## 2 Materials and Methods
### Data Preprocessing Methods
#### 2.2.3 intervals_errors Algorithm  
The intervals_errors algorithm (Algorithm 4) is designed to identify intervals where errors or corruptions are suspected to be in the signal. It uses wavelet decomposition, peak detection, and pair generation to determine these intervals.  
```
1:Input:signal--one-dimensional array
2:Input:wavelet--wavelet type (e.g., 'db20')
3:Input:\(level=3\)--level of decomposition
4:Input:\(epsilon=2\)
5:Input:\(min\_threshold=0.1,\ max\_threshold=25\)
6:Input:\(h=1e{-6}\)
7:Output: List of intervals (pairs) where errors are suspected
8:\(coeffs\leftarrow\) WaveletDecomposition(\(signal\), \(wavelet\), \(level\))\(\triangleright\) See Algorithm 1
9:\(cf\_lvl\gets coeffs[level]\)\(\triangleright\) Coefficients at a given level (detailing or approximating - depends on the library)
10:\(peaks\leftarrow\) detect_faults(\(cf\_lvl\), \(min\_threshold\), \(max\_threshold\))
11:Build a list of \(eps\_peaks\) by \(peak-epsilon\) for each \(peak\in peaks[0]\)
12:\(intervals\leftarrow\) combinations(\(eps\_peaks\), \(2\))
13:return\(intervals\)
```  
**Algorithm 4** intervals_errors  
Main Steps of the Algorithm  
Input data: The algorithm takes as input a one-dimensional array signal, wavelet type wavelet, decomposition level level (default 3), parameter \(\epsilon\) (epsilon, default 2), prominence threshold parameters min_threshold and max_threshold, and filter threshold h (default \(1\times 10\)\({}^{-6}\)).  
Wavelet decomposition: In the first step, the algorithm applies wavelet decomposition to the input signal using the WaveletDecomposition algorithm (Algorithm 1), the selected wavelet wavelet, and the decomposition level level. The result is a list of coefficients coeffs.  
Selection of level coefficients level: The algorithm selects coefficients at a given decomposition level from a list of coeffs. In pseudocode, this is the string cf_lvl \(\leftarrow\) coeffs[level].  
Detect peaks in coefficients: The algorithm applies the detect_faults function (Algorithm 3) to the selected wavelet coefficients cf_lvl with the given prominence thresholds min_threshold and max_threshold. The result is a list of peak indices of peaks in coefficients.  
Building \"extended\" peaks: For each peak index peak of peaks[0], an \"extended peak\" is created from peaks[0] eps_peaks by subtracting the value \(\epsilon\) from the peak index (peak-epsilon). The purpose of this step is to possibly expand the area around the peak to determine the error interval.  
Generation of intervals (pairs): The algorithm uses the combinations function to generate all pairs of combinations from a list of \"extended\" peaks eps_peaks of length 2. Each pair represents an interval potentially containing an error.  
Output: The algorithm returns a list of intervals containing index pairs representing the intervals where the errors in the original signal are suspected to be located.  
Remarks  
The intervals_errors algorithm combines wavelet analysis with peak detection and combinatorics to determine error intervals. Using wavelet decomposition allows one to analyze the signal at different frequency levels and possibly better isolate features associated with faults. The \(\epsilon\) parameter and pair generation can be associated with trying to identify intervals around peaks where errors are most likely to occur. Adjusting the wavelet parameters, decomposition level, prominence and filter thresholds, and the \(\epsilon\) value is critical to the effectiveness of the algorithm in specific applications.