Classification Algorithms for Virtual Metrology · Vahid Partovi Nia GERAD Research Center Department of Mathematical and Industrial Engineering Montreal Polytechnique Email:...

Classification Algorithms for Virtual Metrology

Shaima Tilouche

Department of Mathematical and

Industrial Engineering

Montreal Polytechnique

Email: [email protected]

Samuel Bassetto

Department of Mathematical and

Industrial Engineering

Montreal Polytechnique

Email: [email protected]

Vahid Partovi Nia

GERAD Research Center

Department of Mathematical and

Industrial Engineering

Montreal Polytechnique

Email: [email protected]

Abstract—Virtual metrology in quality control deals withdrifts in product quality that occur during non-sampling periods.This approach enables a hundred percent control and improvesthe precision of statistical control, specially while there is nosampling activity in manufacturing process. The main challengein virtual metrology is inaccurate predictions. As such, the choiceof an appropriate algorithm for prediction is crucial. We compareseveral algorithms that can be used for prediction in virtualmetrology. The comparison over different prediction algorithmsis made on a simulated data inspired from virtual metrologyapplication.

I. INTRODUCTION

Semiconductor manufacturing production line includesmany steps and requires a high level of accuracy in eachstep. Several sensors are placed in different locations in orderto detect the defects to stop following the manufacturingprocess for already defected items. Information collected fromthese sensors are, then, used in quality control. The controlis performed in many levels to ensure the stability and theperformance of the production process. [2] enumerates somelevels that should be taken into account during the controlprocess. These levels are mentioned as layers of control fromtools to product. At the product level, electric tests verifyproper functionality of chips. These tests are performed atthe end of the manufacturing stage [13] [18]. Some otherintegration tests are applied during the fabrication to verifythe properties of technological modules. Tests like verifyingthat transistor’s shape is correctly processed, or examining thatelectrical properties of several layers of materials are in thedesired specifications. After each step, usually it is possibleto perform some tests to monitor any abnormal variations oftools that operate a particular process. Finally, at the tool level,numerous sensors are employed to regulate and to monitor theprocess.

All the tests and the monitoring steps described above arebased on few samples of products, except for the electricalwafer sort being the final product. As a consequence, a drift inproduction can occur in the non-sampling period and this driftis hardly detected [6]. Virtual metrology (VM) algorithms havebeen suggested as an alternative to 100% wafer measurement,in order to support wafer-to-wafer control [12]. VM canincrease metrology data availability, reduce send-ahead wafers,improve quality guarantee levels, and reduce cycle time. Themain challenge in using VM is the poor prediction accuracy.Thus, the choice of the algorithm used in VM method is ofa great importance. VM refers to classification of products(correct or defected) using some auxiliary information col-

lected during the manufacturing process. We briefly review andcompare several classification algorithms that can be used inVM. Section II presents a literature review on VM. Section IIIdescribes the data simulation setup, used to compare theclassification algorithms, Section IV introduces classificationalgorithms and Section V discusses the numerical comparison.

II. VIRTUAL METROLOGY

Virtual metrology has been introduced to employ mathe-matical models on accessible measurements from an operatingprocess with the aim of predicting some variables of interest.This methodology allows to predict relevant variables usingequipment measurement, without physically conducting qual-ity measurement [21] [14]. [6] observed a strong correlationbetween the tool history and the wafer measurement. Theyfound the coefficient of determination R2 > 0.97 can beachieved with more than 500 wafers on deposition as theresponse and equipments as the predicting variables. Thisstrong correlation suggests that a wafer-to-wafer control canbe quickly enabled using an existing lot-to-lot control system.This indirect technique, called virtual metrology, provides anefficient and economical alternative to wafer-to-wafer control.

Several algorithms have been proposed in the VM litera-ture. [14] suggested a linear regression model to predict thestate of the wafer in real time. They proposed to find thecoefficients using least squares. The univariate output of theirmodel was the wafer measurement and the inputs were theequipments’ measurement. The model was updated as new out-put measurement was available. [3] studied a virtual metrologymodel using partial least squares. This model predicts chemicalvapor deposition oxide thickness for an Inter Metal Dielectricdeposition process.

Many VM algorithms have been developed based on neu-ral networks [22]. Neural network is an implicit nonlinearmodel fitting. Different versions of neural networks havebeen considered in the VM litterature. [15] established a VMmodel using radial basis network. The effectiveness of theproposed VM system was tested on chemical vapor depositionprocesses in a practical semiconductor manufacturing. Theirresult confirmed that neural network can be used effectively toconstruct a predictive model. [14] adopted a system using backpropagation neural network for establishing a model for theetching process in semiconductor manufacturing. [7] comparedthe performance of the radial basis network and the backpropagation neural network on the thin-film transistor liquidcrystal display industry. The radial basis function network andthe back propagation neural network produced quite similar

results. Some other versions of neural models are proposedto detect wafer anomaly such as polynomial neural network,piecewise linear neural network, and fuzzy neural network, seefor instance [5], [4], [11], and [20].

Kernel approaches, specifically support vector machines,are a powerful tool to predict wafer drifts based on equipmentmeasurements [16]. [7] reports that the support vector ma-chines approach give a better prediction accuracy comparedwith the radial basis function network and also compared withthe back-propagation approach, see also [1].

Genetic algorithm is a powerful optimization tool for modelfitting [8]. A kernel adjustment is proposed to deal withoverfitting problems. [7] combines the support vector machinesand the genetic algorithm to construct a virtual metrology sys-tem for the chemical vapor deposition process. [23] suggestsprincipal components axes to reduce the dimensionality ofplasma dimensions after etching process. It is well-known thatuncorrelated features improve the estimation and the predici-tion of statistical models. The principal components are linearcombinations of the inputs and are mutually uncorrelated.

Existence of a large list of different classification algo-rithms makes the choice of an appropriate algorithm difficult.We aim to compare different classification algorithms proposedin the literature on a simple simulated example motivated froma VM problem.

III. DATA SIMULATION

We tried to simulate the data according realistic conditionsappearing in VM. The inputs, say x, represent the equipmentmeasurements. In practice, the inputs can be power, pressure,temperature, etc. Some of such inputs are intercorrelated andsome others are independent from each other. We simulatedthe total of 10 inputs. The output variable, say y, is a binaryvariable that represents the final product’s state, being corrector defected. The following matrix describes the data structure

x11 x12 · · · x110 y1x21 x22 · · · x210 y2

......

......

xn1 xn2 · · · xn10 yn

,

where each row corresponds to a measurement of on a wafer.Each column xj is a continuous value of an input variable, sayequipment j, and the last column y shows the binary output.The number of rows, n, is the total number of wafers. Thematrix entries xij represents the measurement of wafer i onequipment j and yi is the final state of wafer i.

We simulated the data with the following structure. Inputvariables x1 and x2 are a block of intercorrelated variables;another block of correlated variables contains x3, x4, andx5. The other inputs x6, x7, x8, x9 and x10 are generatedindependently, all irrelevant to classify the output. The latterblock does not affect the output variabl e, but they contribute inthe classification error generated by the measurement system.Table I illustrates the dependence structure of the simulated in-put variables in which Np(µ,Σ) denotes a p-variate Gaussiandistribution with mean µ and variance-covariance Σ.

block Inputs Correlation Distribution

1 (x1, x2) yes N2

[(

00

)

,

(

1 0.90.9 1

)]

2 (x3,x4,x5) yes N3

000

,

1 0.9 0.90.9 1 0.90.9 0.9 1

3 x6, . . . , x10 no N1(0, 1)

TABLE I: The generated input data structure. Three blocks ofinput variables are generated of size n = 100 observations.

First, we simulated a binary output yi being generated asa function of only three input variables xi1, xi3, and xi4

{

yi = 1 if a′zi ≥ 0,

yi = 0 if a′zi < 0,

(1)

where a = (a1, a2, a3) and zi = (xi1, xi3, xi4). This modelproduces observations that only x1, x3, and x4 are useful forclassification, and the other variables are noise. Second, wesimulated a binary output using a quadratic function of x1,x2,and x4 is generated

{

yi = 1 if a′zi + z

′iAzi ≥ 0,

yi = 0 if a′zi + z

′iAzi < 0.

(2)

The elements of the vector a are sampled independentlyand uniformly from {−6,−3, 3, 6}, and the elements of thesymmetric matrix A are sampled from {−6,−3, 0, 3, 6}. Wegenerated n = 100 observations as the training set. A datasetof the same size is generated as the validation set. The modelis fitted on the training set, and the precision of the resultingclassification is evaluated on the validation set. The total of 20Monte Carlo simulations have been run.

IV. CLASSIFICATION ALGORITHMS

Several classification algorithms listed below are used topredict the output as a function of the inputs.

A. k-Nearest-Neighbours

The k-nearest-neighbours is a model-free algorithm thatpredicts the output based on its k nearest neighbours. Thenearest neighbours are found using a distance, often theEuclidean distance, computed over the corresponding inputvariables. Suppose N(x) is the neighbourhood at point xwhere the k data fall into, then

ŷ(x) =1

k

∑

xi∈N(x)

yi. (3)

This technique gives a step function approximation to theclassification function, see Fig. 1. The tunning parameter k ischosen manually or is estimated using cross-validation.

B. Logistic Regression

The logistic regression is a generalization of the linearregression where the output variable is binary. This techniqueis used to predict a binary outcome based on one or morecontinuous predictor variables. The logistic regression esti-mates the coefficients of a linear classifier using the conditional

Input

Out

put

Fig. 1: An illustrative example of a 3-nearest-neighbours algo-rithm. The circles are observations from an unknown function.The three green blobs are the data that fall in the neighborhoodof x. The vertical red line represents x and the horizontal blueline shows the neighbourhood of size 3, denoted by N(x) in(3). The output is predicted by the average of the closest 3points, denoted by the red blob.

distribution of yi | xi. Since the logistic regression uses aprobabilistic model to estimate the classification function, theprobability of (y = 1 | x) can be extracted after the fitting fora given x. In order to produce a binary predict, this estimatedprobability is cut at a certain point, usually 0.5. The probabilityof yi | xi is expressed as

Pr(yi = 1|xi) =exp(β0 + x

′iβ)

1 + exp(β0 + x′iβ),

where x′i = (xi1, xi2, .., xi10) and β = (β1, . . . , β10)′. The

regression coefficients are estimated using maximum likeli-hood. The log likelihood function of the Bernoulli distributionis maximized using iterative reweighted least squares. TheBernoulli log likelihood, say ℓ(β), is expressed as

ℓ(β) =n∑

i=1

log

[

{

exp(β0 + x′iβ)

1 + exp(β0 + x′iβ)

}yi{

1−exp(β0 + x′iβ)

1 + exp(β0 + x′iβ)

}1−yi]

Like linear regression, logistic regression suffers from over-fitting and produces unstable estimation of coefficients whilea many of noise variables is added in the model. As a remedythe penalized logistic regression is fitted. The penalty term,penalizes large absolute values of model coefficients. Themaximizing function is

ℓ(β)− λ||β||22

where ||β||22 =∑10

j=1 β2j is the squared Euclidean norm of the

regression coefficients. Here λ is a positive tuning parameter,usually estimated by cross-validation.

C. Neural Network

A neural network is a set of simple but highly intercon-nected processing elements, called neurons, to fit a highlynonlinear model, see Fig. 2. Neural network has been evaluatedfor different number of hidden layers with different weights.The most predictive number of layers is chosen. This approachhelp regularizing this algorithm and avoids overfitting.

Fig. 2: Schematic representation of a neural network model.

D. Linear Discriminant

Linear discriminant analysis separates data into differentclasses (two classes for a binary output) using a linear hyper-plane, see Fig. ?? (left panel). However, linear discriminantcoefficients are sensitive to the correlation between the inputvariables. In order to improve the classification performance,it is proposed to perform the classification on the principalcomponents of data [17]. We applied linear discriminant onfour principal components.

An alternative to improve the performance in the presenceof correlated variables is penalization. A penalized discrimi-nant analysis is suggested in [9]. Absolute norm penalty, alsocalled the lasso penalty, is applied to the discriminant vectorsto encourage variable selection simultaneously.

E. Quadratic Discriminant

Quadratic discriminant analysis, as its name indicates, pro-poses quadratic boundaries to separate data. This algorithm issimilar to the linear discriminant, except it allows for quadraticcoefficients as well, see Fig. ?? (right panel). We applied thisalgorithm on the principal components of data also.

F. Mixture Discriminant

Polynomial boundaries such as linear and quadratic func-tions are too restrictive for complex data. Mixture of discrimi-nant functions covers a flexible class of classification functions.This algorithm is called mixture discriminant analysis [10]. AGaussian mixture model for the kth class has density

Pr(X |G = k) =Rk∑

i=1

πkrφ(X,µkr ,Σkr),

where the mixing proportions πkr sum to one. This has Rkprototypes for the kth class, and in our specification, thecovariance matrix Σkr is used as the metric throughout. Givensuch a model for each class, the class posterior probabilities

0 2 4 6 8 10

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Fig. 3: Scatter plot of the quadratic simulated data over Input1 and Input 2, see also Table I. Linear discriminant (topleft panel), quadratic discriminant (top right panel), mixturediscriminant (bottom left panel) and neural networks (bottomright panel) are used to find the decision boundaries.

are given by

Pr(X |G = k) =

∑Rkr=1 πkrφ(X,µkr ,Σkr)πk

∑K

l=1

∑Rkr=1 πlrφ(X,µlr,Σlr)πl

,

where πl represents the class prior probabilities. The param-eters of mixture discriminant are estimated using maximumlikelihood. The classification obtained through Mixture dis-criminant is compared with linear and quadratic discriminantand also with neural networks as shown in Fig. 3. We canconclude that mixture discriminant gives better results thanlinear and quadratic discriminant and basically, as good resultsas neural networks. This good classification result is due to theflexibility of mixture discriminant boundaries.

G. Kernelized Support Vector Machines

A support vector machine constructs a hyperplane in a highdimensional space. Intuitively, a good separation is achievedby the hyperplane that has the largest distance to the nearesttraining data point of any class, so-called functional margins,see Fig. 4. Like the other methods, after computing thehyperplane, the data are categorized into 2 classes. Instead ofthe linear support vector machines, we tested a more flexibleversion called kernelized support vector machines. The kernelfunction transforms the classification problem into a new spacedefined by the kernel (inner product) on the input variables.We used the radial basis kernel also called Gaussian kernel.

V. NUMERICAL RESULTS

A Monte Carlo simulation study over the linear and thequadratic models (1) and (2) is summarized in Table II.

−2 −1 0 1 2

−3

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Fig. 4: Linear support vector machines shown on a separableillustrative example. The dashed lines show the margins andthe data that fall on the margin, shown by triangles, are calledthe support vectors.

Linear Quadratic

Algorithm p̂L p̂ p̂U p̂L p̂ p̂UNeural Network 94 94 94 83 84 84

Kernel SVM 88 88 88 82 82 83

MDA-PCA 86 86 87 81 82 82

QDA-PCA 86 87 87 81 82 82

KNN 85 86 85 82 82 82

LDA-PCA 87 87 88 75 76 77

Penalized LDA 87 88 88 76 76 77

Penalized LR 90 91 91 66 67 68

LR 88 89 89 63 64 64

TABLE II: The estimated correct classification rates for dif-ferent algorithms in percentages, p̂, and their respective 95%confidence lower and upper bounds, p̂L and p̂U . The resultsare demonstrated once for the linear simulated data (left) andonce for the quadratic simulated data (right).

The simulation codes are written the statistical programminglanguage R [19]. Simulations are performed using a 2.30 GHzIntel core i5-2410m processor and 6.00 Go RAM, takingaround 2 minutes to run all algorithms. Datasets and the Rcodes are available and will be provided upon request. Thecorrect classification rates of the output variable is summarizedin Table II.

Neural network outperforms all other algorithms for bothlinear and quadratic data. The logistic regression (LR) andthe penalized logistic regression are, also, good classifiersfor the linear data. However, they give significantly inferiorcorrect classification rates for the quadratic data. The penalizedlogistic regression (Penalized LR) improves the rate of correctclassification compared to the logistic regression, particularlyfor the quadratic output. It achieves an increase of 3% (from64% to 67%). The quadratic discriminant combined with theprincipal components (QDA-PCA) shows better results thanthe linear discriminant (LDA-PCA). The QDA-PC on thequadratic output shows 6% increase of the correct classificationrate compared to LDA. The mixture discriminant method(MDA) gives results similar to the quadratic discriminant, butbetter than the LDA-PCA and the Penalized LDA, for the linearoutput. The kernelized support vector machines (Kernel SVM)gives accurate predictions for both linear and quadratic outputs.

VI. CONCLUSION

We briefly reviewed the existing literature on qualitycontrol with an emphasis on virtual metrology. We insistthat the choice of a proper classification algorithm is ofgreat importance in this area. Therefore, we studied severalalgorithms that could be used for VM on some simulated data.These algorithms perform differently depending on the outputs(linear or quadratic). However, neural network outperforms allothers in both cases. This suggests to keep neural networkmethod as a strong potential candidate for modelling in VM.

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