## 3 Motivation and Research Problem  
In many real-world applications, such as predictive maintenance, it is crucial to classify equipment's health state and predict its Remaining Useful Life (RUL). However, obtaining labeled data for training supervised models is often challenging due to the excessive cost and effort involved in manual labeling. This creates a significant gap in effective monitoring and maintenance of equipment.  
While several unsupervised learning techniques have been proposed for similar problems, comprehensive methodologies that combine autoencoders and GMMs for classification and RUL prediction are lacking. GMMs assume that data points are generated from a mixture of Gaussian distributions. Unlike K-means, GMMs accommodate clusters of varying shapes and sizes, making them suitable for complex industrial data. K-means and hierarchical clustering struggle with overlapping clusters and high-dimensional data. GMMs' probabilistic framework addresses these issues by modeling covariance structures.  
Among the most common recent techniques applied for equipment RUL estimation and prediction are the following:  
* LSTM networks to capture temporal dependencies in sensor data [51, 52].
* Hybrid models to combine CNNs for spatial features and LSTMs for temporal dynamics.
* Physics-based models use domain knowledge but require explicit definitions of failure modes.
* Despite these advancements, several challenges remain:
* Unified frameworks: Most existing work treats health state classification and RUL prediction as separate tasks. A unified model that integrates both functions within an unsupervised framework is rare.
* Data limitations: Real-world datasets lack run-to-failure labels, making supervised training infeasible. Existing studies often rely on simulated datasets like CMAPSS, which do not reflect the complexity of actual operations.
* Interpretability: Industrial adoption requires that clustering results and RUL predictions be explainable to engineers and decision-makers [53, 54].  
These gaps highlight the need for an end-to-end unsupervised solution that bridges classification and prognostics without labeled data.  
This article addresses these limitations by proposing a hybrid framework that synergizes autoencoders for feature compression, GMM for unsupervised health-state classification, and LSTMs for state-specific RUL prediction, implemented and validated on both real-life and simulated turbofan engine data. By unifying these components, the methodology eliminates reliance on labeled training data, offering a scalable solution for industrial applications where failure histories are incomplete or unavailable.  
The methodology is implemented in MATLAB, and the results are analyzed to demonstrate its effectiveness. The specific objectives are as follows:  
1. Develop a hybrid model integrating autoencoders for feature extraction and GMMs for clustering.
2. Provide a mathematical and algorithmic foundation for each component.
3. Implement and validate the framework using MATLAB R2024b on the NASA CMAPSS Turbofan Engine Degradation Dataset, as well as the real-life engine operational data.
4. Compare performance against supervised benchmarks.