coefficient are close to 0.