# Progress of Process Monitoring for the Multi-Mode Process: A Review
## 4 Steady-State Mode Process Monitoring Methods
### Unsupervised Process Monitoring Method
#### 4.2.1 Gmm  
GMM is a weighted combination of multiple single Gaussian probability density functions, the number of which is equal to that of the modes. The GMM shows excellent noise processing ability and robustness in dealing with motion trajectory problems, and, as a result, is widely used in motion graphics processing [49]. Additionally, this model is used to divide the Gaussian component of data to determine the confidence limit and has been applied to fan icing monitoring [50]. The GMM does not rely on prior knowledge and can automatically obtain multiple Gaussian model information through the learning and training of process data. However, its ability to extract non-Gaussian data is weak.  
There are several overall modeling and multiple-model modeling approaches to GMM modeling. When all modes obey Gaussian distribution, an overall GMM can be established for the process. [PERSON] et al. [51; 52] proposed a process monitoring method combining GMM with PCA and Fisher discriminant analysis. A GMM was first established in the low-dimensional feature space, and then PCA was used for process monitoring. [PERSON] et al. [53] proposed a non-optimal state cause tracing method based on the partial derivative of variables combining GMM and Bayesian reasoning. A unified GMM was established for the data of the same operating state, and a monitoring control limit was constructed through the partial derivative contribution index of dominant variables with great state influence to judge the non-optimal cause. This method avoided modal misclassification. [PERSON] et al. [54] established an improved variational inference Gaussian mixture model based on locality preserving projection (IVIGMM-LPP). The model preprocesses data using locality preserving projection (LPP), introduces the eigenvalue discrimination condition of the covariance matrix and the standard deviation vector to reduce the influence of initial parameters and unify data, and builds an overall GMM model for monitoring. However, this model is not suitable for monitoring small faults. [PERSON] et al. [55] used an expectation-maximization (EM) algorithm and Bayesian Yin-Yang (BYY) algorithm to estimate the parameters and number of modes in the GMM, respectively. They also used PCA to reduce the dimensions of the data and built a multi-mode GMM to improve the monitoring performance of the multi-mode process. [PERSON] et al. [56] proposed a Gaussian mixture variational autoencoder (GMVAE) model. Using a latent variable space projection method, the model, which is combined with a multi-variable normal distribution model and a variational posterior model, makes the modes obey the Gaussian distribution. Overall modeling requires that the covariance structure of each mode data should be consistent, otherwise, the covariance of the process data will be strange. Evidently, overall modeling has significant limitations.  
[PERSON] et al. [57] established the GMM for different stable and transition modes for process monitoring. [PERSON] et al. [58] combined LPP and a multi-model GMM to monitor the multi-mode process. The LPP algorithm was used to reduce the dimensions of multi-mode data, then the GMM was established for each mode in low-dimensional subspace, and the local probability index was integrated into the global Bayesian rule probability index for monitoring.