![]() Placing outliers such that they are hidden is even more unclear. To our knowledge, the general question of how to place outliers in datasets has not been studied explicitly so far. Here, our concern is the effective and comprehensive placement of subspace outliers. ![]() The design and assessment of such evaluation schemes is one of our long-term research efforts and is beyond the scope of this current article. An evaluation scheme based on our approach would also allow differentiating between different kinds of hidden outliers and might be more systematic than one depending on the outliers which have been found so far. Placed hidden outliers are known to be outliers in certain subspaces, and one can quantify how well they are found. However, these outliers are not necessarily subspace outliers, in contrast to hidden outliers generated with our approach. Current schemes for evaluation often either use an already existing minority class as outlying or downsample the data to one rare class. 1.1.3 Evaluation of subspace outlier detectionīeing able to hide outliers is expected to help evaluate subspace outlier detection methods. The attacker then identifies regions of the data space where outliers are hidden from the detection method and designs fraudulent transactions whose representation falls into these regions. More specifically, the bank uses a subspace-search method that is confined to subspaces consisting of few attributes only. he knows that the bank checks for fraud by means of an outlier-detection method on a high-dimensional representation of the credit-card transactions. Think of a criminal intending to commit credit-card fraud. The example also shows that hidden outliers pose a risk, and it is worthwhile to quantify this risk. While Example 1 is extreme for the sake of illustration, it is our running example due to its intuitiveness. studying the adverse behaviour in order to shield against it. Example 1 shows that the situation here is analogous, including the motivation, i.e. Research on classifier evasion studies the behaviour of an adversary attempting to ‘vanquish’ a learner. If hidden outliers were detected, a domain expert could inspect them and assess how detrimental they are. Hidden outliers represent combinations of values that remain undetected with existing models. However, data objects representing these states usually do not exist or are extremely rare. Since faults of infrastructures that are critical may be catastrophic, any action preventing such faults pays off. Outliers are unusual system states which may represent any kind of fault or a state preceding a fault. Think of data objects each representing a state of a critical infrastructure. 1.1.1 Reliability of critical infrastructures We now elaborate on these points one by one. We see three reasons why studying the issue is necessary, namely (1) increasing the reliability of critical infrastructures, (2) coping with attacks, and (3) systematic evaluation of outlier detection algorithms. In this article, we examine how to place hidden outliers in high-dimensional data spaces, and we quantify the risk of the data owner that such outliers can be placed in the data. It can only be detected when looking at the two-dimensional subspace. The outlier in the figure is hidden when looking at each one-dimensional subspace in isolation. Figure 1 displays a low-dimensional illustrative example. Hence, the characteristic whether an outlier is hidden or not depends on the subspaces where one is looking for outliers. A hidden outlier exhibits its outlier behaviour only in subspaces where no outlier detection takes place. Depending on the subspaces inspected, the outlier detection method used and the distribution of the data, so-called hidden outliers may occur. Thus, most approaches only inspect a subset of the set of all subspaces. In high-dimensional spaces, it is not feasible to inspect all subspaces for outliers, since their number grows exponentially with the dimensionality. ![]() Such outliers are referred to as subspace outliers. Due to the curse of dimensionality, outliers often occur in attribute subspaces. fraud detection, depend on the effective and efficient identification of outliers. Many applications in different domains, e.g.
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