## 4 Analyses
### Discussion
#### 4.1.3 Local Error Approach  
In this section we will address the information loss directly, conducting an analysis on the bare differences between the raw signal and the signals aggregated over different aggregation intervals.  
Methodology  
We defined the information loss as the difference between the original signal and its down-sampled signal, i.e., the local error, described by the loss function \(\Lambda(t)_{\Delta tagg}\). The  
\begin{table}
\begin{tabular}{c c c} \hline \(p_{\text{min}}\geq 0.98\) & \(0.98>p_{\text{min}}\geq 0.05\) & \(p_{\text{min}}\leq 0.05\) \\ \hline Active power & Ambient temperature & Temperature inside nacelle \\ Generator speed & Current phase A & Temperature of generator bearing 1 \\ Grid frequency & Current phase B & Temperature of generator bearing 2 \\ Pitch angle blade 1 & Current phase C & Temperature of generator stator \\ Pitch angle blade 2 & Temperature of gearbox oil & Top box temperature \\ Pitch angle blade 3 & Temperature of gearbox shaft bearing 1 & Voltage phase B \\ Power factor & Temperature of gearbox shaft bearing 2 & Wind direction \\ Rotor speed & Temperature of main bearing & Wind speed \\  & Voltage phase A & Yaw angle \\  & Voltage phase C & \\ \hline \end{tabular}
\end{table}
Table 3: Signals sorted into bins of \(p_{\text{min}}\). Here, \(p_{\text{min}}\) is the minimal \(p\)-value of a signal for all aggregation resolutions and turbines as already carried out in Figure 3.  
calculation was carried out by further applying the absolute value on the difference, as described in Equation (3):  
\[\Lambda(t)_{\Delta t_{\text{agg}}}=|s(t)_{\Delta t_{\text{agg}}}-s(t)_{\Delta t_{ \text{org}}}| \tag{3}\]  
In this equation \(\Delta t_{\text{org}}\) is the original or native resolution of the data before aggregation. The original signal values are defined as \(s(t)_{\Delta t_{\text{org}}}\), the aggregated values as \(s(t)_{\Delta t_{\text{agg}}}\) correspondingly. Note that \(\Delta t_{\text{org}}=1\,\text{s}\) for the present dataset. Within a sampling window all values \(s(t)_{\Delta t_{\text{agg}}}\) were equal to the averaged raw signal of this window, also known as \"value hold\". Thus, the information loss \(\Lambda(t)_{\Delta t_{\text{agg}}}\) always has the same resolution as the original data. A representation of the original and aggregated signal is provided in Figure 4.  
Due to the diverse nature of physical quantities measured by the SCADA system, the local errors needed to be normalized to be compared across signals effectively. Normalization also facilitated the interpretation of the results, as it allowed to reason about information losses in relative terms, bypassing the necessity to know the typical operating range of a signal. In our analyses, the interquartile range (IQR) of the values of the raw time series was used as the normalization basis throughout the following parts of the paper if not declared otherwise. The calculated IQR values are listed in Table 4. We chose this range over other common approaches, e.g., the minimum to maximum distance, to deal with the presence of unavoidable outliers. At the same time the IQR normalization could deal with multi-modal signals that might alternate between two regime states, rarely taking in-between values. Here, working with the the standard deviation could lead to obscure results.  
The analysis was performed on a set of 1000 hours of operation, i.e., approximately 0.8% of the entire data set, that was randomly sampled from the complete time series of all wind turbines. For this method the time resolutions varied from 5 to 600 s, the latter being the typical timescale of SCADA data for commercial wind farms.