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Beznia ims3tw 2011
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  • 1. Parametric test metrics estimation using non-Gaussian copulas Kamel Beznia† , Ahcène Bounceur† , Salvador Mir‡ , Reinhardt Euler† † Lab-STICC Laboratory - European University of Britanny - University of Brest 20, Avenue Victor Le Gorgeu 29238, Brest, France Email: {Kamel.Beznia, Ahcene.Bounceur, Reinhardt.Euler} ‡ TIMA Laboratory 46, Avenue Félix Viallet 38031, Grenoble Cedex, France Email: 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, namely performances and test measurements. In this paper, we will use Copulas theory for deriving such a model. A copulas-based model separates the dependencies between these output parameters from their marginal distributions, providing a complete and scale-free description of dependence that is more suitable to be modeled using well known multivariate parametric laws. Previous works have used Gaussian copulas for modeling the dependencies between the output parameters for some types of devices (e.g RF LNA). This paper will illustrate the use of Archimedean copulas for modeling non-Gaussian dependencies. In particular, a Clayton copula will be used to model the dependencies between the output parameters of a case-study test technique for CMOS imagers. Parametric test metrics such as pixel false acceptance a nd false rejection will be estimated using the derived model. I. INTRODUCTION The estimation of parametric test metrics at the design stage is an essential task in a design-for-test flow for analog/RF circuits. These circuits are susceptible to the combined devia- tions of multiple input parameters which can lead to a faulty behavior. However, providing a comprehensive parametric fault model is a challenging task. Most often, an arbitrary finite set of parametric faults (typically, single parameter deviations) have been considered. Clearly, this is insufficient to model the variety of defective devices that can result from the combination of multiple parameter deviations. The problem of estimating parametric test metrics such as defect level and yield loss is equivalent to the problem of yield estimation for analog/RF designers. This is typically done us- ing design-of-experiments and regression-based models. Given the large dimensionality of the input parameter space (design and technology parameters), designers must typically select most critical input parameters for building yield estimation models. This approach relies on a deep understanding of the device under test, and is thus very difficult to automate in practice. An alternative approach for analog/RF circuits is to work only on the space of output parameters which is typically of much lower dimensionality than the input space. The cumulative distribution function (CDF) of the output parameters corresponds to a statistical model that embodies all the 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 fault model of the device under test. Several techniques have been considered in the past for deriving this type of statistical model [1] [2] [3]. The data for extracting this model has been obtained through Monte Carlo simulation of the device under test, typically considering a quick run of a few thousand circuits. However, for high yield designs, parametric faulty devices are rare events. Thus, a major difficulty lies in accurately estimating the distribution tails, where these rare events are found. There is little data, if any, about the distribution tails in the initial set of Monte Carlo data. Although techniques exist for generating data at the distribution tails with a limited run of Monte Carlo simulations [4], the application of such techniques for a multidimensional output parameter space will be extremely time consuming as the 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-based model separates the modeling of the dependencies between the output parameters from the model of the marginal distribution of each output parameter. As a consequence, the model of the parameter dependencies and the model of the marginal distribution of each parameter are estimated separately. The model of the data dependencies (a copula) is a complete and scale-free description of dependence that is more suitable to be obtained from well known multivariate parametric laws. For estimating the copula, we will use in this paper the data from a quick Monte Carlo simulation run. In previous works, we have used Gaussian copulas for modeling the dependencies between the output parameters for some types of devices (e.g RF LNA [2]). In this paper, we will illustrate the use of Archimedean copulas for modeling non-Gaussian output parameter dependencies. We will first review 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
  • 2. used to model a case-study test technique for CMOS imagers presented in Section IV. Parametric test metrics such as pixel false acceptance and false rejection will be estimated using the derived model in Section V. Finally, we conclude the paper with some directions of future work. II. PREVIOUS WORKS The direct calculation of test metrics from circuit-level Monte Carlo simulation is misleading since there is insufficient data to represent properly faulty devices (or devices out of specifications). In order to get a precision of parts-per-million (ppm), it is necessary to generate a population of at least one million devices from an initial small population. An approach for generating this large number of devices is to first estimate the joint probability density function (PDF) of the output parameters from a small population of devices, and next 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 the design stage, considering in all cases an initial sample of data obtained through a quick Monte Carlo circuit simulation. A first approach considered fitting a multivariate normal function [1]. This parametric approach is straightforward, but it is limited to the case of output parameters that have Gaussian distributions linearly correlated with each other. A more general nonparametric approach has been considered in [3], using an adaptive Kernel Density Estimation (KDE). While this method does not make any assumption on the true form of the density, it suffers from increasing inaccuracy as the dimensionality of the output space increases. Finally, the use of Copulas theory has been considered in [2], using a Gaussian copula to model the data dependencies. In this approach, the marginal distributions of the output parameters can have an arbitrary form. However, the data dependencies among the marginal laws must be Gaussian. This has been shown to be a 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 population using density estimation. Recently, an approach that uses Extreme Value Theory (EVT) and the statistical-blockade technique [4] to fit a density to the distribution tail of a single output parameter, has been presented in [5]. This approach allows for a rigorously accurate estimation of test metrics with ppm precision when a single output parameter is considered. However, the extension of this approach for a multivariate case is not well known. Some recent results have looked at Copulas theory for multivariate extreme value analysis in the financial [6] and hydrology fields [7]. While the exploration of this field will be considered for further research, this paper is aimed at illustrating the use of non-gaussian copulas for dependence modeling and test metrics estimation. III. ARCHIMEDEAN COPULAS THEORY A. Sample generation using copulas We will use Copulas theory to generate a large sample of devices from an initial small one obtained from a quick Monte Carlo circuit simulation. We will not describe Copulas theory formally in this paper. Basic definitions and properties of copula functions are presented in [8]. A succinct introduction to copulas for test metrics estimation is given in [2]. To illustrate the use of Copulas theory for sample generation, we will 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 two random variables from their marginal distributions, we apply the transformation ui = Fi(xi), where each initial sample point (x1, x2) is transformed into a new point (u1, u2), using the marginal CDF of each variable (F1 for x1 and F2 for x2). The result of this transformation is the bivariate random vector U = (U1, U2) shown in the lower left corner (blue colour). This new sample distribution corresponds to an empirical copula for which the marginal distributions are uniform. This complete and scale-free description of dependence is more suitable to be fitted to well known multivariate parametric laws called copulas. Once a parametric form of the copula has been fitted, we can use it for the generation of a large sample of data. We can sample an arbitrary large number of points from the copula density, and each point can be transformed back to the initial distribution using the inverse CDF of each marginal
  • 3. variable xi = F−1 i (ui). The sample data generated will then have the same joint PDF as the initial sample of data. This is illustrated in Figure 3. As in [2], we have considered a Gaussian copula for this example that can be easily recognized by his eye-like elliptical form (in blue colour). To sample the Gaussian copula, we first transform the initial empirical copula into a multinormal distribution using standardised marginal distributions. We next fit a multinormal distribution to this data which is easy to sample using readily available techniques (in black colour). Each generated point in the multinormal distribution is transformed back to the copula using the stan- dardized marginals (in blue colour). Finally, each copula point is transformed to the initial distribution using the inverse CDF of 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 of Archimedean copulas. Archimedean copulas include a large variety of copula families that can be easily constructed to model non linear dependencies and non elliptical distributions. For example, Archimedean copulas can describe asymmetric dependencies, where the dependence coefficients in the upper and the lower tails are different. In this work, we will use as dependence coefficient the Kendall’s τ instead of the classical linear correlation factor ρ. An estimator ˆτ of this coefficient is calculated as follows: ˆτ = 2 n(n − 1) i<j sgn[(xi−xj)(yi−yj)], i, j = 1, ..., n (1) where sgn(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)) if ϕ(u1) + ϕ(u2) ≤ ϕ(0) 0 otherwise (2) where ϕ is called the generator function of the Archimedean copula. Notice that the generator function allows to write the copula 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. This equation can be generalized to d dimensions. Using the generator function ϕ, the Kendall’s τ of an archimedean copula can be written as follows : τ = 1 + 4 0 1 ϕ(u) ϕ (u) du (3) It is estimated using Equation (1). C. Clayton Copula The Clayton copula [9][10] is an archimedean copula whose generator function is defined as: ϕ(u) = 1 θ (u−θ − 1) (4) with, θ ∈] − 1, 0[∪]0, ∞[. For the two dimensional case its function C(u1, u2) can be written as follows: C(u1, u2) = max u−θ 1 + u−θ 2 − 1 − 1 θ , 0 (5) The generalized form of this equation is given in [8]. The parameter θ depends on Kendall’s τ and is calculated as follows: θ = − 2τ τ − 1 (6) Figure 4(a) shows the CDF of the Clayton copula with a parameter θ = −0.63 and Figure 4(b) shows a set of 1000 samples generated from this copula. (a) (b) Fig. 4. (a) CDF of a Clayton copula with θ = −0.63, (b) 1000 samples generated from the Clayton copula.
  • 4. D. Simulating a Clayton copula To generate N samples from the bivariate Clayton copula we use the Devroye algorithm [11]. The generation of one sample (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 distribution y −→ 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 samples from the bivariate Clayton copula. IV. TEST VEHICLE Our case-study will be a BIST technique for a CMOS imager presented in [12]. The analog and mixed-signal parts of the imager include a large pixel matrix, the column amplifiers and the data converters. The pixel matrix usually composed of millions of pixels is typically read line by line through the column amplifiers. Figure 5 shows the pixel structure composed by PMOS transistors and a photodiode. This type of pixel gives a logarithmic relationship between the output voltage (Vph) and the incoming light (represented as the photogenerated current Iph). Fig. 5. Logarithmic pixel structure. The main performance measured for the pixel matrix and the column amplifier are the Fixed Pattern Noise (FPN). The FPN represents the difference that exists between two pixels (or two column amplifiers) under constant illumination. Different light sources are used to measure this noise. Two major kinds of FPN are measured: Pixel Response Non Uniformity (PRNU) that is obtained by using light sources and Dark Signal Non Uniformity (DSNU) that is obtained under dark conditions. The BIST technique consists of the application of a voltage pulse at the anode of the photodiode, and measuring the output voltage VA of the pixel. The whole analog ground of the pixel is externally pulsed (not shown in Figure 5). This electrical measurement is performed very fast, and the output measurement is thus not dependent on the incoming light. This BIST measurement is intended to capture the major sources of DSNU, such as transistor mismatches. In this work we will consider only the DSNU performance which 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 part par 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 test metrics 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 using relative 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-level simulation of the pixel with 1000 instances. Each output parameter has a Gaussian marginal distribution. The marginal distributions have been validated using the classical univariate Kolmogorov-Smirnov goodness-of-fit test. The parameters of these 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 of each marginal we obtain the empirical copula of Figure 6(b). The resulting distribution does not have an elliptical form typical of a gaussian dependence. Instead, this distribution has the same form as the bivariate Clayton copula of Figure 4(b). In order to formally verify this, we use the goodness-of-fit test presented in [13] that uses as statistic the Cramer-von Mises test. This test implemented in R software compares the empirical copula with a parametric estimate of the copula. The copula is indeed a Clayton copula which is characterized by the parameter θ. This parameter is estimated using Equation 6. Note that to calculate ˆθ we need to estimate the Kendall’s τ. Hence, ˆθ = −0.77 for an estimated ˆτ = −0.63. B. Generation of a large sample With the estimated value of ˆθ, we can generate a large sample of devices using the Clayton copula. For example, Figure 7 shows the result of generating 16000 samples using the Clayton copula (in black) and the initial 1000 samples obtained via Monte Carlo simulation (in gray). C. Test metrics estimation For the estimation of test metrics, we generate now a sample of 1 million pixels. The generated sample is used to fix the test limits on the VA measurement, in order to achieve desired
  • 5. (a) (b) Fig. 6. (a) Initial sample of 1000 pixels obtained from circuit-level Monte Carlo 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 false rejection (FR). Figure 8 shows the values of FA and FR for different 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 to 0.06V with a step of 0.001V. A value of FA=FR=2124 ppm is obtained. VI. CONCLUSIONS AND FUTURE WORK This paper has illustrated the use of a Clayton copula for modeling the nonlinear dependencies between DSNU and BIST measurement of a case-study CMOS imager. The copulas-based model has been used for the setting of test limits of the BIST measurement and the estimation of test metrics such as pixel false acceptance and false rejection. The obtained result of FA=FR=2124 ppm already indicates that further work is required in order to propose a more accurate BIST measurement to replace DSNU tests. Further work will also be considered to explore multivariate extreme value analysis using Copulas theory, in order to increase the accuracy of the test metrics estimation approach and to integrate the developed tools in an existing mixed-signal CAT platform [14]. Fig. 8. Test metrics vs. test limits estimated form the set of 106 circuits generated from the Clayton copula. REFERENCES [1] A. Bounceur, S. Mir, E. Simeu, and L. Rolíndez, Estimation of test metrics for the optimisation of analogue circuit testing, Journal of Electronic Testing: Theory and Applications, vol. 23, no. 6, pp. 471-484, 2007. [2] A. Bounceur and S. Mir. Estimation of Test Metrics at the Design Stage Using Copulas, In International Mixed-Signals, Sensors and Systems Test Workshop (IMS3TW’08), Vancouver, Canada, June 2008. [3] H. Stratigopoulos, S. Mir, and A. Bounceur. Evaluation of analog/RF test 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 to Memory 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 PPM accuracy. 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 Theory and Its Usefulness in Understanding Risk, North American Actuarial Journal, 2006, 10(4), 1-27. [7] B. Renard, M. Langa, Use of a Gaussian copula for multivariate extreme value analysis: Some case studies in hydrology, Advances in Water Resources 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 and its application in epidemiological studies of familial tendency in chronic disease incidence. Biometrika, 65 :141-151. [10] Genest, C. and MacKay, R. J. (1986). Copules archimédiennes et familles 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 Validation of 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 for copulas: A review and a power study. Insurance: Mathematics and Economics, 44, 2009, pp. 199-214. [14] A. Bounceur, S. Mir, L. Rolíndez and E. Simeu. CAT platform for analogue and mixed-signal test evaluation and optimization, 2007, in IFIP International Federation for Information Processing, Volume 249, VLSI-SoC : Research trends in VLSI and Systems on Chip, eds. De Micheli, G., Mir, S., Reis, R., (Boston : Springer), pp. 281-300.