Where ( ) and ( ) are the mean and standard deviation vectors learned by the encoder network, and denotes the encoder parameters. A latent sample is obtained using the reparameterization trick: = ( )+ ( )∙ , ~Ν(0,I) (3) This formulation enables gradient-based optimization while maintaining stochastic sampling in the latent space. The key contribution of the variational component is the imposition of a prior distribution over the latent variables, typically: ( ) = Ν(0, I) (4) This constraint regularizes the latent space, forcing the encoded representations to follow a continuous and structured manifold. As a result, the latent space becomes smooth and continuous, similar inputs are mapped to nearby latent regions, overfitting is reduced, the representation becomes more robust to noise variability. This probabilistic regularization is particularly relevant in industrial process data, where nonlinear interactions and operational variability generate complex high-dimensional patterns. Figure 43. Variational Autoencoder estructure. 3.3.2. Loss function and Evidence Lower bound (ELBO) The model is trained by maximizing the Evidence Lower Bound (ELBO), which combines reconstruction error and Kullback–Leibler (KL) divergence regularization. The KL term enforces alignment between the learned latent distribution and a Gaussian prior, producing a structured latent space suitable for probabilistic clustering, which corresponds to minimizing the following objective function: ℒ = ( | )[log ( | )] − ( ( | )‖ ( )) (5) 3.3.3. Convolutional architecture for multivariate time-series
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