# A Novel Fuzzy Kernel Extreme Learning Machine Algorithm in Classification Problems
## 4 Discussion  
The current work presents a hybrid method that will bring a new perspective to the problem of facial expression classification. The structure of the hybrid method includes fuzzy rules, wavelet structure, and a kernel-based ELM. We call this structure FWMHKELM for short. The key advantage of FWMHKELM lies in its simplicity and its high accuracy and generalization performance. Building on this theoretical foundation, the mathematical framework of the FWMHKELM model was used to classify facial expressions. The facial expression classification helps build a flexible facial expression identification model in terms of mapping behavioral features against physiological biometric features. People have distinct physiognomic expressions associated with emotions such as happiness, sadness, shock, and fear. In the classification system, these expressions are identified through established matching templates based on the geometry of the face. The mathematical basis of the structure is presented in the paper. Taking advantage of the fact that the Mexican hat wavelet is an acceptable ELM kernel, it is discussed how to integrate a fuzzy wavelet into a fuzzy wavelet neural network and into a kernel-based extreme learning machine algorithm. The theoretical equivalence between the fuzzy wavelet structure and the Mexican hat (MH) kernel-based ELM, as demonstrated in Equations (19) and (26), ensures that the model captures both the local and global features of facial expressions through wavelet decomposition and fuzzy rule-based reasoning. The high classification accuracies (94.55% with LBP, 94.27% with HOG) empirically validate this synergy--the MH kernel's edge-detection properties (Equations (11) and (12)) enhance feature discrimination, while fuzzy rules handle inherent uncertainties in expression data. For instance, the confusion matrices (Tables 1 and 2) reflect the model's ability to leverage wavelet-transformed features (e.g., eyebrow/mouth movements) via the MH kernel, with misclassifications primarily occurring between expressions with overlapping geometric cues (e.g., fear/disgust). This alignment between theory and results underscores how the hybrid structure's mathematical design--combining fuzzy adaptability, wavelet multiresolution analysis, and the ELM's efficient optimization--directly translates into superior performance metrics (Figure 12). Future work could further dissect this relationship by visualizing feature maps or kernel responses for specific expressions. Figure 12: Comparative classification analysis of ELM, WNN, and FWMHKELM methods. In this study, the coding for the algorithm and all other comparison algorithms was performed in MATLAB (R2021b). The combination of the algorithm with feature extraction algorithms involved in the classification of facial expression datasets (Log Gabor filters, local binary patterns, and histograms of oriented gradients) tends to produce highly successful classes. The integration of feature extraction techniques such as LBP and HOG further improves the classification performance by capturing critical geometric and textural features of facial expressions. Comparison with existing techniques in the literature, such as ELM and WNN, shows the superiority of the proposed method, especially in handling the intrinsic uncertainty associated with human emotions and expressions. In the experiments, it was observed that the hybrid algorithm failed to classify fear and disgust expressions. One of the main reasons why the FWMHKELM model misclassifies some facial expressions is the feature overlap problem between expressions such as \"fear\" and \"disgust\". These expressions can show highly correlated features because they involve similar movements of facial muscles. For example, both expressions have similar visual cues, such as nose wrinkling, eyebrows pulling in, and lips stretching. This can limit the discrimination ability of the model, especially when using classical feature extraction methods.