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Appendix A: Statistical properties of the TSI reconstructions

Panels (a) to (d) of Fig. A.1 display the probability distributions of the 10-year averaged TSI values. Each panel shows the probability distribution functions (PDF) for the normal distribution computed employing the respective mean and standard deviation values that are also given in each panel. In order to verify if the reconstructed TSI values are normally distributed, we performed a test of the default null hypothesis that the values come from a normal distribution, against the hypothesis that they do not come from this distribution. The test statistic is (A.1)where ecdf(TSI) is the empirical cumulative distribution function (CDF) estimated from the TSI distribution and ncdf(TSI) is the normal CDF with mean and standard deviation equal to the mean and standard deviation of the TSI distribution. Recall that the value of the ecdf at a given value of the TSI is obtained from the PDF by integrating the latter from its lower end up to that particular TSI value. The T values are indicated in the respective panels. The null hypothesis is rejected at a given significance level when T is greater than a critical value. The test employed here uses a table of critical values computed using Monte-Carlo simulations, where the critical value for the significance level of 0.05 is 0.0547. We found that the null hypothesis of all reconstructions is rejected at this significance level, i.e., the distributions do not correspond to normal distributions. This is supported by a visual inspection which reveals the strong asymmetry in all the the empirical distributions. Besides the main peak located above 1365 W/m² there is also a lower (and statistically less significant) secondary peak below 1365 W/m², which corresponds to the relatively common grand minima in this period of time.

Figure A.1e compares the median values (red lines). Additionally, the upper and lower boundaries of the boxes display the respective quartile values. The dashed lines show the regions extending from the boundaries of each box to the most extreme values within 1.5 times the interquartile range. As expected, the VDM reconstructions display higher median values

than the VADM reconstructions. Furthermore, as also suggested by the empirical CDFs, the VDM distributions are shifted in relation to the VADM distributions, and also display a larger range of irradiance values.

We employ the Wilcoxon test to verify if two reconstructions come from identical distributions with equal medians, against the alternative that they do not have equal medians. Note that the distributions need not necessarily be normal. Table A.1 summarizes the results. The reconstructions from the same group (VADM or VDM) have the same median values. However, the reconstructions belonging to different groups disagree from each other.

The statistical properties of the TSI reconstructions for the whole Holocene based on the paleomagnetic reconstructions by Knudsen et al. (2008), Genevey et al. (2008), and Korte & Constable (2005) are presented in panels (a) − (c) of Fig. A.2, respectively. As in Fig. A.1, the red lines represent the probability distribution functions for the normal distribution. Note that each time series covers a different time span (see Table 3). Tests of the null hypothesis that the values come from a normal distribution, as discussed in Sect. 3.2.1, indicate that it is not possible to reject this hypothesis at the significance level of 0.05 for the reconstructions based on the Genevey et al. (2008) paleomagnetic reconstruction. A visual inspection shows that the reconstruction based on the paleomagnetic model by Korte & Constable (2005), that extends from 5000 BC to 1700, gives a slightly asymmetrical distribution. This behavior reflects the more frequent occurrence of grand minima in the later part of the time series, as can be gathered by comparing with the far more asymmetric distributions in Fig. A.1.

Fig. A.1 Comparison between the statistical properties of the TSI reconstructions for the period 1000 BC to 1700 AD. Panels a) to d) present the distributions of the TSI reconstructions. The mean, median and standard deviation values are indicated in the panels. The red lines represent the probability distributions functions for the normal distribution. Panel e) allows a visual comparison between the distributions. The median (red line), lower and upper quartile values of the distributions (lower and upper ends of the blue boxes) are plotted in each box. The black lines extend from the end of each box to the most extreme values within 1.5 times the interquartile range. Note that the interquartile range, which is also a measure of the statistical dispersion of the data, is equal to the difference between the third and first quartiles.

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Fig. A.2 Comparison between the statistical properties of the TSI reconstructions during the Holocene. The time coverage of each time series is presented in Table 3. Panels a) to d) present the distributions of the TSI reconstructions. The red line represents the normal distribution computed employing the sample mean and standard deviation values of as the overplotted sample (the values are indicated in each panel). Additionally, the test statistic (T) and the critical (Tcritical) values are listed.

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