# Outlier Detection for Multivariate Time Series Using Dynamic Bayesian Networks
## 1 Introduction  
In recent times, the machine learning community has boomed coupled with the always-expanding desire to acquire maximum benefit from collected data, apparent in sectors such as biomedicine, socio-economics, and industry. [PERSON] [1] has defined anomalies as observations that deviate appreciably from the sample in which they occur. In the current study, an outlier is described as a data element or segment which there is no explanation for it, being suspected of not have been generated by the data's underlying processes. Outliers can mislead analysts to altogether different insights. However, their discovery is crucial in acquiring a better understanding of the behavior of the data, leading to the development of more efficient methods. Multivariate time series (MTS) are defined as sets of observations measured along time, being a representation for time series analysis. Each observation depicts a collection of variables, which the combined evolution over time is the object of study. In this context, we propose METEOR--MultivariatE Time sEries OutlieR--an outlier detection method to identify abnormal entities among real-world MTS datasets. Outlier detection algorithms in MTS are typically not found in existing literature, which solely considers univariate case [2, 3], overlooking anomalies arising from inter-variable temporal contexts. Within the vast world of anomaly detection [4; 5] extensity and versatility are craved traits. In _time series_ (TS) data, temporal trends play a crucial role in anomaly discovery, where data patterns are not assumed to change abruptly through time. Most of the existing techniques do not take these temporal dependencies into account, leaving them less effective. When time is taken into consideration, mostly univariate temporal data is considered [6]. An example is autoregressive models, extensively used in TS data. In such cases, data points typically depend linearly on previous values and a stochastic term, representing a random process. Alternative procedures consider that outliers are high residual entities with respect to a model expressing the time-varying process [5; 7]. Outlierness can be evaluated according to a distance or similarity measure. Anomalies are usually considered to be isolated from the rest of the data. An example is to use the distance of an element to its \(k\)-th nearest neighbor as a score [4]. Such reasoning can be applied to measure the distance between discrete sequences [8], which can easily represent TS data. Similarly, certain methods create a boundary between an anomalous and normal class. Data instances are scored given their distance to the boundary, typical in clustering and classification methods [9; 10]. Recent efforts have been invested trying to satisfy the existing gap in MTS anomaly detection [11; 12; 13]; however, a complete and available implementation of such approaches is non-existing. This forces analysts to use typical univariate strategies. Temporal dependencies within and between variables can be modeled using _dynamic Bayesian networks_ (DBN) which extend traditional Bayesian networks to temporal processes described by MTS. These are probabilistic graphical methods capable of encoding conditional relationships of complex MTS structures via transition networks. A modeling technique, so-called tree-augmented DBN (tDBN) [14], is used to provide a network possessing optimum inter and intra-time slice dependencies between discrete variables for each transition network, verified to outperform existing methods in the literature. DBNs already proved to benefit anomaly detection [15] and gene expression data modeling [16]. Analogous to unsupervised learning, our fully automatic system, called METEOR, encrypts the need for any prior knowledge. The latter resides in the statistical paradigm, providing a tDBN representing a normality standard for anomaly detection, where observations are scored using the transition networks. Both stationary and non-stationary tDBNs are studied.