Beznia ims3tw 2011

Parametric test metrics estimation usingnon-Gaussian copulas

Kamel Beznia†, Ahcène Bounceur†, Salvador Mir‡, Reinhardt Euler††Lab-STICC Laboratory - European University of Britanny - University of Brest

20, Avenue Victor Le Gorgeu29238, Brest, France

Email: {Kamel.Beznia, Ahcene.Bounceur, Reinhardt.Euler}@univ-brest.fr‡TIMA Laboratory

46, Avenue Félix Viallet38031, Grenoble Cedex, FranceEmail: [email protected]

Abstract— The evaluation of parametric test metrics for ana-log/RF test techniques requires an accurate multivariate statisti-cal model of output parameters of the device under test, namelyperformances and test measurements. In this paper, we will useCopulas theory for deriving such a model. A copulas-based modelseparates the dependencies between these output parametersfrom their marginal distributions, providing a complete andscale-free description of dependence that is more suitable to bemodeled using well known multivariate parametric laws. Previousworks have used Gaussian copulas for modeling the dependenciesbetween the output parameters for some types of devices (e.gRF LNA). This paper will illustrate the use of Archimedeancopulas for modeling non-Gaussian dependencies. In particular,a Clayton copula will be used to model the dependencies betweenthe output parameters of a case-study test technique for CMOSimagers. Parametric test metrics such as pixel false acceptancea nd false rejection will be estimated using the derived model.

I. INTRODUCTION

The estimation of parametric test metrics at the design stageis an essential task in a design-for-test flow for analog/RFcircuits. These circuits are susceptible to the combined devia-tions of multiple input parameters which can lead to a faultybehavior. However, providing a comprehensive parametricfault model is a challenging task. Most often, an arbitraryfinite set of parametric faults (typically, single parameterdeviations) have been considered. Clearly, this is insufficientto model the variety of defective devices that can result fromthe combination of multiple parameter deviations.

The problem of estimating parametric test metrics such asdefect level and yield loss is equivalent to the problem of yieldestimation for analog/RF designers. This is typically done us-ing design-of-experiments and regression-based models. Giventhe large dimensionality of the input parameter space (designand technology parameters), designers must typically selectmost critical input parameters for building yield estimationmodels. This approach relies on a deep understanding of thedevice under test, and is thus very difficult to automate inpractice. An alternative approach for analog/RF circuits isto work only on the space of output parameters which istypically of much lower dimensionality than the input space.The cumulative distribution function (CDF) of the output

parameters corresponds to a statistical model that embodies allthe required information for parametric test metrics estimation.It is a comprehensive model of device parametric behaviour.Thus, it can also be used as a comprehensive parametric faultmodel of the device under test.

Several techniques have been considered in the past forderiving this type of statistical model [1] [2] [3]. The datafor extracting this model has been obtained through MonteCarlo simulation of the device under test, typically consideringa quick run of a few thousand circuits. However, for highyield designs, parametric faulty devices are rare events. Thus,a major difficulty lies in accurately estimating the distributiontails, where these rare events are found. There is little data,if any, about the distribution tails in the initial set of MonteCarlo data. Although techniques exist for generating data at thedistribution tails with a limited run of Monte Carlo simulations[4], the application of such techniques for a multidimensionaloutput parameter space will be extremely time consuming asthe dimensionality of the output space increases.

One approach for tackling this problem is to consider Cop-ulas theory for building the statistical model. A copulas-basedmodel separates the modeling of the dependencies between theoutput parameters from the model of the marginal distributionof each output parameter. As a consequence, the model ofthe parameter dependencies and the model of the marginaldistribution of each parameter are estimated separately. Themodel of the data dependencies (a copula) is a complete andscale-free description of dependence that is more suitable to beobtained from well known multivariate parametric laws. Forestimating the copula, we will use in this paper the data froma quick Monte Carlo simulation run.

In previous works, we have used Gaussian copulas formodeling the dependencies between the output parameters forsome types of devices (e.g RF LNA [2]). In this paper, wewill illustrate the use of Archimedean copulas for modelingnon-Gaussian output parameter dependencies. We will firstreview some previous works in this field in Section II. Next,Section III will introduce the theory of Archimedean copulas,describing in particular the Clayton copula. This copula will be

used to model a case-study test technique for CMOS imagerspresented in Section IV. Parametric test metrics such as pixelfalse acceptance and false rejection will be estimated using thederived model in Section V. Finally, we conclude the paperwith some directions of future work.

II. PREVIOUS WORKS

The direct calculation of test metrics from circuit-levelMonte Carlo simulation is misleading since there is insufficientdata to represent properly faulty devices (or devices out ofspecifications). In order to get a precision of parts-per-million(ppm), it is necessary to generate a population of at leastone million devices from an initial small population. Anapproach for generating this large number of devices is tofirst estimate the joint probability density function (PDF) ofthe output parameters from a small population of devices, andnext sample this density to generate the devices.

As shown in Figure 1, several approaches have been con-sidered in the past to obtain this density function at thedesign stage, considering in all cases an initial sample ofdata obtained through a quick Monte Carlo circuit simulation.A first approach considered fitting a multivariate normalfunction [1]. This parametric approach is straightforward,but it is limited to the case of output parameters that haveGaussian distributions linearly correlated with each other. Amore general nonparametric approach has been consideredin [3], using an adaptive Kernel Density Estimation (KDE).While this method does not make any assumption on the trueform of the density, it suffers from increasing inaccuracy as thedimensionality of the output space increases. Finally, the useof Copulas theory has been considered in [2], using a Gaussiancopula to model the data dependencies. In this approach, themarginal distributions of the output parameters can have anarbitrary form. However, the data dependencies among themarginal laws must be Gaussian. This has been shown to bea reasonable approach for devices such as RF LNAs.

Copulas based model

Gaussian [2]

Archimedean [this work]

Non parametric model

[3]

Multivariate Gaussian model

[1]

Large population generated

from the statistical model

Small population generated

from the circuit simulation

Fig. 1. Statistical methods for the generation of a large circuit populationusing density estimation.

Recently, an approach that uses Extreme Value Theory(EVT) and the statistical-blockade technique [4] to fit a densityto the distribution tail of a single output parameter, has beenpresented in [5]. This approach allows for a rigorously accurateestimation of test metrics with ppm precision when a singleoutput parameter is considered. However, the extension of thisapproach for a multivariate case is not well known. Somerecent results have looked at Copulas theory for multivariate

extreme value analysis in the financial [6] and hydrologyfields [7]. While the exploration of this field will be consideredfor further research, this paper is aimed at illustrating the useof non-gaussian copulas for dependence modeling and testmetrics estimation.

III. ARCHIMEDEAN COPULAS THEORY

A. Sample generation using copulas

We will use Copulas theory to generate a large sample ofdevices from an initial small one obtained from a quick MonteCarlo circuit simulation. We will not describe Copulas theoryformally in this paper. Basic definitions and properties ofcopula functions are presented in [8]. A succinct introductionto copulas for test metrics estimation is given in [2]. Toillustrate the use of Copulas theory for sample generation, wewill consider the bivariate example of Figure 2.

Fig. 2. Calculating the copula from a population.

The scatter plot of a bivariate random vector X = (X1, X2)is shown in the upper right corner (purple colour) of Figure 2.In order to separate the dependencies between these tworandom variables from their marginal distributions, we applythe transformation ui = Fi(xi), where each initial samplepoint (x1, x2) is transformed into a new point (u1, u2), usingthe marginal CDF of each variable (F1 for x1 and F2 for x2).The result of this transformation is the bivariate random vectorU = (U1, U2) shown in the lower left corner (blue colour).This new sample distribution corresponds to an empiricalcopula for which the marginal distributions are uniform. Thiscomplete and scale-free description of dependence is moresuitable to be fitted to well known multivariate parametric lawscalled copulas.

Once a parametric form of the copula has been fitted, wecan use it for the generation of a large sample of data. Wecan sample an arbitrary large number of points from thecopula density, and each point can be transformed back tothe initial distribution using the inverse CDF of each marginal

variable xi = F−1i (ui). The sample data generated will thenhave the same joint PDF as the initial sample of data. Thisis illustrated in Figure 3. As in [2], we have considered aGaussian copula for this example that can be easily recognizedby his eye-like elliptical form (in blue colour). To sample theGaussian copula, we first transform the initial empirical copulainto a multinormal distribution using standardised marginaldistributions. We next fit a multinormal distribution to thisdata which is easy to sample using readily available techniques(in black colour). Each generated point in the multinormaldistribution is transformed back to the copula using the stan-dardized marginals (in blue colour). Finally, each copula pointis transformed to the initial distribution using the inverse CDFof each marginal variable (purple colour).

Fig. 3. Sample generation using a Gaussian copula.

B. Archimedean copulas

In this work we will illustrate the use for test metrics estima-tion of another kind of copula which belongs to the family ofArchimedean copulas. Archimedean copulas include a largevariety of copula families that can be easily constructed tomodel non linear dependencies and non elliptical distributions.For example, Archimedean copulas can describe asymmetricdependencies, where the dependence coefficients in the upperand the lower tails are different.

In this work, we will use as dependence coefficient theKendall’s τ instead of the classical linear correlation factorρ. An estimator τ̂ of this coefficient is calculated as follows:

τ̂ =2

n(n− 1)

∑i<j

sgn[(xi−xj)(yi−yj)], i, j = 1, ..., n (1)

wheresgn(z) =

{1 if z ≥ 0−1 if z < 0

where, (x1, y1), . . . , (xn, yn) are n observations from a vector(X,Y ) of continuous random variables.

Archimedean copulas have the following form:

C(u1, u2) =

ϕ−1(ϕ(u1) + ϕ(u2)) ifϕ(u1) + ϕ(u2) ≤ ϕ(0)

0 otherwise(2)

where ϕ is called the generator function of the Archimedeancopula. Notice that the generator function allows to write thecopula as a sum of functions of the marginals. For 0 ≤ u ≤ 1,ϕ is defined as ϕ(1) = 0, ϕ′(u) < 0 and, ϕ′′(u) > 0. Thisequation can be generalized to d dimensions.

Using the generator function ϕ, the Kendall’s τ of anarchimedean copula can be written as follows :

τ = 1 + 4

∫0

1ϕ(u)

ϕ′(u)du (3)

It is estimated using Equation (1).

C. Clayton Copula

The Clayton copula [9][10] is an archimedean copula whosegenerator function is defined as:

ϕ(u) =1

θ(u−θ − 1) (4)

with, θ ∈] − 1, 0[∪]0,∞[. For the two dimensional case itsfunction C(u1, u2) can be written as follows:

C(u1, u2) = max

{(u−θ1 + u−θ2 − 1

)− 1θ , 0

}(5)

The generalized form of this equation is given in [8]. Theparameter θ depends on Kendall’s τ and is calculated asfollows:

θ = − 2τ

τ − 1(6)

Figure 4(a) shows the CDF of the Clayton copula with aparameter θ = −0.63 and Figure 4(b) shows a set of 1000samples generated from this copula.

(a) (b)

Fig. 4. (a) CDF of a Clayton copula with θ = −0.63, (b) 1000 samplesgenerated from the Clayton copula.

D. Simulating a Clayton copula

To generate N samples from the bivariate Clayton copulawe use the Devroye algorithm [11]. The generation of onesample (u1, u2) requires the following steps:• Sample two random values x1 and x2 from the uniform

distribution xi(i=1,2) −→ U [0, 1].• Sample a random value y from the Gamma distributiony −→ Γ[0, 1].

• Calculate : u1 = (1 + x1

y )−θ and u2 = (1 + x2

y )−θ.By executing these steps N times we can generate N samplesfrom the bivariate Clayton copula.

IV. TEST VEHICLE

Our case-study will be a BIST technique for a CMOSimager presented in [12]. The analog and mixed-signalparts of the imager include a large pixel matrix, the columnamplifiers and the data converters. The pixel matrix usuallycomposed of millions of pixels is typically read line by linethrough the column amplifiers. Figure 5 shows the pixelstructure composed by PMOS transistors and a photodiode.This type of pixel gives a logarithmic relationship betweenthe output voltage (Vph) and the incoming light (representedas the photogenerated current Iph).

Fig. 5. Logarithmic pixel structure.

The main performance measured for the pixel matrix and thecolumn amplifier are the Fixed Pattern Noise (FPN). The FPNrepresents the difference that exists between two pixels (or twocolumn amplifiers) under constant illumination. Different lightsources are used to measure this noise. Two major kinds ofFPN are measured: Pixel Response Non Uniformity (PRNU)that is obtained by using light sources and Dark Signal NonUniformity (DSNU) that is obtained under dark conditions.

The BIST technique consists of the application of a voltagepulse at the anode of the photodiode, and measuring theoutput voltage VA of the pixel. The whole analog ground ofthe pixel is externally pulsed (not shown in Figure 5). Thiselectrical measurement is performed very fast, and the outputmeasurement is thus not dependent on the incoming light. ThisBIST measurement is intended to capture the major sourcesof DSNU, such as transistor mismatches.

In this work we will consider only the DSNU performancewhich has a non-linear relationship with the BIST measure-ment VA. The specification is DSNU ∈ [−0.032, 0.032] V,fixed at 3σ which leads to a yield of ' 3000dppm (defect partpar million, i.e. 3000 pixels out of specs in 1 million).

V. TEST METRICS ESTIMATION WITH A CLAYTON COPULA

For a CUT with output parameters X = (X1, X2, ..., Xn),the general procedure for the estimation of parametric testmetrics is as follows:• Run a circuit-level Monte Carlo simulation to obtain m

samples of X.• Fit a parametric copula to the empirical copula obtained.• Use a goodness-of-fit test to verify that the m samples of

X follow the fitted parametric copula.• Sample the copula to generate N (N � m) new obser-

vations of X.• Given the performance specifications, set test limits as a

trade-off between parametric test metrics calculated usingrelative frequencies in the generated large sample of X.

A. Fitting a Clayton copula

Figure 6(a) shows a scatter plot of the output parameters(DSNU,VA) that results from a Monte Carlo circuit-levelsimulation of the pixel with 1000 instances. Each outputparameter has a Gaussian marginal distribution. The marginaldistributions have been validated using the classical univariateKolmogorov-Smirnov goodness-of-fit test. The parameters ofthese Gaussians are given as follows:• DSNU : µ=1.7mV , σ=10mV• VA : µ=513mV , σ=9mV

By transforming each sample point via the CDF function ofeach marginal we obtain the empirical copula of Figure 6(b).The resulting distribution does not have an elliptical formtypical of a gaussian dependence. Instead, this distribution hasthe same form as the bivariate Clayton copula of Figure 4(b).In order to formally verify this, we use the goodness-of-fittest presented in [13] that uses as statistic the Cramer-vonMises test. This test implemented in R software compares theempirical copula with a parametric estimate of the copula. Thecopula is indeed a Clayton copula which is characterized bythe parameter θ. This parameter is estimated using Equation6. Note that to calculate θ̂ we need to estimate the Kendall’sτ . Hence, θ̂ = −0.77 for an estimated τ̂ = −0.63.

B. Generation of a large sample

With the estimated value of θ̂, we can generate a largesample of devices using the Clayton copula. For example,Figure 7 shows the result of generating 16000 samples usingthe Clayton copula (in black) and the initial 1000 samplesobtained via Monte Carlo simulation (in gray).

C. Test metrics estimation

For the estimation of test metrics, we generate now a sampleof 1 million pixels. The generated sample is used to fix thetest limits on the VA measurement, in order to achieve desired

(a) (b)

Fig. 6. (a) Initial sample of 1000 pixels obtained from circuit-level MonteCarlo simulation, and (b) the empirical copula of the initial sample.

Fig. 7. 1000 pixels generated from circuit-level Monte Carlo simulation(gray) vs. 16000 generated from the Clayton copula (black).

trade-offs between the pixel false acceptance (FA) and falserejection (FR). Figure 8 shows the values of FA and FR fordifferent test limits of the test measurement VA in the range[0.51 − k, 0.51 + k]V where the factor k is varied from 0 to0.06V with a step of 0.001V. A value of FA=FR=2124 ppmis obtained.

VI. CONCLUSIONS AND FUTURE WORK

This paper has illustrated the use of a Clayton copulafor modeling the nonlinear dependencies between DSNUand BIST measurement of a case-study CMOS imager. Thecopulas-based model has been used for the setting of test limitsof the BIST measurement and the estimation of test metricssuch as pixel false acceptance and false rejection. The obtainedresult of FA=FR=2124 ppm already indicates that furtherwork is required in order to propose a more accurate BISTmeasurement to replace DSNU tests. Further work will alsobe considered to explore multivariate extreme value analysisusing Copulas theory, in order to increase the accuracy of thetest metrics estimation approach and to integrate the developedtools in an existing mixed-signal CAT platform [14].

Fig. 8. Test metrics vs. test limits estimated form the set of 106 circuitsgenerated from the Clayton copula.

REFERENCES

[1] A. Bounceur, S. Mir, E. Simeu, and L. Rolíndez, Estimation of test metricsfor the optimisation of analogue circuit testing, Journal of ElectronicTesting: Theory and Applications, vol. 23, no. 6, pp. 471-484, 2007.

[2] A. Bounceur and S. Mir. Estimation of Test Metrics at the Design StageUsing Copulas, In International Mixed-Signals, Sensors and Systems TestWorkshop (IMS3TW’08), Vancouver, Canada, June 2008.

[3] H. Stratigopoulos, S. Mir, and A. Bounceur. Evaluation of analog/RFtest measurements at the design stage. IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems, 28(4), April 2009, pp.582-590.

[4] Amith Singhee, Rob A. Rutenbar: Statistical Blockade: Very Fast Statisti-cal Simulation and Modeling of Rare Circuit Events and Its Application toMemory Design. IEEE Trans. on CAD of Integrated Circuits and Systems,28(8): 1176-1189 (2009)

[5] H. Stratigopoulos and S. Mir. Analog test metrics estimates with PPMaccuracy. IEEE International Conference on Computer-Aided Design(ICCAD), San Jose, USA, November 2010, pp. 241 - 247.

[6] Debbie J. Dupuis and Bruce L. Jones, Multivariate Extreme Value Theoryand Its Usefulness in Understanding Risk, North American ActuarialJournal, 2006, 10(4), 1-27.

[7] B. Renard, M. Langa, Use of a Gaussian copula for multivariate extremevalue analysis: Some case studies in hydrology, Advances in WaterResources 30, 897-912.

[8] R.-B. Nelsen. An Introduction to Copulas. Lecture Notes in Statistics.Springer, New York, (1999).

[9] Clayton, D. G. (1978). A model for association in bivariate life tables andits application in epidemiological studies of familial tendency in chronicdisease incidence. Biometrika, 65 :141-151.

[10] Genest, C. and MacKay, R. J. (1986). Copules archimédiennes etfamilles de lois bidimensionnelles dont les marges sont données. Canad.J. Statist., 14 :145-159.

[11] Luc Devroye. Non-Uniform Random Variate Generation. New York:Springer-Verlag, 1986.

[12] Livier Lizarraga, Salvardor Mir, Gilles Sicard, Experimental Validationof a BIST Technique for CMOS Active Pixel Sensors, VTS, pp.189-194,2009 27th IEEE VLSI Test Symposium, 2009.

[13] C. Genest, B. Remillard and D. Beaudoin. Goodness-of-fit tests forcopulas: A review and a power study. Insurance: Mathematics andEconomics, 44, 2009, pp. 199-214.

[14] A. Bounceur, S. Mir, L. Rolíndez and E. Simeu. CAT platform foranalogue and mixed-signal test evaluation and optimization, 2007, inIFIP International Federation for Information Processing, Volume 249,VLSI-SoC : Research trends in VLSI and Systems on Chip, eds. DeMicheli, G., Mir, S., Reis, R., (Boston : Springer), pp. 281-300.