Table 1 Dictionary of notations.
Notation | Meaning |
|---|---|
\(R_{k}\) | The \(k{\text{th}}\) belief rule |
\(W\) | Number of input attributes in the fault diagnosis model |
\(a_{1} ,...,a_{W}\) | Total \(W\) input attributes |
\({\text{F}}_{1} ,...,{\text{F}}_{W}\) | Reference values corresponding to the \(W\) input attributes |
\(H_{1} ,...,H_{N}\) | \(N\) possible troubleshooting results |
\(D_{1} ,...,D_{N}\) | The belief degree associated with each outcome under the \(k{\text{th}}\) belief rule |
\(\theta_{k}\) | Rule weights for the \(k{\text{th}}\) belief rule |
\(\updelta _{1}^{{}} {,}...\updelta _{w}^{{}}\) | Attribute weights of the \({\text{W}}\) input attributes |
\({\text{a}}_{1} \left( t \right){\text{,a}}_{2} \left( t \right)...,{\text{a}}_{w} \left( t \right)\) | Data features of \({\text{W}}\) attributes extracted in a time interval |
\(\mu ()\) | Computational function for extracting features from the raw sensor data obtained from WSN |
\(\Lambda\) | Parameters involved in the process of extracting data features |
\(X\) | Raw data collected by the sensor over a period of time interval |
\(t\) | At a certain point in the time interval |
\(\delta_{{\text{j}}}^{{\text{i}}}\) | Adaptive attribute weights (BRB-AAW) |
\(K\) | The total number of belief rules in the model |
\(\Xi\) | Expert knowledge for initializing adaptive attribute weights |
\(g()\) | Calculation function of adaptive attribute weights |
\(S\left( t \right)\) | Predictive fault states for troubleshooting systems |
\(\tau\) | Other parameters involved in the fault result diagnosis function |
\(f()\) | Calculation function of fault diagnosis results |
\(\tau_{{{\text{best}}}}\) | Optimized parameters after optimization algorithm |
\(\beta\) | Parameters in the optimization algorithm |
\(h()\) | Parameter optimization algorithm |
\(Q\) | A time point of the node being diagnosed |
\(T\) | Length of time |
\(\overline{{\text{a}}}\) | The average of the collected data features over the specified time interval |
\(\upalpha ^{4}\) | The standard deviation of the collected data features over the specified Time interval |
\(\uprho _{{\text{i}}}^{{\text{j}}}\) | The degree of matching of the \(i{\text{th}}\) attribute in the \(j{\text{th}}\) reference value |
\({\text{F}}_{{\text{i}}}^{{\text{k}}}\) | the \({\text{ith}}\) attribute's \(k{\text{th}}\) reference value |
\(\rho_{{\text{k}}}^{{\text{i}}}\) | the match of the \(i{\text{th}}\) attribute in the \(k{\text{th}}\) belief rule |
θi | Weight of the belief rule \(i\) |
\(\mho_{i}\) | The activation weight of the \(i{\text{th}}\) belief rule |
N | The framework for identifying fault diagnosis models includes \(N\) levels |
\(D_{j,i}\) | Belief degree of the \(j{\text{th}}\) fault diagnosis result in the \(i{\text{th}}\) rule |
\(p\left( {H_{i} } \right)\) | The utility of the \(i{\text{th}}\) fault diagnosis |
g | The \(g{\text{th}}\) generation optimization algorithm iteration |
\(\tau_{{\text{i}}}^{{{\text{g + }}1}}\) | The optimized parameters of the \(i{\text{th}}\) group in the \((g + 1){\text{th}}\) generation |
\(\tau_{{{\text{best}}}}^{{\text{g}}}\) | Average value of optimized parameters in the \(g{\text{th}}\) generation |
\(\varepsilon_{{}}^{{\text{g}}}\) | Step size of the \(g{\text{th}}\) generation |
\({\mathbb{R}}\) | Normal distribution |
\(C_{{}}^{{\text{g}}}\) | Covariance matrix of the \(g{\text{th}}\) generation |
\(\lambda\) | Number of offspring iterated |
\(E_{{\text{e}}}\) | Parameter vector |
\({\text{n}}_{{\text{e}}}\) | Restricted variables in \(\tau_{{\text{i}}}^{{\text{g + 1}}}\) |
\(j\) | The number of restricted variables in \(\tau_{{\text{i}}}^{{\text{g + 1}}}\) |
\({\text{h}}_{{\text{i}}}\) | Weighting factor |
\(\tau_{{{\text{i:}}\lambda }}^{{\text{g + 1}}}\) | The \(i{\text{th}}\) solution in the \(\left( {g + 1} \right){\text{th}}\) generation of the total \(\lambda\) group of optimization parameters |
\(\sigma\) | Number of solutions in the offspring |
