Beelen, T.G.J.

Dohmen, J.J.

Tasic, B.

ter Maten, E.J.W.

English

NARCIS 

Fitting generalized Gaussian distributions for process capability index

Pure OAI Repository

 Fitting generalized Gaussian distributions for processcapability indexCitation for published version (APA):Beelen, T. G. J., Dohmen, J. J., ter Maten, E. J. W., & Tasic, B. (2016). Fitting generalized Gaussiandistributions for process capability index. (CASA-report; Vol. 1620). Technische Universiteit Eindhoven.Document status and date:Published: 01/10/2016Document Version:Publisher’s PDF, also known as Version of Record (includes final page, issue and volume numbers)Please check the document version of this publication:• A submitted manuscript is the version of the article upon submission and before peer-review. There can beimportant differences between the submitted version and the official published version of record. Peopleinterested in the research are advised to contact the author for the final version of the publication, or visit theDOI to the publisher's website.• The final author version and the galley proof are versions of the publication after peer review.• The final published version features the final layout of the paper including the volume, issue and pagenumbers.Link to publicationGeneral rightsCopyright and moral rights for the publications made accessible in the public portal are retained by the authors and/or other copyright ownersand it is a condition of accessing publications that users recognise and abide by the legal requirements associated with these rights.            • Users may download and print one copy of any publication from the public portal for the purpose of private study or research.            • You may not further distribute the material or use it for any profit-making activity or commercial gain            • You may freely distribute the URL identifying the publication in the public portal.If the publication is distributed under the terms of Article 25fa of the Dutch Copyright Act, indicated by the “Taverne” license above, pleasefollow below link for the End User Agreement:www.tue.nl/taverneTake down policyIf you believe that this document breaches copyright please contact us at:openaccess@tue.nlproviding details and we will investigate your claim.Download date: 16. Nov. 2023                                          EINDHOVEN UNIVERSITY OF TECHNOLOGY Department of Mathematics and Computer Science            CASA-Report 16-20 October 2016   Fitting generalized Gaussian distributions for process capability index  by  T.G.J. Beelen, J.J. Dohmen, E.J.W. ter Maten, B. Tasić          Centre for Analysis, Scientific computing and Applications Department of Mathematics and Computer Science Eindhoven University of Technology P.O. Box 513 5600 MB Eindhoven, The Netherlands ISSN: 0926-4507        Fitting Generalized Gaussian Distributions forProcess Capability IndexTheo G.J. Beelen1, Jos J. Dohmen2, E. Jan W. ter Maten1,3, and Bratislav Tasić2Abstract The design process of integrated circuits (IC) aims at a high yield as wellas a good IC-performance. The distribution will not be standard Gaussian anymore.In fact, the corresponding probability density function has a more flat shape thanin case of standard Gaussian. In order to optimize the yield one needs a statisticalmodel for the observed distribution. One of the promising approaches is to use theso-called Generalized Gaussian distribution function and to estimate its definingparameters. We propose a numerical fast and reliable method for computing theseparameters.1 IntroductionWe assume N independent samples xi in some given interval [U,V ] and based onsome empirical density function. To define a quality measure index we are nowinterested in the ‘best’ fitting function within the family of Generalized GaussianDensity (GGD) distributions as shown in Fig.1 and given by the expressionf (x) =β2α Γ (1/β )exp(−(|x−µ|α)β), (1)T.G.J. Beelen, E.J.W. ter MatenEindhoven University of Technology, the Netherlands,e-mail: {T.G.J.Beelen,E.J.W.ter.Maten}@tue.nlJ.J. Dohmen, B. TasicNXP Semiconductors, Eindhoven, the Netherlands,e-mail: {Jos.J.Dohmen,Bratislav.Tasic}@nxp.comE.J.W. ter MatenBergische Universität Wuppertal, Germany,e-mail: Jan.ter.Maten@math.uni-wuppertal.de12 Theo G.J. Beelen, Jos J. Dohmen, E. Jan W. ter Maten, and Bratislav Tasićwhere α,β > 0, µ ∈ R and Γ (z) =∫∞0 tz−1e−tdt, for z > 0, is the Gamma func-tion [8, 14]. The mean and the variance of the GGD (1) are given by µ andα2Γ (3/β )/Γ (1/β ), respectively. Hence after expressing α =σ√Γ (1/β )/Γ (3/β )we get that, for all β , the variance is σ2. We note that for β = 2 one has Γ (1/2) =√π , Γ (3/2) = 0.5√π and then α = σ√2; i.e., the GGD becomes the Gaussian dis-tribution. The parameter β determines the shape. For β = 1 the GGD correspondsto a Laplacian distribution; for β → +∞ the pdf in (1) converges to a uniform dis-tribution in (µ−√3σ ,µ +√3σ), and when β ↓ 0 we get a degenerate distributionin x = µ (but with a finite variance). For some graphical impression, see Fig. 1. Weare interested in the cases when β ≥ 2.The parameters of the ‘best’ fitting distribution function can be found by maximiz-ing the logarithm of the likelihood function L = ln(L ) = ∑Ni=1 f (xi). The necessaryconditions are∂L∂α= 0 : α =(βNN∑i=1|xi−µ|β)1/β, (2)∂L∂β= 0 :1β+Ψ(1/β )β 2− 1NN∑i=1∣∣∣xi− µ̂α∣∣∣β ln∣∣∣xi− µ̂α∣∣∣= 0, (3)∂L∂ µ= 0 : ∑xi≥µ|xi−µ|β−1− ∑xi<µ|xi−µ|β−1 = 0. (4)Fig. 1 Generalized Gaussian density functions with µ = 0 and α = 1.Fitting Generalized Gaussian Distributions for Process Capability Index 3Algorithm 1 Averaged Generalized Gaussian Distribution Fit1: procedure AGGDF(X, N, M, µ̂)2: Determine the empirical pdf f̂ (x) from the trimmed data X. . See Fig. 3 and [8]3: Compute the cumulative distribution function F̂(x) =∫ x−∞ f̂ (t)dt.4: for k = 1, . . . ,M do5: Generate random values {xki | i = 1, ...,N} using F̂−1.6: Compute the zero β̂k of g(β ) = 0, using these xki -values and µ̂ . . See (5)7: Compute âk. . See (2)8: end for9: Average β̂ = 1M ∑Mi=1 β̂k, α̂ =1M ∑Mi=1 α̂k.10: return α̂ , β̂ . . In this Algorithm µ̂ is unchanged11: end procedureHere Ψ is the Digamma function Ψ(x) = ddx ln(Γ (x)) = Γ′(x)/Γ (x) [14]. When weassume that µ = µ̂ is known then we can ignore (4). Several papers consider esti-mates for α and β [2–4, 11, 12, 16] to solve the equations (2)-(3), but they assumethat the sample size is large enough and/or that β ≤ 3, motivated by the variousapplication areas. We note that [6, 7] also consider the case for a small sample size.In general, the resulting estimators can be biased [8].We exploit the explicit elimination of α in (3) after which only one additional equa-tion remainsg(β ) = g(β ; µ̂) = 0, (5)in which µ = µ̂ is now a given parameter. The analytical formulae for g(β ) andg′(β ) are given by (see also [1, 5, 7, 14])g(β ) =1+Ψ(1/β )β− ∑Ni=1 |xi− µ̂|β ln |xi− µ̂|∑Ni=1 |xi− µ̂|β+ln(βN ∑Ni=1 |xi− µ̂|β)β,g′(β ) =−Ψ(1/β )β 2−Ψ′(1/β )β 3+1β 2− ∑Ni=1 |xi−µ|β (ln |xi−µ|)2∑Ni=1 |xi−µ|β+(∑Ni=1 |xi−µ|β ln |xi−µ|∑Ni=1 |xi−µ|β)2+∑Ni=1 |xi−µ|β ln |xi−µ|β ∑Ni=1 |xi−µ|β−ln(βN ∑Ni=1 |xi−µ|β)β 2.(6)Clearly, (5) can be solved by any (nonlinear) iterative method, for example by New-ton’s method using the expressions in (6). We have outlined our algorithm in Alg. 1.We consider the resulting density function f (x; µ̂, α̂, β̂ ) as best fit to the measureddata (see [1]). We make the following observations [1].1. It might happen that (5) has no zero for some particular parameter choices. Nu-merical experiments (see Fig.2) indicate that (5) has a (unique) zero as long asx̄ 6= µ̂ , and no zero in case of x̄ = µ̂;4 Theo G.J. Beelen, Jos J. Dohmen, E. Jan W. ter Maten, and Bratislav TasićFig. 2 Sensitivity of g(β ) with respect to the endpoint of x-interval.2. When M = 1, β̂ is very sensitive to the choice of the interval [U,V ] where thexi are located. In fact, β̂ strongly depends on the difference |µ̂ − x̄| where x̄ =1N ∑Ni=1 |xi− µ̂|.The first observation comes from the simplification by fixing µ = µ̂ , but is easy tomeet.For the second observation we introduced M-times the steps 5-7 within a loop andfinally taking averages, see Alg. 1. By this, in practice, also N can be taken smaller.We observe that due to the large value of ∂α/∂β averaging the α̂k gives betterresults for α than by using (2) on β̂ .2 Numerical resultsIn circuit design one aims to reduce faults and to increase yield [10]. Specially addedelectronic control is applied to obtain narrow tails in empirical probability densityfunctions. This process is called (electronic) ’trimming’. It has no relation to statisti-cal techniques like Winsoring (in which one clips outliers to a boundary percentile),or Trimming (in which one simply neglects outliers). Here it is an electronic tun-ing, f.i., by a variable resistor. Assume that a some measurement point a circuithas a DC solution V (R, p), that depends on a resistor R and an uncertain parame-ter p. The circuit design aims to satisfy a performance criterion VLow ≤ V ≤ VUp.Fitting Generalized Gaussian Distributions for Process Capability Index 5Fig. 3 Measured data (left) and the associated empirical probability density function f̂ (right).Now for each p we can determine how V depends on R. An optimal R(p) assuresthat V (R(p), p) = VRef ∈ [VLow,VUp]. R(p) can determined by some nonlinear so-lution techniqe, involving solving the circuit equations several times. We note thatR(p) can also be found by exploiting the expansion series in generalized polynomialchaos for Uncertainty Quantification using R and p as two parameters [9, 15]. TheUQ facilities provide sensitivities to R and to p as library functionality in post pro-cessing. This allows that R(p) can be determined quite efficiently for every realiza-tion of p. In practice a table can be made from which R(p) can easily be determinedor approximated.We applied Alg. 1 (with M = 50) to ‘trimmed’ data from first NXP IC-measurements(Fig. 3). The computed values βk and their mean β̂ are shown in Fig.4-a. The com-puted density function f as well as the initially fitted (non-symmetrical) densityfunction f̂ are given in Fig. 5. Note that even the tails are very well approximated inFig.4-b. To get an impression of the sensitivity of the computed density w.r.t. α̂ wevaried the computed value of α̂ with +/- 10%, plotted the corresponding densitiesand computed the Mean Square Error (MSE). See Fig. 5 and [1].3 A quality measure index for a Generalized GaussiandistributionAssuming an underlaying distribution being standard Gaussian, the capability of amanufacturing process can be measured using some process capability indices like6 Theo G.J. Beelen, Jos J. Dohmen, E. Jan W. ter Maten, and Bratislav TasićFig. 4(left): The computed βk with mean β̂ = 3.27 and |βk − β̂ |< 20%.(right): The empirical probability function (solid) and the final fitted GGD (dashed).Fig. 5 Sensitivity of the density function f w.r.t. α̂.MSE = (14.31, 56.94, 91.95) for α̂ = (454, 499, 409)∗10−5.Cp =U−L6σand Cpk =min(U−µ,µ−L)3σ, (7)where [L,U ] is the specification interval, µ is the process mean and σ is the processstandard deviation and a process is said to be capable if the process capability indexexceeds a value k≥ 1, where usually k = 4/3. In case of a GGD (1) we can introduceFitting Generalized Gaussian Distributions for Process Capability Index 7a capability index Cpkg similar to the standard Gaussian case asCpkg =min(U−µ,µ−L)3σ, (8)where 2σ2 = αβ . L and U are the lower and upper tolerance levels, respectively.They can be determined as described below.Notice that if x≤ µ then the cumulative distribution function F(x) corresponding tothe GGD (1) is given byF(x) =β2αΓ (1/β )∫ x−∞exp(−(|y−µ|α)β)dy=12Γ (1/β )∫∞((µ−x)/α)βexp(−z)dz.(9)By using the Complementary Incomplete Gamma function defined byΓ (a,x) =∫∞xta−1exp(−t)dt (10)we can rewrite (9) asF(x) =Γ(1/β ,(µ−xα)β)2Γ (1/β )(11)This can be further simplified using the Upper Incomplete Gamma function [13,14]for which standard software is available. For x > µ a similar expression holds.4 ConclusionsWe have shown that measured IC chip production data can adequately be modelledby a Generalized Gaussian distribution (GGD). We developed a new robust numer-ical procedure for computing the parameters of such GGD. The GGD did fit veryaccurately. Using the GGD a quality measure can be defined analogously to the CPKindex for standard Gaussian distributions.Acknowledgements The authors acknowledge support from the projects CORTIF (http://cortif.xlim.fr/): Coexistence Of Radiofrequency Transmission In the Future, a CATRENEproject (Cluster for Application and Technology Research in Europe on NanoElectronics, http://www.catrene.org/) and nanoCOPS (http://fp7-nanocops.eu/): NanoelectronicCOupled Problems Solutions, FP7-ICT-2013-11/619166.8 Theo G.J. Beelen, Jos J. Dohmen, E. Jan W. ter Maten, and Bratislav TasićReferences1. T.G.J. Beelen, J.J. Dohmen: Parameter estimation for a generalized Gaussian distribu-tion. CASA Report 15-40, TU Eindhoven, 2015. Online: http://www.win.tue.nl/analysis/reports/rana15-40.pdf.2. L. Bombrun, F. Pascal, J.-Y. Tourneret, Y. Berthoumieu: Performance of the maximum like-lihood estimators for the parameters of multivariate generalized Gaussian distributions.Proc. ICASSP-2012, IEEE. Int. Conf. on Acoustics, Speech, and Signal Processing, Kyoto,Japan, 3525-3528, 2012. https://hal.archives-ouvertes.fr/hal-00744600.3. G. González-Farı́as, J.A. Domı́nguez-Molina, R.M. Rodrı́guez-Dagnino: Efficiency of theapproximated shape parameter estimator in the generalized gaussian distribution. IEEETrans. on Vehicular Technology 58:8, 4214–4223, 2009.4. K. Kokkinakis, P.C. Loizou: Advances in modern blind separation algorithms – Theory andapplications. Synthesis Lectures on Algorithms and Software in Engineering #6, Morgan &Claypool Publishers, 20105. K. Kokkinakis, A.K. Nandi: Exponent parameter estimation for generalized Gaussian proba-bility density functions with application to speech modeling. Signal Processing 85, 1852–1858,2005.6. R. Krupiński: Approximated fast estimator for the shape parameter of generalized Gaussiandistribution for a small sample size. Bull. of the Polish Academy of Sciences, Technical Sci-ences 63:2, 405–411, 2015.7. R. Krupiński, J. Purczyński: Approximated fast estimator for the shape parameter of general-ized Gaussian distribution. Signal processing 86, 205–211, 2006.8. W.L. Martinez, A.R. Martinez: Computational statistics handbook with Matlab. Chapman &Hall/CRC, 2002.9. E.J.W. ter Maten, R. Pulch, W.H.A. Schilders, H.H.J.M. Janssen: Efficient calculation of Un-certainty Quantification. In: M. Fontes, M. Günther, N. Marheineke (Eds): Progress in Indus-trial Mathematics at ECMI 2012. Series Mathematics in Industry Vol. 19, Springer, 361–370,2014.10. E.J.W. ter Maten, O. Wittich, A. Di Bucchianico, T.S. Doorn, T.G.J. Beelen: Importancesampling for determining SRAM yield and optimization with statistical constraint. In:B. Michielsen, J.-R. Poirier (Eds.): Scientific Computing in Electrical Engineering SCEE2010. Series Mathematics in Industry Vol. 16, Springer, 39–48, 2012.11. O.M.M. Mohamed, M. Jaı̈dane-Saı̈dane: On the parameters estimation of the GeneralizedGaussian Mixture Model. Proc. EUSIPCO/EURASIP 2009, 17th European Signal ProcessingConference, Glasgow, Scotland, 2273–2277, 2009.12. A.A. Roenko, V.V. Lukin, I. Djurović, M. Simeunović: Estimation of parameters for General-ized Gaussian Distribution. Proc. ISCCSP-2014, 6th Int. Symp. on Communications, Controland Signal Processing, Athens, Greece, 376–379, 2014.13. N.M. Temme: Computational aspects of Incomplete Gamma Functions with large complexparameters. Int. Series of numerical Mathematics 119, 551–562, 1994.14. Wikipedia: Generalized Normal Distribution. https://en.wikipedia.org/wiki/Generalized_normal_distribution; Gamma function. https://en.wikipedia.org/wiki/Gamma_function; Digamma function. https://en.wikipedia.org/wiki/Digamma_function; Incomplete Gamma Function.https://en.wikipedia.org/wiki/Incomplete_gamma_function. 2016.15. D. Xiu: Numerical methods for stochastic computations - A spectral method approach. Prince-ton Univ. Press, Princeton, NJ, USA, 2010.16. S. Yu, A. Zhang, H. Li: A review of estimating the shape parameter of Generalized GaussianDistributions. J. of Comput. Information Systems, 8:12, pp. 9055–9064, 2012.PREVIOUS PUBLICATIONS IN THIS SERIES:    Number Author(s) Title Month 16-16     16-17     16-18     16-19     16-20 A.S. Tijsseling      R. Beltman J.H.M. ten Thije Boonkkamp W.L. IJzerman  I.N. Zwaan M.E. Hochstenbach    N.K. Yadav J.H.M. ten Thije Boonkkamp W.L. IJzerman  T.G.J. Beelen J.J. Dohmen E.J.W. ter Maten B. Tasić An overview of fluid-structure interaction experiments in single-elbow pipe systems  A least-squares method for the inverse reflector problem in arbitrary orthogonal coordinate systems  Multidirectional subspace expansion for one-parameter and multiparameter Tikhonov regularization  A least-squares method for the design of two-reflector optical systems   Fitting generalized Gaussian distributions for process capability index July ‘16     August ‘16     August ‘16     September ‘16     October ‘16                                Ontwerp: de Tantes, Tobias Baanders, CWI                                           

Fitting generalized Gaussian distributions for process capability index

Abstract

Similar works

Full text

Available Versions

NARCIS

Pure OAI Repository