# Causality Analysis with Different Probabilistic Distributions Using Transfer Entropy
## 1 Introduction  
In recent years, the field of causality has experienced significant development in its fundamental and theoretical aspects as well as in various applications. Nevertheless, it is uncommon to make a distinction, as suggested by [1], between two prominent problem areas within the field. The first problem area concerns evaluating the probable effects of interventions that have been applied or are being considered. This is known as the \"Effects of Causes\" (EoC) problem. The second problem area involves determining whether an observed outcome in an individual case was caused by a prior intervention or exposure. This is referred to as the \"Causes of Effects\" (CoE) problem. These considerations mainly concern the humanities, in particular philosophy [2]. With growing interest in the subject, research has moved to other areas of interest, e.g., education [3], medical science [4], economics [5], epidemiology [6], meteorology [7], and even environmental health [8]. While the engineering sciences are not omitted, here the nature of causality is understood in a different way. The purpose of such analysis is to find the cause, in the sense of the root cause error (or fault), while the phenomenon of the \"effect\" is known or predicted using other available methods. In engineering applications, causality is always accompanied by the concept of relationships. No analysis would make sense without data generated artificially (simulations) or obtained from industrial plants (processes). The second case deals with very complex systems, resulting in a huge amount of collected data or observations. It is important to be aware that control engineers understand the subject of causality in a specific narrow sense that is very close to the task of fault detection and diagnosis, which is a very popular aspect. On this basis, we may distinguish the mainstream research on fault detection (or groups of faults), in which the task of determining the cause seems to bea secondary issue. Moreover, the majority of existing approaches use the flow structure of the process, which can be obtained in two ways: first, by available process knowledge and modeling, and second, by data-based causality models [9]. From this perspective, causality analysis can provide support for fault detection. Multi-loop control systems generate another challenge, which is witnessed not by maintenance teams but by control engineering or daily operations staff. In cases where it is necessary to tune a multi-loop system, it is crucial to know which loop is the so-called problem generator that causes poor performance for itself and degrades other loops as a consequence. In such a case, control engineers must find that particular loop (the cause) in order to avoid the need to tune other loops. Similar challenges appear in case of alarm systems. In the case of an alarm, the operator is quite often overwhelmed by a cascade of successive derivative alarms. To recover the situation, it becomes necessary to dig into the cause. From this perspective, causality analysis may help in the design of so-called intelligent alarm management systems [10]. This paper addresses the first challenge, i.e., poor control propagation and cause detection. Over the years, many experimental methods have been developed to deal with this issue. They can be generally divided into those requiring a model of the tested object and the so-called model-free approaches. In practice, in root cause analysis the appropriate method should be selected carefully based on process dynamics, the available data, and the type of error. In the literature, there may be found many examples of often conditional solutions, i.e., Granger causality and its extensions [11], cross-correlation analysis [12], information-theoretical methods [13], frequency domain methods [14], and Bayesian nets [15], and it is important to consider which of these methods are sufficient. The Transfer Entropy (TE) method [16] is a model-free approach, which means it does not require a mathematical model.