# A Review of Kernel Methods for Feature Extraction in Nonlinear Process Monitoring

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## Abstract

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## 1. Introduction

## 2. Motivation for Using Kernel Methods

#### 2.1. Feature Extraction Using Kernel Methods

#### 2.2. Kernel Methods in the Machine Learning Context

**Supervised learning**Classification: Given data samples labeled as normal and faulty, find a boundary between the two classes; or, given samples from various fault types, find a boundary between the different types.Regression: Given samples of regressors (e.g., process variables) and targets (e.g., key performance indicators), find a function of the former that predicts the latter; or, find a model for predicting the future evolution of process variables whether at normal or faulty conditions.Ensemble methods: Find a strategy to combine results from several models.**Unsupervised learning**Dimensionality reduction: Extract low-dimensional features from the original data set that can enable process monitoring or data visualization.Clustering: Find groups of similar samples within the data set, without knowing beforehand whether they are normal or faulty.Density Estimation: Find the probability distribution of the data set.

#### 2.3. Relationship between Kernel Methods and Neural Networks

## 3. Methodology and Results Summary

#### 3.1. Methodology

#### 3.2. Results Summary

## 4. Review Findings

#### 4.1. Batch Process Monitoring

#### 4.2. Dynamics, Multi-Scale, and Multi-Mode Monitoring

#### 4.3. Fault Diagnosis in the Kernel Feature Space

#### 4.3.1. Diagnosis by Fault Identification

#### 4.3.2. Diagnosis by Fault Classification

#### 4.3.3. Diagnosis by Causality Analysis

#### 4.4. Handling Non-Gaussian Noise and Outliers

#### 4.5. Improved Sensitivity and Incipient Fault Detection

#### 4.6. Quality-Relevant Monitoring

#### 4.7. Kernel Design and Kernel Parameter Selection

#### 4.7.1. Choice of Kernel Function

#### 4.7.2. Kernel Parameter Selection

#### 4.8. Fast Computation of Kernel Features

#### 4.9. Manifold Learning and Local Structure Analysis

#### 4.10. Time-Varying Behavior and Adaptive Kernel Computation

#### 4.11. Multi-Block and Distributed Monitoring

#### 4.12. Advanced Methods: Ensembles and Deep Learning

## 5. A Future Outlook on Kernel-Based Process Monitoring

#### 5.1. Handling Heterogeneous and Multi-Rate Data

#### 5.2. Performing Fault Prognosis

#### 5.3. Developing More Advanced Methods and Improving Kernel Designs

## 6. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## Abbreviations

AI | Artificial Intelligence | MI | Mutual Information |

ANN | Artificial Neural Network | MSPM | Multivariate Statistical Process Monitoring |

CNKI | China National Knowledge Infrastructure | PSE | Process Systems Engineering |

DTW | Dynamic Time Warping | RKHS | Reproducing Kernel Hilbert Space |

EWMA | Exponentially Weighted Moving Average | SCADA | Supervisory Control and Data Acquisition |

FVS | Feature Vector Selection | SDG | Signed Digraph |

GA | Genetic Algorithm | SFM | Similarity Factor Method |

GLRT | Generalized Likelihood Ratio Test | SOM | Self-organizing Maps |

GP | Gaussian Processes | SPA | Statistical Pattern Analysis |

KDE | Kernel Density Estimation | SVDD | Support Vector Data Description |

kNN | k-Nearest Neighbors | SVM | Support Vector Machine |

KPCA | Kernel Principal Components Analysis | UCL | Upper Control Limit |

AMD | Augmented Mahalanobis distance | ICA | Independent components analysis |

C-PLS | Concurrent partial least squares | K-means | K-means clustering |

CCA | Canonical correlation analysis | LLE | Local linear embedding |

CVA | Canonical variate analysis | LPP | Locality preserving projections |

DD | Direct decomposition | LS | Least squares |

DISSIM | Dissimilarity analysis | MVU | Maximum variance unfolding |

DL | Dictionary learning | NNMF | Non-negative matrix factorization |

DLV | Dynamic latent variable model | PCA | Principal components analysis |

ECA | Entropy components analysis | PCR | Principal component regression |

EDA | Exponential discriminant analysis | PLS | Partial least squares |

ELM | Extreme learning machine | RPLVR | Robust probability latent variable regression |

FDA | Fisher discriminant analysis | SDA | Scatter-difference-based discriminant analysis |

FDFDA | Fault-degradation-oriented FDA | SFA | Slow feature analysis |

GLPP | Global-local preserving projections | T-PLS | Total partial least squares |

GMM | Gaussian mixture model | VCA | Variable correlations analysis |

AEP | Aluminum electrolysis process | HGPWLTP | Hot galvanizing pickling waste liquor |

AIRLOR | Air quality monitoring network | treatment process | |

BAFP | Biological anaerobic filter process | HSMP | Hot strip mill process |

BDP | Butane distillation process | IGT | Industrial gas turbine |

CAP | Continuous annealing process | IMP | Injection moulding process |

CFPP | Coal-fired power plant | IPOP | Industrial p-xylene oxidation process |

CLG | Cyanide leaching of gold | MFF | Multiphase flow facility |

CPP | Cigarette production process | NE | Numerical example |

CSEC | Cad System in E. coli | NPP | Nosiheptide production process |

CSTH | Continuous stirred-tank heater | PCBP | Polyvinyl chloride batch process |

CSTR | Continuous stirred-tank reactor | PenSim | Penicillin fermentation process |

DMCP | Dense medium coal preparation | PP | Polymerization process |

DP | Drying process | PV | Photovoltaic systems |

DTS | Dissolution tank system | RCP | Real chemical process |

EFMF | Electro-fused magnesia furnace | SEP | Semiconductor etch process |

FCCU | Fluid catalytic cracking unit | TEP | Tennessee Eastman plant |

GCND | Genomic copy number data | TPP | Thermal power plant |

GHP | Gold hydrometallurgy process | TTP | Three-tank process |

GMP | Glass melter process | WWTP | Wastewater treatment plant |

RBF | Gaussian radial basis function kernel | HK | Heat kernel |

POLY | Polynomial kernel | SIG | Sigmoid kernel |

COS | Cosine kernel | NSDC | Non-stationary discrete convolution kernel |

WAV | Wavelet kernel |

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**Figure 2.**Basic steps of typical Multivariate Statistical Process Monitoring (MSPM) methods to achieve fault detection. Here, the feature extraction step shows only a linear transformation of data.

**Figure 3.**Illustration of kernel nonlinear transformation. These were generated with code available in https://uk.mathworks.com/matlabcentral/fileexchange/65232-binary-and-multi-class-svm.

**Figure 4.**Machine learning methods relevant to process monitoring (from the authors’ perspective). Those with (*) have versions that belong to the family of kernel methods.

**Figure 6.**(

**a**) Commonly used kernelized methods found in the review; (

**b**) Breakdown of the type of case studies found in the review.

**Figure 8.**(

**a**) Number of papers that cited the use of which kernel functions; (

**b**) Number of papers that cited the use of which kernel parameter selection routes. Note: Papers can appear in more than one column, hence, the numbers will not add to 230 (the total number of reviewed papers).

**Figure 9.**Illustration of manifold learning: (

**a**) S-curve data set; (

**b**) 2-D Kernel principal components analysis (PCA) projection using radial basis function (RBF) kernel, $c=10$; (

**c**) 2-D Local linear embedding (LLE) using kNN, $k=15$. See [319] for more details.

Year | Reference | Remark |
---|---|---|

2012 | Qin [25] | Discusses the general issues and explains how basic data-driven process monitoring (MSPM) methods work. |

2012 | MacGregor and Cinar [26] | Reviews data-driven models not only in process monitoring, but also in optimization and control. |

2013 | Ge et al. [6] | Reviews data-driven process monitoring using recent MSPM tools and discusses more recent issues. |

2014 | Yin et al. [27] | Reviews data-driven process monitoring but from an application point of view; it also provides a basic monitoring framework. |

2014 | Ding et al. [28] | Reviews data-driven process monitoring methods with specific focus on dynamic processes. |

2014 | Qin [15] | Gives an overview of process data analytics, in which process monitoring is only one of the applications. |

2015 | Yin et al. [29] | Reviews data-driven methods not only in industrial processes, but also in smart grids, energy, and power systems, etc. |

2015 | Severson et al. [30] | Gives an overview of process monitoring in a larger context than just data-driven methods, and advocates hybrid methods. |

2016 | Tidriri et al. [31] | Compares physics-driven and data-driven process monitoring methods, and reviews recent hybrid approaches. |

2016 | Yin and Hou [32] | Reviews process monitoring methods that used support vector machines (SVM) for electro-mechanical systems. |

2017 | Lee et al. [9] | Reviews recent progresses and implications of machine learning to the field of PSE. |

2017 | Ge et al. [11] | Reviews data-driven methods in the process industries from the point of view of machine learning. |

2017 | Ge [33] | Reviews data-driven process monitoring methods with specific focus on dealing with the issues on the plant-wide scale. |

2017 | Wang et al. [34] | Reviews MSPM algorithms from 2008 to 2017, including both papers and patents in Web of Science, IEEE Xplore, and the China National Knowledge Infrastructure (CNKI) databases. |

2018 | Md Nor et al. [35] | Reviews data-driven process monitoring methods with guidelines for choosing which MSPM and machine learning tools to use. |

2018 | Alauddin et al. [36] | Gives a bibliometric review and analysis of the literature on data-driven process monitoring. |

2019 | Qin and Chiang [16] | Reviews machine learning and AI in PSE and advocates the integration of data analytics to chemical engineering curricula. |

2019 | Jiang et al. [37] | Reviews data-driven process monitoring methods with specific focus on distributed MSPM tools for plant-wide monitoring. |

2019 | Qui$\tilde{\mathrm{n}}$ones-Grueiro et al. [38] | Reviews data-driven process monitoring methods with specific focus on handling the multi-mode issue. |

This paper | Reviews data-driven process monitoring methods that applied kernel methods for feature extraction. |

Label | Name of Issue | No. of Papers That Addressed It |
---|---|---|

A | Batch process monitoring | 30 |

B | Dynamics, multi-scale, and multi-mode monitoring | 72 |

C | Fault diagnosis in the kernel feature space | 100 |

D | Handling non-Gaussian noise and outliers | 41 |

E | Improved sensitivity and incipient fault detection | 39 |

F | Quality-relevant monitoring | 37 |

G | Kernel design and kernel parameter selection | 30 |

H | Fast computation of kernel features | 34 |

I | Manifold learning and local structure analysis | 20 |

J | Time-varying behavior and adaptive kernel computation | 26 |

K | Multi-block and distributed monitoring | 15 |

L | Advanced methods: Ensembles and Deep Learning | 8 |

**Table 3.**Summary of papers: The issues they addressed and the kernel method, case studies, and kernel functions they used.

Year | Reference | Kernelized | Issues Addressed | Case Studies | Kernel/s Used | ||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

Method/s | A | B | C | D | E | F | G | H | I | J | K | L | |||||

1 | 2004 | Lee et al. [24] | PCA | First application | NE, WWTP | RBF | |||||||||||

2 | 2004 | Lee et al. [71] | PCA | ✓ | PenSim | POLY | |||||||||||

3 | 2004 | Choi and Lee [85] | PCA | ✓ | NE, WWTP | RBF | |||||||||||

4 | 2005 | Choi et al. [86] | PCA | ✓ | NE, CSTR | RBF | |||||||||||

5 | 2005 | Cho et al. [87] | PCA | ✓ | NE, CSTR | RBF | |||||||||||

6 | 2006 | Yoo and Lee [88] | PCA | ✓ | ✓ | NE, WWTP | RBF | ||||||||||

7 | 2006 | Lee et al. [89] | PCA, PLS | ✓ | BAFP | RBF | |||||||||||

8 | 2006 | Zhang et al. [90] | ICA | ✓ | FCCU | - | |||||||||||

9 | 2006 | Deng and Tian [91] | PCA | ✓ | ✓ | CSTR | RBF | ||||||||||

10 | 2007 | Zhang and Qin [72] | PCA, ICA | ✓ | ✓ | NPP | RBF | ||||||||||

11 | 2007 | Cho [74] | FDA | ✓ | ✓ | PCBP, PenSim | POLY | ||||||||||

12 | 2007 | Cho [92] | FDA | ✓ | TEP | RBF | |||||||||||

13 | 2007 | Sun et al. [93] | PCA | ✓ | ✓ | NE, Rot. Machines | RBF | ||||||||||

14 | 2008 | Choi et al. [94] | PCA | ✓ | ✓ | CSTR | RBF | ||||||||||

15 | 2008 | Tian and Deng [95] | PCA | ✓ | ✓ | TEP | RBF | ||||||||||

16 | 2008 | Wang et al. [96] | PCA | ✓ | ✓ | NPP | RBF | ||||||||||

17 | 2008 | Lee et al. [97] | ICA | ✓ | NE, TEP | RBF | |||||||||||

18 | 2008 | Cui et al. [98] | FDA | ✓ | NE, TEP | RBF, POLY | |||||||||||

19 | 2008 | Cui et al. [99] | SDA | ✓ | TEP | POLY | |||||||||||

20 | 2008 | Zhang and Qin [100] | ICA | ✓ | ✓ | TEP, WWTP, PenSim | RBF | ||||||||||

21 | 2008 | Lu and Wang [101] | PLS | ✓ | ✓ | ✓ | TEP | - | |||||||||

22 | 2008 | He et al. [102] | FDA | ✓ | ✓ | TEP | RBF | ||||||||||

23 | 2008 | Cho [103] | FDA | ✓ | TEP | POLY | |||||||||||

24 | 2008 | Li and Cui [104] | SDA | ✓ | ✓ | TEP | POLY | ||||||||||

25 | 2009 | Li and Cui [105] | FDA | ✓ | ✓ | ✓ | TEP, PenSim | POLY, COS | |||||||||

26 | 2009 | Zhang [106] | ICA | ✓ | ✓ | ✓ | TEP | RBF | |||||||||

27 | 2009 | Zhang and Zhang [107] | ICA, PLS | ✓ | ✓ | TEP, PenSim | RBF | ||||||||||

28 | 2009 | Shao et al. [108] | PCA | ✓ | ✓ | NE, TEP | RBF | ||||||||||

29 | 2009 | Shao and Rong [109] | MVU | ✓ | ✓ | TEP | Manifold | ||||||||||

30 | 2009 | Shao et al. [110] | LPP | ✓ | ✓ | NE, TEP | Manifold | ||||||||||

31 | 2009 | Tian et al. [73] | ICA | ✓ | ✓ | ✓ | PenSim | RBF, POLY | |||||||||

32 | 2009 | Liu et al. [111] | PCA | ✓ | ✓ | ✓ | NE, BDP | RBF | |||||||||

33 | 2009 | Ge et al. [112] | PCA | ✓ | ✓ | NE, TEP | RBF | ||||||||||

34 | 2009 | Zhao et al. [113] | DISSIM | ✓ | NE, TEP | RBF | |||||||||||

35 | 2009 | Zhao et al. [114] | ICA | ✓ | ✓ | ✓ | TTP, PenSim | RBF | |||||||||

36 | 2010 | Jia et al. [115] | PCA | ✓ | ✓ | NE, PenSim | RBF | ||||||||||

37 | 2010 | Cheng et al. [116] | PCA | ✓ | ✓ | ✓ | NE, TEP | RBF | |||||||||

38 | 2010 | Alcala and Qin [117] | PCA | ✓ | CSTR | RBF | |||||||||||

39 | 2010 | Zhu and Song [118] | FDA | ✓ | TEP | RBF | |||||||||||

40 | 2010 | Zhang et al. [119] | PLS | ✓ | ✓ | CAP | RBF | ||||||||||

41 | 2010 | Zhang et al. [120] | PCA | ✓ | ✓ | ✓ | NE, PenSim | RBF | |||||||||

42 | 2010 | Xu and Hu [121] | PCA | ✓ | ✓ | TEP | RBF | ||||||||||

43 | 2010 | Ge and Song [122] | PCA | ✓ | TEP | RBF | |||||||||||

44 | 2010 | Wang and Shi [123] | ICA (CCA) | ✓ | WWTP, TEP | RBF | |||||||||||

45 | 2010 | Sumana et al. [124] | SDA | ✓ | ✓ | NE, TEP | RBF | ||||||||||

46 | 2011 | Sumana et al. [125] | PCA | ✓ | TEP | RBF | |||||||||||

47 | 2011 | Khediri et al. [126] | PCA | ✓ | NE, TEP | RBF | |||||||||||

48 | 2011 | Zhang and Ma [127] | PCA, PLS | ✓ | CAP, EFMF | RBF | |||||||||||

49 | 2011 | Zhang and Hu [128] | PLS | ✓ | ✓ | CAP, PenSim | RBF | ||||||||||

50 | 2011 | Zhang and Hu [129] | PLS | ✓ | ✓ | ✓ | NE, PenSim, EFMF | RBF | |||||||||

51 | 2011 | Zhu and Song [130] | FDA | ✓ | TEP | RBF | |||||||||||

52 | 2011 | Yu [75] | FDA | ✓ | ✓ | ✓ | PenSim | RBF | |||||||||

53 | 2012 | Khediri et al. [131] | K-means | ✓ | NE, SEP | RBF | |||||||||||

54 | 2012 | Rashid and Yu [82] | ICA | ✓ | ✓ | ✓ | ✓ | PenSim | RBF | ||||||||

55 | 2012 | Zhang et al. [132] | PCA | ✓ | ✓ | CAP, PenSim | RBF | ||||||||||

56 | 2012 | Zhang and Ma [133] | ICA | ✓ | ✓ | ✓ | CAP | RBF | |||||||||

57 | 2012 | Zhang et al. [134] | PCA | ✓ | ✓ | NE, TEP, EFMF | RBF | ||||||||||

58 | 2012 | Zhang et al. [135] | PLS | ✓ | ✓ | ✓ | ✓ | PenSim | - | ||||||||

59 | 2012 | Yu [136] | GMM | ✓ | ✓ | ✓ | WWTP | RBF | |||||||||

60 | 2012 | Guo et al. [137] | PCA | ✓ | ✓ | TEP | WAV | ||||||||||

61 | 2012 | Jia et al. [84] | PCA | ✓ | NE, PenSim | RBF, POLY, SIG | |||||||||||

62 | 2012 | Sumana et al. [138] | PCA | ✓ | TEP | POLY | |||||||||||

63 | 2012 | Wang et al. [139] | PCA | ✓ | ✓ | ✓ | PenSim | POLY | |||||||||

64 | 2013 | Liu et al. [140] | ICA | ✓ | ✓ | ✓ | CLG | RBF | |||||||||

65 | 2013 | Peng et al. [141] | T-PLS | ✓ | NE, TEP, HSMP | RBF | |||||||||||

66 | 2013 | Peng et al. [79] | T-PLS | ✓ | ✓ | HSMP | RBF | ||||||||||

67 | 2013 | Wang et al. [142] | PCA | ✓ | ✓ | ✓ | ✓ | PenSim | POLY | ||||||||

68 | 2013 | Jiang and Yan [143] | PCA | ✓ | ✓ | NE, CSTR, TEP | RBF | ||||||||||

69 | 2013 | Jiang and Yan [144] | PCA | ✓ | NE, TEP | RBF | |||||||||||

70 | 2013 | Zhang et al. [145] | ICA | ✓ | ✓ | CAP | RBF | ||||||||||

71 | 2013 | Zhang et al. [146] | PLS | ✓ | ✓ | NE, EFMF | RBF | ||||||||||

72 | 2013 | Zhang et al. [147] | PCA | ✓ | ✓ | PenSim, EFMF | - | ||||||||||

73 | 2013 | Zhang et al. [76] | VCA | ✓ | ✓ | EFMF | RBF | ||||||||||

74 | 2013 | Deng and Tian [148] | PCA | ✓ | ✓ | ✓ | NE, TEP | RBF | |||||||||

75 | 2013 | Deng and Tian [149] | LPP | ✓ | ✓ | CSTR | RBF | ||||||||||

76 | 2013 | Deng et al. [150] | PCA | ✓ | ✓ | TEP | RBF | ||||||||||

77 | 2013 | Rong et al. [151] | LPP, FDA | ✓ | ✓ | ✓ | TEP, WWTP | RBF | |||||||||

78 | 2013 | Hu et al. [152] | PLS | ✓ | ✓ | PP, PenSim | RBF | ||||||||||

79 | 2013 | Hu et al. [153] | PLS | ✓ | ✓ | NE, TEP | RBF | ||||||||||

80 | 2014 | Fan and Wang [66] | ICA | ✓ | ✓ | ✓ | TEP | RBF | |||||||||

81 | 2014 | Fan et al. [154] | ICA | ✓ | ✓ | ✓ | ✓ | NE, TEP | RBF | ||||||||

82 | 2014 | Zhang et al. [155] | ICA | ✓ | ✓ | EFMF | - | ||||||||||

83 | 2014 | Zhang and Li [156] | PCA | ✓ | ✓ | EFMF | RBF | ||||||||||

84 | 2014 | Cai et al. [157] | ICA | ✓ | ✓ | NE, TEP | RBF | ||||||||||

85 | 2014 | Wang and Shi [158] | PLS | ✓ | ✓ | TEP | - | ||||||||||

86 | 2014 | Elshenawy and Mohamed [159] | PCA | ✓ | TEP | RBF | |||||||||||

87 | 2014 | Mori and Yu [160] | PCA, ICA, PLS | ✓ | ✓ | ✓ | PenSim | RBF | |||||||||

88 | 2014 | Castillo et al. [161] | PCA | ✓ | Air Heater | RBF | |||||||||||

89 | 2014 | Vitale et al. [69] | PCA, PLS, FDA | ✓ | ✓ | NE, PP, DP | RBF, POLY | ||||||||||

90 | 2014 | Peng et al. [162] | PCA | ✓ | ✓ | CSTR | RBF | ||||||||||

91 | 2014 | Zhao and Xue [163] | T-PLS | ✓ | ✓ | ✓ | TEP | RBF+POLY | |||||||||

92 | 2014 | Godoy et al. [164] | PLS | ✓ | ✓ | ✓ | NE | RBF | |||||||||

93 | 2014 | Kallas et al. [165] | PCA | ✓ | NE, CSTR | RBF | |||||||||||

94 | 2015 | Ciabattoni et al. [166] | CVA | ✓ | Microgrid | RBF | |||||||||||

95 | 2015 | Vitale et al. [81] | PCA | ✓ | ✓ | NE, DP, RCP | RBF, POLY | ||||||||||

96 | 2015 | Li and Yang [167] | PCA | ✓ | ✓ | NE, TEP | RBF | ||||||||||

97 | 2015 | Liu and Zhang [168] | PLS | ✓ | ✓ | NE, PenSim | RBF | ||||||||||

98 | 2015 | Md Nor et al. [169] | FDA | ✓ | ✓ | TEP | - | ||||||||||

99 | 2015 | Yao and Wang [170] | PCA | ✓ | ✓ | PenSim | RBF | ||||||||||

100 | 2015 | Wang and Yao [171] | PCA | ✓ | ✓ | NE, SEP | RBF | ||||||||||

101 | 2015 | Huang et al. [172] | CVA | ✓ | ✓ | TEP | RBF | ||||||||||

102 | 2015 | Zhang et al. [173] | PLS | ✓ | NE, EFMF | RBF | |||||||||||

103 | 2015 | Zhang et al. [174] | SFA | ✓ | NE, TEP | RBF | |||||||||||

104 | 2015 | Zhang et al. [175] | SFA, FDA | ✓ | CSTR | RBF | |||||||||||

105 | 2015 | Zhang et al. [176] | C-PLS | ✓ | PenSim | - | |||||||||||

106 | 2015 | Samuel and Cao [177] | CVA | ✓ | TEP | RBF | |||||||||||

107 | 2015 | Samuel and Cao [178] | CVA | ✓ | TEP | RBF | |||||||||||

108 | 2015 | Chakour et al. [179] | PCA | ✓ | ✓ | ✓ | TEP | RBF | |||||||||

109 | 2015 | Jiang and Yan [180] | PCA | ✓ | NE, TEP | RBF | |||||||||||

110 | 2015 | Cai et al. [181] | CCA | ✓ | ✓ | NE, TEP | RBF | ||||||||||

111 | 2015 | Luo et al. [182] | GLPP | ✓ | ✓ | NE, TEP | RBF, HK | ||||||||||

112 | 2015 | Tang et al. [77] | VCA | ✓ | ✓ | ✓ | PenSim | RBF | |||||||||

113 | 2015 | Bernal de Lazaro et al. [183] | PCA, FDA | ✓ | ✓ | TEP | RBF | ||||||||||

114 | 2016 | Bernal de Lazaro et al. [184] | PCA, ICA | ✓ | ✓ | ✓ | TEP | RBF | |||||||||

115 | 2016 | Ji et al. [185] | PCA | ✓ | NE | RBF | |||||||||||

116 | 2016 | Xu et al. [186] | PCA | ✓ | ✓ | ✓ | NE, TEP | - | |||||||||

117 | 2016 | Luo et al. [187] | GLPP | ✓ | NE, TEP | RBF | |||||||||||

118 | 2016 | Zhang et al. [188] | ICA | ✓ | ✓ | TEP | - | ||||||||||

119 | 2016 | Taouali et al. [189] | PCA | ✓ | CSTR | RBF | |||||||||||

120 | 2016 | Fazai et al. [190] | PCA | ✓ | CSTR, TEP | RBF | |||||||||||

121 | 2016 | Jaffel et al. [191] | PCA | ✓ | ✓ | TEP | RBF | ||||||||||

122 | 2016 | Mansouri et al. [192] | PCA | ✓ | NE, CSTR | - | |||||||||||

123 | 2016 | Botre et al. [193] | PLS | ✓ | CSTR | - | |||||||||||

124 | 2016 | Samuel and Cao [194] | PCA | ✓ | ✓ | TEP | RBF | ||||||||||

125 | 2016 | Ge et al. [195] | FDA | ✓ | CSTH, TEP | RBF | |||||||||||

126 | 2016 | Jia et al. [196] | PLS | ✓ | ✓ | ✓ | NE, HGPWLTP | RBF | |||||||||

127 | 2016 | Jia and Zhang [197] | PLS | ✓ | ✓ | NE, TEP | RBF | ||||||||||

128 | 2016 | Jiang et al. [198] | PCA | ✓ | ✓ | TEP, CSTR | RBF | ||||||||||

129 | 2016 | Peng et al. [199] | PLS, Fuzzy C-means | ✓ | ✓ | ✓ | ✓ | HSMP | RBF | ||||||||

130 | 2016 | Xie et al. [200] | PCA | ✓ | ✓ | NE, BDP | RBF | ||||||||||

131 | 2016 | Wang et al. [201] | PCR | ✓ | NE | RBF | |||||||||||

132 | 2016 | Huang and Yan [202] | PCA | ✓ | NE, TEP | RBF | |||||||||||

133 | 2016 | Xiao and Zhang [203] | PCA, ICA | ✓ | ✓ | TEP | RBF | ||||||||||

134 | 2016 | Feng et al. [204] | FDA | ✓ | ✓ | ✓ | TEP | RBF | |||||||||

135 | 2016 | Sheng et al. [205] | C-PLS | ✓ | NE, TEP | RBF | |||||||||||

136 | 2016 | Zhang et al. [206] | PLS, PCA | ✓ | ✓ | ✓ | CAP | RBF | |||||||||

137 | 2017 | Jaffel et al. [207] | PCA | ✓ | ✓ | CSTR, TEP | RBF | ||||||||||

138 | 2017 | Lahdhiri et al. [208] | PCA | ✓ | ✓ | NE, CSTR, AIRLOR | RBF | ||||||||||

139 | 2017 | Lahdhiri et al. [209] | PCA | ✓ | ✓ | NE, CSTR | RBF | ||||||||||

140 | 2017 | Mansouri et al. [210] | PLS | ✓ | ✓ | CSEC, GCND | RBF | ||||||||||

141 | 2017 | Mansouri et al. [211] | PCA | ✓ | CSEC | - | |||||||||||

142 | 2017 | Sheriff et al. [212] | PCA | ✓ | CSTR | RBF | |||||||||||

143 | 2017 | Cai et al. [213] | ICA | ✓ | ✓ | NE, TEP | RBF | ||||||||||

144 | 2017 | Zhang et al. [214] | ECA | ✓ | ✓ | ✓ | TEP | RBF | |||||||||

145 | 2017 | Zhang et al. [215] | SFA | ✓ | ✓ | ✓ | NE, PenSim | RBF | |||||||||

146 | 2017 | Zhang and Tian [216] | SFA | ✓ | ✓ | PenSim | POLY | ||||||||||

147 | 2017 | Zhang et al. [217] | PCA | ✓ | EFMF | - | |||||||||||

148 | 2017 | Zhang et al. [218] | PCA, LLE | ✓ | EFMF | - | |||||||||||

149 | 2017 | Zhang et al. [219] | PCA | ✓ | NE, SEP | RBF | |||||||||||

150 | 2017 | Deng et al. [220] | PCA | ✓ | ✓ | TEP | RBF | ||||||||||

151 | 2017 | Deng et al. [221] | PCA | ✓ | ✓ | NE, CSTR | RBF | ||||||||||

152 | 2017 | Deng et al. [222] | PCA, FDA | ✓ | ✓ | NE, CSTR | RBF | ||||||||||

153 | 2017 | Tan et al. [223] | CVA | ✓ | MFF | - | |||||||||||

154 | 2017 | Shang et al. [224] | CVA | ✓ | ✓ | CSTR | RBF | ||||||||||

155 | 2017 | Li et al. [225] | DLV | ✓ | HSMP | RBF | |||||||||||

156 | 2017 | Wang and Jiao [226] | LS | ✓ | NE, TEP | RBF | |||||||||||

157 | 2017 | Wang et al. [227] | DD | ✓ | NE, TEP | RBF | |||||||||||

158 | 2017 | Wang et al. [228] | EDA | ✓ | ✓ | PenSim | RBF | ||||||||||

159 | 2017 | Jiao et al. [229] | PLS | ✓ | NE, TEP | RBF | |||||||||||

160 | 2017 | Huang and Yan [230] | PCA | ✓ | NE, TEP | RBF | |||||||||||

161 | 2017 | Yi et al. [231] | PLS | ✓ | ✓ | TEP, AEP | - | ||||||||||

162 | 2017 | Md Nor et al. [232] | FDA | ✓ | ✓ | TEP | RBF | ||||||||||

163 | 2017 | Du et al. [233] | ICA | ✓ | EFMF | - | |||||||||||

164 | 2017 | Zhang and Zhao [234] | PCA, Fuzzy C-means | ✓ | ✓ | TEP, MFF | RBF | ||||||||||

165 | 2017 | Zhou et al. [235] | RPLVR | ✓ | ✓ | NE, TEP | - | ||||||||||

166 | 2017 | Gharahbagheri et al. [236] | PCA | ✓ | DTS, FCCU, TEP | RBF | |||||||||||

167 | 2017 | Gharahbagheri et al. [237] | PCA | ✓ | NE, FCCU, TEP | RBF | |||||||||||

168 | 2017 | Fu et al. [68] | PCA, PLS | ✓ | NE, GMP, BDP, Mixing | RBF | |||||||||||

169 | 2017 | Galiaskarov et al. [238] | FDA | ✓ | ✓ | Pyrolysis gas furnace | POLY | ||||||||||

170 | 2017 | Zhu et al. [239] | ICA | ✓ | ✓ | ✓ | TEP | RBF, POLY, SIG | |||||||||

171 | 2017 | Zhu et al. [240] | CCA | ✓ | TEP | RBF | |||||||||||

172 | 2018 | Liu et al. [241] | CCA | ✓ | ✓ | ✓ | ✓ | CAP | RBF | ||||||||

173 | 2018 | Wang and Jiao [242] | PLS | ✓ | NE, TEP | RBF | |||||||||||

174 | 2018 | Wang [243] | PLS | ✓ | ✓ | NE, CSTR | RBF | ||||||||||

175 | 2018 | Huang and Yan [244] | PCA | ✓ | NE, TEP | RBF | |||||||||||

176 | 2018 | Huang and Yan [245] | PCA | ✓ | ✓ | ✓ | NE, TEP, IPOP | RBF | |||||||||

177 | 2018 | Fezai et al. [246] | PCA | ✓ | ✓ | NE, TEP | RBF | ||||||||||

178 | 2018 | Fezai et al. [247] | PCA | ✓ | ✓ | ✓ | ✓ | AIRLOR | RBF | ||||||||

179 | 2018 | Mansouri et al. [248] | PCA | ✓ | ✓ | NE, CSEC | RBF | ||||||||||

180 | 2018 | Jaffel et al. [249] | PCA | ✓ | ✓ | ✓ | CSTR | RBF | |||||||||

181 | 2018 | Lahdhiri et al. [250] | PCA | ✓ | ✓ | ✓ | NE, TEP | RBF | |||||||||

182 | 2018 | Tan and Cao [251] | PCA | ✓ | NE, TEP | RBF | |||||||||||

183 | 2018 | He et al. [252] | LPP | ✓ | ✓ | ✓ | PenSim, HSMP | RBF | |||||||||

184 | 2018 | Navi et al. [253] | PCA | ✓ | ✓ | ✓ | IGT | RBF | |||||||||

185 | 2018 | Chakour et al. [254] | PCA | ✓ | TEP, Weather station | RBF | |||||||||||

186 | 2018 | Deng and Wang [255] | PCA | ✓ | NE, TEP | RBF | |||||||||||

187 | 2018 | Deng et al. [256] | PCA | ✓ | ✓ | ✓ | NE, TEP | RBF, POLY | |||||||||

188 | 2018 | Deng et al. [257] | PCA | ✓ | ✓ | NE, TEP | RBF | ||||||||||

189 | 2018 | Deng et al. [258] | FDA | ✓ | ✓ | ✓ | TEP | RBF | |||||||||

190 | 2018 | Zhang et al. [259] | SFA | ✓ | ✓ | ✓ | NE, CSTR | RBF | |||||||||

191 | 2018 | Shang et al. [260] | AMD | ✓ | NE, TEP | POLY | |||||||||||

192 | 2018 | Jiang and Yan [261] | PCA | ✓ | NE, CSTR | - | |||||||||||

193 | 2018 | Feng et al. [262] | ICA | ✓ | ✓ | EFMF | RBF | ||||||||||

194 | 2018 | Zhao and Huang [263] | PCA, DISSIM | ✓ | TPP, CPP | RBF | |||||||||||

195 | 2018 | Zhai et al. [264] | NNMF | ✓ | PenSim | - | |||||||||||

196 | 2018 | Ma et al. [265] | ICA | ✓ | ✓ | TEP | RBF | ||||||||||

197 | 2018 | Lu et al. [266] | CVA, LPP, FDA | ✓ | ✓ | ✓ | TEP | HK | |||||||||

198 | 2018 | Li et al. [267] | PCA | ✓ | ✓ | NE, CPP | - | ||||||||||

199 | 2018 | Chu et al. [268] | PLS | ✓ | ✓ | DMCPP | RBF | ||||||||||

200 | 2019 | Zhai and Jia [269] | NNMF | ✓ | NE, PenSim | RBF | |||||||||||

201 | 2019 | Fezai et al. [270] | PCA | ✓ | ✓ | ✓ | PV | RBF | |||||||||

202 | 2019 | Fazai et al. [271] | PLS | ✓ | ✓ | ✓ | TEP | RBF | |||||||||

203 | 2019 | Deng and Deng [272] | PCA | ✓ | ✓ | NE, TEP | RBF | ||||||||||

204 | 2019 | Cui et al. [273] | PCA | ✓ | ✓ | NE, TEP | RBF, Manifold | ||||||||||

205 | 2019 | Pilario et al. [67] | CVA | ✓ | ✓ | ✓ | ✓ | NE, CSTR | RBF+POLY | ||||||||

206 | 2019 | Lahdhiri et al. [274] | PCA | ✓ | ✓ | ✓ | ✓ | AIRLOR | RBF | ||||||||

207 | 2019 | Liu et al. [275] | ICA | ✓ | ✓ | ✓ | GHP | RBF | |||||||||

208 | 2019 | Liu et al. [276] | ICA | ✓ | ✓ | ✓ | TEP | RBF | |||||||||

209 | 2019 | Yu et al. [277] | CCA | ✓ | ✓ | ✓ | NE, TEP | RBF | |||||||||

210 | 2019 | Guo et al. [278] | PCA | ✓ | ✓ | NE, TEP | RBF | ||||||||||

211 | 2019 | Wu et al. [279] | PCA | ✓ | ✓ | ✓ | NE, TEP | RBF | |||||||||

212 | 2019 | Harkat et al. [280] | PCA | ✓ | NE, TEP | RBF | |||||||||||

213 | 2019 | Ma et al. [281] | CVA, EDA | ✓ | ✓ | ✓ | ✓ | HSMP | - | ||||||||

214 | 2019 | Zhang et al. [282] | ELM | ✓ | NE, CSTR | RBF | |||||||||||

215 | 2019 | Peng et al. [83] | ECA | ✓ | ✓ | ✓ | NE, PenSim | RBF | |||||||||

216 | 2019 | Peng et al. [283] | ICA, EDA | ✓ | ✓ | ✓ | ✓ | ✓ | ✓ | TEP | - | ||||||

217 | 2019 | Yan et al. [284] | PCA, PLS | ✓ | ✓ | NE, TEP | RBF | ||||||||||

218 | 2019 | Huang et al. [285] | DL | ✓ | NE, CSTH, AEP | RBF | |||||||||||

219 | 2019 | Li and Zhao [80] | FDFDA | ✓ | ✓ | ✓ | NE, IMP, CFPP | RBF | |||||||||

220 | 2019 | Zhou et al. [286] | PCA | ✓ | NE, TEP | RBF | |||||||||||

221 | 2019 | Deng et al. [287] | PCA | ✓ | ✓ | TEP | RBF | ||||||||||

222 | 2019 | Wang et al. [288] | PCA | ✓ | ✓ | ✓ | CSTR, HSMP | RBF | |||||||||

223 | 2019 | Zhu et al. [289] | PLS | ✓ | ✓ | ✓ | TEP | RBF | |||||||||

224 | 2019 | Xiao [290] | CVA, LPP | ✓ | ✓ | TEP | HK | ||||||||||

225 | 2019 | Xiao [291] | CVA | ✓ | ✓ | TEP | RBF | ||||||||||

226 | 2019 | Shang et al. [292] | PCA | ✓ | TEP | RBF | |||||||||||

227 | 2019 | Geng et al. [293] | PCA | ✓ | ✓ | TEP | RBF | ||||||||||

228 | 2019 | Md Nor et al. [294] | FDA | ✓ | ✓ | TEP | - | ||||||||||

229 | 2019 | Tan et al. [295] | PCA | ✓ | ✓ | NE, MFF | NSDC | ||||||||||

230 | 2019 | Tan et al. [296] | PCA | ✓ | ✓ | NE, MFF | NSDC |

© 2019 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Pilario, K.E.; Shafiee, M.; Cao, Y.; Lao, L.; Yang, S.-H. A Review of Kernel Methods for Feature Extraction in Nonlinear Process Monitoring. *Processes* **2020**, *8*, 24.
https://doi.org/10.3390/pr8010024

**AMA Style**

Pilario KE, Shafiee M, Cao Y, Lao L, Yang S-H. A Review of Kernel Methods for Feature Extraction in Nonlinear Process Monitoring. *Processes*. 2020; 8(1):24.
https://doi.org/10.3390/pr8010024

**Chicago/Turabian Style**

Pilario, Karl Ezra, Mahmood Shafiee, Yi Cao, Liyun Lao, and Shuang-Hua Yang. 2020. "A Review of Kernel Methods for Feature Extraction in Nonlinear Process Monitoring" *Processes* 8, no. 1: 24.
https://doi.org/10.3390/pr8010024