## 5 Results  
This section will present the details of the application of the framework introduced in the selected industrial case. In the first block, a variable analysis was performed. In this case, the results of the correlation analysis show that two groups of variables with high correlation indices could be differentiated. The first group corresponds to the vibration features that present significant interdependence between them but very little correlation with the rest of the variables. The temperature values integrate the second group, with a significant correlation index. However, other features, such as flow or pressure, show a lower dependency on the rest of the features. The poor correlation detected between vibration and temperature can be explained by the mechanisms that physically govern these parameters. Vibration signals are primarily reflections of mechanical defects such as shaft imbalance, bearing wear, or misalignment caused by friction and mechanical stress in the long term. However, temperature measurements tend to reflect thermal load and pump and motor efficiency under operating conditions, which may vary with process demand or cooling system efficiency. Since mechanical degradation and thermal load can increase individually, especially in earlier stages of failure, their poor statistical correlation is to be expected. Furthermore, temperature fluctuations may cause delays in vibration abnormalities because of heat transfer time delays or effective thermal conduction. This delimitation confirms the clustering observed in the correlation analysis and justifies the need for multi-sensor monitoring schemes. For bearing temperature, the graph indicates a high correlation with multiple variables. The Pearson correlation coefficient between motor winding temperature and bearing temperature in Figure 5a is 0.83. In fact, the correlation with intensity is also relevant, with a Pearson correlation of 0.79 (see Figure 5a). Furthermore, Figure 5b presents a histogram of the temperature variables targeted, providing a visual understanding of their distribution. This analysis is critical to assess the presence of data imbalances or extreme values that could affect the predictive modeling process. This visualization supports the identification of relevant pre-processing steps to ensure that the models are trained on representative and unbiased datasets. Then, the next step in the first block is the selection of models. In this case, our objective is to predict the evolution of numerical variables. This is a supervised regression-type problem. Regression modeling using supervised techniques requires preparing two datasets, corresponding to the input and output variables, with the output variables being the temperature of the pump motor assembly (two variables). These datasets must also be divided into training and testing sets: the primary purpose of the former is to enable the learning process, while the test set will be used to measure the accuracy of the entire algorithm with respect to unseen data. Since a non-stratified cross-validation strategy was adopted, the sizes of the training and test populations were already fixed. In addition, future data must never be used to make predictions of past events, so the sets were not be shuffled at any time [61]. Once the relationship between the input and target variables is known, the next step is to select the features necessary for the development of the model. In accordance with [69], to reduce the degrees of freedom of the algorithm, which reduces the number of unnecessary features, regularization models such as Lasso, ElasticNet, and Ridge were considered. These models help to reduce overfitting and the complexity of the model. In addition, multicolinearity analysis was applied. The next block deals with the construction of the models. Once the training and test sets have been generated, the training data set is ready to be applied in the model to make predictions.