# Sensor-Data-Driven Prognosis Approach of Liquefied Natural Gas Satellite Plant
## 4 Methodology
### Sensor-Data-Driven Prognosis Methods
#### 4.2.1 Fuzzy-Inductive-Reasoning Methodology  
Fuzzy-inductive-reasoning (FIR) methodology is a sensor-data-driven approach based on general systems theory proposed by [38]. It is a modelling and qualitative-simulation methodology that allows to obtain qualitative relations between variables that compose a system, and to infer its future behavior. It can also describe systems that cannot easily be described by classical mathematics (e.g., differential equations), that is, systems for which underlying physical laws are not well-understood. FIR consists of four main processes, namely, fuzzification, qualitative model identification, fuzzy forecast, and defuzzification [39; 40].  
* The fuzzification process converts quantitative data obtained from the system into fuzzy data that consist of a triplet containing class, membership, and side values.
* The qualitative-model identification process is responsible for finding causal and temporal relations between variables, therefore allowing to obtain the best model (or optimal mask) that represents the system. The optimal mask is selected from a candidate matrix using quality criteria.
* Once the FIR model is available, a prediction can be performed using the FIR inference engine, called the fuzzy-forecast process, which is a specialization of the \(k\)-nearest-neighbor rule commonly used in the pattern-recognition field.
* Finally, defuzzification is the inverse process of fuzzification. It allows converting the qualitative predicted output into quantitative values that can then be used as inputs to an external quantitative model.  
For better understanding, Table 1 shows an example of a mask with a depth of 2 (2-depth). Each column corresponds to one variable, and each row corresponds to mask depth or the causal-time relation between variables, where \(k\) is the sample time. The negative elements is called \(m\)-input (mask input) (enumeration of m-inputs in the optimal mask has no relevance). The single positive value denotes the m-output. In this example, the mask is equivalent to the causal relation given by \(\hat{y}(k)=f\left(u_{1}(k-2),u_{2}(k),u_{4}(k-2),y(k-1)\right)\) with \(f(\cdot)\) representing the fuzzy-inductive-reasoning function and \(\hat{y}(k)\) the predicted value. The optimal mask is obtained from a candidate mask performing exhaustive research; checking with of them allows to obtain the best quality criteria.  
Assuming that \(n\) is the mask depth, the causal relation can be represented by \(\hat{y}(k)=f\left(\mathbf{U}(k,n),\mathbf{x}(k,n)\right)\), where \(\mathbf{U}(k,n)\) is a window of known inputs, and \(\mathbf{x}(k,n)\) is a window of past outputs \(\mathbf{y}(k-1,n)\) considering samples from \(k-n\) to \(k\).  
The FIR model can be used to forecast the RUL only in case the \(m-\)input is either known in advance or can be estimated (or predicted). Given initial time \(k_{0}\), horizon of prediction \(h\), and function \(g\) that allow to calculate future input values \(\mathbf{U}(k_{h},n)\) with \(k_{h}=k_{0}+h\), the RUL can be computed following Algorithm 1. This computes the predicted state using the FIR model in simulation, and the RUL is determined when the predicted state reaches the predefined failure threshold (FT).  
\begin{table}
\begin{tabular}{c c c c c c} \hline \hline \(k\setminus x\) & \(\mathbf{u_{1}}\) & \(\mathbf{u_{2}}\) & \(\mathbf{u_{3}}\) & \(\mathbf{u_{4}}\) & \(\mathbf{y}\) \\ \hline \(k-2\) & \(-4\) & \(0\) & \(0\) & \(-3\) & \(0\) \\ \(k-1\) & \(0\) & \(0\) & \(0\) & \(0\) & \(-2\) \\