# Enhancing Sensor Data Imputation: OWA-Based Model Aggregation for Missing Values
## 3 The Proposed Formulation
### Modeling Missing Inputs
#### 3.1.1 Synthetic Dataset  
In this section, we assume that M is the observed data and that N samples are used for each observation. We assume that the observations have an independent and identical distribution value, where each \(\mathbf{x}\) is a vector of Gaussian distribution numbers, it can be written as \(x\sim\mathcal{N}(\mu,\sigma^{2})\), where \(\mu\) is the mean of the distribution while \(\sigma^{2}\) is the variance of the distribution. Since the scaling of the OWA weighting vector has directly affected the aggregation decision, we consider different values of the OWA operators set, such as a soft-min. In the missing data problem, we consider the absence of the input value, denoted \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Type of Measurement** & \(\mathbf{w_{1}}\) & \(\mathbf{w_{2}}\) & \(\mathbf{w_{3}}\) & \(\mathbf{w_{4}}\) \\ \hline Soft-Min & 0.1 & 0.2 & 0.3 & 0.4 \\ Soft-Max & 0.4 & 0.3 & 0.2 & 0.1 \\ Mean & 0.25 & 0.25 & 0.25 & 0.25 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: OWA’s value. Figure 1: OWA approach for incomplete dataset aggregation. by the \"-\" symbol. Thus, using Gaussian distribution input, the following OWA operator test of the data aggregation can be expressed as in Equation (4). Moreover, \(\mathbf{x}\) is a synthetic dataset with a real zero-mean random value with a unity covariance matrix. Tractable results are obtained under these typical assumptions; that is, the problem becomes finding a straightforward way to bring robust learning of OWA operators that can be utilized for the real dataset to determine missing values. In general, we specified the different groups of OWA operators, and each group's sum must be based on the OWA constraints. The main goal of this paper is to regress the missing value using other available datasets. We assume that four datasets are observed and one variable is missing; for this case, the estimated number of OWA values is three-dimensional because one variable is missing. Those new OWA weighting vectors are learned using a quadratic form using the dataset with missing values and the original labels. All the inputs are independent and identically distributed (i.i.d) random variables, following a Gaussian distribution with a mean of zero. According to the modeling formula mentioned in Equation (5), aggregation measurements can be obtained by  
\[C(o_{j})=\mathbf{b}_{j}^{T}\hat{\mathbf{w}} \tag{8}\]  
where \(C(o_{j})\) is the label corresponding to inputs \(\mathbf{b}_{j}\), which is then a column vector, and \(\hat{\mathbf{w}}\) is the estimated OWA weighting vector which is supposed to be calculated in this step. \[E_{1}(O,\mathbf{u}) =\sum_{j=1}^{M}(C(o_{j})-y_{j})^{2}=\sum_{j=1}^{M}(\mathbf{b}_{j} ^{T}\hat{\mathbf{w}}-y_{j})^{2}\] \[\quad=\sum_{j=1}^{M}(\hat{\mathbf{w}}^{T}\mathbf{b}_{j}\mathbf{b }_{j}^{T}\hat{\mathbf{w}}-2y_{j}\mathbf{b}_{j}^{T}\hat{\mathbf{w}}+y_{j}^{2}). \tag{9}\]  
Thus, the least-squares minimization solver can be described as follows:  
\[\text{(OP1)}\ \min_{\hat{\mathbf{w}}}f_{O}(\mathbf{w})=\hat{\mathbf{w}}^{T} \mathbf{H}\hat{\mathbf{w}}+\mathbf{d}^{T}\hat{\mathbf{w}},\]  
subject to  
\[\sum_{j=1}^{n}\hat{\mathbf{w}}_{j}=1 \tag{10a}\] \[0\leq\hat{\mathbf{w}}_{j}\leq 1, \tag{10b}\]  
where  
\[\mathbf{H}=\sum_{j=1}^{M}\mathbf{b}_{j}\mathbf{b}_{j}^{T}\text{, and }\mathbf{d}=-2\sum_{j=1}^{M}y_{j}\mathbf{b}_{j}.\]  
OWA measurement is determined at the location of missing data points. After obtaining a new OWA weighting vector, the model can be determined based on the relationship between the original values of the OWA operators and the new OWA operators. In this case, we consider four variables with one missing input to model the relationship between the predicted values of the OWA weighting vector and the original OWA weighting vector. The coefficient of this model is estimated using the quadratic optimization technique.