## 4 Improved Retrieval Method for CBR
### Local Similarity Measure  
In general, there are various data types in CBR systems. In our case base, the data attributes are divided into three types: categorical, numerical, and semantic. Categorical attributes include the aeroengine model, aeroengine category, aeroengine operation state, thrust performance, temperature performance, rotational performance, aeroengine shutdown, and other anomalies. Numerical attributes contain flight height and speed. The semantic attributes contain the fault part and mode in aeroengines. The similarity calculation methods for the three types of data attributes can be found in our previous study [28]. 1. Categorical Attributes. In our CBR system, the similarity of categorical attributes is calculated using Equation (3). \[sim_{a(a_{j}^{nc},a_{j}^{c_{i}})}=\left\{ \begin{array}{l l}0&\quad if\quad a _{j}^{nc}\
eq a_{j}^{c_{i}}\\ 1&\quad if\quad a_{j}^{nc}=a_{j}^{c_{i}}\end{array} \right.\] (3) where \(a_{j}^{nc}\) is the value of attribute \(j\) in new case \(nc\), and \(a_{j}^{c_{i}}\) is the value of attribute \(j\) in case \(c_{i}\). \(i\in\{1,2,\ldots,I\}\), where \(I\) is the total number of cases. \(j\in\{1,2,\ldots,I\}\), where \(I\) is the total number of attributes. Figure 1: Framework of the proposed case-based reasoning (CBR) system. 2. Numerical Attributes. The similarity between numerical attributes is defined as the normalized distance between two attribute values, calculated using Equations (4) and (5). \[\mathit{sim}_{b(b^{nc}_{j},b^{c_{i}}_{j})}=1-DIST(b^{nc}_{j},b^{c_{i}}_{j})\] (4) \[DIST(b^{nc}_{j},b^{c_{i}}_{j})=\frac{\left|b^{nc}_{j}-b^{c_{i}}_{j}\right|}{b^ {\max}_{j}-b^{\min}_{j}}\] (5) where \(b^{nc}_{j}\) is the value of attribute \(j\) in \(nc\), \(b^{c_{i}}_{j}\) is the value of attribute \(j\) in \(c_{i}\), while \(b^{\max}_{j}\) and \(b^{\min}_{j}\) represent the maximum and minimum values of attribute \(j\), respectively. 3. Semantic Attributes. The semantic similarity based on tree is used to measure the semantic attribute similarity of the aeroengine fault part and fault mode, which defines the association of the fault mode with the fault part. It consists of two elements: tree structure of the fault part and semantic diagram of the fault mode. Figure 2 shows a schematic diagram of the fault part tree structure of an aeroengine, wherein parts are differentiated and structured to calculate their similarities under the same or different fault modes. Further, a fault mode semantic diagram is developed, as shown in Figure 3. Finally, the association of fault mode with fault part is defined by combining the fault part tree structure and the fault mode semantic diagram, as shown in Figure 4.  
Figure 3: Semantic diagram of fault mode. Figure 2: Tree structure of fault part. The measure of semantic similarity is based on the following three constructs. 1. In the semantic graph, two nodes with shorter distance are more similar than two nodes with longer distance. The shortest path between nodes \(n_{i}\) and \(n_{j}\) is denoted as _Dist_ (\(n_{i}\), \(n_{j}\)). For example, _Dist_ (\"_Fracture_ (_Compressor Blade_)\", _\"_Wear_ (_Compressor Blade_)\") = 4 implies that the shortest path between _Fracture_ (_Compressor Blade_) and _Wear_ (_Compressor Blade_) is 4. 2. The nearest shared parent node \(n_{k}\) of nodes \(n_{i}\) and \(n_{j}\) is represented by _Nspin_ (\(n_{i}\), \(n_{j}\)). For example, _Nspin_ (\"_Crack_ (_Compressor Vane_)\", \"_Crack_ (_Compressor Blade_)\") = _Compressor_, which means _compressor_ is the nearest shared parent node of _Crack_ (_Compressor Vane_) and _Crack_ (_Compressor Blade_). 3. The distance from a node to the root node is defined as the depth of the node, denoted as _Depth_(\(n_{k}\)). For example, _Depth_ (_Compressor Vane_) = 2 indicates that the depth of node _Compressor Vane_ from root node _Aeroengine_ is 2. Then, the semantic similarity is defined as a function of the node location in the taxonomy, which is calculated using Equation (6).