Discussion Papers In Economics
And Business

Random Correlation Matrix and De-Noising
Ken-ichi Mitsui and Yoshio Tabata

Discussion Paper 06-26

Graduate School of Economics andOsaka School of International Public Policy (OSIPP)
Osaka University, Toyonaka, Osaka 560-0043, JAPANRandom Correlation Matrix and De-Noising
Ken-ichi Mitsui and Yoshio Tabata

Discussion Paper 06-26
September 2006
この研究は「大学院経済学研究科・経済学部記念事業」
基金より援助を受けた、記して感謝する。
Graduate School of Economics andOsaka School of International Public Policy (OSIPP)
Osaka University, Toyonaka, Osaka 560-0043, JAPANRandom Correlation Matrix and De-Noising
Ken-ichi Mitsui ∗and Yoshio Tabata†

Abstract
In Finance, the modeling of a correlation matrix is one of the important problems.
In particular, the correlation matrix obtained from market data has the noise. Here weapply the de-noising processing based on the wavelet analysis to the noisy correlation
matrix, which is generated by a parametric function with random parameters. Firstof all, we show that two properties, i.e. symmetry and ones of all diagonal elements,of the correlation matrix preserve via the de-noising processing and the efficiency ofthe de-nosing processing by numerical experiments. We propose that the de-noisingprocessing is one of the effective methods in order to reduce the noise in the noisy
correlation matrix.

JEL classification: C51, C61, C63, G32

keyword correlation matrix, calibration, rank reduction, de-noising, wavelet analysis∗ Doctor Candidate of Osaka University, E-mail:cfatk406@jtw.zaq.ne.jp† Graduate School of Business Administration, Nanzan Univeristy, E-mail:tabata@nanzan-u.ac.jp

1

Introduction

One of the important problems in finance and risk management is how to specify a correlation
matrix. For instance, calibrating BGM model which is also called LIBOR forward rate model(LFM), one must estimate the correlation matrix of forward rate of each index.
It is one of the methods to specify the correlation matrix to fit the (noisy) elements of themarket correlation matrix by means of some parametric function. In De Jong et al. (2004),
using USD historical interest rate data they estimated the following function ρ of correlation
matrix:ρ(Ti , Tj ) = exp

− γ1 |Ti − Tj | −ppγ2 |Ti − Tj |− γ4 | Ti − Tj |γmax(Ti , Tj ) 3

(1)

where γ1 , γ3 , γ4 > 0, and Ti is the expiry date.
There exists another method to specify the correlation matrix based on the decomposition
using eigenvalues and eigenvectors of matrices. This is often called rank-reduction in financeand is implemented by some researchers; “zeroing eigenvalues”( Spectral decomposition,
also known as principle component analysis) in Rebonato and Jäckel (1999), “2 and 4 rankreduction” in Brigo (2002), “eigenvalue zeroing by iteration” in Morini and Webber (2006),
application to multi-factor lognormal forward rate model in Rebonato (1999).
Moreover, Grubišić and Pietersz (2005) considered the problem to find nearest-lowrank correlation matrix and made numerical comparisons with several algorithms where
the correlation matrix was generated by (1) with γ-parameters assumed to be distributed
normally (capped at 0 for γ1 , γ3 , γ4 ) with mean and standard errors1 estimated by De Jonget al. (2004).
Let Sn be the set of n × n R-valued symmetric matrices and X ∗ be the transpose of X.
For X ∈ Sn , denote by X  0 that X is positive semidefinite. Denote C ∈ Sn a correlation
matrix, e.g. that of observations in market. Then the problem to specify the correlation
matrix by k rank-reduction is given by the following optimization problem;

1 The estimated values; γ , γ , γ , γ (standard errors in parentheses) are 0.000 ( - ), 0.480 (0.099), 1.511
1 2 3 4(0.289), 0.186 (0.127), respectively.

1

Correlation

Correlation

1

1

0.8

0.8

0.6

0.6

0.4

0.4

0.2

0.2

0

16

0

14

12

10
Tj

8

6

4

2

0 0

2

4

6

8

10

12

14

16
16

Ti

14

12

10
Tj

8

6

4

2

0 0

2

4

6

8

10

12

14

16

Ti

Figure 1: 4 reduced-rank angle parameterization.

Find X ∈ Snsuch that

minX 21 kC − Xk2(2)subject to rank(X) = k; {Xii } = 1, i ∈ {1, . . . , n};X  0 (X = BB∗ , B ∈ Rn×k ).
In particular, Rebonato (1999), Rebonato and Jäckel (1999) suggested the reduced-rankangle parameterization that the matrix B in (2) is approximated by means of trigonometricfunctions. We will briefly view the procedure.
Let us implement the reduced-rank angle parameterization.2 As generating the correlation
matrix C by (1) with constant γ-parameters estimated by De Jong et al. (2004), which wecall the non-noisy correlation matrix, and applying the reduced-rank angle parameterizationof 4 rank-reduction, we have the calibrated X such as Figure 1(left). And Figure 1(right)
is in the random elements case where γ-parameters assumed to be distributed normally(capped at 0 for γ1 , γ3 , γ4 ) with mean and standard errors estimated by De Jong et al.
(2004), which we call the noisy correlation matrix.
From these figures, we can see the calibrated matrix X preserving noise through the
reduced-rank angle parameterization of the 4 rank-reduction. It is natural for us to wish tohave a calibrated correlation matrix of which the noise is reduced. Then in this paper we
propose a procedure of rank-reduction method with a noise reduction technique for a noisy
2 In (1) we set T = {0.5, 1.0, . . . , 16.0} and the dimension of the correlation matrix is 32 × 32. Throughi
Section 1-3, we implicitly use this setting.

2

correlation matrix, which is called “de-noising” given by Donoho (1995), Donoho (1993),
Donoho and Johnstone (1994) based on Wavelet analysis. To put it more precisely, we solve
the optimization problem (2) after applying “de-noising” to a noisy correlation matrix C.
The procedure of “de-noising”, which will be explained in Section 2, is (i)to apply C ∈ Sn
to the wavelet transform, (ii) to implement the noise reduction procedure called “softthresholding” for high frequency signal and (iii) to take the inverse wavelet transform.
Then two questions arise: let C̄ be C to which the processing of “de-noising” is achieved,(I) C̄ ∈ Sn ?
(II) diag{C̄}={1,. . . ,1}?
As discussing the two questions below, we previously state the answers: The answer to(I) is yes, that is, the property of symmetry of the correlation matrix preserves through the“de-noising” processing. As for (II), by a numerical example we will show that all diagonal
elements of the correlation matrix processed do not feature ones. And we will deal with thisproblem by means of some procedures.
This paper is organized as follows: In Section 2 we give preliminaries; rank-reduction,
wavelet analysis (multi-resolution analysis), and de-noising. In Section 3, we discuss above
two questions and propose a procedure of rank-reduction method with a noise reduction
technique. Section 4 contains numerical experiments. Finally in Section 5 we conclude.

2

Preliminaries

In this section we briefly review the rank-reduction method, the wavelet analysis (multiresolution analysis) and the “de-noising” procedure.

2.1

Rank-reduction

Let C be a correlation matrix in Rn×n . If one estimates the correlation matrix, the number ofparameters to be estimated is n(n − 1)/2. Since the number of parameters of the correlation
matrix depends on n, the estimation problem becomes more burdensome as n is increasing.
For the large number of n, Rebonato and Jäckel (1999), Rebonato (1999) suggested arank-reduction method. To see the procedure of the rank-reduction, assume that C ∈ Sn
3and C  0, i.e. C is a positive semi-definite symmetric matrix. Denote the eigenvectors andeigenvalues of C by Λ and P, respectively. ThenC = PΛP∗

where P is a real orthogonal matrix, i.e. PP∗ = PP = I.
As denoting Λ := A∗ A and B := PA, we have

C = PAA∗ P∗ = BB∗ .

(3){λ2i }ni=1 denotes the elements of the eigenvalues Λ such that λ21 > λ22 > . . . > λ2n . If one
choose k rank-reduction (k < n), then he only has to set the eigenvales as λ2k+1 = λ2k+2 =· · · = λ2n := 0.
Using the trigonometric function, Rebonato (1999) specified the i-th row vector of B ∈Rn×k bybi,1=cos(θi,1 )

bi,k

=

cos(θi,k )k−1
Ysin(θi,l ), for 1 < k < n,

l=1

bi,n

=

k−1
Y

sin(θi,l ).

l=1

There exists the advantage in that the correlation matrix C can be expressed by the angles
θ of n × k elements. That is, we can reduce the number of the parameters estimated fromn × (n − 1)/2 to n × (k − 1) for (n + 1)/2 > k.
Let X̄(θ) be the approximated correlation matrix of C by the trigonometric function with
angles θ. We can obtain the angles θ by solving the optimization problem

min
θ

n
X

i,j=1

|{X̄(θ)}ij − {C}ij |2 .

However in order to estimate the angles θ, for convenience we here iteratively calibrate
θi,j from θi,j−1 , θi,j−2 , . . ., θi,1 , j < k using the decomposed matrix B.3 The point to notice
3 Since we take such procedure in order to estimate angles θ, the case where the elements of B do not4

is that it is not necessary to feature ones in all diagonal elements of C in (3) via above
decomposition, while all diagonal elements of the correlation matrix constructed by bi,k areones (Rebonato (1999)). Then in order to feature ones in diagonal we process the rescalingcij{crescaling }ij := √cii cjj(4)

where {crescaling }ij denotes (i, j) element of rescaled C, and cij is that of C (Brigo (2002)).

2.2

Wavelet analysis

In order to view the concept of the wavelet analysis, first of all we start with the multiresolution analysis. Subsequently in accordance with the non-standard form (see pp153-157 of
Dahmen et al. (1997)), we explain two-dimensional multiresolution analysis.
We introduce multiresolution analysis, which is the important concept. Let Vj be a vector
space indexed by j ∈ Z. Then a multiresolution analysis of L2 (R) is a collection of Vj suchthat
1. · · · V2 ⊂ V1 ⊂ V0 ⊂ V−1 ⊂ · · ·,
∞
2
2. ∩∞
j=−∞ Vj = 0, ∪j=−∞Vj is dense in L (R),

3. f(x) ∈ Vj ⇔ f(2x) ∈ Vj−1 ,4. f(x) ∈ V0 ⇔ f(x + k) ∈ V0 ,5. ∃ a scaling function φ ∈ V0 such that {φ(x − k)}k∈Z is a Riesz basis of V0 .
As denoting the orthogonal space of Vj by Wj , then we have Vj ⊕ Wj = Vj−1 . We define
two operators: Pj : L2 (R) → Vj , Qj : L2 (R) → Wj .
We give a concrete (discrete transform) example of the multiresolution analysis of order
1 (at resolution level 1), which is Mallat Algorithm (see e.g. Resnikoff and Wells (1998),
Press et al. (1992)).
Example 2.1 (4-coefficient case) Let {hi }i=0,1,2,3 be the filter coefficients4 and define{gi }i=0,1,2,3 such that g0 = h3 , g1 = −h2 , g2 = h1 and g3 = −h0 . Let ñ = 2n , n ∈ N.
enclose the range of bi,k by any angle θ happens. We skip such case in numerical experiments of Section 4.
4 This is the case where the genus is 4.

5

Define the transformation matrices H, H̄ ∈ R2


h0





H := 




h1

h2

h3

h0

h1

n−j

×ñ

such that


h2





,




h3
...







G := 




h2

h3

h0

g0

g1

g2
g0

h1



g3
g1

g2

g3
...

g2

g3

g0

g1





,




where j(∈ N) is the number of the resolution level.
˜Let f(x) be a function in L2 (R) and f(x)be the corresponding discrete data in Rñ . Thenthe relationship of the multiresolution analysis and the discrete wavelet transform of order
1, i.e. j = 1 is˜ ∈ Rñf(x)−→(P1 f)(x)˜
Gf(x)∈ Rñ/2⊕(Q1 f)(x)

∩

∩

V0

V1

∼
=

f(x)

⊕

∼
=

˜
H f(x)∈ Rñ/2

∼
=

−→

∩
⊕

W1 .

Now define the Non-Standard form (NS-form for short) for two-dimensional multiresolution analysis. We refer to Dahmen et al. (1997).
Let Pj and Qj be projectors on subspaces Vj and Wj , respectively. Define an operator Tof the NS-form by
T =X(Qj T Qj + Qj T Pj + Pj T Qj ).

j∈Z

If resolution level J is finite, then

T0 =

J
X

(Qj T Qj + Qj T Pj + Pj T Qj ) + PJ T PJ .

j=1

6

For convenience, denote T0 = {{Aj , Bj , Γj }Jj=1 TJ } such that
Aj=

Qj T Qj : Wj → Wj

Bj=

Qj T Pj : Vj → Wj

Γj=

Pj T Qj : Wj → Vj

TJ=

PJ T PJ : VJ → VJ .

And we denote the operator of the two-dimensional discrete wavelet transform by T0 .

B2

T2

B1
A2

Γ2

Γ1

A1

Figure 2: NS-form at resolution level 2.

Figure 2 shows two-dimensional multiresolution analysis by the operator T0 with J = 2.
Notice that though the structure of elements in this figure is different from that in Figure 5of pp.155 in Dahmen et al. (1997), the elements of each NS-form are identical. The difference
is in the arrangement of elements and the dimensions of the NS-forms: If the dimensions of
the original data matrix are ñ × ñ, the dimension of our structure is same to the original
matrix but that in Dahmen et al. (1997) is 2ñ × 2ñ.
Let us clarify the discrete two-dimensional wavelet transform through an example.

7

Example 2.2 Let us consider the 4-coefficient case same as in Example 2.1. As applying
the two-dimensional wavelet transform of order 1 to f(x) ∈ L2 (R2 ), we have (T0 )f(x) ={(A1 )f(x), (B1 )f(x), (Γ1 )f(x), (T1 )f(x)}. And let C be a discrete data matrix in Rñ×ñ
corresponding to f(x).

Using the transform matrices of H and G, we have (T0 )C ={GCG∗ , HCG∗, GCH ∗ , HCH ∗ } and the following relationship;

2.3(A1 )f(x) ∼= GCG∗ ,(B1 )f(x) ∼= HCG∗ ,(Γ1 )f(x) ∼= GCH ∗ ,(T1 )f(x) ∼= (T1 )C = HCH ∗ .

De-noising

Donoho (1995), Donoho (1993) and Donoho and Johnstone (1994) proposed a method to
extract an unknown function f(ti ) on [0, 1] from noisy data

di = f(ti ) + σzi , i = 0, . . . , ñ − 1(5)where ti = i/ñ, zi is a Gaussian White noise and σ is a noise level.
In accordance with Donoho (1995), the processing of “de-noising” is to find fˆ which is atleast as smooth as f with high probability by minimizing the mean square error

ñ−1

ñ−1
X

i=0;

ˆ
E(f(i/ñ)− f(i/ñ))2 .

ñ−1
Let us apply the de-noising processing to {di }i=0
. Assume ñ = 2n , n ∈ N. First, applyingñ−1
the discrete wavelet transform at resolution level 1 to d := {di }i=0
∈ Rñ , we haved → Hd + Gd.
(ñ−1)/2

Secondly implement the following “soft-thresholding” to Hd := {Hdi }i=0{Hd} i − t0{Hd}new
:=
0
i





{Hd} i + t0
8

for di > t0
for |di | 6 t0
for di < −t0;√where t0 := MAD 2 log ñ and MAD is the median absolute deviation

MAD :=

median(||Hd| − median(|Hd|)|).
0.6745ñ−1Here median(d) is the median of {di }i=0
. Thirdly processing the inverse discrete wavelet(ñ−1)/2transform with {Hd}new := {{Hd} new}i=0i(ñ−1)/2

in place of {Hdi }i=0

, i.e.

dnew ← {Hd}new + Gd,
we obtain the de-noised data dnew .
Similarly we can apply the de-noising processing to the two-dimensional data structure.
Let us consider the NS-form in Figure 2. This data is processed via the discrete wavelet
transform at resolution level 2. In this case, all we have to do is to process the softthresholding to A1 , B1 , Γ1 , A2 , B2 and Γ2 , respectively. And with these data, the de-noised
correlation matrix can be obtained via the inverse discrete wavelet transform.

3

Properties of the correlation matrix processed by denoising

In this section, we discuss the two questions as mentioned above. To begin with, we proveC̄ ∈ Sn . Therefore we first show a symmetric matrix preserves symmetry of the processed
matrix through both the wavelet transform and the inverse wavelet transform. Secondly the
de-noising processing is confirmed not to ruin symmetry of the matrix. Then as for question(II), we give a concrete example and observe that all diagonal elements of a processed
symmetric matrix are not ones. For this problem, we make ones all diagonal elements of thematrix by means of rescaling. Finally we apply the rank-reduction method with de-noising
to a noisy correlation matrix.
Let C be a noisy correlation matrix in Sñ , ñ = 2n , n ∈ N. T0 denotes the operator of thetwo-dimensional discrete wavelet transform defined in Section 2 and we denote (T0 )C that it
is obtained by the processing of the soft-thresholding to (T0 )C. And let C be the de-noised
correlation matrix obtained by the processing of the inverse discrete wavelet transform to9(T0 )C.
Figure 3 shows the series of flow of de-noising.

C−→wavelet transform(T0 )C↓C̄

soft − thresholding←−(T0 )C
inverse wavelet transform

Figure 3: The series of flow of the de-nosing processing to a correlation matrix.

In accordance with Figure 3, question (I) is whether each processing; “wavelet transform”,“soft-thresholding” and “inverse wavelet transform”, preserves symmetry of the input data,or not. So we divide two parts in order to prove C̄ ∈ Sñ : Part 1 is that the wavelet
transform and inverse wavelet transform preserve symmetry, i.e. (a)C ∈ Sñ ⇒ (T0 )C ∈ Sñ
and (b) (T0 )CSñ ⇒ C̄ ∈ Sñ . And part 2 is that the processing of soft-thresholding preserves
symmetry of the matrix, i.e. (T0 )C ∈ Sñ ⇒ (T0 )CSñ .
We give the two lemmas that these processes preserve symmetry of the matrix.
Lemma 3.1 Let C be in Sñ . T0 denotes the operator of the discrete wavelet transform andT0−1 denotes that of the inverse discrete wavelet transform. Then (T0 )C ∈ Sñ where (T0 )C
has the NS-form as in Figure 2. Moreover (T0−1 )C ∈ Sñ .
Proof.

Let (T0 )C be in the NS-form. We prove (T0 )C ∈ Sñ for 4-coefficient case of

the discrete wavelet transform at resolution level 1 as in Example 2.1. We use notation in
Example 2.1. For another coefficient case the property of symmetry of the matrix can beshown by similar way. For the higher resolution level, we can iteratively show (Tj )C, for
j > 1 with (Tj )C ∈ Sñ . And for the inverse discrete wavelet transform, it can be shownwith recalling H ∗ = H −1 and G∗ = G−1 .
{cij }ñi,j=1 and {GCG∗}ij , ij = {1, . . . , ñ/2} denote the elements of C ∈ Sn and GCG∗ ,respectively. Then

{GCG∗ }k,l

=

3
X
j=0

gj

3
X
i=0

c(i+1)+2(l−1),(j+1)+2(k−1)gi , for k, l = {1, . . . , ñ/2 − 1},
10

{GCG∗ }k,ñ/2{GCG∗}ñ/2,l

=

=

1
X
i=0

3
X

1
X

3
X

j=0

gi

c(i+1)+(ñ−2),(j +1)+2(k−1)gi +

gi

3
X

gj

i=2

j=0

gj

3
X

for k = {1, . . . ñ/2},c(i+1)+2(l−1),(j+1)+(ñ−2) gi +c(i+1)−2,(j +1)+2(k−1)gi ,3
Xc(i+1)+2(l−1),(j+1)−2 gi ,

j=0

j=2

i=0

3
X

i=0

for l = {1, . . . ñ/2}.
Therefore by cij = cji , GCG∗ ∈ Sñ/2 . As for HCH ∗ , the property of symmetry holds by
the similar way.
It is easy to show GCH ∗ = (HCG∗ )∗ . Since C ∈ Sñ , i.e. C = C∗ , this equality holds
from(HCG∗ )∗=GC∗ H ∗=GCH ∗Thus we have that C ∈ Sñ ⇒ (T0 )C ∈ Sñ .

2

Lemma 3.2 Let C be in Sñ . T0 denotes the operator of the discrete wavelet transform and(T0 )C denotes (T0 )C to which the soft-thresholding is processed. Then (T0 )C ∈ Sñ .
Proof.

Let (T0 )C be in the NS-form and (T0 )C denotes (T0 )C processed via soft-thresholding.

Since C ∈ Sñ , (T0 )C ∈ Sñ by Lemma 3.1. Since the result at resolution level 1 can be iteratively applied to the higher resolution level case, we will prove for the case where theresolution level is 1. {GCG∗}ij , {GCH ∗ }ij and {HCG∗ }ij denote the elements of GCG∗ ,GCH ∗ and HCG∗ . Note that {GCG∗ }ij = {GCG∗ }ji , {GCH ∗ }ij = {(HCG∗ )∗ }ij .
As processing soft-thresholding to {GCG∗ }ij , then we have∗
 {GCG∗}ij − t0GCG ,{GCG∗}new
:=
0,
ij




 {GCG∗}ij + tGCG∗ ,0∗where t0GCG is MADGCG∗
ation

p

for {GCG∗ }ij > t0GCG∗

for |{GCG∗}ij | 6 t0GCG
for {GCG∗ }ij < t0GCG∗∗

2 log (ñ/2 × ñ/2) and MADGCG∗ is the median absolute devi-

MADGCG∗ :=median(||GCG∗ | − median(|GCG∗ |)|).
0.674511

Since {GCG∗ }ij = {GCG∗ }ji , it is obvious that {GCG∗ }new
= {GCG∗ }newijji .
It is clear that {GCH ∗ }new
= {(HCG∗ )∗ }newholds because of {GCH ∗ }ij = {(HCG∗ )∗ }ij .
ij
ij
Thus we have (T0 )C ∈ Sñ .

2Thus we have the following proposition from Lemmas 3.1 and 3.2.
Proposition 3.1 Let C be in Sñ . As applying the de-noising processing to C ∈ Sñ , thenwe have C̄ ∈ Sñ .
By Proposition 3.1, C ∈ Sñ ⇒ C̄ ∈ Sñ of question (I) holds, i.e. the de-nosing processing
preserves symmetry of the correlation matrix.
Now we will consider question (II): Are all of the diagonal elements of the correlation
matrix via de-noising ones? Let us consider the example of Figure 1(right) in Section 1 again.
This figure is an implementation of the 4-rank reduction of angle parameterization. The
noisy correlation matrix to be calibrated is generated by (2) with γ-parameters estimatedby De Jong et al. (2004) which are normally distributed. Figure 4(left) shows such a noisy
correlation matrix.
1.4

1.2
Correlation

1

1

0.8

0.8

0.6

0.6

0.4

0.4

0.2

0.2

0

16

1
Correlation

Correlation

0.8

0.6

0

14

12

10
Tj

8

6

4

2

0 0

2

4

6

8

10

12

14

Ti

16
16

14

12

10
Tj

8

6

4

2

0 0

2

4

6

8

10

12

14

Ti

16

0.4

0.2

0
2

4

6

8
Ti

10

12

14

Figure 4: Noisy correlation matrix, de-noising and diagonal.

Applying the processing of de-noising, where we use genus 2 of Daubechies and takeresolution lavel 2, to the noisy matrix we have the noise reduced matrix, Figure 4(center),and the right of this figure shows the diagonal elements of it. Though this example is oneof many cases, it is clear that the processing of de-noising loses the property that all of
diagonal elements of the correlation matrix are ones.
Then we may hit upon the rescaling as in Section 2, i.e.

ρij := √12c̄ijc̄ii c̄jj(6)16

Correlation

Correlation

1

1

0.8

0.8

0.6

0.6

0.4

0.4

0.2

0.2

0

16

0

14

12

10
Tj

8

6

4

2

0 0

2

4

6

8

10

12

14

16
16

Ti

14

12

10
Tj

8

6

4

2

0 0

2

4

6

8

10

12

14

16

Ti

Figure 5: De-noised correlation matrices.

where c̄ij is the (i, j)-element of the correlation matrix C̄ applied de-noising to C, and ρij isthe (i, j)-element of the correlation matrix by the rescaling process (6). Figure 5(left) shows
the rescaled correlation matrix of that in Figure 4(center).
We propose the following procedure in place of the rescaling for the property of non-onesof diagonal elements in the correlation matrix. Our proposing procedure is based on taking
into account the form (1) and (5).
To begin with, taking the logarithm of the noisy correlation matrix C we have from (1)log cij = −γ1 |Ti − Tj | −ppγ2 |Ti − Tj |− γ4 | Ti − Tj |.
γ
3max(Ti , Tj )(7)

Then, comparing (7) with (5), the term corresponding to the noise level in (7) depends on
Ti and Tj while that in (5) is constant. It should be divided by some function of Ti and Tj ,p√e.g. |Ti − Tj |, | Ti − Tj | and so on. However from (7) it is difficult to specify a function toadjust (7). We leave the matter open for further research. For convenience we here divide(7) by |Ti − Tj |, i.e.
log cij ad :=

log cij|Ti − Tj | 0

for i 6= j

(8)for i = j.

We apply the soft-thresholding processing to log cij ad . Denoting the elements of de-noisedlog cij ad by log cij ad , we again adjust log cij ad using |Ti − Tj |, i.e.

13c̄ij ad := log cijad × |Ti − Tj | 0

for i 6= j

(9)for i = j.

Finally taking the exponential of c̄ij ad , we obtain the de-noised correlation matrix.
The important point to note is that the exponential values of c̄ij ad for i = j are ones sincec̄ij ad are set to equal zeros for i = j in above procedure. Therefore the problem of nonzero diagonal elements of the de-noised correlation matrix is solved. We furthermore would
like to lay special emphasis on achieving more effective in the processing of de-noising.
The effectiveness of de-noising comes from the adjustment in log cij ad , i.e. reducing thedependence of Ti and Tj in the noise level. Figure 5(right) shows the de-noised correlation
matrix via our proposing procedure. The following table is the results of these errors ofthe de-noised correlation matrices; by the rescaling process (6) (Figure 5(right)) and by our
proposing procedure (Figure 5(left)). Here the error is defined byError :=

Xi>j({cnon−noise }ij − {c̄noise }ij )2(10)

where {cnon−noise }ij is (i, j)-element of the correlation matrix generated by (1) with constantγ-parameters by De Jong et al. (2004), and {c̄noise }ij is (i, j)-element of the correlation matrix, which is similarly generated but γ-parameters are assumed to be normally distributedwith mean and standard error estimated by De Jong et al. (2004), processed via each the
de-noising procedure. We will use the error defined as (10) in the next section.

Table 1: Comparison of errors of de-noised correlation matrices.
Rescaling processing Our proposing procedureError
7.691851274
1.362358664

Finally we give Figure 6 that the de-noising processing (genus 2 of Daubechies and resolution level 2) with 4 rank-reduction of angle parameterization is applied to a noisy correlation
matrix; Figure 5(left).
Comparing Figure 6 with Figure 1(right), we can find it that the noise of the correlation
matrix with 4 rank-reduction is reduced by the de-noising processing. In the next section,

14

Correlation
1
0.8
0.6
0.4
0.2
0

16

14

12

10
Tj

8

6

4

2

0 0

2

4

6

8

10

12

14

16

Ti

Figure 6: The de-noised correlation matrix with 4 rank-reduction of angle parameterization.

we investigate accuracy of the de-noised correlation matrix by numerical experiments.

4

Numerical results

We experiment on the de-noising processing for the numerical efficiency with different casesof dimensions of the correlation matrix (We will select 32 × 32 and 64 × 64. For 64 × 64,
Ti = {0.5, 0.25, . . ., 16}.), wavelet basis, resolution levels and the number of reduced rank.
Before proceeding, we confirm the arbitrary parameters.
First, it is the kind of wavelet bases. Wavelet bases used in numerical experiments are
Daubechies 2-10, Coiflet 2-10(only the even number) and Symlet 4-10, where the numbers
are genus of wavelet bases. Secondly, resolution levels. Resolution levels depend on thedimension of data implemented the processing of the discrete wavelet transform. Thirdly,
for rank reduction of angle parameterization we can choose the reduction number. It goes
without saying that as the number of reduction increases, the rank reduced correlation
matrix becomes closer to original one.
Tables 2, 3 and 4 show the average error, which is the average of errors defined as (10) by
ten trials, of between the non-noisy correlation matrix and the noisy one implemented thede-noising processing. We attempt the de-noising processing of both our proposed procedureand the processing of soft-thresholding directly to the noisy matrix with rescaling (the errorsof this procedure are in parentheses), respectively.
From these tables we can see that each wavelet basis has an effect on reduction of noise
15

Table 2: The average errors of ten trials in the case of Daubechies
Dimension
Resolution level
Genus

32 × 32
64 × 64
1
2
3
1
2
3
2
3.44358
1.368740.751105
13.7606
5.05773
3.24893(6.5766)(7.00861)(14.0139)(20.6087)(15.7337)(21.4448)
33.50705
1.186420.858654
13.8425
5.08308
3.10786(6.38463)(6.84171)(13.7677)(21.1469)(14.8616)(20.6742)
43.64409
1.14975
0.832121
12.6055
4.913672.79003(6.5261)(6.62765)(12.037)(21.5777)(15.0325)(18.7512)53.53333
1.42193
0.868147
13.9286
4.89595
2.86826(6.39687)(5.66469)(10.6696)(22.0682)(13.8777)(18.4925)
63.78049
1.32183
0.862698
13.6346
4.58636
3.09941(5.80856)(5.72751)(11.3211)(20.4376)(13.3408)(19.879)73.52203
1.17397
0.816016
13.83124.847613.08698(5.94017)(6.28963)(11.8559)(20.442)(15.8285)(20.7714)
8
3.388
1.32895
0.896223
16.5219
5.11483
3.15518(6.38468)(6.72878)(12.1746)(20.8376)(15.8776)(20.8797)
93.55838
1.71375
1.65457
13.3096.76205
5.94225(5.90228)(8.22615)(15.2378)(21.4901)(18.4647)(25.0892)
10
3.387831.240190.860728
13.9113
4.93779
3.02772(6.65682)(5.8396)(9.44729)(22.4168)(14.353)(17.853)
N otes. The average errors of ten trials without de-noising are 15.5953 for 32 × 32 dimensions and59.822 for 64 × 64 dimensions.

43.18586(41.5408)
2.88396(55.5278)
13.5351(38.7879)
3.08182(32.0768)
2.8359(40.6603)
2.99891(42.1737)
2.96394(44.2803)
7.30971(50.9098)
2.92374(36.9585)

Table 3: The average errors of ten trials in the case of Symlet
Dimension
Resolution level
Genus

32 × 32
64 × 64
1
2
3
1
2
3
4
3.86259
1.16994
0.895216
13.65034.706933.01925(6.17135)(6.50023)(14.3212)(21.0281)(15.2748)(21.5359)
5
3.37395
1.35071
0.926237
22.1464.68016
2.87207(5.75714)(5.57946)(9.95477)(20.1559)(13.2243)(17.6088)
6
3.45378
1.29530.865213
14.30424.85847
2.77971(6.61443)(6.33027)(12.4705)(21.7418)(16.3727)(20.7341)
7
3.79725
1.245480.752502
13.79074.64732
3.03287(5.48011)(6.32576)(12.0127)(20.9631)(14.5937)(20.0962)
8
3.19487
1.39074
0.944772
17.7634
4.538512.73749(6.48661)(5.48471)(11.4298)(20.8045)(14.0669)(17.8917)
93.49675
1.26362
0.7732113.02094.790313.02266(6.59085)(6.56399)(11.0917)(21.5547)(14.6114)(19.5484)
10
3.67471.33780.828756
15.0832
4.92973
2.94068(6.44662)(6.70892)(12.0674)(22.215)(15.8075)(19.2987)
N otes. The average errors of ten trials without de-noising are 15.5953 for 32 × 32 dimensions and59.822 for 64 × 64 dimensions.

42.81157(48.5764)
2.58169(30.6046)
2.61081(44.5585)
2.93934(36.0471)
2.46935(40.9124)
3.14536(36.4282)
2.76981(42.6612)

Table 4: The average errors of ten trials in the case of Coiflet
Dimension
Resolution level
Genus

32 × 32
64 × 64
1
2
3
1
2
3
2
3.69033
1.27274
0.705419
14.3172
4.70438
2.89604(6.54103)(7.32041)(19.1372)(21.0537)(16.0893)(25.0673)
43.20396
1.23853
0.838013
14.01034.49569
2.86704(6.58319)(6.35604)(12.5084)(20.8765)(14.8778)(19.9739)
6
3.74978
1.234590.904902
14.56034.807492.91008(6.28458)(6.4922)(12.0739)(21.1872)(14.4249)(18.3925)
73.64536
1.349030.857927
13.56
4.87452.77451(6.49138)(6.30138)(12.305)(20.9527)(14.8684)(17.6072)
10
3.595071.22305
0.788097
14.15454.827912.92645(6.25314)(5.85984)(11.4895)(21.6836)(14.1711)(19.306)
N otes. The average errors of ten trials without de-noising are 15.5953 for 32 × 32 dimensions and59.822 for 64 × 64 dimensions.

16

4
2.52319(63.1392)
2.63752(46.1159)
2.5838(37.9046)
2.50775(40.4683)
2.76616(44.7241)

in the correlation matrix. Moreover, as increasing the resolution level we can achieve theeffect more. The exceptions however exist, e.g. the case of Daubechies 4 and 5 (64 × 64)
at resolution level 3-4 in Table 2. Since the errors without the de-noising processing of our
proposed procedure or the processing of soft-thresholding with rescaling are apparently lager
than those with de-noising (see the notes of each table), there is no need to go into details
about the case of no de-nosing.
Table 5: The errors of one trial in the case of Daubechies(32 × 32)
Number of reduction
Resolution level
Genus

2
3
4
5
6
7
8
9
10

2
1

2

4
1

2

3.47839(7.80334)
3.35458(5.54594)
3.53995(7.11109)
4.00682(7.21688)
3.87462(6.07728)
3.40078(5.50847)
3.42811(6.18565)
3.13907(6.58181)
2.90535(7.14704)

1.2304(7.47017)
1.43929(5.3571)
1.46991(8.02536)
1.09067(6.41524)
1.47681(4.64816)
1.22714(6.62746)
1.28955(5.29226)
1.66327(7.43516)
1.103(5.77488)

3.47839(7.80334)
3.43044(7.79298)
3.22274(8.11481)
3.35526(7.89056)
3.43223(6.15129)
4.10106(7.599)3.8106(7.34069)
3.10539(7.89338)
3.09497(7.60268)

1.24315(5.9989)
1.26945(5.66786)
1.24773(5.58232)
1.70721(5.84447)
1.15183(6.35335)
1.27902(6.03748)
1.29159(5.49672)
1.73053(6.70895)
1.26236(5.38444)

N otes. The errors of one trial without de-noising are 88.3573 for 2
rank-reduction and 27.966 for 4 rank-reduction.

Table 6: The errors of one trial in the case of Coiflet(32 × 32)
Number of reduction
Resolution level
Genus

2
1
2
4
6
8
10

3.3737(4.82535)
3.13909(5.26267)
5.42238(6.39225)
3.33172(5.80168)
3.72734(5.85368)2
1.20694(7.32102)
0.901501(5.3324)
1.32268(6.11669)
1.0402(6.94001)
1.56455(7.111)4
12.94696(10.6183)
2.89812(10.4358)
3.08198(7.0152)
3.49045(7.23875)
3.39523(6.40654)

2
1.12896(8.77185)
1.59106(6.25513)
0.988084(7.10367)
0.941876(4.41349)
0.880426(6.43521)

N otes. The errors of one trial without de-noising are 88.3573 for 2
rank-reduction and 27.966 for 4 rank-reduction.

Let us now attempt to apply the de-noising processing with rank reduction of angle
parameterization. Tables 5-10 show the error of one trial for 2 and 4 rank reduction of angle
parameterization via each de-noising processing; our proposing procedure and the processing

17

Table 7: The errors of one trial in the case of Symlet(32 × 32)
Number of reduction
Resolution level
Genus

4
5
6
7
8
9
10

2
1

2

4
1

2

3.62293(7.09927)
3.71419(3.86964)
3.30247(5.58069)
3.33233(6.07235)
3.6064(5.39474)
3.73318(5.47919)
2.82371(5.24888)

1.85502(6.63074)
1.44007(6.48222)
1.38123(6.64417)
1.66218(7.43886)
1.57959(5.78264)
1.17182(7.4134)
1.0937(6.94403)

2.91825(10.8838)
3.55601(6.60093)
3.47745(8.54499)
3.50047(8.39405)
2.73993(6.69617)
3.53883(8.53637)
3.50745(8.69077)

1.09082(7.78382)
1.254(4.46325)
1.291(7.13329)
1.39193(5.7028)
1.31866(4.5321)1.25953(6.96826)
1.23799(7.0935)

N otes. The errors of one trial without de-noising are 88.3573 for 2
rank-reduction and 27.966 for 4 rank-reduction.

Table 8: The errors of one trial in the case of Daubechies(64 × 64)
Number of reduction
Resolution level
Genus

2
3
4
5
6
7
8
9
10

2
1
12.1616(19.5914)
15.6802(22.2374)
12.8896(22.0133)
12.8131(17.852)
14.4926(21.2014)
40.1463(17.5169)
17.0413(19.9906)
13.5226(21.5648)
14.1428(23.8334)2

35.26446(14.7071)
4.35574(13.8672)
4.9642(13.7415)
4.79937(13.472)4.70052(14.9421)
4.35982(16.0471)
4.83928(18.1415)
6.51037(21.1974)
4.98524(15.1974)

2.82076(21.5356)
2.98269(22.2439)
3.02964(16.6519)
3.03124(17.3104)
2.48094(23.9511)
3.25772(24.6019)
2.44311(21.8668)
5.194(21.1485)
2.57266(17.7665)

4112.1616(19.5914)
15.6802(22.2374)
11.0168(20.1757)
10.9582(20.3416)
14.1549(18.5432)
11.4282(20.0715)
11.7313(20.1699)
21.6994(22.6609)
11.7691(20.4305)

2

34.91042(14.0582)
4.7495(12.5364)
5.45062(15.2537)
4.81322(14.1903)
3.96311(11.532)
5.78058(15.8102)5.24397(15.8513)
5.71822(14.5948)
5.12459(13.339)

3.10275(24.5382)
3.76557(21.2842)
3.53037(21.7249)
2.324(20.164)4.09055(20.0147)
3.4782(21.2178)
2.60227(24.1446)
7.44931(24.2116)
2.65957(19.8569)

N otes. The errors of one trial without de-noising are 345.772 for 2 rank-reduction and 122.603
for 4 rank-reduction.

Table 9: The errors of one trial in the case of Coiflet(64 × 64)
Number of reduction
Resolution level
Genus

2
3
4
5
6

2
1
13.5877(19.9902)
14.3766(21.611)13.6058(22.789)
14.7043(21.6336)
13.952(19.1076)

2

34.35398(17.6652)
5.44463(14.4066)
3.97281(12.4538)
5.37858(15.3365)
4.38408(11.9616)

3.14338(25.7925)
3.22989(17.3562)
2.72442(21.9992)
3.00026(18.3604)
2.4456(18.7356)

4115.3794(19.7249)
11.2677(20.7702)
14.2765(18.8921)
11.1599(20.5655)
14.1915(19.0094)

2

3

4.46501(13.1525)
4.92094(15.409)4.25495(10.8864)
4.92286(14.9866)
4.21693(11.0611)

3.51686(26.8393)
2.41073(19.2783)
3.62265(17.196)
2.54106(17.1398)
3.67346(16.135)

N otes. The errors of one trial without de-noising are 345.772 for 2 rank-reduction and 122.603
for 4 rank-reduction.

18

Table 10: The errors of one trial in the case of Symlet(64 × 64)
Number of reduction
Resolution level
4
5
6
7
8
9
10

2
1
15.1263(19.6266)
13.7677(17.8774)
14.1219(21.8168)
14.168(19.929)
13.0946(20.1913)
14.6731(20.4017)
13.3157(19.6267)

2

3

4.26181(18.3206)
5.19445(17.5095)
4.35096(14.0008)
4.71341(17.7445)
4.60801(12.0684)
5.20236(15.8373)
4.60539(17.2603)

3.50585(21.7973)
1.89293(17.2179)
4.10913(17.502)2.92352(22.6889)
3.40682(18.8404)
2.35333(20.3934)
2.99442(18.3559)

41
11.1341(20.9973)
11.531(19.8187)
14.549(19.5489)
10.9073(20.871)
15.8596(18.8492)
11.1207(20.4596)
15.1549(19.6054)

2

35.89072(14.4947)
4.80997(13.5594)
4.36362(11.7588)
5.13088(15.3294)
4.21399(10.7254)
5.09365(16.4008)
4.30419(12.2258)

3.53155(21.7264)
2.3099(20.6842)
3.72636(20.5307)
2.88013(19.9687)
3.72223(16.4405)
2.71046(18.5191)
3.72316(17.9967)

N otes. The errors of one trial without de-noising are 345.772 for 2 rank-reduction and 122.603
for 4 rank-reduction.

of soft-thresholding with rescaling (in parentheses). Tables 5-7 are for 32 × 32 dimensions,
Tables 8-10 for 64 × 64 dimensions, respectively. We take resolution level to 2 for 32 × 32
dimensions and 3 for 64 × 64 dimensions. Since there exist exceptions of the increasing error
at resolution level 3 in Table 2, we omit the case of resolution level 4 for 64 × 64 dimensions
and 3 for 32 × 32 while such exceptions occur only for Daubechies case of 64 × 64.
The results of the numerical experiments for 2 and 4 rank reduction of angle parameterization with the de-noising processing are excellent as expected it from Tables 2-4. There exists
no apparent difference in errors by the kind of wavelet bases and genus. And the decreasing
tendency in errors exists as the resolution level increases. However the exception of thistendency arises for Daubechies coefficient (64 × 64, resolution level 9 and 4 rank-reduction)case in Table 8.

5Conclusion

In this paper, we proposed a simple procedure based on the de-noising processing using
wavelet analysis in order to reduce the noise in the noisy correlation matrix. And supposing
the elements of the noisy correlation matrix as (1) with random γ-parameters, we madenumerical experiments on the reduction of the noise by this procedure. The result of numerical experiments showed good performance. Subsequently, the rank reduction method withde-noising was also effective in modeling the approximated correlation matrix. Though it

19depends on the assumption of the distribution of noise in the correlation matrix, these numerical results lead to us that our proposed procedure of de-noising works well in modelingthe correlation matrix from the noisy correlation data.
Lastly we remark the further research on improving the efficiency of de-noising and theapplication to finance.
First as mentioned in Section 3, the problem is how to adjust in (8) and (9). We divided(7) simply by |Ti − Tj |. This means that from the noisy correlation data we substantiallyextract the unknown fˆij such that{correlation data}ij = exp{fˆij + σ|Ti − Tj |zij }(11)

where zij ∼iid N (0, 1) and σ is the noise level. Therefore we presumed (11) instead of (1),alternatively. Then we should establish the processing of de-noising or soft-thresholdingto (1) in order to improve the effectiveness of de-noising, if (1) is the good parametricfunction for the market correlation. If not so, we should consider another de-noising orsoft-thresholding for better parametric function.
Secondly, it is the practical application to the calibration of LFM. The calibration of LFM
is usually accomplished as follows (Refer to, e.g. London (2004)): Let σi (a, b, c, d, θ) be avolatility model. vmarket denotes the market volatilities. Then the parameters a, b, c, d, θare estimated by solving

mina,b,c,d,θ

X  vmarket − σ 2 (a, b, c, d, θ) 2ii

vmarket.

Using the Newton-Raphson, gradient-decent or Brent’s method solves the minimization
problem. For instance, in such iterative calculus one can apply the de-noising processing asfollows: Implement the de-noising processing to the angles obtained by the iterative routine.
Set the de-noised angles as the initial values and execute the iteration again. Though wewill have to discuss the convergence by such procedure, it is likely to become very effective
to solve the minimization problem in the case of clarifying of effectiveness of it.

20

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22