# Biophotons: A Hard Problem
## 3 Statistical Analysis of the Experimental Data
### The Diffusion Entropy Method
In detail: \(\eta=\mu-1\) for \(1<\mu<2\); \(\eta=\frac{1}{\mu-1}\) for \(2<\mu<3\); and \(\eta=0.5\) for \(\mu>3\). If complexity is generated by crucial events, the region \(\mu>3\) corresponds in any case to stationary fluctuations and is interpreted as the manifestation of ordinary equilibrium statistical physics; conversely, the region with \(\mu<3\) is the area of nonstationary behavior, either temporary when \(2<\mu<3\) or permanent for \(1<\mu<2\). The experimental data showed extreme variability in terms of observed intensity; for this reason, we decided to divide the total acquisition time of about 72 h into six regions, the first five having a length of 10 h, while the last one was greater, being equal to 22 h. The idea behind this way of analyzing data was to understand if \(\mu\) changes with time during the germination process, its value at different stages of germination, and how these compare with the values obtained in the analysis of some other physiological processes, like heartbeats [45] and brain dynamics studied via EEG recording [47]. In Figure 7, we show the six regions chosen for the analysis, separated by vertical black lines. For each of these regions, we used DEA with and without stripes to determine the various scaling factors. A similar analysis was also conducted for the dark counts to enable a comparison with the experimental data coming from a case without any seeds. We found that the dark count yields the ordinary scaling, thereby showing that no temporal complexity of either kind may occur in the absence of any seed in the chamber. In the presence of seeds in the chamber, anomalous scaling emerged. The analysis with no stripes yielded a scaling significantly larger than the scaling obtained by DEA supplemented with stripes. Furthermore, while the scaling factor remained practically constant in all the six different regions when obtained by using the DEA without stripes, there was a significant time dependence for the scaling factor obtained by the DEA with stripes. In  
\[x(t)=\int_{0}^{t}\xi(t^{\prime})dt^{\prime}+x(0) \tag{8}\]2 Number of photons emitted during the germination of lentil seeds (violet points). The black lines represent the six different regions used for the DEA. The different spectral components of the signal relating to four spectral windows are also indicated with different colors in the inset at the top of the figure. the first phase of germination, within the three first temporal regions, the value of \(\mu\) was significantly less than three, a value that increased with the progress of the germination process. Without stripes, the \(\eta\) value increased from 0.694 (\(\mu\) = 2.44) to 0.796 (\(\mu\) = 2.25), while with stripes, it increased from 0.496 (\(\mu\) = 3.01) to 0.596 (\(\mu\) = 2.67). For brevity, we do not report here the complete analysis (see Ref. [17] for details), but we present a summary using an average procedure on the different scaling factors. These numbers related to the different temporal regions are divided into two sets: in the first one, the average is determined using the scaling factors obtained in temporal regions #1-3, while in the second set, the average is determined using the scaling factors coming from regions #4-6 (see Table 2 in Ref. [17] for the numbers). Using DEA without stripes, the average scaling index related to the first 30 h (regions #1-3) was \(\eta\) = 0.77 \(\pm\) 0.03, which corresponds to a mean \(\mu\) = 2.30 \(\pm\) 0.05, while in the second three regions (regions #4-6), the average scaling index was \(\eta\) = 0.72 \(\pm\) 0.02, corresponding to a mean \(\mu\) = 2.39 \(\pm\) 0.04. Essentially, there is no time dependence of the scaling factor throughout the whole germination process. In contrast, when applying the DEA with stripes, the results showed a clear and significant time dependence.