Multivariate Mixtures of Normal Distributions: Properties, Random Vector Generation, Fitting, and as Models of Market Daily Changes

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DOI: 10.1287/ijoc.2014.0616
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Abstract
Mixtures of normal distributions provide a useful modeling extension of the normal distribution—both univariate and multivariate. Unlike the normal distribution, mixtures of normals can capture the kurtosis (fat tails) and nonzero skewness often necessary for accurately modeling a variety of real-world variables. An efficient analytical Monte Carlo method is proposed for considering multivariate mixtures of normal distributions having arbitrary covariance matrices. The method consists of a linear transformation of a multivariate normal having a computed covariance matrix into the desired multivariate mixture of normal distributions. The computed covariance matrix is derived analytically. Among the properties of the multivariate mixture of normals that we demonstrate is that any linear combination of mixtures of normal distributions is also a mixture of normal distributions. Methods of fitting mixtures of normal distributions are briefly discussed. A motivating example carried throughout this paper is the use of multivariate mixtures of normals for modeling daily changes in market variables.
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INFORMS Journal on Computing
Vol. 27, No. 2, Spring 2015, pp. 193–203
ISSN 1091-9856 (print) óISSN 1526-5528 (online) http://dx.doi.org/10.1287/ijoc.2014.0616
© 2015 INFORMS
Multivariate Mixtures of Normal Distributions:
Properties, Random Vector Generation, Fitting, and
as Models of Market Daily Changes
Jin Wang
Department of Mathematics and Computer Science, Valdosta State University, Valdosta, Georgia 31698,
jwang@valdosta.edu
Michael R. Taaffe
Department of Industrial and Systems Engineering, Virginia Polytechnic Institute and State University,
Blacksburg, Virginia 24061, taaffe@vt.edu
Mixtures of normal distributions provide a useful modeling extension of the normal distribution—both uni-
variate and multivariate. Unlike the normal distribution, mixtures of normals can capture the kurtosis
(fat tails) and nonzero skewness often necessary for accurately modeling a variety of real-world variables. An
efficient analytical Monte Carlo method is proposed for considering multivariate mixtures of normal distribu-
tions having arbitrary covariance matrices. The method consists of a linear transformation of a multivariate
normal having a computed covariance matrix into the desired multivariate mixture of normal distributions.
The computed covariance matrix is derived analytically. Among the properties of the multivariate mixture of
normals that we demonstrate is that any linear combination of mixtures of normal distributions is also a mixture
of normal distributions. Methods of fitting mixtures of normal distributions are briefly discussed. A motivat-
ing example carried throughout this paper is the use of multivariate mixtures of normals for modeling daily
changes in market variables.
Keywords: Monte Carlo simulation; mixture of normals; kurtosis and skewness; EM algorithm
History : Accepted by Winfried K. Grassmann, Area Editor for Computational Probability and Analysis;
received May 2013; revised April 2014; accepted May 2014.
1. Introduction and Motivation
The normal distribution is the most commonly used
model of daily changes in market variables, such as
the daily volatilities of stock-option prices from the
financial market. However, many studies (Duffie and
Pan 1997; Hull and White 1998; Venkataraman 1997;
Wilson 1993, 1998; Wirjanto and Xu 2009; Zangari
1996) show that the distributions of daily changes,
such as daily changes in equity, foreign exchanges, and
commodity markets, frequently depart from the normal
distribution shape by their asymmetry (skewness) and
fat tails (kurtosis). Thus the assumption of normality
is often inappropriate.
Mixtures of normals are a more general and flexible
distribution for fitting phenomena exhibiting heavy
tails and nonzero skewness, such as daily changes in
market data. Mixtures of normals can properly fit the
kurtosis and skewness often found in market vari-
ables. And of course, the normal distribution is a spe-
cial case of the mixture of normal distributions.
Mixtures of normal distributions provide an
extremely flexible method for modeling a wide vari-
ety of random phenomena (see McLachlan and Peel
2000, for example). These mixtures have received
increasing attention and have been successfully
applied in many fields including economics, market-
ing, and finance (Clark 1973, Duffie and Pan 1997,
Hull and White 1998, Venkataraman 1997, Wang 2000,
Zangari 1996). For example, recently, mixtures of nor-
mal distributions have become a popular modeling
choice for the distribution of daily changes in market
variables having fat tails. Wirjanto and Xu (2009) pro-
vide a literature survey on the application of mixture
of normals in finance.
State-of-the-art algorithms for constructing an
appropriate mixture of normals distribution are, how-
ever, also problematic. Existing methods for fitting
and generating mixtures of normals with a desired
covariance matrix are neither efficient nor accurate.
There is no closed-form expression for converting the
desired covariance matrix into the input covariance
matrix—the covariance matrix that will be used to
generate random vectors in a Monte Carlo simulation
experiment. Existing methods for creating the input
covariance matrix require that it be estimated numer-
ically. In our algorithm we use the desired covariance
193
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Wang and Taaffe: Multivariate Mixtures of Normal Distributions
194 INFORMS Journal on Computing 27(2), pp. 193–203, © 2015 INFORMS
matrix as the input covariance matrix. We show that
our method is both efficient (linear) and accurate.
Although other ways to handle the fat-tail issue
exist, they are problematic. The multivariate-tdis-
tribution is one such model (and in fact is a so-
called heavy-tailed distribution) and is a widely used
distribution to fit fat-tailed distributions (Alexander
and Baptista 2002, Glasserman et al. 2002, Wilson
1998, Zangari 1996). However, as pointed out by
Glasserman et al. (2002), one shortcoming of the
multivariate-tdistribution is that all multivariate-t
distributions have all of their marginal distributions
possessing equally fat tails (sharing the same degrees
of freedom). Glasserman et al. (2002) propose the use
of a copula to extend the multivariate-tmodel to allow
for different degrees of freedom for the marginal dis-
tributions. In computer simulation, copula functions
are used for modeling and generating dependent ran-
dom vectors. Although in theory and in general, cop-
ula functions are applicable for any distribution, they
do require use of a computationally costly nonlinear
inverse transformation. In addition, the multivariate-t
model cannot capture nonzero skewness, such as in
market variables for example, because, like the nor-
mal distribution, the tdistribution is symmetric and
thus is not appropriate for use as a modeling distri-
bution for skewed phenomena. Our proposed algo-
rithms, by contrast, require only an efficient linear
transformation, and can model nonsymmetric skew-
ness, and heavy-tailed kurtosis distributions.
In practice, the most commonly used method for
generating random-vector random variates emanat-
ing from a multivariate distribution with arbitrary
marginal distributions and covariance matrix is the
so-called three-step method (Biller and Nelson 2003,
Hull and White 1998, Schmeiser 1991). It is a spe-
cial case of using the copula idea. In the three-step
method, the input-covariance matrix has to be esti-
mated correctly to get the desired algorithm output-
covariance matrix. There is no general method to
derive the input-covariance matrix theoretically. The
NORTA (NORmal To Anything) method by Biller and
Nelson (2003) provides a fast general method for gen-
erating random vectors with given marginal distri-
butions and given correlation matrix. However, the
NORTA method requires adjustment (numerically) to
the input-covariance matrix to match the covariance
matrix for the desired marginal distributions. Ghosh
and Henderson (2003) developed a method to improve
the NORTA method when the NORTA fails to work
with an increasingly large proportion of correlation
matrices.
Biller and Ghosh (2006) provide a detailed sum-
mary on generating random vectors. Early work
on this topic can be found in Li and Hammond
(1975) and Mardia (1970). Putting their algorithms into
Schmeiser’s framework:
Step 1.Generate a multivariate normal with an
input covariance matrix.
Step 2.Transform this multivariate normal into a
multivariate uniform distribution on 40115.
Step 3.Transform this multivariate uniform distri-
bution on 40115into the desired multivariate distribu-
tion via inverse functions.
Both the transformations in Steps 1 and 2 are non-
linear. The fundamental difficulty is that there is no
general analytic way to determine the input-covariance
matrix for Step 1. The method requires a solution to
a set of nonlinear equations to derive a numerical
approximation of the required input-covariance matrix.
The most difficult part is to find the inverse marginal
distributions. It is often time consuming, and state-of-
art methods offer no alternative.
We propose an efficient Monte Carlo method for
generating random-vector random variates from a
multivariate mixture of normal distributions having
arbitrary covariance matrices. Our method is differ-
ent from the three-step method. We linearly transform
a multivariate normal having an input covariance
matrix into the desired multivariate mixture of nor-
mal distributions where the input-covariance matrix can
be derived analytically. In our method the resulting
marginal distributions are mixtures of normals and
may each have a different number of components.
So, for example, our method can appropriately model
market daily changes because our marginal distribu-
tions may each have a different skewness and each
may have a different fat tail. Furthermore, our method
is more efficient than the state-of-the-art method
because it requires only a linear transformation.
In this paper we also prove that linear combinations
of mixtures of normals are closed; thus, every linear
combination of mixtures of normals are themselves a
mixture of normals.
We also discuss parameter fitting for mixtures of
normal distributions. Fitting parameters to a mixture-
of-normal distributions is one of the oldest estima-
tion problems in the statistical literature, and a variety
of approaches have been used to estimate mixture-of-
normal distributions (see McLachlan and Peel 2000,
Titterington et al. 1985 for details). Most of the related
literature is focused on estimating parameters for mix-
tures of normal distributions (Cohen 1967; Dempster
et al. 1977; McLachlan and Peel 1998, 2000; Pearson
1894). For a general mixture of k-normals, a fun-
damental question is “what is the optimal value
for k?” In all of these cited studies, the value of k
is a fixed given number. The most powerful fitting
method is the expectation maximization (EM) algo-
rithm (McLachlan and Peel 1998), which requires
initial parameter values. In practice, the k-mean clus-
tering method (Hamerly and Elkan 2003) is used to
derive initial input parameters for the EM algorithm.
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Wang and Taaffe: Multivariate Mixtures of Normal Distributions
INFORMS Journal on Computing 27(2), pp. 193–203, © 2015 INFORMS 195
Compared to the normal mixture parameter-fitting
problem, there is substantially less work on generat-
ing random-vector variate values using a multivari-
ate mixture of normal distributions. The difficulty
in generating random-vector random variates is in
accurately modeling correlations among the marginal
random variables. For example, oil-related stocks are
positively correlated most of the time. Securities, gold,
and the U.S. dollar are negatively correlated for the
majority of the time.
The main contribution of this paper is an efficient
Monte Carlo method for generating a multivariate
mixture of normal distributions having an arbitrary
covariance matrix, and a closed-form exact ana-
lytic expression for the algorithm’s input-correlation
matrix.
Also in this paper we carry forward an extended
example use of our methods in financial asset portfolio
analysis. We provide an efficient analytic computation
method for so-called value-at-risk (VaR) computations.
This paper proceeds as follows. Section 2 discusses
fat tails and skewness, such as is found in daily
change distributions. We derive general results for the
kurtosis and skewness of mixtures of normal distribu-
tions. In an example, we compare the normal distribu-
tion having a given mean and variance with a mixture
of two normals having the same mean and variance.
Figure 1 shows the significant kurtosis and skew-
ness differences between these two densities. Section 3
0
0.1
0.2
0.3
0.4
0.5
–4 –2 2 4
f(x)
x
Mixture of two normals
Standard normal
Figure 1 Probability Density Functions of the Standard Normal and a Mixture of Two Normals Having Equal Means and Variances, But Differing
Kurtosis and Skewness
discusses random vector generation for multivari-
ate mixtures of normals having arbitrary covariance
matrices. The marginal distribution of each random
variable is also a mixture of normals, but the num-
ber of components in the mixture may be different for
each marginal random variable. In the univariate case,
we introduce a random variate-generating method for
a univariate mixture of normals. In the multivariate
case, we derive an efficient method to generate a mul-
tivariate mixture of normals. A detailed algorithm is
given. Section 3.3 provides an example using our algo-
rithm and discusses an efficient analytic method for
calculating portfolio VaR. Use of the multivariate mix-
ture of normals results in portfolio VaR computations
that are as simple to use as the simple normal distribu-
tion is to use. Section 4 briefly reviews some methods
for fitting a mixture of normal distributions. Two pop-
ular methods are discussed: the method of moments
and the maximum likelihood method. An easy and
implementable EM algorithm is given with a detailed
formulation. Section 5 is a summary of conclusions
with remarks.
1.1. Example: Value at Risk—VaR
An example use of our methods is in value at risk (VaR)
analysis. Our work is related to VaR analysis described
by Li (1999), Hull and White (1998), Venkataraman
(1997), and Zangari (1996). Zangari (1996) proposed
the so-called RiskMetrics™ (Morgan 1995) method,
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Wang and Taaffe: Multivariate Mixtures of Normal Distributions
196 INFORMS Journal on Computing 27(2), pp. 193–203, © 2015 INFORMS
which allows for a more realistic model of financial-
return tail distribution. Their RiskMetrics™ assump-
tion is that returns follow a conditional normal
distribution. They used a mixture of two distribu-
tions to model fat tails. They use a Gibbs sampler
for estimating the return distribution. Venkataraman
(1997) used the same mixture of two normals, but
his estimation technique is the quasi-Bayesian maxi-
mum likelihood approach, which was first proposed
by Hamilton (1991). Hull and White (1998) proposed
a maximum-likelihood method to estimate param-
eters of the mixture of two normal distributions.
Their market data was rescaled by its standard devi-
ation into four categories. They then estimated dis-
tribution parameters by fitting the quantiles of the
distribution. In addition, they described a method
for generating multivariate distributions having arbi-
trary marginal distributions. Their idea is basically the
same as the Schmeiser three-step method. Li (1999)
used the theory of estimating functions to construct
an approximate confidence interval for calculation
of VaR. Kurtosis and skewness are explicitly used in
his study.
2. Mixture of Normal Distributions
Before we introduce the mixture of normal distribu-
tions, we first discuss kurtosis and skewness. We also
provide an example of the mixture of two normal dis-
tributions and compare the exact result with a normal
distribution. Figure 1 demonstrates the significant dif-
ference in the kurtosis and skewness between the two
densities even though they have the same mean and
variance.
2.1. Kurtosis and Skewness
Kurtosis is a measure of how fat the tails of a distri-
bution are using the fourth central moment. In models
of market daily changes, for instance, kurtosis is very
sensitive to extremely large market moves. Skewness
is a function of the third central moment and is a mea-
sure of asymmetry. Skewness measures the degree of
difference between positive deviations from the mean
and negative deviations from the mean. In general,
the normalized skewness and kurtosis of a random
variable Xare defined as follows, respectively (see
Alexander and Baptista 2002):
Å3E6XÉå73
ë3and Å4E6XÉå74
ë41(1)
where åE6X7and ë2E6XÉå72. Standardized kur-
tosis is defined as
Å0
4E6XÉå74
ë4É31
and is a relative measure used for comparison with
the normal density. All normally distributed random
variables have kurtosis of three and skewness of zero.
2.2. Mixture of Normal Distributions
In this section, we describe the univariate mixture of
knormal distributions and derive its basic properties.
The mixture of two normal distributions, as a simple
example, is discussed. We compare this mixture to the
standard normal distribution, having the same mean
and variance, but different kurtosis and skewness. Fig-
ure 1 shows the significant difference between the two
densities.
In general, the cumulative distribution function
(cdf) of X, a mixture of kindependent normal random
variables, Xi, for i=11210001k, is defined by
F4x5=
k
X
j=1
pjÍxÉåj
ëj1(2)
where Íis the cdf of N40115. The associated probabil-
ity density function (pdf) of Xis
f4x5=
k
X
j=1
pjîj4x3 åj2
j51 (3)
where, for j=110001k,
îj4x3 åj2
j5=1
p2èë
j
eÉ4xÉåj52/42ë2
j51
0pj11
k
X
j=1
pj=10
After a direct calculation using the definitions of (1),
we derive the following basic properties.
Proposition 1. If Xis a mixture of kindependent nor-
mals with pdf (3), then the mean, variance, skewness, and
kurtosis of Xare:
å=
k
X
j=1
pjåj1
ë2=
k
X
j=1
pj2
j+å2
j5Éå21
Å3=1
ë3
k
X
j=1
pjjÉå53ë3
j+jÉå521
Å4=1
ë4
k
X
j=1
pj3ë4
j+6jÉå52ë2
j+jÉå540
In the following example, we show that the model
of a normal distribution is quite inappropriate for fit-
ting market data, since its density does not have suffi-
ciently fat tails and it has skewness of zero.
Example 1. A mixture of two independent normal
distributions.
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Wang and Taaffe: Multivariate Mixtures of Normal Distributions
INFORMS Journal on Computing 27(2), pp. 193–203, © 2015 INFORMS 197
Table 1 Standard Normal vs. Mixture of Normals
Distribution Mean Variance Skewness Kurtosis
Standard normal 0 1 0 3
Mixture of normals 0 1 0.9544 3.625
We consider a mixture of two independent normal
distributions (3) with the following parameters:
p1=0051p
2=005
1=É005
2=0051
ë1=0051and ë2=101180
We use Proposition 1 to compute its skewness and kur-
tosis. Table 1 provides a comparison to the standard
normal distribution.
We compare the two densities in Figure 1. The den-
sity of the standard normal is symmetric with skew-
ness of 0 and kurtosis of 3 whereas the density of this
mixture of two independent normals is asymmetric
with skewness of 009544 and kurtosis of 30625; how-
ever, both distributions have the same mean and vari-
ance. This indicates that the density of the mixture of
independent normals is skewed to the left and has a
fat tail.
Duffie and Pan (1997) provide a daily-change-
market-data example exhibiting nonnormal skewness
and kurtosis. They show that the S&P 500 daily
returns from 1986 to 1996 have an extremely high sam-
ple kurtosis and negative skewness. Our Figure 1 is
similar to Duffie and Pan (1997, Figure 3). Clearly,
compared to the normal distribution, the mixture of
independent normals is a more appropriate model for
daily change market data because of its ability to bet-
ter model the data skewness and kurtosis.
3. Generating Mixtures of
Normal Variates
In this section, we propose a new method for generat-
ing a multivariate mixture of (dependent) normal dis-
tributions having arbitrary covariance matrices. Our
method is different from the general Schmeiser three-
step framework. Our method contains two steps. In
step 1, we generate a multivariate normal having
an input covariance matrix. Later we show how this
covariance matrix is derived analytically. In step 2,
we linearly transform this multivariate normal into
the desired multivariate mixture of normals having
any given arbitrary covariance matrix. The linearity of
our transformation is a key difference and advantage
of this method over other general methods, such as
the NORTA. Our discussion starts with the univariate
case, and the multivariate case follows.
3.1. The Univariate Case
We discuss a generation method for single daily
change random variates using a mixture of kunivari-
ate (independent) normal distributions. The following
key result provides a feasible procedure:
Proposition 2. If YN40115,UU40115, and Y
and Uare independent of each other, then
X=
k
X
j=1
jY+åj5I8PjÉ1
l=1plU<
Pj
l=1pl9
is a mixture of kindependent normals with cdf of the
form (2), where I8·9is the indicator function, and P0
l=1pl0.
Proof. By definition, the cdf of Xis
F4x5=P4Xx5=
k
X
j=1
PXx1
jÉ1
X
l=1
plU<
j
X
l=1
pl
=
k
X
j=1
PXx
jÉ1
X
l=1
plU<
j
X
l=1
plPjÉ1
X
l=1
plU<
j
X
l=1
pl
=
k
X
j=1
PjY+åjx5pj=
k
X
j=1
PYxÉåj
ëjpj
=
k
X
j=1
pjÍxÉåj
ëj0É
As a direct application of Proposition 2, generat-
ing random variates from a mixture of independent
normals having cdf (2) can be easily accomplished as
follows.
Algorithm 1
1. Generate Yfrom N40115.
2. Generate Ufrom U40115, independent of Y.
3. Return X=Pk
j=1jY+åj5I8PjÉ1
l=1plU<
Pj
l=1pl9, where
P0
l=1pl0.
3.2. The Multivariate Case
We assume that X=4X110001X
n50is a random vector,
where the marginal distribution of each component Xi
is a univariate mixture of kiindependent normals hav-
ing pdf:
fXi4x5=
ki
X
h=1
pih
1
p2èëih
eÉ44xÉåih52/42ë2
ih551(4)
where
0pih11h=110001k
i1
ki
X
h=1
pih=11i=110001n0
Define the covariance matrix of Xas
ËXij 4X571 (5)
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