Download Slides 13: Multivariate Derivatives (PDF, 136 KB)

Stochastic Models
Multivariate Distributions
Walt Pohl
Universität Zürich
Department of Business Administration
April 24, 2013
Multivariate Distributions
A multivariate distribution is the probability distribution
for a family of random variables X1 , . . . , Xn . To define
the distribution, we must specify, for a family of ranges
[a1 , b1 ], . . . , [an , bn ],
P(a1 ≤ X1 ≤ b1 , . . . , an ≤ Xn ≤ bn ).
Walt Pohl (UZH QBA)
Stochastic Models
April 24, 2013
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Independence
An important special case is when the random variables
are independent. Then
P(a1 ≤ X1 ≤ b1 , . . . , an ≤ Xn ≤ bn )
= P(a1 ≤ X1 ≤ b1 ) · · · P(an ≤ Xn ≤ bn ).
More challenging is modelling dependent distributions.
Walt Pohl (UZH QBA)
Stochastic Models
April 24, 2013
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Defining Multivariate Distributions
The methods for univariate distributions generalize.
Discrete distribution – a probability mass function,
f,
P(X1 = x1 , . . . , Xn = xn ) = f (x1 , . . . , xn )
Continuous distribution – a multivariate density, p
P(a1 ≤ X1 ≤ b1 , . . . , an ≤ Xn ≤ bn )
Z b1
Z bn
=
p(x1 , . . . , xn )dx1 · · · dxn
···
a1
Walt Pohl (UZH QBA)
an
Stochastic Models
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Defining Multivariate Distributions, cont’d
In practice, very few multivariate distributions are defined
this way. Instead, we rely heavily on transformations.
One (partial) exception: the multivariate normal
distribution.
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Stochastic Models
April 24, 2013
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Multivariate Normal Distribution
The multivariate normal distribution depends on
A vector of means, µ = (µi ), where E (Xi ) = µi .
A matrix of covariances, Σ = (σij ), where
Cov (Xi , Xj ) = σij .
Its density is
1
− 12 (x−µ)0 Σ−1 (x−µ)
e
(2π)k/2 (det Σ)1/2
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Stochastic Models
April 24, 2013
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Multivariate Normal Distribution, cont’d
The covariance matrix must be symmetric positive
definite.
Symmetric: σij = σji , or Σ0 = Σ.
Positive definite: v 0 Σv > 0 for v 6= 0. This is
because v 0 Σv is the variance of v 0 X .
Walt Pohl (UZH QBA)
Stochastic Models
April 24, 2013
7 / 15
Simulating the Multivariate Normal
To simulate the multivariate normal, we usually simulate
n independent normal random variables, = (i ), and
use a linear change of variables,
X = µ + M,
where µ is a vector, and M = (mij ) is a matrix.
How should we pick M?
Walt Pohl (UZH QBA)
Stochastic Models
April 24, 2013
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Simulating the Multivariate Normal
If Xi =
P
j
mij j , then
X
X
Cov (Xi , Xi 0 ) = Cov (
mij j ,
mi 0 j 0 j 0 )
j0
j
=
XX
=
X
j
mij mi 0 j 0 Cov (j , j 0 )
j0
mij mi 0 j .
j
Walt Pohl (UZH QBA)
Stochastic Models
April 24, 2013
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Simulating the Multivariate Normal,
cont’d
The last sum can be written in matrix form as MM 0 . So
we just need to find a matrix M such that MM 0 = Σ.
But how do we find such an M?
Walt Pohl (UZH QBA)
Stochastic Models
April 24, 2013
10 / 15
Simulating the Multivariate Normal,
cont’d
There are two main methods of finding such an M.
Cholesky decomposition.
Symmetric square root.
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Stochastic Models
April 24, 2013
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Cholesky decomposition
Any symmetric positive-definite matrix Σ can be written
as
Σ = CC 0 ,
where C is upper triangular.
This is called the Cholesky decomposition of Σ.
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Stochastic Models
April 24, 2013
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Symmetric square root
A symmetric positive-definite matrix has all positive
eigenvalues, and can be diagonalized as
Σ = O 0 DO,
where O is an n-dimensional rotation (also known as an
orthogonal matrix). Orthogonal matrices have the
property that O 0 O = OO 0 = I , the identity matrix.
This is known as the singular value decomposition.
Walt Pohl (UZH QBA)
Stochastic Models
April 24, 2013
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Symmetric square root
Let D 1/2 be a diagonal matrix formed by taking the
square roots of the diagonal elements of D. Then the
symmetric square root, Σ1/2 is
Σ1/2 = O 0 D 1/2 O.
This matrix is indeed symmetric, and
Σ1/2 Σ1/2 = (O 0 D 1/2 O)(O 0 D 1/2 O)
= O 0 DO
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Stochastic Models
April 24, 2013
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Comparison
Cholesky decomposition – faster to compute.
Symmetric square root – numerically more accurate.
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Stochastic Models
April 24, 2013
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