# Enhancing Sensor Data Imputation: OWA-Based Model Aggregation for Missing Values
## 3 The Proposed Formulation
### Modeling Missing Inputs
#### 3.1.1 Synthetic Dataset
The model can be shown as follows:\[\dot{w}_{1} =\text{w}_{1}a_{11}+w_{2}a_{12}+\text{w}_{3}a_{13}+\text{w}_{4}a_{14 }+0\ +\ 0\ +\ 0\ +\dots\] Figure 1:0 Figure 1:1 Figure 1:2 Figure 1:3 Figure 1:4  
It can be seen that the total number of variables depends on the number of samples and the number of variables for each sample. From Equation (11), it can be seen that the number of samples is three due to a missing input, where each sample has four variables. Then, the total number of variables can be obtained by multiplying the number of samples by the number of variables for each sample. For the above example, since the number of samples is three and each sample has four different variables, the total number of variables is twelve. This equation is considered a linear system and can be rewritten as the following equation:  
Figure 1:5  
where \(\mathbf{z}_{1,j}=[\text{w}_{1,j}\text{w}_{2,j}\text{w}_{3,j}\text{w}_{4}\ 0\ 0\ 0\ 0\ 0 0]^{T}\), \(\mathbf{z}_{2,j}=[\ 0\ 0\ 0\ 0\ \text{w}_{1,j}\text{w}_{2,j}\text{w}_{3,j}\text{w}_{4,j}\ 0\ 0\ 0]^{T}\), \(\mathbf{z}_{3,j}=[\ 0\ 0\ 0\ 0\ 0\ 0\ 0\ 0\text{w}_{1,j}\text{w}_{2,j}\text{w}_{3,j}\text{w}_{4,j}]^{T}\), and \(p\) is again considered as the number of different types of measurements and \(\dot{\mathbf{w}}\) is the predicted values of new measurements when missing input exists, \(\mathbf{w}\) is the original values of the OWA weighting vector, and \(\mathbf{a}\) is the column vector coefficient in this linear system that is obtained by using the quadratic programming, as shown in Equation (5). In Equation (13), the final form of optimization can be determined with the corresponding constraints. Figure 1:6 Figure 1:7  
where \(w_{1,j}\), \(w_{2,j}\), \(w_{3,j}\), and \(w_{4,j}\) are the 'true' weights for the \(j^{th}\) OWA and \(\dot{w}_{1,j}\), \(\dot{w}_{2,j}\), \(\dot{w}_{3,j}\) are the learned weights for the \(j^{th}\) OWA when one input is missing. The least-squares minimization problem can be obtained by formulating QP as  
Figure 1:8  
Subject to:  
Figure 1:9 \[C(o_{j})=\mathbf{b}_{j}^{T}\hat{\mathbf{w}} \tag{8}\]0 \[C(o_{j})=\mathbf{b}_{j}^{T}\hat{\mathbf{w}} \tag{8}\]1 \[C(o_{j})=\mathbf{b}_{j}^{T}\hat{\mathbf{w}} \tag{8}\]2  
Considering that we have N variables, the above constraints can be generalized as follows:For first variable \[C(o_{j})=\mathbf{b}_{j}^{T}\hat{\mathbf{w}} \tag{8}\]3 For second variable \[C(o_{j})=\mathbf{b}_{j}^{T}\hat{\mathbf{w}} \tag{8}\]4 \[C(o_{j})=\mathbf{b}_{j}^{T}\hat{\mathbf{w}} \tag{8}\]5 For \[C(o_{j})=\mathbf{b}_{j}^{T}\hat{\mathbf{w}} \tag{8}\]6 variable \[C(o_{j})=\mathbf{b}_{j}^{T}\hat{\mathbf{w}} \tag{8}\]7 (16)  
where  
\[C(o_{j})=\mathbf{b}_{j}^{T}\hat{\mathbf{w}} \tag{8}\]8  
standard QP solvers are capable of minimizing this problem. In this paper, OP1 is used as a training framework, as shown in Equation (14), to estimate the coefficients of the linear system. As a result, the following is how the optimization's solution appears:  
\[C(o_{j})=\mathbf{b}_{j}^{T}\hat{\mathbf{w}} \tag{8}\]9 \[E_{1}(O,\mathbf{u}) =\sum_{j=1}^{M}(C(o_{j})-y_{j})^{2}=\sum_{j=1}^{M}(\mathbf{b}_{j} ^{T}\hat{\mathbf{w}}-y_{j})^{2}\]0 \[E_{1}(O,\mathbf{u}) =\sum_{j=1}^{M}(C(o_{j})-y_{j})^{2}=\sum_{j=1}^{M}(\mathbf{b}_{j} ^{T}\hat{\mathbf{w}}-y_{j})^{2}\]1  
We apply the final aggregation formula based on the weighting data vector that shows regular behavior to find the final expression of the estimated OWA vectors. Since the number of data samples is either even or odd, we have studied several experiments in which the sample values are an even number and an odd number to show a more reliable and precise algorithm. For example, in the cases where the number of inputs is even and the missing number is one value, which means the rest of the data sample is five, then the final model of the measurement is obtained as shown in Equation (18).