# Enhancing Sensor Data Imputation: OWA-Based Model Aggregation for Missing Values
## 3 The Proposed Formulation  
### Modeling Missing Inputs  
This work's primary goal is to learn the OWA weighting vector where the missing data samples exist, i.e., the estimated OWA weighting vector can be modeled related to the OWA weighting vector with the existing complete data point. In other words. After generating all data, arrange the data values in descending order to provide a decision maker based on the newly assigned OWA weighting vector. Upon assuming the values of the OWA weighting vector such as the softmax OWA weighting vector, the data point labels are obtained according to Equation (4). Next, assign the percentage of missing data volume. For example, 10% or 20% of the total observation dataset has missing values within each data point. These values can be randomly picked for each data point. Since the data point, including the missing samples and the labels, is available, new weighted vector values are produced using the data point with the missing data samples. This label was determined when the complete dataset was available earlier. The estimated weighting vector rs is now determined, which has the same dimension as in the incomplete dataset. Since two sets of OWA operators are obtained, one where the complete dataset exists and another with an incomplete dataset, the new expression is obtained between the estimated OWA operator and the original OWA weighting vector. This expression can be solved using the quadratic form with specific constraints, as shown in Equation (5), to determine the coefficient parameters. Each estimated OWA operation has an expression in terms of the original values of the OWA weighting vector, and this expression can be used to predict the missing data sample. The missing values within the data point will be regressed on other variables based on the relationship between the estimated OWA weighting vector and the original operators. The missing values will then be replaced by the predictive values obtained. Similarly, each data point with missing values can replace its missing values with predicted ones.  
Figure 1 shows more details on the proposal approach. Three steps can be seen in this figure. The first step shows the learning expression to obtain the missing data. Subsequently, the missing values are derived based on the other variables. Ultimately, the mean squared error (MSE) is obtained to analyze the performance of the proposed approach.  
Next, the synthetic datasets are utilized in this experiment. Synthetic datasets are generally used because it is thought to be very difficult to identify real-world applications of many complex scenarios. However, it is much easier to build OWA weighting vectors in synthetic data with varying complexity. The second reason is that synthetic datasets allow us to see how the learning approach would work with noise-free datasets. Moreover, carrying out tests also yields computational complexity. The mean square error performance is also obtained in various settings by applying the learned OWA weighting vector and the expected test labels. Table 1 represents the OWA weighting vector with independent variables.