## 2 Review of Related Scientific Works
### Time Series Anomaly Detection  
One of the open research questions regarding GTSMLFM is, when do such models provide acceptable forecasts [3]? Such response can be obtained from anomaly detection algorithms and models, which can alert on point anomalies. In this section, we provide an overview of anomaly detection techniques, focusing on the ones developed and applied in a time series setting. We use the terms anomaly, outlier, or deviant interchangeably [29], to denote \"observations that deviate so much from others as to arouse suspicion that it was generated by another mechanism\" [30]. In particular, outliers related to time series data must also consider the behavior across time [31]. Outliers can be characterized into many types. The first characterization of this type can be found in [32], who introduced the concepts of two time series' outlier types: those (a) that affect a single observation (_type I_), and those (b) that affect an observation and the subsequent ones (_type II_). More recently, [31] distinguishes between _point outliers_ (single point in the time series, which has an unusual value when compared to the whole time series or the neighboring points) and _subsequence outliers_ (points which may not be outliers by themselves, but the sequence arrangement is anomalous). _Subsequence outliers_ are further classified into _contextual anomalies_ (they are anomalous in the context of the surrounding observations) or _pattern anomalies_ (they are anomalous regardless of the surrounding observations) [33]. In this research, we limit ourselves to point outliers. Anomaly detection techniques are frequently classified into three categories: statistical, distance-based approaches, and model-based approaches [34]. The first anomaly detection techniques were developed in statistics and remained among the most frequently used ones. Non-parametric techniques usually allow fast computations and are adopted where such speed is of primary importance. Among them, we find the histogram-based approaches, which assume feature independence and determine the outliers based on the histogram distribution [35; 36; 37; 38]. Other non-parametric approaches are bitmap time series anomaly detectors, which compute the relative frequency of its features to create a bitmap and identify anomalous time series [39; 40], and statistical methods that allow estimating outliers based on a kernel density estimation by using a kernel function. Such kernel functions can provide a probability estimate, given that function is a probability density function [34; 41; 42; 43]. Among the parametric methods, we find the Gaussian methods, such as box-plot anomaly detection [44], the Gaussian process [45; 46], or regression approaches such as least squares regression [47; 48; 49]. Statistical anomaly detection methods cannot be applied on datasets with an unknown distribution [38]. Different approaches were developed to overcome this issue. When considering the distance between data points, we find the k-nearest-neighbors (kNN), which determines the outliers based on the kNN distance. While approaches were developed to determine the best k parameter [50], some techniques attempted to avoid dependence on the k-value. One such method is the local outlier factor (LOF) method, which computes the distance from a point to all other points in the dataset [51]. A similar approach was followed at the outlier detection using indegree number (ODIN), which computes the number of instances that contain a given point in their neighborhood. However, a parameter is required to determine the outlier threshold [52]. A different method was developed by [53], who introduced the multi-granularity deviation factor to identify local density variations that lead to isolated outliers or outlying clusters. Variations to the LOF method were developed to solve some of its shortcomings.