Virtual metrology as an approach for product quality estimation in Industry 4.0: a systematic review and integrative conceptual framework

Figures & data

Figure 1. Publications on virtual metrology, from the first paper in 2005 until November 2020.

Figure 2. Selection of relevant articles as per the research methodology.

Table 1. Summary of technical group categories.

Category name	Short explanation	Papers	Review
Preprocessing	Data preprocessing before a quality estimation.	149	Q
Quality estimation	Quality estimation based on machine learning.	168	Q
Drift detection	Detection of concept drift.	22	EXH
Sampling decision system	Based on the detection of drift, this system adapts the measurement frequency, enabling the quality estimator to be updated and ensuring system reliability.	9	EXH
Updatability feature	Variation of the quality estimator, enabling it to be updated.	31	Q
Adaptability feature	Variation of the quality estimator, enhancing its data efficiency.	11	EXH
Multi-stage architecture	Architecture of the interconnections between different implementations of VM.	5	EXH
Machine control	Machine control based on quality estimation.	25	Q
Fab-wide architecture	VM implementation in industrial environments.	13	EXH

Q: quantitative review. EXH: exhaustive review.

Table 2. Summary of manufacturing application categories.

Category name	Short explanation	Papers	Review
Semiconductor manufacturing	All studies of VM operations involving semiconductors.	158	Q
Other manufacturing domain	All studies of VM operations involving other manufacturing domains.	12	Q

Q: quantitative review. EXH: exhaustive review.

Figure 3. Virtual metrology framework.

Figure 4. Virtual metrology terminology chart.

Table 3. Outlier removal algorithms.

Methods name	Algorithms name	Description	Advantage	Limitation
Manual outlier detection	Basic plot, t-SNE	Handmade based on visualisation approaches.	Simple	Only offline
Statistics-based methods	Hotelling’s $T^2$, Grubb’s test	Statistical algorithm for testing the distribution distance between two sets of data.	Simple	Statistical assumptions
Principal component analysis (PCA)-based methods	PCA	Distance monitoring of each data point projected into another space.	Simple	Linear transformation
Specific method	ART2	Specific clustering algorithm.
Novelty detection methods	SVDD, Gauss, MoG, KMC, (k-NN), GMM, Parzen window, SOM	Clustering or one-class classification algorithm that detects novelty.	Nonlinear	Computationally demanding

All of the citations are available in Table 2 of the Annex.

Table 4. Feature selection algorithms based on a similar table in Vallejo et al. (2019).

Methods name	Algorithms name	Description	Advantages	Limitations
Filter method	Pearson, input covariance, ANOVA	Correlation-based approach	Computationally efficient Simple Interpretable	Mostly linear Not associated with the inference model
Wrapper method	Forward, backward, or stepwise selection, genetic algorithm	Iterative approach	Nonlinear Interpretable Associated with the inference model	Computationally demanding Difficult to generalise Can overfit
Embedded method	Lasso, trees	Inference algorithm with in-built feature selection	Can be nonlinear Two in one Interpretability	Depends on the inference algorithm

All of the citations are available in Table 3 of the Annex.

Table 5. Feature-extraction algorithms.

Algorithm name	Description	Advantages	Limitations
Principal component analysis (PCA)	Input features projected on the eigenspace, removing the smallest eigenvector	Computationally efficient	Linear Unsupervised
Partial least squares (PLS)	Like PCA but matches both projection of the input and the output	Computationally efficient	Linear Interpretability
Convolutional neural network (CNN)	Inference neural network algorithm with in-built feature extraction	Nonlinear Two in one	Computationally demanding Large learning set
Autoencoder	Neural network-based feature extraction	Nonlinear	Computationally demanding Unsupervised Large learning set

All of the citations are available in Table 3 of the Annex.

Table 6. Quality estimation algorithms based on a similar table in Vallejo et al. (2019).

Method name	Algorithm name	Advantages	Limitations
Linear models	Multiple linear regression (MLR)	Low complexity Computationally and data-efficient Interpretable	Linear Prone to instability Sensitive to outliers
	Partial least square (PLS)	Computationally and data-efficient Feature extraction	Linear Interpretability Risk of omitting real correlations
	Lasso	Computationally and data-efficient Feature selection Interpretable	Linear Low performance with highly correlated variables Low performance if the number of descriptors exceeds the number of observations
Neural networks	Multiple layer perceptron (MLP)	Nonlinear Ability to generate new features from a subset of features located in the training dataset	Computationally demanding Large data set required Black box
	Convolutional neural network (CNN)	Nonlinear Feature extraction More robust	Computationally demanding Large data set required Black box
	Recursive neural network (RNN)	Nonlinear Time memory: effective for time series	Computationally demanding Large data set required Black box
	Bayesian neural network (BNN)	Nonlinear Causal Prior-knowledge injection	Computationally demanding Large data set required Black box
Kernel methods	Gaussian process regression (GPR)	Nonlinear Data-efficient In-built uncertainty quantification	Computationally demanding Poorly scalable Hard to tune Assumes Gaussianity
	Support vector regression (SVR)	Non-linear Data-efficient Less prone to overfitting	Poorly scalable Difficult to tune
Ensemble methods	Bagging	Nonlinear Easily updatable In-built uncertainty quantification Reduces variance	Can result in high bias
	Stacking	Nonlinear Enhances accuracy	Depends on the learners that are used

All of the citations are available in Table 4 of the Annex.

Figure 5. SDS strategies.

Table 7. Sampling decision system strategies.

Method name	Algorithm name	Description	Advantages	Limitations
Time-based SDS	Statistical process control	Optimisation of the fixed frequency	Simple No need for VM	Under-optimised quality and efficiency
Event-based SDS	Uncertainty sampling	Update when an event occurs		Hidden-context sensitivity
Hybrid SDS	Intelligent sampling decision system	Update at the fixed frequency or when an event occurs	Partially optimised SDS	Room for optimisation
	Automated sampling decision system	Update at the optimised frequency or when an event occurs	Optimised SDS	Room for optimisation Complex

Table 8. Updatability algorithms.

Algorithm name	Algorithm variation	Description	Advantages	Limitations
Moving window (MW)	Moving window	Keeps the youngest points in memory	Simple SPD depending on window size	Can forget meaningful points Bad against recurrent drift
	Weighted moving window	The same but with an emphasis on the youngest point	The same Increases plasticity
Just-in-time (JIT) learning		Keeps the full set, trains on a subset on the fly	Deals with recurrent drift Fast inference Good SPD	No forgetting capability Only applicable to fast learners

All of the citations are available in Table 5 of the Annex.

Table 9. Adaptability approaches.

Approach name	Description	Advantages	Limitations
System identification	Analytically learns the difference between two products to compensate for it	Simple Keeps the same model	Very low applicability
Transfer learning	Uses an existing VM of a similar product to pretrain a new product VM	Faster learning time Smaller learning set	Needs similar products
Multi-task learning	Trains two similar tasks’ VMs simultaneously	Faster learning time Smaller learning set	Detrimental if the tasks are not similar
Semisupervised learning	Generates training data from an unlabelled point	Smaller learning set	Sensitive to concept drift

Figure 6. Multi-stage architecture diagrams.

Table 10. Multi-stage architectures.

Architecture name	Description	Advantages	Limitations
Serial architecture	Serially interconnected VM points.	Efficient	Information loss
Hub architecture	Use an existing VM of a similar product to pre-train a new product VM.	Fewer VM points Raw information	Requires high computational power
Cascade architecture	Mix between serial and hub architecture.	Depends the exact architecture	Depends the exact architecture

Table 11. Control algorithms.

Algorithm name	Description	Advantages	Limitations
Exponentially weighted moving average (EWMA)	Low pass filter with linear controller.	Simple Reduce noise	Sensitive to trends (slow drift) Linear controller
Double EWMA (dEWMA)	Same as EWMA, with a second EWMA for trends.	Same as EWMA Deals with trends	Linear controller
Model predictive control (MPC)	Trajectory optimisation based on the process model.	Nonlinear controller Optimal	Requires a precise process model

All of the citations are available in Table 6 of the Annex.

Table 12. Fab-wide architectures.

Vallejo, Marcela, Carolina De La Espriella, Juliana Gómez-Santamaría, Andrés Felipe Ramírez-Barrera, and Edilson Delgado-Trejos. 2019. “Soft Metrology Based on Machine Learning: A Review.” Measurement Science and Technology 31 (3): 032001. doi:https://doi.org/10.1088/1361-6501/ab4b39.

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