# Progress of Process Monitoring for the Multi-Mode Process: A Review
## 4 Steady-State Mode Process Monitoring Methods
### Supervised Process Monitoring Method
#### 4.1.5 K-Nearest Neighbor (KNN) Based Method  
The K-nearest neighbor method is often used as a multi-mode process monitoring statistic to judge the system state by observing the distance relationship between sample data and their K-nearest neighbors. [PERSON] et al. [39] proposed fault detection method using the K-nearest neighbor (FD-KNN) method based on K-nearest neighbor multi-mode process monitoring. Taking the square sum of the Euclidean distance between sample data and their K-nearest neighbor \(D^{2}\) as the monitoring statistics index, this method can increase the computational burden of the system. Additionally, when the variance structure of multi-mode data is different, the statistical index cannot correctly reflect the differences between samples, resulting in false or missing reports. [PERSON] et al. [40] constructed K-nearest neighbor monitoring statistics in the residual space to improve the fault resolution of the system. [PERSON] et al. [41] replaced the original [PERSON]'s \(T^{2}\) monitoring statistics with the monitoring statistics of K-nearest neighbors and the Mahalanobis distance of kernel entropy projection principal component space to improve the rate of early warning and accuracy of fault monitoring. [PERSON] et al. [42] defined the monitoring statistics as the mean of the sum of squares of the original distance \(D^{2}\) based on the FD-KNN algorithm to improve the monitoring performance in case of large modal differences.  
To optimize the calculation amount of the traditional KNN and improve the monitoring accuracy, [PERSON] et al. [43] proposed a variable moving window K-nearest neighbor (VMW-KNN) method. A variable-length sliding window is used to determine the K-nearest neighbor search range of the sample, and then KNN is used to preprocess the sample data to calculate the load matrix. Finally, the corresponding sub-PCA model is selected for monitoring. [PERSON] et al. [44] combined kernel density estimation with the K-nearest neighbor algorithm to construct a local density function, i.e., density estimation of K samples, reducing the sensitivity of window width parameters and eliminating multi-modal differences in data. [PERSON] [45] proposed a principal component difference method based on the K-nearest neighbor to obtain the K-nearest neighbor augmented matrix of each sample. The loading and score matrices using PCA were then used to calculate the data sample, estimate scores and monitor the system by the new monitoring statistics determined by the covariance matrix of the differences between them.  
To improve the variance structure difference of data, [PERSON] et al. [46] proposed a principal component and weighted K-nearest neighbor (PC-WKNN) method and obtained the weighted distance statistic \(D\) to reduce the variance difference of data by calculating the distance between the training sample and the K-nearest neighbor and the reciprocal of its local nearest neighbor in the principal component space. [PERSON] et al. [47] proposed a KNN based on local relative probability density (LRPD-KNN), which combined KNN with kernel density estimation, to establish a local probability density estimation function, and used the local relative probability density to improve data variance.  
In addition, [PERSON] et al. [48] proposed a multi-mode process monitoring method based on distance space statistics analysis (DSSA). This method detects the presence of a fault at the current time by analyzing the difference between the K-nearest neighbor distance of normal samples and fault samples. The method is suitable for cases where the variance structure of data is quite different in different modes.