Noise in the correlation matrix: A simulation approach
Saba Sharifi
Atid Shamaie
Martin Crane
School of Computer Applications
Dublin City University
The results of applying Random Matrix Theory (RMT) in finance suggest that the historical correlation matrix, C, carries
a large amount of noise. The purpose of RMT is to compare the statistical properties of C with those of a random matrix.
By applying RMT we study how C responds to the different volumes of noise in data.
20
1
10
Value
0.5
Value
We generate a set of 450 random
sinusoidal time series and add some
random noise normally distributed
with zero mean and a particular
standard deviation. The volume of
noise is controlled by its standard
deviation.
0
-10
-0.5
-1
0
-20
0
500
1000
1500
Observations
-30
0
500
1000
Observations
1500
Distrubution
1.2
1
Maximum
eigenvalue deviated
from random band
0.8
0.6
0.4
0.2
0
0
5
10
15
20
25
Eigenvalues
30
35
1
Distribution
0.8
Maximum
eigenvalue
0.6
The distributions of the eigenvalues in historical and random
cases are plotted. A large part of the historical graph is similar to
the random one. This part that is carrying noise corresponds to the
noisy band of C. There are some eigenvalues that deviate from
the random graph. These eigenvalues and the corresponding part
of C have information and are known as the non-noisy band. To
estimate the exact effect of the added noise we increase the
volume of noise gradually. Starting from 0.02, 11 different values
of standard deviation are examined and the number of deviated
eigenvalues is estimated.
0.4
0.2
0
0
0.5
1
1.5
Eigenvalues
2
2.5
3
Number of deviated eigenvalues
30
25
20
15
At the beginning, by increasing the standard deviation an
10
increase in the number of deviated eigenvalues is
observed. But for the standard deviation varying from
5
0.08 to 4 no dramatic change is observed. Therefore, it
0
indicates that from a particular point onward, there is no
0
1
2
3
4
Noise
standard
deviation
relationship between the volume of noise and the
number of non-noisy eigenvalues or in the other words,
amount of noise has no effect on the number of noisy
eigenvalues.
Consequently, we conclude that the volume of noise in the data has almost no effect on the genuine information part of
the eigenvalues of the correlation matrix. This represents that except for very small volumes of noise involved in data,
RMT result is the same for different amounts of noise. In the case of stock markets, we conclude that RMT result does
not depend on the different volumes of noise involved in stocks’ prices. Whatever noisy the stocks’ prices are, RMT
estimates the same percentage of the deviated eigenvalues of C from the noisy bound.
_____________________________________________________________________________________________________________________________
Contact ssaba@computing.dcu.ie
23rd Conference on Applied Statistics in Ireland, May 2003.