# Gaussian Processes for Signal Processing and Representation in Control Engineering
## 4 Anomaly Detection in Industrial Processes  
Gaussian Process is growing in popularity of usage in domains connected with statistics and machine learning. It is an especially efficient solution, while we are dealing with a problem of obtaining data e.g., for faulty cases or just an overall lack of comprehensive data. The main areas where GP is used are: regression, prediction, classification and identification. Most of solutions, which are currently used in practice, in whole or in part, is classified to at least one of the mentioned areas. The usage of statistics and machine learning in the industrial process is not anything new. There is a large debate about which approach could give the best results on which issues. However, on the subject of detecting faulty states, GP seems to be widely used and appreciated. In industrial processes, it is key to guaranteeing continuous workflow without any disturbances and simultaneously taking care of production quality [19]. Any failures can lead to significant costs, so assuring the reliable anomaly detection method has a great impact on the world's economy. In \"Outlier detection based on Gaussian process with application to industrial processes\" [19] we are encountering the problem of the detection of anomalies called outliers. They agree that the the definition of an outlier can vary depending on the problem, but basically there are \"patterns in data that do not conform to a well defined notion of normal behavior\". In the case of industrial processes, outliers are not so easy to measure and contaminate, because they can come from different sources such as parameters, sensors and actuators fault or any structural changes. Principal component analysis (PCA) and partial least squares (PLS) are the two most common techniques used in anomaly detection but suffers from practical conditions of gathering abnormal datapoints--outliers. GP stands as an alternative for typical model-based methods in order to handle industrial scales [19]. The authors' main workflow consists of:  
* extending the GP models in order to detect outliers in industrial processes;
* selecting GP specific parameters suitable for problem;
* testing effectiveness of algorithm with synthetic and real-world data. For calculating outlier score both regression and classification frameworks of GP were used. Both were carefully set up with special care for defining specific selections of its parameters. Different approaches for selecting every parameter used in the paper method were discussed. However, in conclusion, the mean for all approaches in the paper was set to zero and the kernel function has been set to squared exponential kernel and to a simple composite. In addition, the authors used a specified likelihood function: Student's t and Laplace distribution and determined special inference methods [19]. At the end they came up with three solutions based on GP regression and one based on GP classification. Each of them has been exhaustively tested on the Tennessee Eastman benchmark process, electric arc furnace process control and wind tunnel process control. Presented methods were compared to well-known competitors such as: Gaussian mixture model (GMM), k-mean, k-nearest neighbor, support vector data description and principal component analysis (PCA). The drawn conclusion is clear: compared with traditional detection methods, proposed scheme has less assumptions and is more suitable for modern industrial processes [19]. Synonymous problems were encountered in article [20], where anomalous situations are detected in Industry 4.0 environment. The main principles of these two projects are similar--diagnosis of faulty states in industrial usability. However, here we encounter Big Data environment with concepts called Digital Twin (DT). DT is defined as \"a set of virtual information constructs that fully describes a potential or actual physical manufactured product from the micro atomic level to the macro geometrical level\". According to [20], the main characteristics of DT are as follows:  
1.