Abstract
In supervised learning it has been shown that a collection of weak classifiers can result in a strong classifier with error rates similar to those of more sophisticated methods. In unsupervised learning, namely in anomaly detection such a paradigm has not yet been demonstrated despite the fact that many methods have been devised as counterparts to supervised binary classifiers. This work partially fills the gap by showing that an ensemble of very weak detectors can lead to a strong anomaly detector with a performance equal to or better than state of the art methods. The simplicity of the proposed ensemble system (to be called Loda) is particularly useful in domains where a large number of samples need to be processed in realtime or in domains where the data stream is subject to concept drift and the detector needs to be updated online. Besides being fast and accurate, Loda is also able to operate and update itself on data with missing variables. Loda is thus practical in domains with sensor outages. Moreover, Loda can identify features in which the scrutinized sample deviates from the majority. This capability is useful when the goal is to find out what has caused the anomaly. It should be noted that none of these favorable properties increase Loda’s low time and space complexity. We compare Loda to several state of the art anomaly detectors in two settings: batch training and online training on data streams. The results on 36 datasets from UCI repository illustrate the strengths of the proposed system, but also provide more insight into the more general questions regarding batchvsonline anomaly detection.
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1 Introduction
Imagine identifying anomalous users in a social network (Šourek et al. 2013), where user’s behavior constantly changes and their numbers are enormous, detecting weirdly behaving computers (frequently an indication of an infection by malware) in a network of large corporation with hundreds of thousands of computers, whose traffic constantly changes (Pevný et al. 2012), or identification of fraudulent card transactions (Akhilomen 2013) realized thorough big credit providers. These domains share similar features, which is processing of enormous number of samples with constantly changing characteristics. Most fast versions of existing anomaly detection methods, especially those based on indexing techniques, require the data to be available in one single batch and to fit in the memory, which is in aforementioned domains clearly impossible. Moreover, data’s nonstationarity forces detector’s models to be continuously updated, which is again very difficult with indexing techniques, as created indexes would need to be recalculated, which is usually expensive. Other methods, such as Bay and Schwabacher (2003) assumes some additional knowledge which might not be available. The presented anomaly detector has been designed with respect to these constraints, and it has been shown to achieve state of the art accuracy measured by area under ROC curve.
The definition of an anomaly, outlier, or novelty is not unified in the literature. Recent book (Aggarwal 2013a) considers all these terms to be equivalent, and defines them as follows: “An outlier is an observation which deviates so much from the other observations as to arouse suspicions that it was generated by a different mechanism.” According to this definition, outliers are generated by the same probability distribution as normal samples, but they are very rare. In this text, we define anomalies to be samples generated by a different probability distribution than normal ones, and their presence in the data to be rare, less than 1–10 % of the data. Oneclass problem is similar to the anomalydetection problem with the difference that training set contains samples exclusively from the normal class and testing set contains may contain samples from other classes at any rate.
An ideal detector would model joint probability of datagenerating processes. Although this goes against the principle of never solving a more difficult process than is needed (density estimation vs. classification), the knowledge of the probability of observed samples is useful^{Footnote 1} information in making a decision about their anomalousness. Modeling joint probability is generally difficult in spaces of many dimensions and in practice some simplifications have to be made. The detector presented here, further called Loda, approximates the joint probability by using a collection of onedimensional histograms, where every onedimensional histogram is constructed on an input space projected onto a randomly generated vector. The rationale behind the use of onedimensional histograms is that they can be efficiently constructed in one pass over data and the query operation needed during classification is simple. Consequently, Loda’s complexity is linear with respect to the number of training samples n and the dimension of the input space d.
Although one onedimensional histogram is a very weak anomaly detector, their collection yields to a strong detector. This phenomenon (collection of weak classifiers result in a strong classifier) is already a well established paradigm in supervised classification (Kuncheva 2004; Freund and Schapire 1996), but has not been demonstrated in unsupervised anomaly detection, as most ensemble systems used in anomaly detection (Aggarwal 2013b; Lazarevic and Kumar 2005; Tan et al. 2011; Liu et al. 2008) use individual detectors of much higher complexity than that of a onedimensional histogram. The comprehensive experimental section (see Sect. 4) demonstrates that the proposed ensemble of onedimensional histograms has accuracy measured by the area under ROC curve competitive with established solutions of much higher complexities. Loda therefore achieves a very good accuracy to complexity ratio and therefore it is well suited for processing large data.
Besides being fast, Loda is able to deal with missing variables and can rank features according to their contribution to sample’s anomalousness. Both of these capabilities are important for practitioners. For example, the knowledge of which features caused the anomaly can serve as guidance for further investigation and can decrease overall cost of anomaly investigation. This knowledge can also be used to cluster similar anomalies together which is again useful during investigation. Interestingly, none of these abilities substantially increase Loda’s computational complexity. Note that both abilities can be achieved with any homogenous ensemblebased anomaly detector, where individual detectors within are diversified by subspace sampling. Detectors with subspace sampling were recently proposed (Keller et al. 2012; Nguyen et al. 2010; Muller et al. 2011) but neither the robustness against missing features or the explanation of the cause of a sample’s anomalousness had been mentioned.
Loda is controlled by two hyperparameters: number of one dimensional histograms and number of histogram bins. Because the hyperparameter setting is particularly cumbersome in most anomaly detection methods where few or no anomalous samples are available (most anomaly detection), a method to determine these parameters solely on the basis of observed samples is presented. This makes Loda hyperparameter free.
Loda’s accuracy is firstly extensively compared to the prior art trained in the batch mode on 36 problems downloaded from the UCI database (Frank and Asuncion 2010) in the category of classification problems with numerical attributes. The reason, why Loda is compared to detectors processing only data presented in one batch is that we wanted to compare Loda to state of the art detectors, even though they are not applicable for intended scenario. The experimental results reveal that Loda’s measured by area under ROC curve is equal to and many times better than that of other algorithms with higher computational complexity requiring batch training (we expect algorithms trained in batch to perform better than online algorithms). Consequently Loda provides a good alternative to the already established solutions on problems with a large number of features and samples that need to be processed efficiently. Secondly, Loda is compared to the prior art on streaming problems and it is shown when it is better to continuously update histogram and when to do so in batches. Finally, Loda’s capability to efficiently process big datasets is demonstrated on the WEB UCI dataset (Ma et al. 2009) with 2.4 million samples and 3.2 million features.
The contribution of this paper is threefold: (i) it demonstrates that an ensemble of very weak anomaly detectors can lead to a strong anomaly detector; (ii) it presents a lightweight anomaly detector suitable for datastreams robust to missing variables and identifying causes of anomalies; (iii) it extensively compares existing and the proposed anomaly detector, which sheds some light on conditions in which a particular anomaly detector excels.
This paper is organized as follows: The next section reviews relevant prior art, shows its computational complexity, and discusses issues related to online learning and classification. Loda is presented in Sect. 3. In Sect. 4, Loda is experimentally compared to the prior art, its low computational requirements are demonstrated on artificial data, and its efficiency is demonstrated on a largescale dataset (Ma et al. 2009). The same section also demonstrates Loda’s ability to handle data with missing values and explain a sample’s anomalousness. Finally Sect. 5 concludes the paper.
For better reproducibility, all source code, results, datasets, and their creation is available at http://agents.cz/~pevnak.
2 Related work
The recent book (Aggarwal 2013a) and survey (Chandola et al. 2009) contain many methods for anomaly and outlier detection. Below, those relevant to this work are reviewed.
2.1 Modelcentered detectors
Basic modelcentered anomaly detectors assume that data follow a known distribution. Detectors (Shyu et al. 2003) based on principal component analysis (PCA) assume that the data fit a multivariate normal distribution. Despite this being rarely true in practice, they frequently outperform more complicated detectors. The complexity of their training is \(O(nd^{3})\), where d is the dimension of the input space and n is the number of samples used for training.
A OneClass Support Vector Machine (Schölkopf et al. 2001) (1SVM) does not make any assumptions about the distribution of data. It finds the smallest area where \(1\nu \) fraction of data are located (\(\nu \) is parameter of the method specifying desired falsepositive rate). This is achieved by projecting the data to a highdimensional (feature) space and then finding a hyperplane best separating the data from the origin. It has been noted that when 1SVM is used with a linear kernel it introduces a bias to the origin (Tax and Duin 2004), which can be removed by using a Gaussian kernel. The Support Vector Data Description algorithm (Tax and Duin 2004) (SVDD) removes the bias in 1SVM by replacing the separation hyperplane by a sphere encapsulating most of the data. The complexity of both methods is superlinear with respect to the number of samples, n, being \(O(n^{3}d)\) in the worstcase.
Isolation Forest (Liu et al. 2008) (IForest) relies on a collection of Isolation Trees grown by using unlabeled data. During growth of a tree, its internal nodes are added until the terminal leafs contain one sample or the maximum depth is reached. The anomaly score is proportional to the level of leaf reached by the sample, as the idea is that anomalies will reach leafs at the base of the tree (close to the root), while legitimate samples reach leafs closer to the root. The complexity of training one tree is \(O(n\log n)\) and of classification \(O(\log n)\), where the authors recommend subsampling of training samples to increase diversity in the ensemble, robustness against anomalies within the data, and simultaneously decrease complexity of the training and classification.
The recently proposed FRAC (Noto et al. 2012) aimed to bridge the gap between supervised and unsupervised learning. FRAC is an ensemble of models, each estimating one feature based on other features (for data of a dimension d, FRAC uses d different models). The rationale behind this is that anomalous samples exhibit different dependencies among features, which can be detected from prediction errors modeled by histograms. FRAC’s complexity depends on the algorithm used to implement individual models, which can be large, considering that a search for possible hyperparameters needs to be undertaken. Because of this, an ordinary linear leastsquare regression is used here leading to the complexity \(O(nd^{4})\). FRAC is not well suited for online learning since an update of models changes the distributions of errors that cannot be estimated from one sample.
The online training of modelcentered detectors is generally difficult as the algorithms used to create the model have nontrivial complexity with respect to the dimension, or models cannot be updated incrementally (e.g. principal component analysis). The online adaptation of 1SVM is discussed in Kivinen et al. (2004), but the solution is an approximation of the solution returned by the batch version. The exact online version of SVDD is described in Tax and Laskov (2003), but the algorithm requires substantial bookkeeping thereby increasing the complexity. The proposed Loda is also modelbased detector and as will be shown in Sect. 4, its one dimensional histograms can be easily updated based on upcoming samples which makes the whole algorithm friendly to online learning.
2.2 Datacentered detectors
Datacentered detectors do not have any builtin model. A sample’s anomalousness is determined according to its distance to all previously observed samples. Consequently, there is no training phase as new samples are just added to the set of already observed samples. However, this increases the complexity of the classification phase, which is a linear function of the number of samples n.
A knearest neighbor (Knorr and Ng 1999) (KNN) is a popular method to identify outliers inspired by the corresponding method from classification. It ranks samples according to their distance to kthnearest neighbor. It has been recently shown that a variant of KNN (Sricharan and Hero (2011)) converges to the optimal densitylevel detector for a given false positive rate. Nevertheless KNN has been criticized for not being able to detect outliers in data with clusters of different densities (Breunig et al. 2000). The local outlier factor (Breunig et al. 2000) (LOF) solves this problem by defining the outlier score as a fraction of sample’s distance to its kthnearest neighbor and the average of the same distance of all its k nearest neighbors. True inliers have a score around one while outliers have much greater score. We refer to Zimek et al. (2012) for a comprehensive review of the prior art.
The complexity of the classification phase of nearestneighbor based detectors is driven by the nearestneighbor search, which is an O(n) operation in the worst case. More efficient approaches based on bookkeeping (Pokrajac et al. 2007), better search structures like KDtrees (Bentley 1975), or approximate search (Andoni and Indyk 2006) have been adopted. Nevertheless, the complexity of all methods depends in some way on the number of training samples n, and better search structures are usually useful only in low dimensions.
2.3 Online anomaly detectors
There are few works in online anomaly detection. The closest work to this is HalfSpace trees (Tan et al. 2011) (HStrees), which is a method similar to Isolation Forest with the difference that decision rules within treenodes are generated randomly. The output of each HStree can be interpreted as a crude estimate of the probability density function, which is further refined by taking a sufficient number of them. It is worth noting that HStrees assume that the data is scaled such that values of features are bounded in [0, 1]. This is in a sharp contrast to Loda only requiring features to have approximately the same scale, which is a more relaxed condition, especially if concept drift occurs. HStrees handle conceptdrift by dividing datastreams into sample batches of size 256, where HStrees trained on a previous batch are used to scrutinize samples in a given batch. Simultaneously new HStrees are learned on a current batch and once all samples from the current batch are processed, new HStrees replaces the old one.
OLINDDA (Spinosa et al. 2009) uses standard kmeans clustering to group previously observed samples into clusters called concepts. Anomalies not belonging to any cluster are grouped into candidate clusters, which based on cohesiveness criteria are either promoted as a new cluster and novel concept is reported, or evicted. Thus OLINDDA focuses on detection of a novel concepts in streams rather than on identification of anomalies.
2.4 Ensembles and random projections
Ensembles have so far been underutilized in anomaly detection. A significant portion of the prior art focuses on a unification of anomaly scores (Gao and Tan 2006; Schubert 2012), because different detectors within one ensemble may provide anomaly scores of different magnitudes (Nguyen et al. 2010). Diversifying detectors by random subsets of features was presented (Lazarevic and Kumar 2005; Keller et al. 2012) to improve the accuracy, especially on high dimensional problems.
Projections of the input space onto the arbitrary subspace have been utilized mainly in distancebased outlier detection schemes to speedup the search of nearest neighbors. de Vries et al. (2010) performs the kthNN search in the LOF first in the random subspace on a larger neighborhood and then it is refined by the search in the original input space. Similarly, Pham and Pagh (2012) estimates the distribution of angles between samples in the input space from the distribution of angles in the subspace (distribution of angles has been proposed in Kriegel and Zimek (2008) as a good anomaly criterion). In experiments in Sect. 4.3 the similar approach is used on prior art to compare it to Loda.
Most prior art employs ensembles and random projections with datacentered detectors, which have high computational complexity and are not well suited for realtime processing. Exceptions are Isolation Forest (Liu et al. 2008) and HalfSpace Trees (Tan et al. 2011) relying on an ensemble of trees, each having a relatively low accuracy. This is similar to Loda which uses even simpler model (onedimensional histogram) with even lower complexity, but their combination is similarly powerful. Random projections in Loda are used to project the input space into a single dimension, which simplifies the complexity of all operations over it. Loda’s sparse random projections can be considered as a subspace sampling method. Unlike the prior art, the subspace method is not used only to increase efficacy, but also to gain robustness against missing variables and the ability to find causes of anomalousness of a given sample.
Interpreting the decision in anomaly detection is very important in practice since it can reduce the cost of subsequent analysis and increases trust of the detector. The work of Knorr and Ng (1999) has focused on finding a minimal subspace in which a given sample is an outlier. Algorithms presented therein are datacentered and suitable only for lowdimensions. HiCS algorithm (Keller et al. 2012) and the algorithm of Dang et al. (2013) both aim to identify a subspace in which a scrutinized sample is an outlier. Both are datacentered algorithms and consequently their computational complexity is prohibitive for processing datastreams.
Loda’s method to identify relevant features bears similarity to HiCS algorithm. But unlike HiCS, Loda’s method is general and can be used on all ensembles diversified by random subspace sampling. Finally, contrary to all prior art, the identification of relevant features does not increase Loda’s computational complexity in big O notation.
3 Description of Loda
Loda is comprised of a collection of k onedimensional histograms \(\{h_{i}\}_{i=1}^{k}\), each approximating the probability density of input data projected onto a single projection vector \(\{w_{i}\in {\mathbb {R}}^{d}\}_{i=1}^{k}\). Projection vectors \(\{w_{i}\}_{i=1}^{k}\) diversify individual histograms, which is an essential requirement for ensemble systems to improve performance of a single classifier / detector. Their initialization is described in detail below in Sect. 3.1.
Loda’s output, f(x), on a sample, x, is an average of the logarithm of probabilities estimated on individual projection vectors. Adopting \(\hat{p}_{i}\) to denote the probability estimated by ith histogram and \(w_{i}\) to denote the corresponding projection vector, Loda’s output f(x) can be written as
which can be reformulated as
where \(p(x^{\mathrm {T}}w_{1},x^{\mathrm {T}}w_{2},\ldots ,x^{\mathrm {T}}w_{k})\) denotes the joint probability of projections. Equation (2) shows that Loda’s output is proportional to the negative loglikelihood of the sample, which means that the less likely a sample is, the higher the anomaly value it receives. This holds under a strong assumption (a) in (2) that probability distributions on projection vectors \(w_{i}\) and \(w_{j}\) are independent \(\forall i,j\in k,\, i\ne j\). Even though this is almost never true in practice, Loda still delivers very good results. We believe that the reasons are similar to those in naïve Bayes classifiers theoretically studied by Zhang (2004), which give conditions under which the effects of conditional dependencies cancel out. These conditions depend on the probability distribution of both classes and they are difficult to be verified in practice as they require an exact knowledge of conditional dependencies among features. Due to Loda’s similarity to the Parzen window detector (Yeung and Chow 2002), the similar argumentation might explain Loda’s surprisingly good performance.
In highdimensional spaces Loda can be related to a PCA based detector (Shyu et al. 2003), because projection vectors \(w_{i}\) and \(w_{j}\), \(i\ne j\) will be approximately orthogonal (this is due to their random construction described below). Assuming again the independence of \(x^{\mathrm {T}}w_{i}\) and \(x^{\mathrm {T}}w_{j}\), the projected data are orthogonal and uncorrelated, which are properties of Principal Component Analysis (PCA). Loda’s histogram on random projections are therefore similar to histograms on principal components.
Loda’s training and classification routines are summarized in Algorithms 1 and 2, respectively. Loda is initialized by generating a set of sparse random projections \(\{w_{i}\}_{i=1}^{k},w_{i}\sim N(0,\mathbf {1}^{d})\) with \(d^{\frac{1}{2}}\) nonzero components and initializing the corresponding set of histograms \(\{h_{i}\}_{i=1}^{k}\). Each histogram is updated by a training sample by projecting the sample onto a vector and then the corresponding histogram bin is updated. The classification procedure follows similar steps, but instead of updating histograms, they return probabilities whose logarithms are averaged and returned. Notice that the construction requires only one pass over the data and can be used on datastreams by first classifying a new sample and then updating all histograms.
The rest of this section describes creation of and rationale behind sparse projection vectors \(w_{i}\), presents how sparse projections enable robustness against missing variables and allow explanation of sample’s anomalousness, and closes by commenting issues related to building online histograms.
3.1 Random projections
Each sparse projection vector \(\{w_{i}\}_{i=1}^{k}\) is created during initialization of the corresponding histogram by first randomly selecting \(d^{\frac{1}{2}}\) nonzero features (d is the dimension of the input space) and then randomly generating nonzero items according to N(0, 1). The choice of normal distribution of nonzero items comes from the Johnson–Lindenstrauss (1984) lemma showing that with this choice, \(L_{2}\) distances between points in the projected space approximate the same quantity in the input space. The \(d^{\frac{1}{2}}\) sparsity is due to Li (2007) showing that \(L_{2}\) distances can be preserved with random projections with only \(d^{\frac{1}{2}}\) nonzero elements. Another justification for the use of sparse random projections is to increase diversity among histograms by making them work on different subsubspaces. This is a popular diversification technique used for example in Random Forest (Ho 1998).
One can ask a question, are random projections actually needed? Would a detector based on an ensemble of histograms of individual features (in this section called perfeature detector) have similar accuracy? According to the experimental results on problems used throughout this paper (described in Sect. 4 in detail), random projections consistently improve the performance (see Fig. 10 in “Appendix”). On four out of five rate of anomalies Loda was better than the perfeature detector. Wilcoxon signedrank test assessing if both detectors delivers the same performance was rejected with a pvalue 0.028. Which means that the Loda is better and the difference is statistically significant. For more details see comments under Fig. 10.
3.2 Missing variables
Sparse projections enable Loda to handle missing variables by calculating the output only from those histograms whose projection vector has a zero on the place of missing variables. To formalize the approach, assume that \(x\in {\mathbb {R}}^{d}\) is a sample with missing variables, and \(\mathcal {J}\) is the indexset of missing variables. Let \(\mathcal {I}(x)\) be a set of indices of histograms whose projections have all entries in \(\mathcal {J}\) zero, i.e. \(\left( \forall i\in \mathcal {I}(x)\right) \left( \forall j\in \mathcal {J}\right) \left( w_{ij}=0\right) \), where \(w_{ij}\) is jth element of the projection vector of ith histogram. Then, the anomaly score for the sample x is calculated as
Since the output of all histograms within Loda have the same meaning there is no need for calibration as in Nguyen et al. (2010). The final output is reliable even if several histograms are omitted. Notice that by using the same mechanism, Loda can be also trained on data with missing variables, as only detectors from \(\mathcal {I}(x)\) are updated upon observing sample x.
3.3 Explaining the cause of an anomaly
Two independent works (Rondina et al. 2014; Somol et al. 2011) propose a feature selection method for a binary classification based on the comparison of scores (e.g. classification of error) of classifiers on randomly generated subspaces. Recognizing that each histogram with sparse projections in Loda provide an anomaly score on a randomly generated subspace, this method can be used to rank features according to theirs contribution to sample’s anomalousness.
Let \(\hat{p_{i}}\) denote the probability estimated by ith histogram on a sample x, and \(\mathcal {I}_{j}/\overline{\mathcal {I}}_{j}\) denote index sets of histograms that use/do not use jth feature. Formally \(\left( \forall i\in \mathcal {I}(x)\right) \left( w_{ij}=0\right) \) and \(\left( \forall i\in \overline{\mathcal {I}}(x)\right) \left( w_{ij}\ne 0\right) \), where \(w_{i}\) is the projection vector of ith histogram. The score function proposed in the prior art calculates the difference of means of \(\log \hat{p_{i}}\) over \(\mathcal {I}_{j}\) and \(\bar{\mathcal {I}}_{j}\), which means that if an anomaly score with a feature being present is much higher than that of without, the feature is important for identification of outliers. We improve this score function by recognizing that it’s main goal is to assert if the contribution of jth feature is statistically significant, for which we use onetailed twosample t test with a test statistic
where \(\mu _{j}/\bar{\mu _{j}}\) is the mean and \(s_{j}^{2}/\bar{s}_{j}^{2}\) variance of \(\log \hat{p_{i}}\) calculated with \(i\in \mathcal {I}_{j}/i\in \bar{\mathcal {I}}_{j}\). The higher the \(t_{j}\) the more important the \(j^{\mathrm {th}}\) feature is. Since the complexity of the contrast function (4) is linear with respect to the number of projections k and number of features d, the calculation of a feature’s contrast increases Loda’s complexity by a constant in big O notation. Experiments demonstrating the feature selection are presented in Sect. 4.6.
3.4 Histogram
The onedimensional histogram on random projections is an important design element in Loda, as it determines its learning mode (batch vs. online). The prior art on histogram construction and the optimal number of bins is very rich. In the following only the tiny subset relevant to online construction on streams is recapitulated.
The most common approach for batch data are equiwidth or equidepth histograms with bins having either the same length or containing the same number of samples. In database research, VOptimal histograms (Poosala et al. 1996) minimizing weighted variance of estimated frequency are popular, but their construction has prohibitive complexity \(O(n^{2})\). Moving to datasteams, approximations of VOptimal histograms have been studied in Guha et al. (2006), but the algorithms therein are quite complex. Simpler Partition Incremental Discretization (Gama and Pinto 2006) constructs a histogram in two steps: (i) create an equiwidth histogram with small bins in a single iteration through the data; (ii) use this fine histogram to return either equiwidth or equidepth histogram with a given number of bins. The advantage is that userprovided partition in step (i) is only indicative, as bins can be split if they already contain too many samples. The disadvantage for Loda is that the second step needs to be triggered before classifying a sample if the fine histogram was updated. DADO algorithm (Donjerkovic et al. 2000) constructs incremental histograms close in quality to Voptimal histograms. The key idea behind this technique is to internally represent every bin by two subbins. Counts in subbins are used for binsplit and binmerge operations and for their triggering. DADO’s biggest advantage for Loda is a fixed number of buckets which implies fixed memory requirements. An interesting alternative was proposed in BenHaim and TomTov (2010) originally for determining splitting points in decision trees. Unlike all previous approaches it does not require any knowledge about the range of values in histogram (see Appendix “Online histogram” for its recapitulation), but according to our results Loda with equiwidth histogram is better.
With the exception of the last algorithm, all above approaches require knowledge of the range of modeled values in advance. This can be usually estimated from a sample of data, but in the case of severe nonstationarity, the histogram’s support can shift outside the initial range. A simple solution for equiwidth histograms is to specify bin width and store bin counts in a hashmap structure with keys \(\kappa =\left\lfloor \frac{x}{\varDelta }\right\rfloor \), where \(\kappa \) is an integer key, \(\varDelta \) is the bin width, and x is the value to be inserted. This approach has O(1) time complexity and if coupled with countminsketch (Cormode and Muthukrishnan 2005) its space complexity can be upper bounded. It also allows the modeling fixed length windows in a stream by keeping samples (indexes of bins) to be later removed in memory. This can have potentially large memory footprint, which can be decreased by using exponential buckets proposed in Datar et al. (2002). Alternatively, two histograms as in HSTrees (Tan et al. 2011) can be used, where the older one is used for classification while the newer one is being constructed on newly arrived samples. Once the construction of the new one is finished, it replaces the old one and the construction of the new one is started. Both constructions are experimentally compared in Sect. 4.2 with interesting conclusions.
The determination of an optimal number of histogram bins b is an important design parameter. For an equiwidth histogram, there exists a simple method to determine optimal number of histograms (Birgé and Rozenholc 2006), which is used in Loda. The method of Birgé and Rozenholc (2006) maximizes penalized maximum likelihood in the form
where \(n_{i}\) is the number of samples that falls in ith bin and \(N=\sum _{i=1}^{b}n_{i}\) is the total number of samples. Penalization factor \(b1+\left( \log b\right) ^{2.5}\) penalizes histograms with too many bins.
3.5 Computational and storage complexity
Loda’s complexity is mainly determined by the type of the histogram. Assuming equiwidth histogram and sparse random projections the time complexity of learning is \(O(nkd^{\frac{1}{2}})\), where n is the number of training samples, d is the dimensionality of the input space, and k is the number of histograms. The time complexity of classification is \(O(kd^{\frac{1}{2}})\). As discussed in the previous subsection the space complexity can be made \(O(k(d^{\frac{1}{2}}+b))\) with b being number of histogram bins.
For a floating window histogram over a length of l the space complexity can be made \(O(k(d^{\frac{1}{2}}+b\log l))\) by using the algorithm of Datar et al. (2002) if space is constrained, otherwise \(O(k(d^{\frac{1}{2}}+b+l))\) for a naïve approach storing all l values for discounting them on expiration.
Alternatively, online histogram can be implemented by using two alternating histograms as in Tan et al. (2011), where the older histogram is used for the classification while the new one is constructed. Once the construction of the new one is finished (it has accommodated l samples), it replaces the older one for the classification and the new construction of the new histogram one is started. The advantage is possibly smaller space complexity \(O(k(d^{\frac{1}{2}}+b))\) and according to the experimental results in Sect. 4.2 better robustness to clustered anomalies. Table 1 shows the time and space complexity of Loda with continuous and two equiwidth alternating histograms and the prior art used in the experimental section.
4 Experiments
Below Loda is compared to relevant state of the art in three different scenarios. The first scenario simulates anomaly detection on stationary problems, where all detectors are first trained on a training set and then evaluated on a separate testing set. This scenario is included because it enables the comparison of Loda to other anomaly detectors, which are designed for batch training and their accuracy should be superior due to more sophisticated training algorithms. The second scenario mimics streamed data, where anomaly detectors first classify a sample and then use it to update their model. The third scenario uses a dataset with millions of samples and features to demonstrate that Loda is able to efficiently handle big data. The section is concluded by an experimental comparison of detectors time complexity in classify and update scenarios and by the demonstration of robustness against missing variables and the identification of causes of anomalies.
4.1 Stationary data
Loda has been compared to the following anomaly detectors chosen to represent different approaches to anomaly detection: PCA based anomaly detector (Shyu et al. 2003), 1SVM (Schölkopf et al. 2001), FRAC (Noto et al. 2012), \(\delta \)version of kthnearest neighbor detector (Harmeling et al. 2006) (KNN) which is equivalent to Sricharan and Hero (2011) with \(\gamma =1\) and \(s=d\) (in notation of the referenced work) if the area under ROC curve (AUC) is used for evaluation, LOF (Breunig et al. 2000), and IForest (Liu et al. 2008).
Benchmarking data were constructed by the methodology proposed in Emmott et al. (2013) converting realworld datasets (Frank and Asuncion 2010) to anomaly detection problems. The methodology produces set of normal and anomalous samples, where samples from different classes have different probability distribution, which is aligned with our definition of anomalous samples. Created problems are divided according to (i) difficulty of finding an anomaly (easy, medium, hard, very hard), (ii) scatter / clusteredness of the anomalies (high scatter, medium scatter, low scatter, low clusteredness, medium clusteredness, high clusteredness), (iii) and finally with respect to the rate of anomalies within the data \(\{0,0.005,0.01,0.05,0.1\}\). The training and testing set were created such that they have the same properties (difficulty, scatter/clusteredness, and rate of anomalies)^{Footnote 2} with clustered anomalies located around the same point. Note that the total number of unique combinations of problem’s properties is up to 120 for each dataset. The construction of individual problems is recapitulated in Appendix “Construction of datasets”.
The quality of the detection is measured by the usual area under ROC curve (AUC). Since from every dataset (out of 36) up to 120 problems with different combinations of problem’s properties can be derived, it is impossible to present all \(7\times 4200\) AUC values. Tables with AUCs averaged over difficulty of detecting anomaly are in the supplemental and cover 15 pages. They are therefore presented in an aggregated form by using critical difference diagrams (Demšar 2006) which show average rank of detectors together with an interval in which Bonferroni–Dunn correction of Wilcoxon signed ranks test cannot reject the hypothesis that the detectors within the interval have the same performance as the best one. The average rank comparing AUC is calculated over all datasets on which all detectors provided the output, following Demšar (2006) lower rank is better. AUCs used for ranking on each dataset are an average over all combinations of problem parameters with fixed parameter(s) of interest. For example, to compare detectors on different rates of anomalies within the data, for each rate of anomalies the average AUC is calculated over all combinations of difficulty and scatter. Average AUCs of different detectors on the same dataset are ranked, and the average of ranks over datasets is the final average rank of a detector on one rate of anomalies within the data.
To decrease the noise in the data caused by statistical variations, experiments for every combination of problem properties were repeated 100 times. This means that every detector was evaluated on up to \(100\times 120\times 35=4.32\times 10^{5}\) problems. Benchmark datasets created as described in Appendix “Construction of datasets” were varied between repetitions by randomly selecting 50 % of normal samples (but maximum of 10,000) to be in the training set and putting the remaining data to the set. Similarly, anomalous samples were first selected randomly from all anomalous samples to reach the desired fraction of outliers within the data, and then randomly divided between training and testing set. The data in the training and testing set have the same properties in terms of difficulty, clusteredness, and fraction of anomalies within.
4.1.1 Settings of hyperparameters
Setting hyperparameters in anomaly detection is generally a difficult unsolved problem unless there is a validation ground truth available for the parameter tuning. Wrong parameters can cause an otherwise well designed anomaly detector to fail miserably. Nevertheless, in this context we do not aim to solve the problem as our intention here is primarily to compare multiple detectors against each other. For this purpose we follow hyperparameter setting guidelines given by the authors of the respective methods. Note that our proposed method does not require manual hyperparameter setting. Employed parameter settings are detailed below and in Table 2.
Settings of the number of nearest neighbors in our implementations of LOF and KNN algorithms followed (Emmott et al. 2013; Breunig et al. 2000) and was set to \(\max \{10,0.03\times n\}\), where n is the number of samples in the training data. 1SVM with a Gaussian kernel used \(\nu =0.05\) and width of the kernel \(\gamma \) equal to an inverse of median \(L_{2}\) distance between training data. SVM implementation has been taken from the libSVM library (Chang and Lin 2011). Our implementation of the FRAC detector used ordinary linear leastsquare estimators, which in this setting does not have any hyperparameters. Our implementation of PCA detector based on principal component transformation used top k components capturing more than 95 % variance. IForest [implementation taken from Jonathan et al. (2014)] used parameters recommended in Liu et al. (2008): 100 trees each constructed with 256 randomly selected samples.
The number of histograms in Loda, k, was determined similarly to the number of probes in feature selection in Somol et al. (2013). Denoting \(f_{k}(x)\) Loda’s output with k histograms, the reduction of variance after adding another histogram can be estimated as
Although \(\hat{\sigma }_{k}\rightarrow 0\) as \(k\rightarrow \infty \), its magnitude is problemdependent making it difficult to set a threshold on this quantity. Therefore \(\hat{\sigma }_{k}\) is normalized by \(\hat{\sigma }_{1}\), and the k is determined as
where \(\tau \) is the threshold. In all experiments presented in this paper, \(\tau \) was set to 0.01. Unless said otherwise, Loda used equiwidth histogram with the number of histogram bins, b, determined for each histogram separately by maximizing penalized maximum likelihood method of Birgé and Rozenholc (2006) described briefly in Sect. 3.4. With b being set, width of histogram bins in equiwidth histograms was set to \(\frac{1}{b}(x_{\mathrm {max}}x_{\mathrm {min}})\).
Setting of hyperparameters is summarized in Table 2.
4.1.2 Experimental results
Figure 1a–e shows the average rank of detectors plotted against rate of anomalies within the training set for different clusteredness of anomalies (AUCs were averaged only with respect to the difficulty of anomalies). Since highly scattered anomalies could not be produced from the used datasets, the corresponding plot is omitted. The number of datasets creating problems with medium scattered anomalies is also low, and we should not draw any conclusions as they would not be statistically sound.
For problems with low number of lowscattered and lowclustered anomalies the datacentered detectors (KNN and LOF) are good. With increasing number of anomalies in data and with their increasing clusteredness, however, the performance of datacentered detectors appears to deteriorate (note LOF in particular). Additional investigation revealed that this effect can be mitigated by hyperparameter tweaking, specifically by increasing the number of neighbours k. In practice this is a significant drawback as there is no way to optimize k unless labeled validation data is available.
The modelcentered detectors (Loda, IForest as well as 1SVM) appear to be comparatively more robust with respect to the increase in number of anomalies or their clusteredness, even if used with fixed parameters (see Sect. 4.1.1. ).
In terms of statistical hypothesis testing the experiments do not show marked differences between the detectors in question. Note that in isolated cases FRAC and 1SVM perform statistically worse than the respective best detector.
Figure 1f shows the detectors time to train and classify samples with respect to the size of the problem measured by the dimension multiplied by the number of samples.^{Footnote 3} On small problems KNN, LOF, and 1SVM are very fast, as their complexity is mainly determined by the number of samples, but Loda is not left behind too much. As the size of problems gets bigger Loda quickly starts to dominate all detectors followed by PCA. Surprisingly IForest with low theoretical complexity had the highest running times. Since its running time is almost independent of the problem size, it is probably due to large multiplication constant absorbed in big O notation.
Comparing detectors by the performance/time complexity ratio, we recommend KNN for small problems with scattered anomalies, while for bigger problems and problems with clustered anomalies we recommend the proposed Loda, as its performance is similar to IForest but it is much faster.
The investigation of problems on which Loda is markedly worse than KNN revealed that Loda performs poorly in cases where the support of the probability distribution of nominal class is not convex and it encapsulates the support of the probability distribution of the anomalous class. A typical example is a “banana” dataset in Fig. 2 showing isolines at false positive rates \(\{0.01,0.05,0.1,0.2,0.4\}\) of Loda and KNN. Isolines show that Loda has difficulty modeling the bay inside the arc. Although the adopted construction of problems from UCI datasets (see Appendix “Construction of datasets” for details) aimed to create these difficult datasets, Loda’s performance was overall competitive to other, more difficult detectors.
Breunig et al. (2000) has criticized KNN, which has performed the best on clean training data, for its inability to detect local outliers occurring in datasets with areas of very different densities. Figure 3 shows again isolines at false positive rates \(\{0.01,0.05,0.1,0.2,\) \(0.4\}\) of Loda and LOF (designed to detect local outliers) on a data generated by two Gaussians with very different standard deviations (a typical example of such data). Loda’s isolines around the small compact cluster (lower left) are very close to each other, whereas isolines of the loose cluster (upper right) are further away which demonstrates that Loda is able do detect local outliers similarly to LOF. LOF’s wiggly isolines also demonstrates the overfitting to the training data.
Figure 4 visualizes the average rank of Loda with equiwidth, equidepth, and online (BenHaim and TomTov 2010) histograms on different rates of anomalies within the data. Although on almost all rates Friedman’s statistical test accepted the hypothesis that all three versions of Loda performs the same, Loda with equiwidth histograms is obviously better. This is important because equiwidth histograms can be efficiently constructed from a stream of data if the binwidth is determined, for example by using sample data.
4.2 Streamed and nonstationary data
This section compares Loda with floatingwindow histogram and two alternating histograms (see Sect. 3.4) to HalfSpace Trees (HSTrees) (Tan et al. 2011) [implementation taken from Tan (2014)]. Online version of 1SVM was omitted from the comparison due to the difficulties with hyperparameter setting and inferior performance when algorithms were compared in the batch learning. The comparison is made on seven datasets: Shuttle, Covertype, HTTP and SMTP data from kddcup 99. HTTP and SMTP data are used in two variants: (i) with all 41 features (called HTTPfull and SMTPfull) and (ii) with 3 features (called HTTP3 and SMTP3) used in Tan et al. (2011).^{Footnote 4}
In every repetition of an experiment the testing data were created by replacing maximum of 1 % of randomly selected normal samples by randomly selected anomalous samples. Order of normal samples has not been permuted in order to preserve data continuity. Thus, the variation between repetitions comes from (i) the position to where anomalous samples were inserted and (ii) from the selection of anomalous samples to be mixed in. To study the effect of clustered anomalies, anomalies were first selected to form clusters in space (as in the previous section) and then they replaced sequences of normal data of the same size. By this process datastreams contained anomalies clustered in time and space of sizes \(\{1,4,16\}\) (this quantity is hereafter called clusteredness). Experiments on each dataset for every value of clusteredness of anomalies were repeated 100 times.
HSTrees used recommended configuration of 50 trees of depth 15 with minimum of 20 samples in leaf node. Loda’s configuration, namely number of histograms and bins were optimized on first 256 samples in every repetition, thus there were no parameters to be set manually. Both algorithms used a window of length 256 samples, which is the hardcoded length in implementation of HSTrees provided by authors.
Table 3 shows AUCs and average time to process samples of Loda and that of HSTrees. We can see that both algorithms deliver nearly the same performance, while Loda is on average 7–8 times faster (the time complexity is treated in more detail in the next subsection). Figure 5 shows AUCs on segments of 120,000 continuous samples on datasets Covertype, HTTP, HTTP3, and SMTP+ HTTP with clusters of 16 anomalies (omitted datasets did not contained enough samples). Since all AUCs of all three detectors remains stable, it can be concluded that all detectors equally well combat the concept drift in datasets.
Further investigation of results reveals that Loda with two alternating histograms (construction similar to that in HSTrees) is more robust to clustered anomalies than Loda with continuous histogram. Our explanation of this phenomenon is that clustered anomalies are classified most of the time by histograms built on clean data without anomalies, while polluted histograms classify most of the time clean data, which is not influenced by anomalies in training data. These results demonstrate that the right choice of histogram for nonstationary data should depend on the type of anomalies (time and space clustered vs. scattered).
4.3 Big data
The demonstration of Loda’s ease of handling big data is done on a URL dataset (Ma et al. 2009) containing 2.4 million samples with 3.2 million features. Normal samples contain features extracted from random URLs, while anomalous samples have features extracted from links in spam emails during 120 days at rate 20,000 samples per day (see Ma et al. 2009 for dataset details). Due to the high number of “anomalous” samples and cleanliness of normal samples, the dataset belongs to a category of oneclass problems where the goal is to separate class of legitimate URLs from others.
Since the data are presented in batches of 20,000 samples per day, all methods evaluated in Sect. 4.1 can be theoretically applied by training them on normal samples from the previous day, i.e. detectors classifying samples from lth day are trained on normal samples from the \((l1)\)th day. The problem is that the computational complexity effectively prevents a direct application of FRAC, 1SVMs, PCA, LOF, and KNN, as methods would not finish in reasonable time. The only methods that can be used without modification is Loda and IForest. Their settings were same as in Sect. 4.1 with the difference that (i) Loda used 500 projections and histograms bins were optimized on data from day zero and then stay fixed; (ii) number of trees within IForest was increased to 1000. Rows labeled \(3.2\times 10^{6}\) in Table 4 show AUCs, average rank over 120 days, and the average time to classify new samples and retrain the detectors. We can see that Loda was not only significantly better (AUC of IForest is close to the detector answering randomly), but also more than 15 times faster.
To reduce the computational complexity such that other algorithms can be used, the original data of dimension 3.2 million were projected onto 500 dimensional space with a randomly generated matrix \(W\in {\mathbb {R}}^{3.2\times 10^{6},500}\), \(W_{ij}\sim N(0,1)\). According to the JohnsonLindenstrauss lemma, \(L_{2}\) distances between points should be approximately preserved, therefore most detectors should work. Hyperparameters of all detectors were set as in Sect. 4.1.1. AUCs of all detectors on every day are shown in Fig. 6, and their summaries again in Table 4. On this new dataset the best algorithm was the KNN algorithm followed by PCA and Loda. These results are consistent with those in Sect. 4.1.2, where KNN detector was generally good if trained on clean data. The good performance of PCA can be explained by projected data being close to multivariate normal distribution, although the Lilliefors’ test has rejected the hypothesis that marginals follow the normal distribution with 85 % rate. Notice that Loda’s execution time was more than 100 times faster than KNN, as its time to classify and update on samples from one day was 1.16s on average, while that of KNN was 140s.
4.4 Time complexity
Loda was primarily designed to have low computational complexity in order to efficiently process large data and datastreams. This feature is demonstrated below, where times to classify one sample with simultaneous update of the detector is compared for PCA, KNN, LOF, HSTrees, and Loda with continuous histogram.^{Footnote 5} All settings of individual algorithms were kept the same as in previous experiments. Loda’s time includes optimization of hyperparameters on the first 256 samples of the stream as described in Sect. 4.1.1 (Loda used on average 140 projections with histograms having 16 bins and sliding window of length 256).
The comparison was made on 100,000 artificially generated samples of dimensions 100 and 10,000. As the interest here is on throughput, the details of data generation are skipped as not being relevant for the experiment. The results are summarized in Fig. 7 showing the average time to classify and update one sample against the number of observed samples. The experiment was stopped if the processing of 1000 samples took more than half an hour causing some graphs to be incomplete. The graph for the PCA detector is missing entirely in the left Fig. 7, because processing of 1000 samples took more than 15 h (109s per one sample). Both graphs demonstrate that Loda’s efficiency is superior to all other detectors, as it is two order of magnitudes faster than the fastest algorithms from the prior art. This experiment highlights Loda’s suitability for efficiently handling datastreams.
4.5 Robustness to missing variables
As mentioned in Sect. 3, Loda with sparse projections can classify and learn from samples with missing variables. This feature was evaluated under the “missing at random” scenario, where missing variables are independent of the class membership. In every repetition of an experiment, before the data was divided into training and testing sets, \(\{0,0.01,0.02,0.04,\ldots ,0.20\}\) fraction of all variables in data matrices were made missing (set to NaN). Consequently, missing variables were equally present in sets on which Loda were trained and evaluated. The robustness was evaluated on datasets created in the same way as in Sect. 4.1 with clean training sets and 10 % of anomalies in the test sets. Loda’s hyperparameters were determined automatically as described in Sect. 4.1.1.
The effect of missing features on detector’s AUC is summarized in Fig. 8 showing the highest missing rate at which Loda had the AUC higher than 99 % of that on the data without missing variables (light grey bars). For a large set of problems the tolerance is high and more than 10 % of missing variables are tolerated with negligible impact on the AUC. There are also exceptions, e.g. magic telescope, pageblocks, wine, etc, where the robustness is very small, but we have failed to find a single cause for this fragility. Interestingly, the robustness is independent on the dimension of the problem and the success with which outliers are detected. The exception is datasets on which the detection of outliers is poor (AUC is around 0.5), which is caused by the fact that missing some variables cannot generally improve or decrease already bad AUC. The same figure also shows the rate of missing variables at which Loda stops to provide output (dark grey bars). We can see that this rate is usually much higher than the rate at which Loda retains its AUC.
4.6 Identification of features responsible for an outlier
Explaining the causes of an anomaly detection is relatively new field and there is not yet an established methodology to evaluate and compare the quality of algorithms, hence, the following is adopted here. It starts by calculating the average of Loda’s feature scores according to (4) over all anomalous samples in the testing set. Once the scores of individual features are known, they are sorted in decreasing order, which means that features in which anomalous samples deviates the most should be the first. Then, Loda with dense projections utilizing only first \(d'\) features is used and its AUC is recorded (\(d'\) has varied from 2 to \(\min (100,d)\)). If the feature relevance score (4) is meaningful, then Loda that uses first couple features (low \(d'\)) should have the same or better AUC than Loda that uses all features. This is because all testing problems were originally classification problems which means that anomalous samples were generated by the same probability and hence similar features should be responsible for their anomalousness. Experiments were performed on clean training datasets and all other experimental settings were kept same as in Sect. 4.1.
Figure 9 shows the average AUC of Loda with dense projections using the most important \(d_{\mathrm {min}}\) features, where \(d_{\mathrm {min}}\) is the least such that AUC by using only first \(d_{\mathrm {min}}\) features is higher than 0.99 times the best AUC. On average only 25 % of features are needed to identify outliers with AUC comparable or better than that of Loda with all features. This result confirms that Loda is able to identify features causing the anomaly.
It is interesting that for some problems (e.g. cardiotocography, glass, spectheart, yeast, etc.) the improvement in the AUC is significant. This can be explained by the curse of dimensionality as anomaly detectors are, by their nature, more sensitive to noise generated by noninformative features.
5 Conclusion
This paper has focused on creating a detector of anomalous samples being able to quickly process enormous amounts of data produced in many contemporary domains. Although the presented detector (Loda) was aimed to process streams of data with a constantly changing behavior, it has been compared to the state of the art in settings where all data are available at once and they can fit into the memory. The rationale behind was to compare Loda under various conditions differing by the rate of anomalies within the data, their clusteredness/scatterness to state of the art detectors, even though they are not applicable for intended scenario. This comparison, which scale is to our knowledge the biggest so far published has shown that Loda’s accuracy measured by area under ROC curve does not lag behind more sophisticated detectors. Moreover, it has revealed conditions under which particular type of detectors with the recommended setting of its parameters excels. Based on it we can conclude that datacentered detectors such as kthnearest neighbor or Local Outlier Factor with recommended setting for k are good for problems with a few number of scattered anomalies. Contrary, for problems with clustered anomalies and/or with higher rate of anomalies within the data (more than 1 %), the modelcentered detectors such as the proposed Loda, 1SVM, and Isolation Forest are more suitable. Loda was also the only evaluated method capable to process data with 3.2 millions features in a reasonable time without any modification. Last but not least, it was shown that Loda is well suited for intended domains, where data are in the form of nonstationary streams with samples observed just once, classified, and the detector is updated. The comparison to state of the art HalfSpace Trees method on seven problems of this type showed that both detectors have nearly the same detection performance, but Loda is on average 7–8 times faster.
Loda has practically demonstrated that an ensemble of anomaly detectors as weak as onedimensional histograms yields to a strong anomaly detector. Loda’s diversification of weak anomaly detectors relies on sparse random projections, which can be viewed as a random subspace sampling. It is shown that this type of diversification gives the ensemble capability to explain why a scrutinized sample is anomalous and to be robust against missing values. A thorough evaluation showed that detectors employing only features explaining anomalies have equal or better performance than detectors using all features, which shows that features explaining anomalies were selected correctly (on average 25 % of features explained the anomaly). The similar evaluation showed that the ensemble keeps on most problems its detection performance even if more than 10 % of features are missing.
To conclude, this paper has shown that there is no single detector excelling on all type of problems, as different detectors are suitable for different types of problems. The biggest advantage of the proposed Loda with respect to others is its overall good performance and very good speed to detection performance ratio, as it is order of magnitude faster than prior art on big problems while its detection performance is comparable to them. Besides, it can explain causes of anomalies and it is robust to missing variables in data.
Notes
The hypothetical knowledge of the probability function generating the data would allow to formulate the problem as a hypothesis test.
The dataset with no anomalies contained no anomalies in the training data and 10 % of anomalies in the evaluation data, which captures the case the case when the anomaly detector is trained on clean data.
Because all algorithms have not used any search for hyperparameters, Loda used fixed number of histogram being equal to \(\sqrt{n}\), where n is the number samples.
SVM and FRAC were omitted, as their online adaptation is not straightforward. Although the implementation of HSTrees is in Java, the binary (jar) reports time to process samples, which was used in this comparison.
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Editor: Joao Gama.
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Appendices
Appendix: Online histogram
The online histogram from BenHaim and TomTov (2010) approximates the distribution of data by using a set of pairs \(\mathbf {H}=\{(z_{1},m_{1})\), \(\ldots ,(z_{b},m_{b})\}\), where \(z_{i}\in {\mathbb {R}}\) and \(m_{i}\in {\mathbb {N}}\), where b is an upper bound on the number of histogram bins. It is assumed that every point \(z_{i}\) is surrounded by \(m_{i}\) points, of which half is to the left and half is to the right to \(z_{i}\). Consequently, the number of points in the interval \([z_{i},z_{i+1}]\) is equal to \(\frac{m_{i}+m_{i+1}}{2}\), and the probability of point \(z\in (z_{i},z_{i+1})\) is estimated as a weighted average.
The construction of the set \(\mathbf {H}\) is described in Algorithm 3. It starts with \(\mathbf {H}=\{\}\) being an empty set. Upon receiving a sample, \(z=x^{\mathrm {T}}w\), it looks if there is a pair \((z_{i},m_{i})\) in \(\mathbf {H}\) such that z is equal to \(z_{i}\). If so, the corresponding count \(m_{i}\) is increased by one. If not, a new pair (z, 1) is added to \(\mathbf {H}\). If the size of \(\mathbf {H}\) exceeds the maximal number of bins b, the algorithm finds the two closest pairs \((z_{i},m_{i})\), \((z_{i+1},m_{i+1})\), and replaces them with an interpolated pair \(\left( \frac{z_{i}m_{i}+z_{i+1}m_{i+1}}{m_{i}+m_{i+1}},m_{i}+m_{i+1}\right) \). Keeping \(z_{i}\) sorted makes all the above operations efficient.
The estimation of the probability density in point \(z=x^{\mathrm {T}}w\) is described in Algorithm 4. Assuming the pairs in \(\mathbf {H}\) are sorted according to \(z_{i}\), the algorithm starts by finding i such that \(z_{i}<z\le z_{i+1}\). If such i exists, then the density in z is estimated as \(\frac{z_{i}m_{i}+z_{i+1}m_{i+1}}{2M(z_{i+1}z_{i})}\), where \(M=\sum _{i=1}^{b}m_{i}\). Otherwise it is assumed that z is outside the estimated region.
Construction of datasets
Benchmark problems for the evaluation of batch anomaly detectors were constructed following the process in Emmott et al. (2013). It advocates the creation of anomaly detection benchmarks from real data, since artificial problems can be too far from realistic problems. 36 source datasets from which all benchmark problems have been created were downloaded from Frank and Asuncion (2010) from a category of classification problems with numerical attributes and without missing variables. The list of downloaded datasets together with informations about the number of normal / anomalous samples is in Table 5
If the source dataset was a binary problem, the larger class was used as a normal class and the smaller as the anomaly class. Multiclass datasets were converted to binary ones as follows:

1.
Train a random forest classifier to solve the multiclass problem in the original source dataset using all samples.

2.
Estimate the confusion matrix \(\mathbf {C}\) with \(\mathbf {C}_{k,j}=P(j\vert x_{i},k)\), where the conditional probability is the probability returned by random forest that the sample \(x_{i}\) belongs to the class j when k is true.

3.
Create a complete graph with vertices being classes from the datasets and edges having weights from confusion matrix as \(\mathbf {C}_{k,j}+\mathbf {C}_{j,k}\).

4.
Compute the maximum weight spanning tree of the graph to identify “mostconfusable” pairs of classes.

5.
Twocolor the maximum spanning tree such that no adjacent vertices has the same color. Each color determines set of classes that make the normal and anomalous class.
Notice that the above construction aims to maximize the difficulty (confusion) between normal and anomalous class.
Anomalous samples were divided into four groups according to the probability that a sample from the anomaly class is assigned to an anomaly class estimated the by the kernel logistic regression with a Gaussian kernel (Zhu and Hastie 2005). Based on this score, all anomalous points were assigned a difficulty category as: easy ([0.84, 1), medium ([0.7, 0.84), hard ([0.5, 0.7), very hard (0, 0.5].
A concrete benchmark problem with a given fraction of anomalies (experiments used fractions 0.005, 0.01, 0.05, 0.1) and difficulty (easy, medium, hard, very hard) was created as follows:

1.
Randomly divide samples into training a testing set allowing maximum of 10000 samples in each set.

2.
If anomalies should be clustered, select randomly a pivot and find sufficient number of nearest points in chosen difficulty category such that the final training and testing sets have the desired number of anomalous samples. If anomalies are not clustered, select anomalous samples randomly.

3.
Divide the selected anomaly samples into training a testing set and mix with normal samples.
The final benchmark problem is attached a clusteredness category according to a fraction of sample variance of normal samples to the sample variance of anomalous samples as: high scatter (0, 0.25), medium scatter [0.25, 0.5), low scatter [0.5, 1), low clusteredness [1, 2), medium clusteredness [2, 4), and high clusteredness \([4,\infty )\).
Created benchmark problems have three basic properties (rate of anomalies within data, difficulty of anomalies, and clusteredness) according to which they can be divided.
Experimental results
See Fig. 10.
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Pevný, T. Loda: Lightweight online detector of anomalies. Mach Learn 102, 275–304 (2016). https://doi.org/10.1007/s1099401555210
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DOI: https://doi.org/10.1007/s1099401555210