Efficient Correlation Matching for Fitting Discrete Multivariate Distributions with Arbitrary Marginals and Normal-Copula Dependence.

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INFORMS Journal on Computing
Articles in Advance, pp. 1–19
issn1091-9856?eissn1526-5528
informs®
doi10.1287/ijoc.1080.0281
©2008 INFORMS
Efficient Correlation Matching for Fitting Discrete
Multivariate Distributions with Arbitrary
Marginals and Normal-Copula Dependence
Athanassios N. Avramidis
School of Mathematics, University of Southampton, Highfield, Southampton,
SO17 1BJ, United Kingdom, a.avramidis@soton.ac.uk
Nabil Channouf, Pierre L’Ecuyer
Département d’Informatique et de Recherche Opérationnelle, Université de Montréal,
Montréal, Québec H3C 3J7, Canada {channoun@iro.umontreal.ca, lecuyer@iro.umontreal.ca}
A
In this approach, known as the NORTA method, the normal distribution function is applied to each coordinate
of a vector Z of correlated standard normals to produce a vector U of correlated uniform random variables
over (0?1); then X is obtained by applying the inverse of the target marginal distribution function for each
coordinate of U. The fitting requires finding the appropriate correlation ? between any two given coordinates
of Z that would yield the target rank or linear correlation r between the corresponding coordinates of X. This
root-finding problem is easy to solve when the marginals are continuous but not when they are discrete. In
this paper, we provide a detailed analysis of this root-finding problem for the case of discrete marginals. We
prove key properties of r and of its derivative as a function of ?. It turns out that the derivative is easier to
evaluate than the function itself. Based on that, we propose and compare alternative methods for finding or
approximating the appropriate ?. The case of discrete distributions with unbounded support is covered as well.
In our numerical experiments, a derivative-supported method is faster and more accurate than a state-of-the-
art, nonderivative-based method. We also characterize the asymptotic convergence rate of the function r (as a
function of ?) to the continuous-marginals limiting function, when the discrete marginals converge to continuous
distributions.
popular approach for modeling dependence in a finite-dimensional random vector X with given univariate
marginals is via a normal copula that fits the rank or linear correlations for the bivariate marginals of X.
Key words: statistics; multivariate distribution; estimation; correlation; copula; simulation
History: Accepted by Marvin Nakayama, Area Editor for Simulation; received November 2006; revised
March 2008; accepted March 2008. Published online in Articles in Advance.
1.
This paper develops methods that support the estima-
tion (fitting) of discrete multivariate distributions. A pow-
erful scheme for modeling multivariate distributions
in general is based on the concept of copula; it per-
mits one to specify separately the marginal distribu-
tions and the stochastic dependence. To put our work
in the proper perspective, we start by recalling basic
facts from copula theory. For a concise introduction to
copulas, see Embrechts et al. (2002) or Joe (1997); for a
more complete treatment, see Nelsen (1999).
A function C? ?0?1?d→ ?0?1? is called a copula
if it is the distribution function of a random vec-
tor in ?dwith U?0?1? marginals (uniform over
the interval ?0?1?). Consider a random vector X =
?X1?????Xd? with joint distribution F and write Fjfor
the marginal distribution of Xj. A copula associated with
Introduction
F ?equivalently, X? is a copula C that satisfies
F?x?=C?F1?x1??????Fd?xd???
x =?x1?????xd?∈?d?
(1)
Given an arbitrary F, a copula C satisfying (1) always
exists. If each Xjis a continuous random variable,
then C is unique, and this uniqueness means that
we have separated the marginals from the depen-
dence structure, which is captured by C. (Otherwise,
there may be more than one C satisfying (1), so the
dependence cannot be uniquely characterized.) We
will shortly specify a class of distributions F via (1) by
specifying the dependence via a d-variate copula C
that is selected after the marginals have been selected.
For given marginals, the choice of copula can have a
dramatic impact; see Embrechts et al. (2003, §7.1) for
an example.
1
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Avramidis, Channouf, and L’Ecuyer: Efficient Correlation Matching for Fitting Discrete Multivariate Distributions
INFORMS Journal on Computing, Articles in Advance, pp. 1–19, ©2008 INFORMS
2
In this paper, we nevertheless restrict our attention
to normal copulas; these are the copulas defined by
taking F as a multivariate normal distribution in (1).
This family of copulas has been suggested by sev-
eral authors, dating back to Mardia (1970). Attractive
features of normal copulas are that they facilitate esti-
mation (as will be explained) and simulation. They
are sufficient and very convenient for a wide range
of applications where fitting only the marginals and
the correlations is a reasonable compromise. In more
than two or three dimensions, estimating the entire
copula in a complicated real-life situation is often an
insurmountable challenge.
Other models of discrete multivariate distributions
can be found, e.g., in Joe (1997, §7.2). A limitation
of several of these models is that the same parame-
ters affect the marginal distributions and the depen-
dence. For example, in model (7.27) of Joe (1997),
the Xis are conditionally independent Poisson with
mean Ai, where the Ai, i = 1?????d, obey some mul-
tivariate continuous distribution, but the upper limit
Corr?Xi?Xj? = 1 is only possible in the limit where
Xi and Xj have identical marginals and Var?Xi?/
Ɛ?Xi?→?; a further limitation is that if one wanted
negative binomial marginals for the Xi, then one
would need the Aito obey a multivariate distribution
with gamma marginals, which is not convenient to
use (Joe 1997, p. 236).
Returning to the normal copula, if we write ?Rfor
the normal distribution with mean the zero vector
and d×d correlation matrix R, and CRfor the associ-
ated copula defined via (1) with F = ?R, we have the
representation
Z=?Z1?????Zd?∼?R?
1???Z1???????F−1
X=?X1?????Xd?=?F−1
where ? is the standard normal distribution function
(with mean zero and variance one) and F−1
by F−1
i
?u? = inf?x? Fi?x? ≥ u? for 0 ≤ u ≤ 1, is the quan-
tile function of the marginal distribution Fi. It is easily
seen that CRis a copula associated with X in (2). This
CRis a normal copula. Model (2) is also known under
the name NORTA (Cario and Nelson 1996, 1997; Chen
2001), an acronym for NORmal To Anything, because
normal variates are transformed to variates with gen-
eral nonuniform marginals.
The main issue here is how to find a matrix R such
that the vector X has the desired rank or linear cor-
relation matrix, either exactly or approximately. The
natural way of doing this is elementwise, so we start
by discussing the bivariate case (d =2). Later, we will
discuss the extension to d >2.
Suppose that d = 2 and that the marginals F1and
F2have been specified. Selecting R in (2) reduces to
d????Zd?????
(2)
i
, defined
selecting the scalar correlation ? = Corr?Z1?Z2?. The
rank correlation between X1and X2is
rX??? = rX???F1?F2?=Corr?F1?X1??F2?X2??
= Corr?F1?F−1
where ?=Corr?Z1?Z2? and “?” denotes function com-
position. We will explain shortly that rXmay depend
on the marginals only if at least one of them is not
continuous. One approach to specifying ? is to require
that rX???F1?F2? equals a given target value ˜ r, which
may be the sample rank correlation computed from
data (observations of X) or determined otherwise.
This leads to the NORTA rank-correlation matching prob-
lem of solving
rX???F1?F2?= ˜ r?
The dependence of rXon the marginals disappears
when F1and F2are both continuous: Fl? F−1
are the identity map, and thus
1
???Z1??F2?F−1
2
???Z2???
(3)
l
, l = 1?2
rX???F1?F2?=Corr???Z1????Z2??=?6/??arcsin??/2??
where the second equality is a well-known prop-
erty of the bivariate normal distribution (references
are given in the proof of Theorem 1 in §2.1). Thus,
solving (3) is trivial if all marginals are continuous,
and the solution is 2sin?? ˜ r/6?; consequently, the solu-
tion poses a problem only when at least one of the
marginals is not continuous.
Another possibility would be to work analogously
with the linear correlation (also called the product-
moment correlation):
?X???F1?F2? = Corr?X1?X2?
= Corr?F−1
which leads to the NORTA linear-correlation matching
equality:
?X???F1?F2?= ˜ ??
where
˜ ? is the sample linear correlation com-
puted from data. Embrechts et al. (2002) give a
detailed account of measures of dependence and
strong arguments that rank correlation is a more
appropriate measure than linear correlation. We
review their Example 5, which illuminates this issue.
Consider the marginals X1∼Lognormal?0?1? and
X2∼Lognormal?0??2? for ? > 0. Under several mea-
sures of dependence discussed there, extreme pos-
itive and negative dependence occurs when X2
is an increasing (decreasing) function of X1, i.e.,
in the stochastic representations ?X1?X2? = ?eZ?e?Z?
and ?X1?X2? = ?eZ?e−?Z?, respectively, where Z ∼
Normal?0?1?. Then, the rank correlation of the
pair ?X1?X2? equals 1 and −1, respectively. On
1
???Z1??F−1
2
???Z2???
(4)
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Avramidis, Channouf, and L’Ecuyer: Efficient Correlation Matching for Fitting Discrete Multivariate Distributions
INFORMS Journal on Computing, Articles in Advance, pp. 1–19, ©2008 INFORMS
3
the other hand, we have Corr?eZ?e?Z? = ?e?− 1?/
?
?
far from 1 and −1 over most of their domain, and
they converge to zero as ? → ?. Here, linear corre-
lation fails to capture well the dependence, and the
failure is dramatic in the limit. Hörmann et al. (2004,
§12.5) give additional examples of this phenomenon
and strongly recommend matching the rank correla-
tions instead of the linear correlations.
When d > 2, (2) is specified by constructing R ele-
mentwise. That is, for each pair (i?j), one has a tar-
get value ˜ ri?j(or ˜ ?i?j) and one sets the (i?j)th element
of R to the solution of (3) with ˜ r = ˜ ri?j(or the solu-
tion of (4) with ˜ ? = ˜ ?i?j). Thus, one needs to solve
d?d − 1?/2 such independent equations. In case the
resulting matrix R is not positive semidefinite, various
authors suggest replacing it by another matrix that is
positive semidefinite and minimizes some measure
of distance from R (Mardia 1970, Cario and Nelson
1997, Lurie and Goldberg 1998, Ghosh and Henderson
2003). According to Ghosh and Henderson (2003), this
appears to work well, in the sense that the minimized
distance was very small in their tests.
Another related setting is the VARTA class of multi-
variate stationary time series (Biller and Nelson 2003),
?Xt= ?X1?t?????Xk?t?? t = 1?2?????, where one spec-
ifies the marginals Flfor l = 1?????k and depen-
dence via the normal copula, i.e., via correlations
between Xi?tand Xj?t−hfor h = 0?1?????p and i?j ∈
?1?2?????k?; the univariate case k = 1 is known as
ARTA (Cario and Nelson 1996). That is, the ith com-
ponent time series is obtained by the transforma-
tion Xi?t=F−1
a k-variate vector autoregressive process of order p
and whose noise vectors are Gaussian; see Biller and
Nelson (2003, §3.1.1). Here, the number of equations
that must be solved is pk2+k?k−1?/2. (The complica-
tions and remedies mentioned earlier have analogs in
the time-series setting.) Because the number of equa-
tions to be solved can be considerable, efficient meth-
ods for solving equations of the form (3) and (4) are
of interest.
We now review past work on NORTA correla-
tion matching. This literature has emphasized linear-
correlation matching (Cario and Nelson 1998, Chen
2001, Biller and Nelson 2003), despite the existing
arguments in favor of rank correlation, and in princi-
ple applies to both continuous and discrete marginals
unless otherwise said. Cario and Nelson (1998)
use root bracketing combined with approximating
?X???F1?F2? (a function of ?) via two-dimensional
numerical integration (Gauss and Kronrod quadra-
ture rules). With discrete marginals, the integrand
has a discontinuity at every support point, so these
general-purpose quadrature rules are not well suited.
?e−1??e?2−1?
?e−1??e?2−1?; these continuous functions of ? are
and Corr?eZ?e−?Z? = ?e−?− 1?/
i
???Zi?t??, where ?Zt?=?Z1?t?????Zk?t? is
Chen (2001) proposed a simulation-based approach.
Biller and Nelson (2003) showed that the restriction
of the marginals to certain Johnson families simpli-
fies the solution. For the case of discrete marginals,
we were unable to find a published or unpub-
lished example of NORTA rank- or linear-correlation
matching.
The main contributions of this paper are a detailed
study of the NORTA correlation matching prob-
lems (3) and (4) and the development of efficient
methods for solving these problems when the marginal
distributions are discrete. We do not address the case
where some marginals are discrete and others are
continuous. Allowing the support to be infinite, we
express rX???F1?F2? as an infinite series, where each
term involves a bivariate normal integral to the north-
east of a bivariate support point. We obtain the
derivative of rXwith respect to ? as a series of terms
that only involve the exponential function. For finite
support, it turns out that the derivative is consid-
erably faster to evaluate than rX, even if one uses
state-of-the-art methods to compute the bivariate nor-
mal integrals. We then develop solution methods
that exploit the derivative. In particular, we pro-
pose a simple Newton-type method, which in numer-
ical experiments is faster and more accurate than
a state-of-the-art, nonderivative-based method. For
unbounded marginals, we propose a method that
does not require evaluating rXand that substitutes an
approximation of the derivative (obtained by truncat-
ing the series), and we provide bounds on the result-
ing error.
Another contribution is an asymptotic upper bound
and convergence result on the L?distance (i.e., the
supremum over ?∈?−1?1? of the absolute difference)
between the rank-correlation function rX???F1?F2? for
given discrete marginals F1and F2and the explic-
itly known analog for continuous marginals, in terms
of the maximum probability masses of F1and F2, as
these masses go to zero. The bound is relevant to the
correlation-matching problem in the following sense.
Suppose that one uses the continuous-marginals solu-
tion, 2sin?? ˜ r/6?, as an approximation. If the bound
was smaller than the desired accuracy, then our algo-
rithms would no longer be needed. In our examples,
the bound was larger than the desired accuracy, so the
discrete-marginals correlation-matching problem had
to be dealt with directly.
Our results and methods for the rank-correlation
problem extend immediately to the linear-correlation
problem, under mild uniform convergence condi-
tions. For reasons given earlier, we emphasize the
rank-correlation problem and discuss only briefly the
extension to the linear-correlation problem.
The remainder of this paper is organized as follows.
Section 2.1 summarizes relevant background. In §2.1,
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Avramidis, Channouf, and L’Ecuyer: Efficient Correlation Matching for Fitting Discrete Multivariate Distributions
INFORMS Journal on Computing, Articles in Advance, pp. 1–19, ©2008 INFORMS
4
we prove key properties of the rank and linear corre-
lations as a function of ?, obtain expressions for their
derivatives, and discuss implications. Section 2.2 pro-
poses an approximation to the derivative, with error
bounds, for the infinite-support case. The convergence
result to the continuous case is proved in §2.3. Sec-
tion 3 specifies the benchmark and the new methods
for bivariate NORTA correlation matching for either
finite or infinite support. In §4, we give numerical
examples. Section 5 contains concluding remarks.
2.Mathematical Properties
2.1.
Theorem 1 below summarizes useful known results
that hold for arbitrary marginals. Let
Background
???x?y? =
1
2??1−?2
·exp?−?x2−2?xy +y2?/?2?1−?2???? (5)
the bivariate standard normal density function with
correlation ?.
Theorem 1. Assume that F1 and F2 are arbitrary
cumulative density functions (c.d.f.s), and define rX??? =
Corr?F1?X1??F2?X2?? and ?X??? = Corr?X1?X2? with
?X1?X2? defined as in (2) with ?=Corr?Z1?Z2?.
(1) The functions rX and ?X are nondecreasing on
?−1?1?. We have rX?0?=0 and ?X?0?=0.
(2) Assumethatthere
Ɛ??X1X2?1+??<? for all ? ∈ ?−1?1?. Then, rXand ?Xare
continuous on ?−1?1?.
(3) If the marginals Flare continuous, then
exists
?>0
such that
Corr?F1?X1??F2?X2?? = 12gC???−3
=6
?arcsin??/2?=?rC???? (6)
where
gC???=
??
−?
??
−???x1???x2????x1?x2?dx1dx2?
Proof. For the linear correlation ?X, parts 1 and 2
are Theorems 1 and 2 of Cario and Nelson (1997),
respectively. These unpublished results are straight-
forward extensions to the case of different marginals
of analogous results published as Theorems 1 and 2
in Cario and Nelson (1996) for the case of identical
marginals. To prove the analogous results for rX, it
suffices to replace the nondecreasing functions F−1
in the proofs of Theorems 1 and 2 of Cario and Nelson
(1997), respectively, by the nondecreasing functions
Fl? F−1
and Cooke (2001), part 3 was obtained by Karl
Pearson in 1907. A more recent reference is Kruskal
(1958).
?
l
??
l
? ? for l = 1?2. According to Kurowicka
Parts 1 and 2 provide the basis for solving (3)
and (4) via root bracketing; see method NI1 in §3.
In §2.3, we provide a theoretical result that estab-
lishes rC??? as a natural approximation of rX???F1?F2?.
The derivative-based solution methods of §3 can work
without this approximation, but the approximation
usually helps increase their speed.
This section develops the basis for the proposed
solution methods. We assume that marginals are dis-
crete and satisfy weak conditions, and we develop
explicit formulae for the derivatives of the functions
rXand ?X.
For l = 1?2, we assume that the positive support
can be (and is) enumerated in increasing order as 0≤
xl?0< xl?1< xl?2< ··· and that the negative support
is enumerated as 0 > xl?−1> xl?−2> ···. Here is an
example of a positive support that is not enumerable
as above: there is a support point x0> 0 such that
there are infinitely many positive support points to
the left of x0and there are support points to the right
of x0. The enumeration is straightforward for most
discrete distributions usually encountered in appli-
cations, e.g., discrete uniform, binomial, geometric,
Poisson, negative binomial, and certainly for many
more, e.g., any finite mixture of any of these. Also
note that a negative support is enumerable as above
if it is obtained by reflection about zero of a conform-
ing (enumerable as above) positive support. From this
practical standpoint, the assumption does not appear
restrictive.
Denote the probability mass of xl?jas pl?j. For any
integer k, the cumulative probability mass is fl?k=
?k
Write zl?k= ?−1?fl?k?, and note that limk→?zl?k=
−limk→?zl?−k= ?. Results are stated below for the
case where each marginal has infinite support. The
finite-support case is an (artificial) special case; to see
this, note that if the probability mass above zero is
concentrated on a finite number of points, then an
increasing sequence of artificial points xl?jwith prob-
ability pl?j= 0 can be added as needed, and similarly
for the probability mass below zero.
2.1.1.Derivative of the Rank Correlation. The
rank correlation between X1and X2is
rX???=Corr?F1?X1??F2?X2??=g???−?1?2
j=−?pl?j. For l = 1?2, limk→?pl?k=limk→?pl?−k=0.
?1?2
?
(7)
where
g??? = Ɛ?F1?X1?F2?X2??
=
−?
??
??
−?F1
?F−1
·???x1?x2?dx1dx2?
1???x1???F2
?F−1
2???x2???
(8)
where ?kand ?kare the known mean and standard
deviation of Fk?Xk?, respectively. Note that rXinvolves
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Avramidis, Channouf, and L’Ecuyer: Efficient Correlation Matching for Fitting Discrete Multivariate Distributions
INFORMS Journal on Computing, Articles in Advance, pp. 1–19, ©2008 INFORMS
5
only shifting and scaling of g by known constants. We
rewrite the double integral in (8) as
g??? =
?
?
?
?
−? ???z1?i−1?z2?j?−? ???z1?i?z2?j−1?+? ???z1?i?z2?j??
?
?
?
?
i=−?
?
?
?
?
j=−?
f1?if2?j
?? z1?i
z1?i−1
?z2?j
z2?j−1
???x1?x2?dx1dx2
?
(9)
=
i=−?
j=−?
f1?if2?j
?? ???z1?i−1?z2?j−1?
=
i=−?
?
?
j=−?
?f1?i+1−f1?i??f2?j+1−f2?j?? ???z1?i?z2?j?
=
i=−?
p1?i+1
?
?
j=−?
p2?j+1? ???z1?i?z2?j??
(10)
which involves the bivariate normal integral? ???x?y?=
??
(9) follows directly from the definition (2); the second
step rewrites each double integral over a square as
the signed summation of four terms involving four
related integrals at the square’s corners; the third step
is a simple rearrangement of the summation. Observe
that in (10), the weight p1?i+1p2?j+1multiplies the value
of? ??at ?z1?i?z2?j?, not at ?z1?i+1?z2?j+1?. If x1?i+1and
x2?j+1are the smallest values with positive probabili-
ties for X1and X2, respectively, then z1?i= z2?j= −?,
so? ???z1?i?z2?j? = 1 and the corresponding term in
(10) is p1?i+1p2?j+1. As a special case, suppose that X1
is degenerate to a single value, say p1?i+1= 1. Then,
(10) yields
x
??
y???z1?z2?dz1dz2?
In thederivationabove,
g??? =
?
?
?
?
j=−?
p2?j+1? ???−??z2?j?=
?
?
j=−?
p2?j+1? ???−1?f2?j??
=
j=−?
p2?j+1?1−f2?j?=Ɛ??F2?X2??
(a constant), where?F2?x? ?= P?X2≥ x?. If both X1and
X2are degenerate, this gives g???≡1.
Proposition 1. The function g??? is infinitely differ-
entiable on the interval ?−1?1?, with first derivative
g????=
?
?
i=−?
p1?i+1
?
?
j=−?
p2?j+1???z1?i?z2?j??
(11)
Proof. We start with the first derivative. We will
exploit the property of the bivariate standard normal
density that for −1<?<1,
d
d????x?y?=
?2
?x?y???x?y?
for any x?y
(12)
(Kendall and Stuart 1977, p. 393, exercise 15.4).
We have
d
d?
xy
??
x
dz1
y
?z2
??
x
dz1
In Steps 1 and 2, the interchange of differentiation
and integration is valid because of the existence and
boundedness of the derivatives over the integration
domain; in Step 2, we used (12); Steps 3 and 4 use the
fundamental theorem of calculus.
Equation (13) shows that the derivative of each
term in the series (10) is the corresponding term in
the series (11). It remains to show the validity of
interchanging the order of differentiation and sum-
mation. A sufficient condition for this is that for each
?0∈ ?−1?1?, there is a neighborhood of ?0, N???0? =
??0−???0+ ?? ⊂ ?−1?1?, such that the series on the
right side of (11) converges uniformly for ? ∈ N???0?
(Rudin 1976, Theorem 7.17). This uniform conver-
gence holds in particular if there is an increasing
sequence of finite sets Sk⊂?2, k ≥0, such that
lim
k→?
?∈N???0?
(Because all the terms in (11) are nonnegative, this
condition is actually a special case of the well-known
Cauchy criterion for uniform convergence (Rudin
1976, Theorem 7.8).) The latter condition is easily
verified if we take Sk as the bounded rectangle
??i?j?? max??i???j??≤k?:
sup
?∈N???0?
1
2??1−?2
? ???x?y? =
??
??
d
d
d????z1?z2?dz2dz1
?? ?
=
?
???z1?z2?dz2
?
dz1
=
d
?−???z1?y??dz1=???x?y?? (13)
sup
?
?i?j?∈?2\Sk
p1?j+1p2?j+1???z1?i?z2?j?=0?
(14)
?
?i?j??max??i???j??>k
p1?i+1p2?j+1???z1?i?z2?j?
≤
∗
??
i??i?>k
p1?i+1+
?
j??j?>k
p2?j+1
?
→0
as k →?? (15)
where ?∗= max??? − ????? + ???. To study the
higher-order derivatives, we note that ???x? y? =
?1 − ?2?−1/2??x????y − ?x??1 − ?2?−1/2? and we change
from coordinates ?x?y? to polar coordinates ?r???, i.e.,
set x = r cos?, y = r sin?, where r ≥ 0 and ? ∈ ?0?2??.
Let ? > 0 and write ??d?
?
for the dth derivative of ??
with respect to ? for ???≤1−?. Differentiation gives
???1?
?1−?2
·?1−?2??2r2a?????−1?cos?+r2a2?????
≤ K1r2exp?−r2b?????/2?
for all r??? and ???≤1−??
??r???? =
??????r cos???
?ra?????
?
2?1−?2?5/2
????
(16)
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