# Enhancing Sensor Data Imputation: OWA-Based Model Aggregation for Missing Values
## 3 The Proposed Formulation
### Experimental Results and Discussion  
The main purpose of conducting this experiment is to show the effectiveness of the proposed approach when utilizing real data and comparing this approach with other essential methods for treating missing data. In Section 3.1.1, OWA weighting vectors were learned where the missing inputs are available in synthetic datasets. Next, we performed a real experiment using the OWA weighting vectors learned in Section 3.1.1. We assume that real datasets have more than one missing input. We utilized the modeling of the OWA weighting vector based on previous learning to replace the missing inputs in the real datasets. Abalone datasets [33] are used in this experiment. The UCI machine learning repository provides this benchmark. For more information about the real datasets that were used for the experiment, Table 2 is set up to summarize the characteristics of the dataset. It shows that each observation has nine attributes that describe the age of the abalone, categorizing it as an infant, a male, or a female. The age of the abalone is obtained by clipping the shell from the cone and staining it, then counting the number of ringsusing a microscope. This process is considered a time-consuming task, especially with missing features. Furthermore, we demonstrate the results with simulated data and illustrate the convergence of our proposed method among different parameter settings. Within the simulation experiment, the constraint of the measurements has been defined as shown in Equation (7a,b). First, we clarify the effect of missing input in data aggregation, where the number of missing data points is in different positions for each observation. As mentioned, the data distribution is assumed to be Gaussian with an i.i.d. correlation feature. For these assumptions, we show the performance of our approach for estimation of the OWA weighting vector from which the model of missing data is obtained. Also, we show its performance when there are a varied number of missing inputs. It can be seen that a Gaussian distribution is utilized to make the approach more precise in data aggregation. As mentioned earlier, this approach considered the number of the observation datasets M to be 500, the number of simulations is 20, while the number of variables (N) is four. At first, a Gaussian distribution is used to model the missing input within the observation dataset. Subsequently, another dataset is generated randomly to produce the missing values within the datasets based on the percentage number of the disappeared inputs, as shown in Table 3. The percentage of disappeared numbers is assumed to be \([10\%,\,20\%,\,\dots,\,50\%]\) and randomly chosen within the observation. The efficiency of the missing data imputation approach is determined by calculating the mean standard error. \[\mathbf{MSE}=\frac{\sum_{i}^{L}(x_{i}-\hat{x}_{i})^{2}}{N} \tag{24}\]  
where the number of measurement values for OWA is assumed to be \(N\), and \(i^{th}\) is assumed to be the index of each element in the measurement vector. The following figure illustrates the performance of the proposed approach, showing the error of missed prediction in terms of missing percentages where the number of missing data points is assumed to be one. The result shows that the error increases slightly as the percentage of missing data increases. It also shows that the error will increase as the number of missing inputs increases, meaning that the learned OWA weighting vector will be affected by a large number of missing inputs. In addition, three different real measurements are used in the proposed approach where missing variables are available. Thus, the rate of missing data in the observation also has a significant effect on the error. These results illustrate that the error is directly proportional to the percentage of missing data. This relation can be clearly seen in Figure 2.