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}@univ-brest.fr
‡
TIMA Laboratory
46, Avenue Félix Viallet
38031, Grenoble Cedex, France
Email: Salvador.Mir@imag.fr
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.
Be the first to comment