Non-parametric Data Analysis of Low-latitude Auroras and Naked-eye Sunspots in the Medieval Epoch

We have studied solar activity by analyzing naked-eye sunspot observations and aurorae borealis observed at latitudes below \(45^{\circ}\). We focused on the medieval epoch by considering the non-telescopic observations of sunspots from AD 974 to 1278 and aurorae borealis from AD 965 to 1273 that are reported in several Far East historical sources, primarily in China and Korea. After setting selection rules, we analyzed the distribution of these individual events following the months of the Gregorian calendar. In December, an unusual peak is observed with data recorded in both China and Japan, but not within Korean data.In extreme conditions, where the collection of events is reduced and discontinuous in some temporal intervals, we used the non-parametric kernel method. We opted for the plug-in approach of Sheather and Jones instead of cross-validation techniques to estimate the probability density functions (pdf) of the events. We obtained optimized bandwidths of 13.29 years for sunspots and 9.06 years for auroras, and 95% confidence intervals. The pdf curves exhibit multiple peaks occurring at quasi-periodic times with a very high positive correlation, \(r_{\mathrm{tt}} = 0.9958\), between the dates of occurrence of the nine extrema of sunspots and auroras. Furthermore, these extrema enabled us to evaluate mean periods at two standard deviations, \(66.77 \pm 7.25~\mbox{years}\) for sunspots and \(65.06 \pm 9.36~\mbox{years}\) for auroras. The accuracy of the average periods, 62.00 years for sunspots and 61.80 years for auroras, was improved by the use of the power spectrum method. The percentage of the total number of non-observed sunspots, using redundant data, from AD 1151 to 1275 was estimated to be greater than or equal to 78%.