# Using Intelligent Edge Devices for Predictive Maintenance on Injection Molds +
## 2 Literature Review
### Fault Trees (FTs) and Generalized Fault Trees (GFTs)  
Reliability analysis is concerned with the failure probability of a given system or part. It relies on statistical data from relevant parameters of a given process. One of the most widely used reliability methodologies is FTA [27], which is exploited by well-known entities such as NASA [28].  
FTA graphically represents a failure event, designated as the top event (TE) using a fault tree (FT). An example of a generic FT is depicted in Figure 2. This structure has two main components, namely, the basic events (BEs) and the gates or operators. BEs are theroot factors responsible for causing a given TE, while gates are logic operators (\(AND\) and \(OR\)) that compute the probability of the TE based on the probability of the BEs according to \(AND(F_{X},F_{Y})=F_{X}F_{Y}\) and \(OR(F_{X},F_{Y})=F_{X}+F_{Y}-F_{X}\,F_{Y}\), where \(X\) and \(Y\) are two BEs and \(F_{X}\) and \(F_{Y}\) are the respective cumulative density functions (CDF) of \(X\) and \(Y\). Note that the \(AND\) gate triggers when both of the input BEs have occurred, while the \(OR\) gate triggers when at least one of the input BEs has occurred. In addition to quantitative analysis enabled by the calculation of TE probability, FTA allows for qualitative analysis of the minimal cut sets (MCS), which are the sets containing the minimum number of BEs to ensure the occurrence of a TE [29] (for example, the minimal cut sets for the FT shown in Figure 2 are \(C1=\{BE1\}\) and \(C2=\{BE2,BE3\}\)).  
There have been several approaches for quantitative analysis of FT proposed in the literature, such as state-space models [30; 31; 32], simulation methods [29; 33; 34], algebraic approaches [35; 36], and commercial software tools and model checkers such as Galileo [37], Coral [38], DFTCalc [39], and Storm [40], among others [41]. Although research effort that has been dedicated to FTA, the methodology does have a number of issues [42; 43]. Typically, the TE represents the failure of a complex system, while BEs represent the failure of parts or components within the main system. This has two limitations: (1) expert knowledge is necessary to build the FT structure according to the possible failure modes of the components, which may be tedious and expensive; and (2) the probability of BEs, which is the failure probability of a given part or component, does not accord with the actual state of degradation of these components, as its calculation is only based on the hazard rate calculated from historical data on the malfunctioning of these components.  
To overcome the aforementioned issues, the GFT methodology was proposed in [42]. Contrary to FTA, the GFT methodology does not rely on previous expert knowledge of the system, as the BEs are generated automatically from data; in addition, the tree structure that best describes a given TE is automatically obtained through a training procedure. The definition of BEs from continuous data can be achieved by discretizing the data, using cutting-edge parameters [42; 43], or using an anomaly detection technique to detect abnormal events which are then used as BEs in the GFT analysis [13]. With respect to the training process, it can be used to either minimize the error between the CDF obtained by the GFT and the real CDF of the TE for a given dataset, or to optimize the maintenance costs.