## 3 Materials and Methods
### Multi-Instances Oriented NHL  
In the multi-relational FCM, each concept represents a useful field name in data resource. In the multi-relationship, one field in main table corresponding to another table has some sub fields. The field in the main table points to a coarse-grained concept in low-level FCM and the fields in another table indicate the fine-grained concepts in the FCM. Thus, the low-level FCM can be used to model the multi-relationship.  
One coarse-grained concept (\(C_{i}\)) in \(j^{th}\) dimension corresponds to some fine-grained concepts (\(C_{i}\), \(C_{2}\), \(\ldots\), \(C_{m}\)). It becomes the key to the state value of get the weights between the coarse-grained concept and some fine-grained concepts in the low-level FCM. The weights express the causality relationship between fine-gain concepts and coarse-grained concept. The weights need to be learned from multi-instances. NHL can be extended to multi-instances oriented mining for the optimistic of the nonlinear units' weights in Figure 2. In the prerequisite, the state states of coarse-grained concepts in high-level FCM can be inferred.  
Multi-instances oriented NHL has two constraints. First constraint is that it maximizes the mathematical expectation of \(A_{j}^{2}\) of all multi-instances as Equation (7).  
\[\text{maximize}\,J=E[\sum_{\mathbf{r}}A_{j}^{2}] \tag{7}\] \[\text{subject to }||\mathbf{w}||=\mathbf{1} \tag{8}\]  
Figure 2: Nonlinear unit in Fuzzy Cognitive Map (FCM).  
Second constraint is that the weight vector \(\mathbf{w}\) has to be limited to stabilize the learning rule as Equation (8), which generates the following nonlinear Hebbian learning rule as Equation (9).  
\[\Delta w_{ij}(t+1)=\eta A_{j}(A_{ji}-w_{ij}(t)A_{j}) \tag{9}\]  
Accordingly, multi-instances oriented NHL is presented in the function of M_NHL. The execution phase of multi-instances oriented NHL (M_NHL) is consisted of the following steps:  
Step 1: Random initialize the weight vector \(\mathbf{w}(t)\), \(t=0\), \(p=0\) and input all instance \(\{\mathcal{A}_{ji}\}\)  
Step 2: Calculate the mathematical expectation of \(A_{j}\)2 of all \(\{A_{ji}\}\)  
Step 3: Set \(t=t+1\), repeat for each iteration step \(t\):  
3.1 Set \(p=p+1\), to the \(p^{th}\) instance:  
3.1.1 Adjust \(\mathbf{w}(t)\) matrix to \(\{A_{ji}\}^{p}\) by Equation (9)  
3.2 Calculate the \(A^{j}\)2 to all \(\{A_{ji}\}\) by Equation (10)  
3.3 Determine whether \(A_{2}^{2}\) is maximum or not at present  
3.4 If \(A_{j}\)2 is maximum, output the optimal \(\mathbf{w}(t)\)  
Step 4: Return the final weight vector \(\mathbf{w}(t)\)  
According to the literature [21], whether NHL clusters better or not is closely related to the activation function \(f\). It means that if cumulative normal distribution function or approximate cumulative logic distribution function is chosen as activation function, output results will show a U-distribution and easily achieve better NHL clustering. In order to avoid integral calculation, the cumulative logic distribution, shown as Equation (11), is selected. The output values (nonlinear unit outputs) can be inferred by Equation (10), where \(A_{ji}\) is the input and \(w_{ji}\) is mined by M_NHL algorithm.  
\[A_{j}(t)=f(\sum_{i\
eq j}A_{ij}(t-1)w_{ji}(t-1)) \tag{10}\]  
\[f(x)=\frac{1-e^{-x}}{1+e^{-x}} \tag{11}\]