Positive semi-definite correlation matrix completion

Positive semi-definite correlation matrix completion Peter J¨ackel∗

Christian Kahl†

First version: 6th May 2009 This version: 10th May 2009

Abstract

of xi and a further standard normal variate i which is independent from all the xj , as given by a pairwise We give an intuitive derivation for the correlation ma- Cholesky decomposition: trix completion algorithm suggested in [KG06]. This leads us to a more general formula for the completion. yi = ηi xi + ηi0 i , (5) The presented extension is positive semi-definite by hxi j i = 0 , (6) construction, but we also give a simplified algebraic proof for its universal validity. with p ηi0 := 1 − ηi2 . (7)

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Introduction

This immediately yields

Since the nature of this note is to present an extenhxi yj i = rij ηj sion to [KG06], we skip the general motivation and background of the problem and refer the reader to whence we choose B := RH with the references [KG06, Kah07]. Given a set of 2n standard normal variates H := diag(η1 , · · · , ηn ) x1 , · · · , xn , and y1 , · · · , yn , and the constraint that the pairwise correlations as in [KG06]. Further, we have hxi xj i = rij

(1)

hxi yi i = ηi

(2)

z := (x1 , · · · , xn , y1 , · · · , yn )>

(3)

cij = hyi yj i = ηi rij ηj + ηi0 hi j iηj0 .

(8)

(9)

(10)

We note that for hi j i = 0 we obtain the structure are pre-specified, we seek a completion of the as yet given in [KG06]. under-specified (auto-)correlation matrix of

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Since the matrices B and C given in the previous section are derived from linear combinations of standard normal variates, the completed matrix   R B > A := hz · z i = (11) B> C

which has the structure 0

r11 B . B . B . B B rn1 B hz · z > i = B B η B 1 B B @ ?

... .. . ...

..

r1n .. . rnn

η1

?

1

? ..

?

..

. ηn

.

?

.

1

C C C C ηn C „ R C C= B> ? C C C C A 1

B C

«

is by construction symmetric positive semi-definite, (4) which we denote as

with rii = 1.

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Positive semi-definiteness

A0.

Pairwise Cholesky construction

(12)

However, for the sake of completeness, we provide below a simple algebraic proof. We start our intuition with the suggestion that each Given two matrices R, E ∈ Rn×n , with R  0, E  of the yi can be represented as a linear combination 0, we set   ∗ R RH OTC Analytics A := (13) † ABN AMRO HR C 1

with C := HRH + H 0 EH 0 , 0

H :=

diag(η10 , · · ·

, ηn0 )

.

(14) (15)

Since the spectrum of A is invariant with respect to the addition of one of its (scaled) rows to any other, and likewise for columns, Gaussian elimination gives us « „ R RH 0 (16) HR C „ „

R 0 R 0

«

RH C − HRH 0 C − HRH

«

0

(17)

0.

(18)

Since R  0, equation (18) holds if C − HRH  0. This, however, follows trivially since C − HRH = HRH + H 0 EH 0 − HRH 0

0

= H EH  0 .

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(19) (20)

Summary

We showed how, given a correlation structure R for n standard normal variates x1 , · · · , xn , and given the correlations hxi yi i = ηi to a second set of standard normal variates y1 , · · · , yn , one can constructively arrive at bij = hxi yj i = rij ηi cij = hyi yj i = ηi rij ηj +

(21) ηi0 eij ηj0

(22)

for an arbitrary correlation matrix E ∈ Rn×n , E  0, as a possible choice for the completed correlation „ « R B matrix A = B> C . We also proved „

R HR

RH HRH + H 0 EH 0

«

0

(23)

by the aid of straightforward Gaussian elimination of rows and columns. It remains to be said that in practice one may wish to use the homogenous parametric form eij = β + (1 − β)δij (24) h i 1 for E, with β ∈ − n−1 , 1 and δ(· ·) being the Kronecker symbol, for the sake of simplicity.

References [Kah07] C. Kahl. Modeling and simulation of stochastic volatility in finance. PhD thesis, Bergische Universit¨ at Wuppertal and ABN AMRO, 2007. Published by www.dissertation.com, www.amazon.com/ Modelling-Simulation-Stochastic-Volatility-Finance/ dp/1581123833/, ISBN-10: 1581123833. [KG06]

C. Kahl and M. G¨ unther. Complete the Correlation Matrix. Working paper, Bergische Universit¨ at Wuppertal, 2006. www.math.uni-wuppertal.de/~kahl/publications/ CompleteTheCorrelationMatrix.pdf.

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