A decomposition of the outlier detection problem into a set of supervised learning problems
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Abstract
Outlier detection methods automatically identify instances that deviate from the majority of the data. In this paper, we propose a novel approach for unsupervised outlier detection, which re-formulates the outlier detection problem in numerical data as a set of supervised regression learning problems. For each attribute, we learn a predictive model which predicts the values of that attribute from the values of all other attributes, and compute the deviations between the predictions and the actual values. From those deviations, we derive both a weight for each attribute, and a final outlier score using those weights. The weights help separating the relevant attributes from the irrelevant ones, and thus make the approach well suitable for discovering outliers otherwise masked in high-dimensional data. An empirical evaluation shows that our approach outperforms existing algorithms, and is particularly robust in datasets with many irrelevant attributes. Furthermore, we show that if a symbolic machine learning method is used to solve the individual learning problems, the approach is also capable of generating concise explanations for the detected outliers.
Keywords
Outlier detection Machine learning Outlier explanations1 Introduction
Outlier or anomaly detection methods are used to identify observations that “appear to deviate markedly from other members of the same sample”, i.e., that “appear to be inconsistent with the remainder of the data” (Barnett and Lewis 1994). In other words, the majority of the data is supposed to follow certain patterns, and outliers do not adhere to those patterns. Typical applications of outlier detection include fraud and network-intrusion detection, or error detection in data (Hodge and Austin 2004).
Many classic outlier detection methods use the notions of density and proximity, i.e., they mainly identify outliers as data points that occur in sparsely populated areas of the dataset, and far away from neighboring points. To do so, they rely on distance measures. Therefore, they struggle from the curse of dimensionality, which render many classic distance measures useless once the dataset is of higher dimensionality (Aggarwal et al. 2001). The underlying problem is that at high dimensionality, most distance measures collapse in a way that all pairs of instances have a similar distance, which makes distance-based data mining approaches fail at such datasets.
More concisely, such algorithms struggle with datasets containing a larger number of attributes that are irrelevant for the outlier detection, since they expose no (or only very weak) meaningful patterns. For example, in a dataset which contains the height and age of children, outliers would be, e.g., children that are unusually tall or short for their age. If the dataset contains a large number of other attributes, such as database identifiers, social security numbers of both the child and its parents, ZIP codes, phone numbers, etc., many distance-based algorithms may yield suboptimal results, given that those attributes are not identified and removed upfront.
In this paper, we propose the attribute-wise learning for scoring outliers (ALSO) approach, which, instead of exploiting density, directly searches for patterns in the data. Such patterns are expected to present themselves as dependencies between the different attributes. To detect those patterns, we decompose the outlier detection problem into a set of supervised learning problems. Learning algorithms solving those problems return both the patterns underlying the data, as well as estimators for the strength of those patterns in each attribute. These strengths can be turned into attribute weights, assigning low weights to attributes exposing no or only very weak patterns. Outliers are then identified as data points which deviate from the patterns found, taking the weights into account when quantifying the deviation.
We show that for numerical datasets, the approach can be used in conjunction with arbitrary regression learning algorithms, that it reliably yields good results using M5’ (regression trees) or isotonic regression as base learners, and that its results are invariant to the adding of irrelevant noise attributes. Furthermore, we demonstrate that when using a symbolic learning algorithm, concise explanations for the outliers found can be generated.
The rest of this paper is structured as follows. Section 2 discusses related work, and shows how ALSO is novel with respect to existing approaches. We introduce the ALSO approach in Sect. 3, and present its evaluation in Sect. 4. Section 5 discusses the generation of explanations for outliers. We conclude with a summary and an outlook on future work.
2 Related work
In the past decades, an abundance of methods have been proposed for outlier detection (Aggarwal 2013; Chandola et al. 2009; Hodge and Austin 2004). Chandola et al. (2009) distinguish three types of approaches: supervised approaches are trained based on labeled examples for both outliers and normal examples, semi-supervised approaches are trained using labeled examples only for normal observations, and unsupervised approaches that are built using no labeled data at all.
In that classification, the ALSO approach discussed in this paper is an unsupervised one—while the outlier detection problem is decomposed into a set of supervised learning problems, the overall approach is still unsupervised.
Classic unsupervised approaches identify outliers based on their distance to the nearest neighbors and/or on the local density around instances. Outlier detection approaches using machine learning have already been proposed, but many of them rely on labeled examples, i.e., they are supervised or semi-supervised. In the following, we provide an overview of unsupervised, machine-learning based approaches.
Clustering-based approaches like CBLOF and LDCOF reformulate the problem of outlier detection as a clustering problem, to be solved with any clustering algorithm. They first identify clusters in the data. Those clusters are considered the model which underlies the data; consequently, data points that are not contained in any cluster (or only in a very small cluster) are considered as outliers. Measures used as outlier scores are, e.g., the distance of a data point to the next cluster centroid (Amer and Goldstein 2012; He et al. 2003).
One-class support vector machines try learn a boundary around a set of training examples, i.e., they can be applied to learn a model in a semi-supervised setting. Robust one-class support vector machines are capable of dealing with datasets that contain outliers, i.e., they learn the boundary of the region where most of the examples are located in, and mark the examples outside that area as outliers (Amer et al. 2013; Xu et al. 2006). Thus, they can be used for unsupervised outlier detection as well.
Abe et al. (2006) propose a method that solves the outlier detection problem by generating artificial instances as outliers, and thus turning the outlier detection problem into a problem of supervised classification, using a sample of the given instances as normal points. The authors propose the use of active learning to optimize the sampling of normal points. Similarly, the work described in Yamanishi and Takeuchi (2001) follows a two-step approach: it first aims at fitting a statistical distribution to the data in order to assign outlier scores. Then, those outlier scores are used to train a rule learning classifier telling outliers from non-outliers, using the data points with the highest outlier scores as positive examples.
Frequent itemset mining finds typical patterns that occur in (usually categorical) data. He et al. (2005) propose an approach that first mine frequent patterns, and then mark those instances as outliers that do not match any frequent pattern, or only very rare patterns. In Padmanabhan and Tuzhilin (2000), a direct approach for mining rare patterns is introduced, which are assumed to describe outliers.
Replicator Neural Networks (RNNs) (Hawkins et al. 2002) are an approach which is close to the one introduced in this paper. The authors propose training a neural network where the training vectors for input and output are identical (i.e., the neural network tries to replicate the training instances as good as possible), and use the prediction error of the neural network as an outlier score for each instance. The ALSO approach described in this paper can be seen as a generalization of that approach, which is capable of using any learning strategy (i.e., it is not limited to neural networks), and it inherently learn weights for each attribute, so that the computation of the prediction error focuses on the more relevant attributes, which increases the robustness of the approach, in particular in higher dimensional cases.
The concept of trying to predict an attribute from the other attributes is not new. In Teng (1999), a noise removal method based on such predictions has been proposed. While the authors’ focus is on noise removal on attribute level, i.e., replacing single attribute values which are likely to be wrong, our focus is on outlier detection, i.e., identifying entire instances which deviate from the majority of the data.
The idea of identifying outliers by comparing actual and expected attribute values is also used in the Correlation Outlier Probabilities (COP) approach (Kriegel et al. 2012), which tries to identify local correlations between attributes. Based on those correlations, deviations between the expected and the actual attribute values can be computed. Similarly, DEMUD computes a singular value decomposition of the data, and computes an outlier score from the per-attribute deviations from the actual data points and the values created from their SVD-based reconstruction (Wagstaff et al. 2013).
Isolation forests rely on a particular kind of decision trees, i.e., isolation trees, to directly learn a model for outliers (in contrast to most of the other approaches discussed above, which try to learn a model for the normal data points). Isolation trees create their splits in a way that each leaf node contains only one instance (or a set of instances of the exact same value). If the trees are cut at a certain height, all instances ending up in leaf nodes can be considered outliers. In Liu et al. (2012), the idea is combined with random forests, training a set of isolation trees on different attribute subsets.
For many outlier detection algorithms, especially distance and density based ones, a high dimensionality of the data can be a problem (Aggarwal and Yu 2001), since outliers in lower dimensional subspaces are likely to be obscured by accidental similarities in other attributes. One recently proposed approach to cope with that problem is to use ensemble methods, which are also popular in machine learning (Zimek et al. 2013). In that approach called Ensemble Subsampling, an ensemble of outlier detection methods is created, where the members of the ensemble are outlier detection methods trained on different subspaces of the original vector space.
In contrast to the approaches above, which usually adopt one machine learning method so that it can be used for outlier detection, the ALSO approach introduced in this paper can use any regression learning algorithm (linear regression, regression/model trees, neural networks, etc.) and exploit it for outlier detection. Furthermore, we do not assume any particular distribution of the data.
Another feature of ALSO is that the approach does not only identify outliers with an outlier score, but is also capable of delivering explanations, a task which is addressed by only a few outlier detection methods so far. One exception is the aforementioned COP, which measures deviations between actual and expected attribute values. Thus, those deviations may also be used for generating explanations for outliers.
ART-E is a clustering-based approach, which uses the distances of outlier points to clusters per attribute to identify those attribute values which make a data point an outlier (Mejía-Lavalle and Sánchez Vivar 2009), i.e., which attribute value contributes most to the distance measure, and provide an explanation based on those attributes. Knorr and Ng (1999) introduce the notion of non-trivial outliers being outliers in a feature space which are not identified as outliers in any subspace of that feature space. This notion helps isolating those attributes which actually contribute to identifying outliers. Similar to that idea, the SOREX tool looks for attributes in subspaces that do well identify outliers (Müller et al. 2010).
While ALSO also exploits deviations on single attributes for creating explanations, the approach is also capable of creating more concise explanations when combined with a symbolic learning algorithm, as we show in Sect. 5.
3 Approach
The core idea of the ALSO approach is that patterns within the underlying data and outliers are two sides of the same coin. More precisely, as outliers are deviations from patterns, the ALSO approach works in two steps: it first learns the patterns for each attribute, and then measures the deviation of the actual data from the patterns found. For computing those deviations, a weighted distance metric is used, with the weights being learned together with the patterns.
3.1 Overall approach
3.2 Assigning weights to dimensions
Further, we want to emphasize deviations on attributes which expose very strong patterns, i.e., that are very well predictable from the others.
If we can find a pattern in an attribute, it means that we can predict the value of that attribute better than by mere guessing. For example, for database IDs, it is unlikely that we find a predictive model that works better than guessing,^{2} while dependencies between attributes (e.g., age and height of children) lead to predictive models that are better than guessing.
Since a RRSE larger than 1 means that the predictive model is worse than guessing, we limit the values at 1 because it is irrelevant how much worse than guessing the learner is—in any case, the learner did not find any meaningful pattern for the attribute.
Given that a noise-tolerant learning algorithm is used for learning the individual models, meaningless attributes have no influence on the final outlier score. For a meaningless attribute, such as an ID or a ZIP code, a reasonable learning algorithm should not learn a model that is different from a default prediction (i.e., always predicting the mean value). Thus, in Eq. 5, all \((i^\prime _j)_k\) would be equal to \(\overline{k}\), thus, the RRSE of such a learner would be 1, which leads to a weight of zero. Since attributes with a weight of zero have no influence on the score defined in (3), adding meaningless attributes to the dataset does not change the outlier score. In turn, this means that only attributes which expose meaningful patterns will be considered for computing the final computing outlier scores. That property of ALSO helps identifying outliers in high-dimensional datasets, which are otherwise obscured.
The use of such weights defines the a central property of the ALSO approach: if an attribute value deviates from its prediction given the learned model, it has a major impact on the final outlier score iff both the deviation as well as the attribute weight are high. This also gives way to a fundamental limitation of our approach: in datasets containing total redundancies, ALSO may fail to reliably identify outliers. As an extreme case, consider a dataset which contains temperature measurements made by a sensor, which are picked from the sensor and stored both in Celsius and Fahrenheit as two attributes. An extreme measurement made by the sensor is likely not to be revealed by ALSO since the two attributes are perfectly predictable from each other. Hence, no matter how far away from the majority of the values the extreme measurement is, it will not be identified as an outlier. While such constellations may happen in theory, we expect them to be rather rare in practical cases.
3.3 Nature of outliers found by ALSO
In our approach, we follow a different notion of outliers than, e.g., traditional distance or density based approaches. While in the latter, outliers are identified based on whether there are other data points close to them, we look at the underlying model instead, and identify outliers based on those models. This can lead to data points in sparse areas not identified as outliers, and vice versa. For example, in Fig. 1, a data point which is close to the regression line, but far away from the rest of the data (e.g., \(\langle 100,100\rangle \)) would not be recognized as an outlier.
Figure 2 illustrates those different notions by comparing ALSO (with isotonic regression as a base learner) and a typical density-based approach, i.e., LoOP (Kriegel et al. 2009). The plots show a two-attribute version of the Auto MPG dataset,^{3} which describes cars and their fuel consumption, reduced to the two attributes weight (x-axis) and MPG (y-axis).
3.4 Algorithm
In order to avoid overfitting to outliers, we train the individual predictive models in different folds, using a cross-validation like scenario. The whole algorithm is shown in Algorithm 1.
The function weight(error) computes the weights from the absolute squared errors by first normalizing them to a RRSE, and then applying Eq. 4. The weights for each attribute are learned on the fly while the algorithm proceeds, allowing it to adapt to the dataset at hand.^{5}
Given that the algorithm is run on a dataset with n attributes and m instances, using f folds, \(f \cdot n\) predictive models have to be trained and applied. With a learner that has a complexity \(C_T(m,n)\) for training the model and \(C_A(m,n)\) for applying the model on one instance, the overall complexity of ALSO is \(O(n \cdot C_T(m,n) + n \cdot m \cdot C_A(m,n))\), since f is a constant factor.
3.5 Interpretation of outlier scores
3.6 Example
- 1.
It produces sensible results also in cases where there is no simple functional dependency between the attributes.
- 2.
Even if the underlying regression models learned are far from perfect (note that the regression trees predict 0 instead of a correct 1 or \(-1\) value in the majority of cases), they are good enough for our purpose of computing outlier scores.
4 Empirical evaluation
To evaluate our approach, we have created a set of datasets, which are derived from real world datasets for classification and regression. On those datasets, we compare ALSO with different base learner to a number of both well-established as well as recent outlier detection methods.
4.1 Datasets
Canonical datasets for evaluating outlier detection are rare. Hence, we follow an approximation as suggested in Emmott et al. (2013): starting from standard classification and regression datasets (which can be found in dataset collections quite frequently), we define a subset of the instances, typically one of the classes or unusually high and low regression target values, as outliers, and sample it to a smaller share.^{6}
We selected twelve real-world datasets from UCI, four of which were also used by Amer et al. (2013). In particular, we use six classification dataests, i.e., Shuttle, Satellite, Wisconsin Breast Cancer, Ionosphere, Glass, and Seismic Bumps, as well as six regression datasets, i.e., Concrete, Parkinsons Telemonitoring, White Wine Quality, Housing, CCPP, and Energy Efficiency. The datasets were selected by looking for classification and regression datasets with only numerical attributes. Furthermore, we aimed at a selection of both smaller and larger datasets, with different numbers of attributes.
As proposed in Emmott et al. (2013), we use the class attribute of the classification datasets to divide the datasets into normal and outlier points, defining one class (usually the smallest) as outliers, and sample the outlier class to a smaller size. The original class attribute is removed and retained for evaluation. For the first three datasets, we used the already preprocessed data also used in Amer et al. (2013).^{7}
For the regression datasets, Emmott et al. (2013) propose to split the dataset into two classes at the mean of the regression target, and treat the datasets like classification datasets. Here, we decided to use extreme values of the regression target as outliers instead, which comes closer to the intended semantics of an outlier. Hence, we use all data points with a regression target within one standard deviation from the mean as normal points, and all data points further than two standard deviations away as outliers. Again, the original regression attribute is removed and retained for evaluation.^{8}
Modified datasets used for evaluation, including final dataset size (in number of instances), percentage of sampled outliers, and the mean \(\mu \) and standard deviation \(\sigma \) of each dataset
Dataset (DS) | Original # inst. | # Att. | Outlier class(es) | Resulting # inst. | Final sampl. outlier pct. (%) | \(\mu \) of att. values | \(\sigma \) of att. values |
---|---|---|---|---|---|---|---|
Satellite | 6435 | 36 | 2,4,5 | 5100 | 1.49 | 86.000 | 18.000 |
Shuttle | 50, 000 | 9 | 2,3,5,6 | 46, 464 | 1.89 | 29.185 | 68.892 |
Breast cancer | 469 | 30 | M | 367 | 2.72 | 870, 423 | \(2.1E+07\) |
Ionosphere | 351 | 32 | b | 233 | 3.40 | 0.292 | 0.520 |
Glass | 214 | 9 | 5,6,7 | 170 | 4.11 | 11.265 | 22.124 |
Seismic bumps | 2584 | 19 | 1 | 2584 | 6.57 | 5885.75 | \(3.5E+09\) |
Concrete | 1030 | 9 | CCS | 711 | 5.63 | 298.6 | \(1.2E+05\) |
Parkinsons tele. | 5875 | 26 | Total_UPDRS | 4170 | 7.53 | 9.55 | 763.75 |
Wine quality | 4898 | 12 | Quality | 3847 | 4.99 | 18.43 | 1726.01 |
Housing | 506 | 14 | MEDV | 334 | 5.09 | 75.82 | 22, 396.09 |
CCPP | 9569 | 4 | PE | 5974 | 2.54 | 290.13 | \(1.7E+05\) |
Energy efficiency | 768 | 8 | Y1 | 492 | 1.44 | 167.98 | 56, 871.52 |
Since we want to investigate the influence of irrelevant attributes on the outlier detection performance, we furthermore created three additional versions of each dataset by adding a certain number (10, 50, and \(100\,\%\) of the original number of attributes) of random noise attributes. The attribute values of each random noise attribute are drawn from a normal distribution, with its mean \(\mu \) and standard deviation \(\sigma \) equal to the mean and standard deviation of the original attribute values of the corresponding dataset. Thus, our evaluation was eventually carried out on 48 datasets.
4.2 Setup
For evaluating the ALSO approach, we use three different base learners, i.e., Linear Regression, Isotonic Regression (Barlow et al. 1972), and M5’ (Quinlan et al. 1992) as base learners.^{10} For all learners, we use the implementation in Weka.^{11} For M5’, we learn pruned regression trees with a minimum leaf size of 4; linear and isotonic regression were run with their respective standard settings.
The choice for those algorithms is to have a larger variety in the types of models that can be learned. Linear regression learns only linear functions, isotonic regression learns an arbitrary monotonically increasing or decreasing function, and M5’ can also cover more complex functions.
The number of folds for learning and evaluating the models for each attribute was fixed to 10.
- 1.
The k-NN global anomaly score (GAS) is the average distance to the k nearest neighbors (Angiulli and Pizzuti 2002), following the intuition that outliers are located in rather sparsely populated areas of the vector space. We use GAS with \(k=10\), \(k=25\), and \(k=50\).
- 2.
The Local Outlier Factor (LOF) is computed from the density of data points around the point under inspection, which in turn is computed from the distances to the k nearest neighbors (Breunig et al. 2000). Similar to our setup for GAS, we use \(k_{min}=10\) and \(k_{max}=50\).
- 3.
The Local Outlier Probability (LoOP) follows a similar idea as LOF, but maps the outlier scores to probabilities in a [0; 1] interval (the scores assigned by other methods are usually unbound) (Kriegel et al. 2009). Like for GAS, we compute LoOP with \(k=10\), \(k=25\), and \(k=50\).
- 4.
The Cluster-based Local Outlier Factor (CBLOF) was used in conjunction with use the X-means algorithm, which restarts k-means with different values for k, in order to find an optimal one (Pelleg et al. 2000), using \(k_{min}=2\) and \(k_{max}=60\).
- 5.
The Local Density Cluster-based Outlier Factor (LDCOF) was used with X-means in the same configuration as above.
- 6.
We use one-Class Support Vector Machines (see Sect. 2 with three different kernels: the standard kernel (1-classSVM\(_1\)) as well as the robust kernel (1-classSVM\(_r\)) and eta kernel (1-classSVM\(_e\)) defined particularly for outlier detection (Amer et al. 2013).
- 7.
For Replicator Neural Networks (RNN), we follow the setup in Hawkins et al. (2002), using three hidden layers (size 35, 3, and 35), and 1000 iterations.
- 8.
For COP, which requires the identification of nearest neighbors, we use \(k=3*d\), where d is the number of dimensions.
- 9.
For ensemble subsampling, we follow the setup in Zimek et al. (2013), using LOF as a base outlier method, with 20 ensemble members, and averaging the outlier scores.
- 10.
As reported in Liu et al. (2012), isolation forests provide stable results if at least 30 trees are learned, and the best results are achieved with a height limit of 1, so we use those values.
For all approaches, we compare the area under the ROC curve (AUC). To that end, outlier scores are computed for each instance, and the instances are ordered by those scores. The ROC curve is then drawn using the outliers as the positive class, plotting the true positive rate (y-axis) against the false positive rate (x-axis).
Since different tools were used for conducting the experiments (as there is no single tool implementing all the approaches at hand), we omit a comparison of runtimes, since such a comparison would be skewed, e.g., by the different programming languages and data storage strategies of the individual tools. However, if any of the approaches was not able to process a dataset within 12 h, we canceled the run. For being fair on this policy, we ran ALSO in single thread mode. All processes and datasets used for the evaluation, as well as the full set of individual experimental results, can be found online.^{17}
4.3 Results
ALSO yields the best results when using M5’ as a base learner, significantly outperforming all of the compared approaches. The results using Isotonic Regression are slightly worse, while using Linear Regression as a base learner is clearly inferior.
Most of the compared approaches yield significantly worse results when adding different amounts of irrelevant attributes. 1-class SVM with a robust kernel, COP, and Ensemble Subsampling are the only approaches that do not show a significant decrease for any of the modified datasets (the increase for COP is not statistically significant). For ALSO, when inspecting the weights learned for random noise attributes, those are always close to 0.
Results reporting the average AUC across all twelve datasets. 0.0 denotes the results on the unmodified datasets; 0.1, 0.5, and 1.0 are the results on the datasets with different amounts of added noise attributes
Algorithm | 0.0 | 0.1 | 0.5 | 1.0 | |||
---|---|---|---|---|---|---|---|
GAS (\(\mathrm{k}=10\)) | .789* | .669** | (\(-\)15.21 %*) | .613** | (\(-\)22.31 %**) | .611** | (\(-\)22.56 %**) |
GAS (\(\mathrm{k}=25\)) | .792* | .651** | (\(-\)17.80 %**) | .617** | (\(-\)22.10 %*) | .652** | (\(-\)17.68 %**) |
GAS (\(\mathrm{k}=50\)) | .777* | .638** | (\(-\)17.89 %*) | .642** | (\(-\)17.37 %*) | .613** | (\(-\)21.11 %*) |
LOF | .777* | .648** | (\(-\)16.60 %*) | .627** | (\(-\)19.31 %*) | .629** | (\(-\)19.05 %*) |
LoOP (\(\mathrm{k}=10\)) | .678** | .580** | (\(-\)14.45 %) | .597** | (\(-\)11.95 %*) | .574** | (\(-\)15.34 %) |
LoOP (\(\mathrm{k}=25\)) | .730** | .586** | (\(-\)19.73 %*) | .586** | (\(-\)19.73 %*) | .597** | (\(-\)18.22 %) |
LoOP (\(\mathrm{k}=50\)) | .727** | .608** | (\(-\)16.37 %**) | .591** | (\(-\)18.71 %*) | .585** | (\(-\)19.53 %) |
CBLOF | .654** | .567** | (\(-\)13.33 %) | .548** | (\(-\)16.21 %**) | .517** | (\(-\)20.9 %*) |
LDCOF | .749** | .642** | (\(-\)14.29 %*) | .579** | (\(-\)22.70 %*) | .605** | (\(-\)19.23 %**) |
1-ClassSVM\(_1\) | .740* | .642** | (\(-\)13.24 %) | .548** | (\(-\)25.95 %*) | .588** | (\(-\)20.54 %*) |
1-ClassSVM\(_r\) | .636** | .579** | (\(-\)8.96 %) | .563** | (\(-\)11.48 %) | .603** | (\(-\)5.19 %) |
1-ClassSVM\(_e\) | .743* | .630** | (\(-\)15.21 %) | .560** | (\(-\)24.63 %**) | .604** | (\(-\)18.71 %*) |
RNN | .731* | .669** | (\(-\)8.48 %*) | .651** | (\(-\)10.94 %) | .652** | (\(-\)10.81 %) |
COP | .704** | .746** | (+5.97 %) | .754** | (+7.10 %) | .728** | (+3.41 %) |
Ensemble | .731** | .643** | (\(-\)12.04 %) | .626** | (\(-\)14.36 %) | .629** | (\(-\)13.95 %) |
iForest | .781* | .765* | (\(-\)2.05 %) | .749** | (\(-\)4.10 %*) | .732** | (\(-\)6.27 %*) |
ALSO (M5’) | .854 | .854 | (\(\pm \)0.00 %) | .853 | (\(-\)0.12 %) | .852 | (\(-\)0.23 %) |
ALSO (Iso) | .848 | .848 | (\(\pm \)0.00 %) | .849 | (+0.12 %) | .836 | (\(-\)1.42 %) |
ALSO (LR) | .749 | .744 | (\(-\)0.67 %) | .745 | (\(-\)0.53 %) | .747 | (\(-\)0.27 %) |
In our experiments, the Shuttle dataset poses scalability problems to many approaches, with COP as well as all three 1-class SVM variants not being able to compute them within 12 h.
Moreover, our experiments have shown that base learners capable of learning non-linear regression functions outperform linear regression, and that M5’ slightly outperforms Isotonic Regression (i.e., an approach restricted to learning monotonic functions). This indicates that learning algorithms which can learn more complex models are usually better suited for ALSO. In practice, a learning algorithm for ALSO should be chosen that (a) is capable of handling irrelevant features, (b) is capable of learning non-linear functions, (c) does not require extensive parameter tuning to work well for a given problem, and (d) is reasonably performant for the amount of models to be learned.
In addition to comparing the AUC values achieved by the different approaches, we also compared the ranks of the approaches across the different datasets. Following Demšar (2006), we first use a Skillings–Mack test (Skillings and Mack 1981), a variant of the Friedman test which can also be applied to data with missing observations^{18}, to confirm that there are significant differences in the approaches’ ranks. The significance of the individual differences has then been determined using a Nemenyi post-hoc test (Nemenyi 1962).
4.4 Experiments on higher dimensional datasets
Modified high dimensional datasets used for evaluation, including final dataset size (in number of instances), percentage of sampled outliers, and the mean \(\mu \) and standard deviation \(\sigma \) of each dataset
Dataset (DS) | Original # inst. | # Att. | Outlier class(es) | Resulting # inst. | Final sampl. outlier pct. (%) | \(\mu \) of att. values | \(\sigma \) of att. values |
---|---|---|---|---|---|---|---|
Comm. and crime | 1994 | 128 | \(>\)2000 | 297 | 6.06 | \(1.0E+04\) | \(6.7E+09\) |
Internet adv. | 3279 | 1558 | Ad | 2107 | 6.12 | 0.138 | 24.269 |
Multiple features | 2000 | 649 | 1–9 | 218 | 8.25 | 114.257 | \(7.5E+04\) |
Urban land cover | 508 | 148 | Shadow | 508 | 5.90 | 346.920 | \(2.2E+06\) |
Results on the four high dimensional datasets: communities and crime (CC), Internet Advertisements (IA), multiple features (MF), and urban land cover (ULC). The approach marked with “–” means did not finish on the respective dataset within 24 h. For the Internet Advertisements dataset, no valid configuration of COP was possible (marked “X”): values for k smaller than d lead to an error, while smaller ones were not accepted as valid values
Approach | CC | IA | MF | ULC | Avg. |
---|---|---|---|---|---|
GAS (\(\mathrm{k}=50\)) | 0.569 | 0.528 | 0.981 | 0.392 | 0.618 |
LOF | 0.568 | 0.466 | 0.980 | 0.377 | 0.598 |
LoOP (\(\mathrm{k}=10\)) | 0.611 | 0.566 | 0.981 | 0.633 | 0.698 |
LoOP (\(\mathrm{k}=25\)) | 0.649 | 0.690 | 0.681 | 0.735 | 0.689 |
LoOP (\(\mathrm{k}=50\)) | 0.596 | 0.629 | 0.974 | 0.716 | 0.729 |
CBLOF | 0.621 | 0.566 | 0.983 | 0.668 | 0.710 |
LDCOF | 0.429 | 0.355 | 0.166 | 0.694 | 0.411 |
1-Class SVM\(_1\) | 0.579 | 0.374 | 0.558 | 0.653 | 0.541 |
1-Class SVM\(_r\) | 0.594 | 0.139 | 0.980 | 0.508 | 0.555 |
1-Class SVM\(_e\) | 0.545 | 0.781 | 0.977 | 0.503 | 0.702 |
RNN | 0.489 | 0.856 | 0.958 | 0.236 | 0.635 |
COP | 0.715 | X | – | 0.641 | 0.678 |
Ensemble | 0.614 | 0.591 | 0.986 | 0.656 | 0.712 |
iForest | 0.742 | 0.649 | 0.911 | 0.774 | 0.769 |
ALSO (M5’) | 0.761 | 0.707 | 0.995 | 0.856 | 0.830 |
The problem with the latter dataset is that it is mostly a sparse dataset. The majority of the dataset are binary variables, most of which are very sparse, i.e., they contain mostly 0s. That, in turn, means that it is likely that a model for one of those attribute is trained on a training set with mostly 0s as labels, and, hence, only a default model is learned. That, in turn, assigns the weight of 0 to most of the attributes. This means that the attribute set is implicitly reduced to the non-sparse attributes, i.e., outliers are only identified based on anomalies of the non-sparse attributes.^{19}
In summary, however, ALSO performs well also on datasets with several hundred attributes, and it is again noteworthy that the result quality is rather stable compared to other approaches. In contrast, RNN and 1-class SVM\(_e\), which outperform ALSO on the Internet Advertisements dataset, achieve a performance of an AUC of only slightly above 0.5, or even below, on others.
5 Generating explanations of outliers
For computing the overall outlier score, ALSO first determines the weighted deviation from the expected value on each attribute. These deviations indicate which attributes contribute most to an instance’s outlier score. If the dataset has been normalized using a standardization, an addend value larger than n means that the actual attribute value is more than \(\sqrt{n}\) standard deviations away from the value that was predicted by the underlying model. Comparing the predicted to the actual values for those attributes, a first understanding why an instance has been marked as an outlier can be gained. Identifying the highest-scoring attributes for an outlier instance is a first step towards explaining outliers, like it is also done, e.g., by COP (Kriegel et al. 2012).
If a symbolic learning approach is used as a base learner, i.e., a learner creating human-interpretable models, the learned model for the respective attribute can be used to create an even more concise explanation why a certain instance has been marked as an outlier.
Outlier scores on the zoo dataset, using ALSO with M5’ as a base learner, and largest addend according to Eq. (3)
Instance | Eggs | Backbone | Milk | Outlier score | Largest summands |
---|---|---|---|---|---|
Scorpion | 0 | 0 | 0 | 1.498 | Eggs (2.033), |
Backbone (2.346) | |||||
Platypus | 1 | 1 | 1 | 1.377 | Milk (2.114) |
Seasnake | 0 | 1 | 0 | 1.135 | Eggs (2.033) |
The scorpion is an outlier because animals not giving milk are expected to lay eggs, and since non-toothed animals with a tail are expected to have a backbone (cf. Fig. 8a, b).
The platypus is an outlier because animals laying eggs are not expected to give milk (cf. Fig. 8c).
The seasnake is an outlier because animals not giving milk are expected to lay eggs (cf. Fig. 8a).
6 Conclusion and outlook
In this paper, we have introduced a novel method of unsupervised outlier detection, which reformulates the problem of unsupervised outlier detection as a set of supervised learning problems. Our approach foresees the training of a predictive model for each attribute in the dataset, using the other attributes as features. In order to compute outlier scores, the algorithm compares the predicted attribute values to the original ones, using attribute weights that are learned on the fly based on the quality of the respective predictive models. These weights allow dealing with irrelevant attributes without losing the power to identify outliers. The ALSO approach can use any learning algorithm to build the predictive models and does not make any assumptions about the distribution of attributes within the dataset.
We have made experiments with different datasets to validate our approach. We have shown that ALSO yields good results, compared to a number of established approaches. The best results have been achieved with using regression tree learning (M5’) and isotonic regression as base learners.
In addition to the quantitative results, ALSO is also capable of delivering interpretations for outliers. When using symbolic base learners, such as tree learners, those interpretations go beyond pointing out the attributes that have the largest contribution on the outlier score, which is the current state of the art for most outlier explanation approaches.
On the downside, ALSO requires training a number of single models for each attributes, which can become time-consuming, depending on the number of attributes. However, the approach is highly parallelizable by design (i.e., each of the predictive models can be trained independently from the the others). Our implementation in RapidMiner is capable to parallelize the model training on a multi-core machine.
Other approaches for improving the runtime are also possible. For example, the weights of the attributes could be estimated based on a small dataset, including only those attributes with high weights in the computation of the outlier scores, which would reduce the number of predictive models to learn on the whole dataset. Furthermore, our approach in principle allows for treating the learning problem as a multi-target regression problem (Aho et al. 2009), which could help improving both the quality and the performance approach.
So far, we have used only datasets with numeric attributes (and regression learning for building the predictive models), but our approach is not limited to that. While a straight forward approach would be converting categorical attributes to numeric ones in a preprocessing step, that approach might not yield the best results. It might be more beneficial to use learning algorithms that are tailored to categorical attributes. However, for mixed datasets containing both categorical and numeric attributes, a suitable definition for a weighted distance function has to be defined first. As a direction for future work, we aim at exploring the possibilities of ALSO for such mixed datasets.
One limitation we have observed, in particular for high dimensional datasets, is the application of ALSO to sparse datasets. Here, the base learners often fail to learn a useful model, if the vast majority of the examples has 0 as a regression target. Future work will examine this limitation and ways to circumvent it, e.g., by preprocessing the data at hand, or using base learners tailored to that type of learning problem.
Another limitation of ALSO are datasets that contain a lot of missing values. While various strategies exist for dealing with missing values (ignoring instances with missing values, filling in attributes with a special code, or average values), analyzing how those strategies affect the performance of the ALSO approach will be an issue of future research.
While the experiments in this paper only consider batch outlier detection, approaches for online outlier detection (Pokrajac et al. 2007) and outlier detection in data streams (Elahi et al. 2008) have been proposed as well. The ALSO approach may also be extended to those classes of problems, e.g., by using incremental learning algorithms as base learners and/or applying windowing techniques. We aim at analyzing those capabilities in more detail in the future.
In summary, ALSO is a method of decomposing the outlier detection problem into a number of supervised learning problems. Given a suitable base learner, it has been shown to be a robust and flexible outlier detection method, which is tolerant to irrelevant attributes and reliable in yielding good results on a large variety of datasets. Furthermore, when used with symbolic learning algorithms, it can deliver concise explanations for outliers.
Footnotes
- 1.
In scenarios where only the ordering by scores is needed, the normalization by the inverse of the sum of weights as well as the square root can be omitted to simplify the computation.
- 2.
That is, unless the IDs show a correlation with other attributes, such as the IDs of orders may be correlated with the order time.
- 3.
- 4.
Note that a high consumption is reflected by a low MPG value.
- 5.
Note that for computing \(\overline{k}\) as in (5), we take the average of the overall dataset, not the average per fold. The rationale is that for a dataset where a feature has many equal values, onefold may by coincidence only contain the same value in the test set, which would result in \(R_k\) being undefined. Moreover, if a feature contains only equal values in the whole dataset, we set its weight to 0, since it is of no use for determining the outlier score of an instance.
- 6.
A typical drawback with this approach is there can be also outliers within the set defined as normal data points, which manifest as unusual attribute values or combinations thereof, but with a majority class label or an average regression target. If an outlier detection algorithm correctly identifies those outliers, they count as false positives. However, since the problem equally exists for all approaches at hand, a fair comparison of approaches is still possible even in the presence of this drawback.
- 7.
Source: http://madm.dfki.de/downloads.
- 8.
For the Parkinsons Telemonitoring and the Energy Efficiency datasets, different regression targets exist. Here, all the original target variables were removed.
- 9.
Note that, although downloaded from the web page by the authors of Amer et al. (2013), the percentage of outliers is different than reported in their papers. Furthermore, the number of attributes in the Ionosphere dataset differs from the paper.
- 10.
We intentionally did not consider support vector machine (SVM) regression, since SVMs require careful parameter tuning, which is difficult to achieve in our case since learning the model for an attribute represents a new learning problem for each attribute, for which the SVM parameters would have to be tuned individually.
- 11.
- 12.
- 13.
- 14.
- 15.
- 16.
- 17.
- 18.
Since not each approach was able to finish on every dataset, missing observations occur in our setting.
- 19.
More precisely: as M5’ is used in its standard configuration, a minimum of four instances are required per leaf node. Thus, to form at least one leaf node with a majority of 1s (i.e., three out of four) in each of the folds, an attribute has to contain a minimum 30 instances with a 1 value in the optimistic case. This, however, is only the case for 130 of the 1558 attributes of the dataset.
- 20.
Notes
Acknowledgments
The work presented in this paper was supported by RapidMiner in the course of the RapidMiner Academia program. The authors would like to thank all the anonymous reviewers, as well as Frederik Janssen from the Knowledge Engineering Group at TU Darmstadt, for their valuable feedback on previous versions of this paper. Moreover, the authors would like to thank Petar Ristoski on his advice and assistance in performing the statistical computations presented in this paper.
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