A survey on pre-processing techniques: Relevant issues in the context of environmental data mining <s> Dimensionality of outliers <s> This paper addresses a common methodological flaw in the comparison of variable selection methods. A practical approach to guide the search or the selection process is to compute cross-validation performance estimates of the different variable subsets. Used with computationally intensive search algorithms, these estimates may overfit and yield biased predictions. Therefore, they cannot be used reliably to compare two selection methods, as is shown by the empirical results of this paper. Instead, like in other instances of the model selection problem, independent test sets should be used for determining the final performance. The claims made in the literature about the superiority of more exhaustive search algorithms over simpler ones are also revisited, and some of them infirmed. <s> BIB001 </s> A survey on pre-processing techniques: Relevant issues in the context of environmental data mining <s> Dimensionality of outliers <s> This work presents advances in the design of a hybrid methodology that combines artificial intelligence and statistical tools to induce a model of explicit knowledge in relation to the dynamics of a wastewater treatment plant. The methodology contributes to problem solving under the paradigm of knowledge discovery from data in which the pre-process, the automatic interpretation of results and the explicit production of knowledge play a role as important as the analysis itself. The data mining step is performed using clustering based on rules by states, which integrates the knowledge discovered separately at each step of the process into a single model of global operation of the phenomenon. This provides a more accurate model for the dynamics of the system than one obtained by analyzing the whole dataset with all the steps taken together. <s> BIB002
Something often ignored is that outliers have a certain dimensionality. This means that sometimes an observation appears as an outlier regarding a certain single variable, but sometimes it appears as outlier in a subspace of higher dimensionality. To clarify, a twodimensional example is shown in Fig. 4 . The same situation can also appear in higher dimensional spaces, although we are not able to visualize in a clear way yet. In Fig. 4 , a scatter plot for COD (Chemical Oxygen Demand) and BOD (Biochemical Oxygen Demand) of a WWTP is shown. In one day, the WWTP inflow has a value of COD concentration equal to 1279 mg/l, and BOD concentration equals 198 mg/l. None of these values are particularly extreme (see the marginal histograms) BIB002 . This means that this day is not considered as an outlier, neither regarding the COD, nor the BOD by themselves. However, looking at the plot, things change. In the bivariate space formed by BOD and COD the point (1279, BIB001 becomes an outlier. The relationship between the two values is the rarity. Thus, individuals appear as outliers or not, according to the subset of variables considered together. Outliers become dangerous only when they perform as influential observations in the analysis. An influential observation is an observation that strongly determines the results of a certain analysis. In the top-right corner of Fig. 4 , a second outlier is much far from the general data cloud than the one marked by the circle. However, the former aligns better with the general association between BOD and COD. Thus the one with the circle in Fig. 4 is more influent. Influent observations are well-known in the context of regression analysis, but they can also distort other non-robust models. This means that they may spoil modeling. When the outlier is far from other observations, but follows the same model, it is not dangerous.