## 4 Analyses
### Discussion
#### 4.1.2 Kolmogorov-Smirnov Test  
Beside comparing descriptive statistics calculated for different levels of aggregation, inferential methods were applied in order to quantify a change of the distribution of the aggregated signal. Here, the Kolmogorov-Smirnov (KS) two sample test was used to ascertain whether the distribution of the temporally aggregated signal differed from the distribution of the non-aggregated signal. Methodology  
We only briefly introduce the idea and procedure of the test, for a detailed description please see [41]. In this work, the KS two sample test was applied and no assumption on the distribution of the underlying data was made. Suppose the data consisted of two independent samples, a first sample \(X_{1}\), \(X_{2}\),..., \(X_{n}\) of size \(n\) and a second sample \(Y_{1}\), \(Y_{2}\),..., \(Y_{m}\) of size \(m\). \(F(x)\) and \(G(x)\) denote their respective, unknown distribution functions. We wanted to test the hypothesis  
\[H_{0}:F(x)=G(x)\quad\text{vs.}\quad H_{1}:F(x)\
eq G(x). \tag{1}\]  
Let \(F_{n}(x)\) be the empirical distribution function based on the random sample \(X_{1}\), \(X_{2}\),..., \(X_{n}\) and let \(G_{m}(x)\) be the empirical distribution function based on the random sample \(Y_{1}\), \(Y_{2}\),..., \(Y_{m}\). Then, the test statistic \(D\) measures the maximum difference between the two empirical distribution functions and is defined as  
\[D=\sup_{x}|F_{n}(x)-G_{m}(x)|. \tag{2}\]  
From the test statistic \(D\) the \(p\)-value was derived, adjusting for the different sample sizes \(n\) and \(m\). As a level of significance the typical value of \(\alpha\) = 0.05 was chosen. If the \(p\)-value was smaller than the level of significance, i.e., \(p<0.05\), \(H_{0}\) was rejected and we had sufficient evidence to say the aggregated data had another underlying distribution. If the \(p\)-value was greater than 0.05, then we did not have sufficient evidence to reject the null hypothesis and we could not draw any further conclusions. Please note that in our case there was an important difference of the test application when compared to its standard use. Here, we compared two samples that are known to come from the same generating process, as the tested signal was the same. Therefore, the test tried to determine whether the aggregation process changed the distribution of the signal such that the aggregated and original distribution no longer appeared similar. Results  
The test statistics \(D\) and the \(p\)-values were calculated for all signals based on incorporating the raw signal and several aggregated signals on different temporal resolutions of 10 s, 60 s, 300 s, and 600 s, equivalent to Section 4.1.1. The test was conducted with 10,000 h of randomly chosen samples of operating data. As the quantity of interest we present the derived \(p\)-values for an exemplary turbine, i.e., turbine 5, containing all investigatedsignals on the left side of Figure 3 and the minimum values of all turbines on the right. Data containing all numerical results are provided in the Supplementary Materials. Regarding all turbines the majority of \(p\)-values for all signals were above the significance level of 0.05. Especially for higher resolutions \(<\)60 s the \(p\)-values even tended to keep around 1, making it very unlikely that both the raw and aggregated value came from two different underlying distributions. There was even a set of signals whose \(p\)-values were always 1. These were the active power generation, grid frequency, generator speed, all pitch angles, power factor, and rotor speed. The turbine on the left of the figure had explicitly been chosen because it featured \(p\)-values below 0.05 for certain channels for aggregated data of 600 s resolution.