## 2 Developed Multiscale PCA-Based NN
### MSPCA-Based NN Fault Diagnosis  
In this study, the MSPCA model was applied with the goal of merging the PCA's capacity to extract the corresponding cross-correlation betwixt variables with the orthonormal wavelets' capacity to split features from noise and avoid the auto-correlation between the obtainable measurements. In order to combine the outcomes at adequate scales, the MSPCA technique processes and computes a PCA model of the wavelet coefficients at each scale. Accordingly, the pertinent attributes are attained for each scenario by selecting only those latent variables that capture the relationship betwixt the variables after the elimination of less important signal features. Thus, for the purpose of fault diagnosis, the most pertinent features are fed into NN classifiers. A wavelet decomposition is utilized to remove errors, decrease noise, and decorrelate the relationship between the stochastic measurements, once measurements representing healthy and various potential defective operating modes in the process are provided. The data gathered while the system is in normal operation is then used to develop a PCA model. The obtained dataset is projected onto a subspace of positive right directions while maintaining the information about the most collected features. The resultant PCA model's structure is described by the directions of the subspace projector, whose dimension is smaller than that of the raw dataset. With the use of this projection technique, various important attributes are gathered for each situation. Subsequently, a bank of various classifiers is trained using a range of attributes as input and their associated labels as the desired output. Otherwise, a comparison between the resulting classifier output and the set of feature labels is evaluated in order to generate an efficient decision.  
The main various tasks of the suggested strategy are illustrated in Figure 6.  
Figure 5: CFNN Architecture of a Cascaded Forward Neural Network.  
Algorithm 1 highlights the key tasks of the MSPCA-based NN algorithm.  
Input: \(n\times m\) data matrix \(X\),  
Collecting the data.  
**Training phase**  
1. Calculate the wavelet decomposition for each column in the data matrix;  
2. Calculate the mean and standard deviation of each process variable, then standardize the dataset matrix;  
3. Decompose each variable into wavelet coefficients;  
- Each scale's wavelet coefficients are formed into a matrix;  
- Each of these scales is subjected to PCA;  
- Determine the \(Q\), \(T^{2}\), and \(\phi\) statistics of each dataset;  
4. Retained wavelet scales and coefficients are used to reconstruct the dataset matrix;  
5. Perform PCA to acquire an estimated dataset matrix and residuals;  
6. Classify the faults through NN classifiers;  
7. Ascertain the classification model;  
**Testing phase**  
1. Standardize the testing dataset by the use of the mean and the standard deviation computed from the fault-free training phase;  
2. Decompose each variable into wavelet coefficients;  
- Each scale's wavelet coefficients are formed into a matrix;  
- Each of these scales is subjected to PCA;  
- Determine the \(Q\), \(T^{2}\), and \(\phi\) statistics of each dataset;  
3. Reconstruct the data matrix by use of the retained wavelet scales and coefficients;  
4. A PCA technique is performed on the reconstructed data matrix;  
5. Classify the faults through NN classifiers;  
6. Establish the prediction model;  
7. Attain the fault diagnosis outcomes.  
Figure 6: MSPCA-based NNs fault diagnosis and detection process.