# Progress of Process Monitoring for the Multi-Mode Process: A Review
## 4 Steady-State Mode Process Monitoring Methods
### Unsupervised Process Monitoring Method
#### 4.2.3 Statistical Monitoring Analysis Methods  
The monitoring statistics analysis method assumes that the sample data is normal data that meets all requirements and starts by analyzing the multivariate spatial distribution characteristics between normal data and fault data. It then establishes the corresponding monitoring statistical indicators by analyzing and comparing the distance relationship between them. The traditional methods often take [PERSON]'s \(T^{2}\) value, which measures data changing in the main space, and the squared prediction error (SPE) value projected on the residual subspace, as the monitoring statistical indicators--if there is significant change, a fault is considered to have occurred. This method does not need to cluster the historical modeling data and can still effectively monitor the multi-mode process in the absence of prior knowledge of the process.  
Multivariate Statistical Process Control (SPC) Methods  
Based on the multivariate statistical analysis method of statistical control charts, a traditional Hotelling \(T^{2}\) monitoring statistic distribution of values is proposed on the assumption that the covariance matrix relates to unchanged conditions. Sample data, from high-dimensional space projection to the low-dimensional feature space, is used via statistical measurement and control to improve product quality and maintain the primitiveness of data as much as possible. Multivariate statistical process control charts are mainly divided into \(T^{2}\) control charts, that control the mean vector, and Wishart distribution \(\Lambda\)-type \(F\) control charts of control covariance matrix promoted by univariate statistical \(x^{2}\) distribution control charts [70]. The SPC method is widely applied and can be combined with a time series model or an artificial intelligence method to produce a classification control chart method to improve algorithm performance under the assumption of data nonlinearity [71; 72].  
[PERSON] et al. [73] proposed a local neighborhood standardization-local density factor (LNS-LNF) method. The local neighborhood standardization (LNS) method is used to preprocess the data to conform to a Gaussian distribution, calculate the local density factor value, and determine the control limit of monitoring statistics by combining the kernel density estimation method to judge the sample state. [PERSON] et al. [74] proposed a global probability index based on local density to replace the traditional monitoring index. The local density in the dataset makes use of the local nearest neighbor features rather than relying on the data distribution, which can better reflect the actual engineering modes. However, the method based on LNS may misguide the data under new operation models.  
Based on a local density factor method, some researchers have proposed a local outlier factor method which defines the fault sample as an outlier factor and assesses whether the training sample is in a fault state by comparing the differences between the training sample and the normal sample. The authors of [75; 76] screened the nearest neighbor samples meeting the distance requirements using weighted distance or Euclidean distance, calculated the local outlier factor by calculating the reachable distance and local reachable density, and combined these to calculate the monitoring control limit to improve monitoring accuracy.