## 3 Algebraic Complexity of the Schumann Resonance Amplitude Data
### The Definition of the Complexity Index Based on H-Ranks  
As mentioned previously, the main objective of this paper is to study the complexity of Schumann resonance amplitudes (SRAs) measured at different magnetometers located throughout the world and to assess possible correlations between those magnetometers. The power of the SRA is measured once per second in different frequency ranges. Since the SRA measurement process is continuous, it would be feasible to introduce an index representing the slow dynamics of the SRA. A natural approach would be to choose an index that represents the variation in the SRA over a time period of 1 h. Thus, the slow dynamics of the SRA would be represented by 24 values of the complexity index during each day. We will illustrate the algorithm for the construction of the complexity index based on H-ranks using the synthetic data generated by the MSC logistic maps. Let us assume that a single iteration of the MSC logistic maps corresponds to one second. Therefore, the complexity index based on H-ranks must be computed from \(60\times 60\) iterations. Note that all computations of H-ranks in the previous section were performed using the Hankel matrix, whose dimension \(d\) is fixed to 150. In other words, the length of the observation window required to fill the 150-dimensional Hankel matrix is 300 time-forward iterations of MSC logistic maps. It is clearly not feasible to construct a single Hankel matrix of dimensions \(1800\times 1800\)\((2\cdot 1800=3600)\). \begin{table}
\begin{tabular}{l l l l l l l} \hline \hline  & \(\Delta=0.1\) & \(\Delta=0.3\) & \(\Delta=0.4\) & \(\Delta=0.425\) & \(\Delta=0.45\) & \(\Delta=0.5\) \\ \hline \(\sigma^{2}\) & 14.6058 & 7.6784 & 3.4373 & 0.2531 & 0.1767 & 0.1064 \\ \hline H-rank & & & & & & \\ \hline \(\varepsilon=0\) & 150 & 150 & 150 & 150 & 150 & 150 \\ \hline \(\varepsilon=0.1\) & 149 & 147 & 136 & 96 & 39 & 2 \\ \hline \(\varepsilon=0.125\) & 149 & 146 & 133 & 84 & 16 & 0 \\ \hline \(\varepsilon=0.15\) & 147 & 146 & **130** & **77** & **2** & **0** \\ \hline \(\varepsilon=0.2\) & 147 & 144 & 125 & 62 & 0 & 0 \\ \hline \(\varepsilon=0.3\) & 145 & 142 & 113 & 29 & 0 & 0 \\ \hline \(\varepsilon=0.4\) & 144 & 139 & 104 & 14 & 0 & 0 \\ \hline \(\varepsilon=0.5\) & 140 & 135 & 96 & 8 & 0 & 0 \\ \hline \(\varepsilon=1\) & 126 & 119 & 65 & 0 & 0 & 0 \\ \hline \(\varepsilon=15\) & 0 & 0 & 0 & 0 & 0 & 0 \\ \hline \end{tabular}
\end{table} 
Table 1: Calibration of \(\varepsilon\) for the H-rank algorithm applied to the time series generated by the MSC logistic maps. Instead, the size of the Hankel matrix is kept at \(150\times 150\), eliminating the need to repeat the calibration procedure of the threshold parameter \(\varepsilon\). All further computations of H-ranks are performed in overlapping observation windows (300 iterations) until the last iteration is reached in the current hour (iteration number 3600). Note that the calibration of \(\varepsilon\) is performed once for the first 300 iterations. Let us denote the H-rank computed in the first observation window as \(Hr_{1}\). Then, the last H-rank computed in the current hour is reached at the last iteration in this hour (iteration number 3600) (Figure 3). Afterward, the complexity index characterizing the algebraic complexity of the investigated time series is computed as the arithmetic average of all computed H-ranks in the current hour. The process is continued without any gaps for the next hour (next 3600 iterations). Note that the threshold \(\varepsilon\) is not recalibrated for the next hour (it is assumed that the process is stationary). Such a computation of complexity indices can be interpreted as an information reduction algorithm (each consecutive hour is represented by a single number). Note that H-ranks are measured in natural numbers, but the complexity index is represented as a real number due to the averaging operation.