On Equivalence of Anomaly Detection Algorithms

Anomaly detection (AD) is ubiquitous in many domains. Some of its applications are: intrusion detection, which refers to detecting malicious activity in a computer-related system; fraud detection such as credit card fraud, insurance fraud, tax fraud; medical anomaly detection, where the anomalies are detected in medical images, or clinical electroencephalography (EEG) records to diagnose or prevent diseases; anomalies in social networks, which refers to irregular and often unlawful behavior patterns of individuals in social networks, some examples of these are scammers, sexual predators, online fraudsters; industrial AD, which refers to detecting anomalies in industrial processes that are carried out countless times; AD in autonomous vehicles to prevent attackers from taking over the vehicle, and so on. [3, 27].

For real-world applications, labeling can be expensive and time-consuming, for this reason, AD is generally cast as an unsupervised learning problem. In this setup, the learning is done directly from patterns naturally occurring in the data. Then, a follow-up question is: out of a set of algorithms, how to determine which AD algorithm is better and in what way without having labels? As it turns out, evaluating an algorithm’s performance under the unsupervised learning setting is not a trivial task. Nonetheless, it is of paramount importance to be able to compare AD algorithms beyond the supervised setting in a toy dataset (i.e., simulated datasets that include labels), and labeling real-world datasets is not a practical or scalable solution for most domains.

An anomaly is defined as a rare observation or an observation that deviates from the norm [3]. Mathematically, it can be described as a low-likelihood data point with respect to some unknown underlying likelihood function \(f:\mathcal {X}\rightarrow \mathbb {R}^+\), where \(\mathcal {X}\) is the spaces where the data lives. Furthermore, the set of anomalies is given by \(a=\lbrace x:f(x)\lt \epsilon \rbrace\) for some small \(\epsilon\). The goal of AD is to approximate f or some form of it, especially in the regions of low likelihood. Note that if \(g=T\circ f\) with T being a increasing transformation, then \(a=\lbrace x:g(x)\lt \delta \rbrace\) using an appropriate \(\delta\) yields exactly the same a as when using f, which is why approximating a “form” of f suffices.

An AD algorithm (sometimes called scoring function) is a mapping \(r:\mathcal {X}\rightarrow \mathbb {R}^+\). Suppose r and s are two different AD algorithms that can detect anomalies \(a_r=\lbrace x:r(x)\lt \delta _r\rbrace\) and \(a_s=\lbrace x:r(x)\lt \delta _s\rbrace\), then the challenge is to meaningfully compare r and s in the context of AD. This meaningful comparison entails special considerations that will be discussed later on. It is important to recall that even though AD can be described mathematically, its motivation is practical, and usually there is interest in detecting only a particular subset of a (the subset relevant to the application in question); Foorthius et al. [13] provide a qualitative description of different types of anomalies of interest. As a consequence, whereas mathematically one can compute a sensible distance between f and r, and f and s to determine which one is closer to f, this is only truly meaningful if there is a high overlap between \(a_r\) and \(a_s\). This reasoning is further demonstrated in Figure 1.

We have distilled the problem of comparing AD algorithms in the unsupervised setting into two tasks: (1) determining to what extent the algorithms are detecting the same anomalies, if they are, (2) choosing a criterion to determine which one is “better”; we use “better” to emphasize that there is no intrinsic better, instead one chooses a criterion that has suitable properties and that criterion is used to evaluate the different AD algorithms. There has been pivotal work on (2) such as the work by Goix et al. [9, 15, 16], who proposed criteria to evaluate the quality of unsupervised AD algorithms based on the MV (Mass–Volume) and EM (Excess–Mass) curves. In a similar fashion, Marques et al. [22, 23] proposed a different criterion that considers the degree of separability of each point with respect to the rest weighted by the anomaly score. We will discuss these approaches and others in more depth in the RelatedWork section.

Our work focuses on task 1), which consists of defining a notion of equivalence between r and p (two AD algorithms) that is meaningful in the context of AD. The fact we are in the context of AD is crucial because it requires the notion of proximity to have certain properties. If this were not the case, a viable notion of proximity would be the Kullback–Leibler (KL) divergence assuming that r and s are converted into probability distributions. Therefore, task (1) consist of designing a notion of distance or equivalence that exhibits certain properties. One of them is that it is more important that two AD algorithms agree over sets of low-likelihood than sets of high-likelihood (which in the case of KL divergence is the opposite because sets of high-likelihood have higher mass, hence higher weight in the calculation).

Our work is complementary to that by Goix et al. and Marques et al. [15, 22] in the quest to evaluate unsupervised AD algorithms. More specifically, any two algorithms could pass first through our algorithm to determine whether they detect the same of kind of anomalies, if so, then [15] or [22] provide criteria that can be used to indicate which one is “better”, if not, then further comparisons to determine which one is “better” are not meaningful; this procedure is illustrated in Figure 2, where we also included a deployment block that uses an additional model to produce of an explanation of that prediction. Moreover, this line of work as standalone is worthy of pursuit as it helps elucidate what each of the AD algorithms is doing, which is particularly useful since modern AD algorithms are often constructed as black boxes [26].

The rest of the article is organized as follows: Section 2 presents a literature review, and also lists our specific contributions, in comparison with those contributions by other works; Section 3 provides the background and the technical motivation for our algorithm; Section 4 describes the algorithm in detail; Section 5 shows and discusses the properties our algorithm has; Section 6 shows empirical evidence of the results our equivalence criterion can achieve on simulated data and on a real-world dataset of the daily energy consumption registered by electrical meters; finally Section 7 provides the conclusions.

The literature on AD is vast, as there are countless AD algorithms [11]. In contrast, the literature on how to compare different AD algorithms is scant. Traditionally, there have been three approaches to compare AD algorithms: the first one is supervised learning on publicly available datasets that are labeled. Among these, there is the work by Hasain et al. [17], where they deal with detecting anomalies in data that is being streamed. Another example is [28], where the application is cybersecurity and network intrusion in emerging technologies such IoT (Internet of Things) or the work in [10] that deals on a similar problem. In this approach, comparing AD algorithms is trivial because the receiving operating characteristic (ROC) curve can be calculated, as well as the area under the ROC curve (AUC), which also serves as comparison criterion. The drawback, though, is that it can only be utilized for applications that have publicly available datasets that have been labeled. Even in those cases, it is necessary to assess whether the data in question has a similar distribution to that of the public dataset, e.g., fraudsters are adaptive [7], then an outdated dataset becomes obsolete for training algorithms for fraud detection.

The second approach to comparing two AD algorithms is supervised learning on simulated data, where labels can be generated. Some of the works using this approach are: Anton et al. [2], where they deal with cybersecurity and intrusion detection; Flach et al. [12], where they want to detect anomalies on Earth observations; or Meire et al. [24], where they want to detect anomalies on acoustic sounds. Comparing algorithms is trivial for the same reason as in the first approach. The drawback with this approach is that it requires an accurate model to simulate non-anomalous and anomalous points, and for many real-world applications even state-of-the-art simulators still fail to close the gap between real-world data and simulated data [31].

Finally, the third approach is unsupervised learning, which is the case when there are no labels. This is the most useful case because it does not require any human labeling and it can be used on any application; the challenge is that it is not trivial to compare different AD algorithms. In fact, despite its usefulness this approach has been vastly understudied, as pointed out by Ma et al. [21]. More specifically, Ma et al. found only three techniques (that we also found independently) that address the problem of algorithm comparison and evaluation in the context of AD in the unsupervised setting. Those three approaches are [15, 22], which we mentioned previously, and [25].

The criterion provided by Goix et al. [15], namely the MV curve, is defined by the following equation \(MV_s(\alpha)=\lbrace \inf _{u\ge 0}\lambda (s\ge u) \, | \, \mathbb {P}(s(X)\ge u)\ge \alpha \rbrace\), where \(\lambda\) is the Lebesgue measure, s is the AD algorithm, and \(\alpha \in [0,1]\). For a given \(\alpha\), \(MV_s(\alpha)\) consists of finding the infimum over u of the set \(\lambda (s\ge u)\) (which amounts to making u as big as possible), such that the probability of \(s(X)\) being greater or equal than u is greater or equal than \(\alpha\), then the \(MV_s(\alpha)\) corresponds to \(\lambda (s\ge u)\). If \(MV_r(\alpha)\le MV_s(\alpha)\), then AD algorithm r is better than s. The EM curve has a similar construction. Intuitively, the idea is that the level sets have minimum volume, hence minimum MV curve, when the generating function, f, is used to generate the MV curve, therefore as an anomaly scoring function r gets closer to f, its MV curve will decrease (we recognize this discussion of that work is rather opaque, but an in-depth discussion is out of the scope. Interested readers may check [9]). We borrow an idea from this work indirectly, which is that you do not have to compare directly the generating function f with the scoring function r, instead you can compare their level sets or even their rankings.

Marques et al. [22] criterion to compare AD algorithms is named IREOS (Internal, Relative Evaluation of Outlier Solutions). It estimates how separable is each point from all the other points by using a maximum-margin classifier, which in their case it is a nonlinear SVM, then this is weighted by the anomaly score of the corresponding point. If the points with largest margins have the highest anomaly scores, then IREOS will be the highest possible. The higher the IREOS, the better the scoring function. Nguyen et al. [25] propose different indexes whose underlying idea is that normal data should form one big cluster whereas anomalies will form a smaller cluster. They focus on two concepts: compactness and separability. Therefore, according to them, a good AD algorithm should separate the data in compact clusters that are far from each other. One of their indexes, root-mean-square standard deviation (SDT) is given by the formula \(RMSSTD=(\sum _{i=1}^{NC}\sum _{x\in C_i} ||x-c_i||^2)/(P\sum _{i=1}^{NC}(n_i-1))\), where NC is the number of clusters, \(C_i\) is the ith cluster, \(c_i\) is the center of \(C_i\), \(n_i\) is the number of objects in \(C_i\), P is the number of attributes in the dataset, and x is the datapoint. The numerator is measuring how compact each cluster is by calculating \(||x-c_i||\) and summing over clusters; the denominator is a normalization constant. Of those three approaches, Ma et al. [21] suggest that none of them is useful in practice, as they select models only comparable to a state-of-the-art detector with random hyperparameters. While we find that claim arguable, because the AD models were not tested as to whether they detect the same anomalies or not, it is clear that significant improvement is needed in this area.

A research area, relevant to AD, that has received significant attention in the last few years is Explainable Artificial Intelligence (XAI) [6]. XAI seeks to equip opaque models with explanations or interpretations that are understood and accepted by a human audience [6, 8]; this process is known as post-hoc explainability [6]. The definition given by Arrieta et al. [6] recites as follows: “post-hoc explainability techniques aim at communicating understandable information about how an already developed model produces its predictions for any given input”. XAI is important for AD because many times flagging a data point as anomaly is not enough; an explanation of why the point was flagged is also needed.

An example of this is the work by Kim et al. [20]. Their work is on AD on maritime engines. For this application, knowing that there is an anomaly is not sufficient, instead one needs to know what sensor is triggering the anomaly. In their work, they use traditional AD algorithms and equip them with XAI models that generate an explanation. Another example is the work by Gnoss et al. [14], which is on detection of erroneous or fraudulent business transactions and corresponding journal entries; and in this case too, when an anomaly detected, an explanation is needed. Furthermore, there is work such as the one by Barbado et al. [4], whose application is AD for fuel consumption in fleet vehicles. Their work, not only generates an explanation why a point was labeled an anomaly, but it also provides a counterfactual recommendation, that is, it provides what could have been done differently about that vehicle to turn it into an inlier. All the works presented in this paragraph employ post-hoc explainability techniques. There are many more examples of AD applications that have incorporated XAI; a more comprehensive list can be found on the review by Yepmo et al. [30].

Another approach to having AD algorithms with explanations is to make the AD algorithms transparent, that is, explainable by themselves; a typical example of a transparent algorithm is linear regression [6]. In this category, Alvarez-Melis et al. [1] propose self-explaining neural networks. A self-explaining neural network is a neural network that behaves smoothly, as a result, it can be approximated by a linear function at any point. Alvarez-Melis et al. enforce the smoothness by adding the term \(\mathcal {L}_\theta =||\nabla _x f(x)-\theta (x)^TJ_x^h(x)||\) to the loss function, where f is the neural network function, x is the datapoint, h is a function that maps from input space to feature space, \(J_x^h(x)\) is the Jacobian of h with respect to x, and \(\theta (x)\) refers to the parameters of the linear approximation at point x. If \(\mathcal {L}_\theta =0\), then the model can be locally approximated by a linear function, thus making the model explainable. A different approach with the same underlying idea was taken by Barbado et al. [5]. Their approach consists of taking a trained one-class SVM and extracting rules from it. The rules are obtained by partitioning the input space in hypercubes that separate anomalous points from non-anomalous points, in other words, the rules are of the form: if x (the datapoint) is inside a hypercube that contains inliers, then x is an inlier. Then, these rules are used for prediction and also serve as explanations.

Above we presented some examples of AD algorithms equipped with XAI post-hoc techniques, and transparent AD algorithms. Our work and post-hoc techniques operate in a complementary fashion. To see this, we consider the definition of post-hoc technique presented above. From that definition, we can notice there are already two differences between our work and post-hoc techniques. First, our work does not say anything about how the models produce predictions, rather our work deals with what predictions one model produces in contrast to another model; second, our work does not require a human audience to interpret explanations. Note that the first and second differences come together because if there is no explanation, then there is no need for an explainee. Another advantage of not using explanations to analyze a model is that one does not have to compare different explanations for the same model to determine which explanation is better. In fact, this is one weakness of XAI, namely, there is no consensus or a rigorous mathematical definition of what constitutes a good explanation [6, 8]. Although there has been progress, such as the work by Hoffman et al. [18] that proposes metrics to quantify the quality of explanations, Arrieta et al. still conclude that more quantifiable, general XAI metrics are needed to support the existing measurement procedures and tools proposed by the community [6].

To sum up, our work and XAI try to achieve the same general goal, namely, to obtain insights from black-box models; but do so in a different way. XAI does so by generating explanations that have to interpreted by a human, whereas our work directly answers the question whether two AD algorithms are equivalent without. XAI and our work can work jointly, as illustrated in Figure 2. In fact, if two algorithms are equivalent as given by our work, then those two algorithms should have the same explanations; the converse should also hold, that is, if two algorithms, for every data point, have the same explanation, then they should be equivalent.

In this work, we propose, to best of the authors’ knowledge, the first equivalence criterion for AD algorithms. More concretely, our work aims to determine to what degree two AD algorithms (scoring functions) are detecting the same kind of anomalies, which is different from designing a criterion to determine which AD algorithm is better. An equivalence measure is crucial because it only makes sense to determine “better” algorithms from a set of AD algorithms that detect the same kind of anomalies, otherwise they are just different and the search for “better” AD algorithms becomes superfluous.

The analysis so far uses rankings, instead of anomaly scores, because the work done in [19, 29] was formulated in terms of rankings. However, Equation (3) and its weighted version can be calculated with anomaly scores because the ranking map \(\mathcal {R}:\mathbb {R}\rightarrow \lbrace 1,2,\ldots ,n\rbrace \in \mathbb {N}\) is order preserving. Moreover, we have that converting \(\mathbf {s}\) into \(\mathbf {r}\) takes a permutation; and a re-scaling map g, that matches the values of \(\mathbf {r}\) with those values of the permuted version of \(\mathbf {s}\). While our equivalence criterion does not concern itself with g, it explicitly uses the anomaly scores, \(\mathbf {r}\) and \(\mathbf {s}\), in the weight calculation.

The proofs of the mathematical statements presented here can be found in the Supplementary Material section at the end of the document.

The results section is divided into two parts: the first part uses simulated data and serves as benchmark for comparing GEC with SEC, as well as other future algorithms dealing with the same problem; in this part, we also showcase the flexibility of GEC as regions can be defined to assign different levels of importance to high anomalous point. In this part, we also validate how GEC matches intuition using a dataset with a Gaussian distribution. The second part uses real data from residential electric meters, where we test GEC in a real-world application. In this part, not only GEC provides insight about the AD algorithms, but about the dataset itself.

We argue for the importance of the overlooked problem of having a criterion to measure equivalence between AD algorithms in the context of unsupervised learning. We devise a set of properties that are desirable for such equivalence criterion as they ensure highly anomalous points will have priority on the calculation of it. We propose, GEC, an equivalence criterion that satisfies all devised properties; in addition, GEC is flexible in that it allows the definition of classes, which serve to assign importance to anomalies; this is useful because different applications assign different values to anomalies (e.g., anomalies in medical applications are more critical than in commercial applications). We mathematically show, and empirically validate that GEC satisfies the properties in simulated data as well as in real-world data.

First, we will illustrate how the Kendall Tau coefficient is calculated with a simple example. Let \(\mathbf {m}\) and \(\mathbf {n}\) be rankings that rank the points \([1,2,3,4,5,6]\) as follows: \(\mathbf {m}=[4,6,3,2,1,5]\) and \(\mathbf {n}=[6,4,2,3,5,1]\), namely element 4 is ranked one according to \(\mathbf {m}\), but two according to \(\mathbf {n}\). Note that the enumeration \([1,2,3,4,5,6]\) is just an indexing; it could have been any other enumeration, e.g., \([a,b,c,d,e,f]\). In fact, we will re-index \(\mathbf {m}\) such that \(\mathbf {m}=[1,2,3,4,5,6]\), which yields \(\mathbf {n}=[2,1,4,3,6,5]\), that is, we rename element 4 to 1, which is the rank given by \(\mathbf {m}\), then 6 to 2, and so on. This way, we can simply use the order of the naturals to find concordances and discordances, as shown in Figure 3. Note that \(\langle \mathbf {m},\mathbf {n}\rangle =\sum _{j\lt i}\text{sgn}(m_i-m_j)\text{sgn}(n_i-n_j)\) is the element-wise multiplication of the subfigures shown in Figure 11 and subsequent summation.

In this particular example \(\langle \mathbf {m},\mathbf {n}\rangle =9\), which comes from 12 concordances, and 3 discordances.

For the SEC example, we will arbitrarily use the following two scores: \(\mathbf {r}=[(a,0.5), (b,1.2), (c,1.6)]\) and \(\mathbf {s}=[(a,1.3),(b,0.2),(c,0.7)]\), where for \(\mathbf {r}\), the tuple \((a,0.5)\) means the point labeled a got a score of 0.5. Recall that SEC, also denoted as \(\sigma\), is given by \(\sigma =4\sigma _n-3\), where \(\sigma _n=\tfrac{\sum _i^n \hat{r}_i\hat{s}_i}{\sum _{i=1}^ni^2}\). The denominator is just a constant, whereas the numerator is calculated as follows: \(\sum _{i=1}^3\hat{r}_i\hat{s}_i=1\cdot 3+2\cdot 1+3\cdot 2=11\). Then, \(\sigma = 4\cdot \frac{11}{14}-3=0.1428\), which is low because \(\mathbf {r}\) and \(\mathbf {s}\) order \(a,b,c\) completely differently.

The pseudocode will be divided into two blocks:

\(C_i,j\) refers to the jth region (subclass) of the anomaly score i (i takes values \(\mathbf {r}\) or \(\mathbf {s}\)).