## 1 Introduction
### Gross Error Types and Examples  
Two types of GEs are taken into account:  
* Type 1 GEs, which are produced by sensor failures and lead to a stable measurement error. Preparation for the detection of type 1 GEs is made. In this case, some user interaction is needed--for example, identifying the number of continuous measurements in type 1 GE. It can succeed for variables such as analyzers, for which clogging may cause such a reaction. In the future, both the number of regular measurements and the dead band must be treated by an expert in this process.
* Type 2 GEs are caused, for example, by sensor faults, which can have brief, spike-like errors in the measurements as a result.  
Figure 2 presents the data from a measured temperature, which was contaminated with outliers, and it also presents vague limits min and max temperature, indicated by thick horizontal lines. Some outliers may be easily deleted by the limits. Figure 2 shows data from a temperature measurement contaminated with outliers and hypothetical limits of the min and max values (horizontal thick lines) of the temperature. The limits can be used to perform an easy check to remove some of the outliers. Obviously, not all of the outliers would be removed using the min/max GE check. Another possible easy check could be implemented with a rate limitation. However, this would involve detailed process knowledge of the time constants and noise levels of the measured variables. The GEDR is to remove and replace all of the shown spikes. After detection and replacement, the clean data is shown in the lower part of the figure using a median absolute deviation (MAD) based filter (see [5]), combined together with the wavelet based peak noise level estimation algorithm. Figure 3 shows a graphical representation of the MAD approach with the construction of the MAD flow pipe. This was determined to be the optimal setting, such that the flow pipe is wide enough to accommodate the presence of the noise. However, it is narrow enough to remove a high percentage of outliers. In addition, it can be  
seen that the \"radius\" of the flow pipe is invariant under outliers and noise, which, in turn, makes it very robust.  
It is clear that not every outlier will be deleted by means of the min/max GE check. There is another possibility of a check which could be applied with a rate limitation. In any case, it requires a deep knowledge on the time constants and noise levels of the variables to be measured. The work presented in [11] is extended by the authors in this contribution. Moreover, some experiments, applying the proposed fault detection method, are considered. This contribution is organized in the following way. Section 2 deals with the problem formulation. Section 4 is devoted to the wavelet based algorithm and some basic definitions of the outliers and evaluation tests. Section 5 considers structured validation evaluation tests of the algorithm. Here, the comparison of the obtained results with another existing algorithm is shown. Finally, industrial application is presented. Finally, conclusions are presented.  
Figure 3: Application of the wavelet based algorithm applied to the mining data.  
Figure 2: Example of contaminated temperature measurements. With outliers (top) and minimum, maximum limits. The clean temperature data is obtained with help of GEDR.