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A determinantal inequality for correlation matrices
Statistics and Probability Letters 88 (2014) 88–90
Contents lists available at ScienceDirect
Statistics and Probability Letters journal homepage: www.elsevier.com/locate/stapro
A determinantal inequality for correlation matrices Ingram Olkin ∗ Departments of Statistics and Education, Stanford University, Sequoia Hall, 390 Serra Mall, Stanford, CA 94305-4065, USA
article
info
Article history: Received 13 December 2013 Received in revised form 10 January 2014 Accepted 13 January 2014 Available online 25 January 2014
abstract Correlations are a source of continuing discoveries. A new inequality is obtained that provides easily computed bounds for the determinant of a correlation matrix. © 2014 Elsevier B.V. All rights reserved.
Keywords: Intraclass correlation matrix Majorization Schur convex functions
Correlations have passed their centennial, and one might assume that all is known about them. However, evidence that this is not the case crops up periodically. In 1960, Kendall wrote a paper entitled ‘‘The evergreen correlation coefficient’’ to emphasize that new insights about correlations continue to surface. The present paper adds a new, and somewhat surprising, result into the literature on correlation matrices. Although the proof is simple, and I believe somewhat elegant, it relies on machinery generally alien to the study of correlations. Let R = (rij ) denote an m×m correlation matrix, and R = (˜rij ), where r˜ii = 1, r˜ij = r¯ for all i ̸= j, and r¯ = i̸=j rij /m(m−1). In a sense R is an ‘‘average’’ correlation matrix. One way to compare R and R is to examine the difference ∆ = R − R. Unfortunately, such a difference can never be positive or negative definite. Instead, we compare a function of R with the same function of R. Let λ(R) = (λ1 (R), . . . , λm (R)) and λ( R) = (λ1 ( R), . . . , λm ( R)) denote the eigenvalues of R and R, respectively, and assume the eigenvalues are ordered λ1 ≥ λ2 ≥ · · · ≥ λm . The main result relates the eigenvalues λ(R) and λ( R). The particular relation is based on majorization, which is discussed in great detail in Marshall et al. (2011) [MOB]. A brief discussion follows. For real vectors x = (x1 , . . . , xm ) and y = (y1 , . . . , ym ) ordered x1 ≥ · · · ≥ xm and y1 ≥ · · · ≥ ym , we write x ≻ y and say x majorizes y if k
xi ≥
k
1
1
m
m
1
xi =
yi ,
k = 1, . . . , m − 1, (1)
yi .
1
The main result is a comparison of λ(R) and λ( R). Theorem. λ( R) ≺ λ(R).
∗
Tel.: +1 6507232628; fax: +1 6507258977. E-mail addresses: [email protected], [email protected], [email protected].
0167-7152/$ – see front matter © 2014 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.spl.2014.01.012
I. Olkin / Statistics and Probability Letters 88 (2014) 88–90
89
Proof. We prove a stronger result. Let P12 , P13 , . . . , Pm−1,m , denote M = m(m − 1)/2 distinct permutation matrices in which Pij permutes the ith and jth rows and columns, and define Rk =
k 1
k
Pi RPi ,
(2)
1
where P1 , . . . , Pk , k ∈ {1, 2, . . . , M }, is a subset of the M permutation matrices. Furthermore, note that Pi2 = I, the identity matrix, and that for any square matrices A and B, the eigenvalues λ(AB) = λ(BA). Consequently,
λ(Pi RPi ) = λ(RPi2 ) = λ(R).
(3)
Now invoke a well-known result due to Ky Fan (see [MOB], Sect. G): if G and H are Hermitian matrices, then
λ(G + H ) ≺ λ(G) + λ(H ). Hence
λ(Rk ) = λ
Pi RPi
k
=
1 k
λ
Pi RPi
k
≺
1 k
λ(Pi RPi ) = λ(R).
(4)
1
The result follows for k = m(m − 1)/2.
A number of consequences follow from the fact that x ≺ y. One consequence is that
g (xi ) ≥
g (yi )
(5)
for all concave functions g. A second consequence is that
ϕ(x) ≥ ϕ(y)
(6)
for all Schur-convex functions ϕ , where the class of Schur-convex functions ϕ is defined by the condition that ϕ is permutation invariant, and that
∂ϕ ∂ϕ − ∂ x1 ∂ x2
(x1 − x2 ) ≥ 0;
(7)
ϕ is Schur-concave if (7) is replaced by ∂ϕ ∂ϕ − (x1 − x2 ) ≤ 0. ∂ x1 ∂ x2
(8)
Suppose that the random variables X1 , . . . , Xm have a correlation matrix R. The function det R serves as an indicator of the degree of correlation among the variables. When det R = 1, that is, R = I, x’s are uncorrelated. When rij = 1 for all i, j, then det R = 0 and x’s are maximally correlated. But det R can be equal to zero under other circumstances, namely when the rank of R is less than m. Thus det R does not completely clarify the degree of correlation among the variables. However, det R can serve as a useful general indicator of the degree of dependence among the variables. Corollary. det R ≤ det R
= (1 − r¯ )m−1 (1 + (m − 1)¯r ) . Proof. The function ϕ(x) = m 1
λi (R) ≤
m
(9)
xi is Schur-concave, and hence from (6):
λi ( R).
1
The eigenvalues of R are well-known to be (1 + (m − 1)¯r , 1 − r¯ , . . . , 1 − r¯ ).
90
I. Olkin / Statistics and Probability Letters 88 (2014) 88–90
Note that the Hadamard inequality yields the bound det R ≤ rii = 1 (see [MOB], pp. 4, 300). Because (1 − r¯ )m−1 (1 + (m − 1)¯r ) ≤ 1, the corollary provides an improved bound given knowledge of the mean r¯ . For example,
R=
R=
R=
1 .5 .2
.5 1
.8
−.3 .5 ,
1
.5 .1 1
.5
det R = 0.23, det R = 0.50;
1
.1
1 .1 −.3 1 .1 .3
.2 .8 ,
det R = 0.62, det R = 0.97;
1
.3 .5 ,
det R = 0.68, det R = 0.78.
1
In two cases, the bound provides a considerable improvement over the Hadamard inequality. The bound is particularly useful for larger-size correlation matrices because the computation of r¯ is relatively simple. We note that many likelihood ratio tests in multivariate analysis are based on determinants for which this result may prove useful. Comment. It is straightforward to show that the theorem holds for any Hermitian matrix. References Kendall, M.G., 1960. The evergreen correlation coefficient. In: Olkin, I., Ghurye, S., Hoeffding, W., Madow, W., Mann, H. (Eds.), Contributions to Probability and Statistics, Essays in honor of Harold Hotelling. Stanford Univ. Press, Stanford, Calif., pp. 274–277. Marshall, A.W., Olkin, I., Arnold, B.C., 2011. Inequalities: Theory of Majorization and its Applications, second edition. In: Springer Series in Statistics, Springer, New York.
Contents lists available at ScienceDirect
Statistics and Probability Letters journal homepage: www.elsevier.com/locate/stapro
A determinantal inequality for correlation matrices Ingram Olkin ∗ Departments of Statistics and Education, Stanford University, Sequoia Hall, 390 Serra Mall, Stanford, CA 94305-4065, USA
article
info
Article history: Received 13 December 2013 Received in revised form 10 January 2014 Accepted 13 January 2014 Available online 25 January 2014
abstract Correlations are a source of continuing discoveries. A new inequality is obtained that provides easily computed bounds for the determinant of a correlation matrix. © 2014 Elsevier B.V. All rights reserved.
Keywords: Intraclass correlation matrix Majorization Schur convex functions
Correlations have passed their centennial, and one might assume that all is known about them. However, evidence that this is not the case crops up periodically. In 1960, Kendall wrote a paper entitled ‘‘The evergreen correlation coefficient’’ to emphasize that new insights about correlations continue to surface. The present paper adds a new, and somewhat surprising, result into the literature on correlation matrices. Although the proof is simple, and I believe somewhat elegant, it relies on machinery generally alien to the study of correlations. Let R = (rij ) denote an m×m correlation matrix, and R = (˜rij ), where r˜ii = 1, r˜ij = r¯ for all i ̸= j, and r¯ = i̸=j rij /m(m−1). In a sense R is an ‘‘average’’ correlation matrix. One way to compare R and R is to examine the difference ∆ = R − R. Unfortunately, such a difference can never be positive or negative definite. Instead, we compare a function of R with the same function of R. Let λ(R) = (λ1 (R), . . . , λm (R)) and λ( R) = (λ1 ( R), . . . , λm ( R)) denote the eigenvalues of R and R, respectively, and assume the eigenvalues are ordered λ1 ≥ λ2 ≥ · · · ≥ λm . The main result relates the eigenvalues λ(R) and λ( R). The particular relation is based on majorization, which is discussed in great detail in Marshall et al. (2011) [MOB]. A brief discussion follows. For real vectors x = (x1 , . . . , xm ) and y = (y1 , . . . , ym ) ordered x1 ≥ · · · ≥ xm and y1 ≥ · · · ≥ ym , we write x ≻ y and say x majorizes y if k
xi ≥
k
1
1
m
m
1
xi =
yi ,
k = 1, . . . , m − 1, (1)
yi .
1
The main result is a comparison of λ(R) and λ( R). Theorem. λ( R) ≺ λ(R).
∗
Tel.: +1 6507232628; fax: +1 6507258977. E-mail addresses: [email protected], [email protected], [email protected].
0167-7152/$ – see front matter © 2014 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.spl.2014.01.012
I. Olkin / Statistics and Probability Letters 88 (2014) 88–90
89
Proof. We prove a stronger result. Let P12 , P13 , . . . , Pm−1,m , denote M = m(m − 1)/2 distinct permutation matrices in which Pij permutes the ith and jth rows and columns, and define Rk =
k 1
k
Pi RPi ,
(2)
1
where P1 , . . . , Pk , k ∈ {1, 2, . . . , M }, is a subset of the M permutation matrices. Furthermore, note that Pi2 = I, the identity matrix, and that for any square matrices A and B, the eigenvalues λ(AB) = λ(BA). Consequently,
λ(Pi RPi ) = λ(RPi2 ) = λ(R).
(3)
Now invoke a well-known result due to Ky Fan (see [MOB], Sect. G): if G and H are Hermitian matrices, then
λ(G + H ) ≺ λ(G) + λ(H ). Hence
λ(Rk ) = λ
Pi RPi
k
=
1 k
λ
Pi RPi
k
≺
1 k
λ(Pi RPi ) = λ(R).
(4)
1
The result follows for k = m(m − 1)/2.
A number of consequences follow from the fact that x ≺ y. One consequence is that
g (xi ) ≥
g (yi )
(5)
for all concave functions g. A second consequence is that
ϕ(x) ≥ ϕ(y)
(6)
for all Schur-convex functions ϕ , where the class of Schur-convex functions ϕ is defined by the condition that ϕ is permutation invariant, and that
∂ϕ ∂ϕ − ∂ x1 ∂ x2
(x1 − x2 ) ≥ 0;
(7)
ϕ is Schur-concave if (7) is replaced by ∂ϕ ∂ϕ − (x1 − x2 ) ≤ 0. ∂ x1 ∂ x2
(8)
Suppose that the random variables X1 , . . . , Xm have a correlation matrix R. The function det R serves as an indicator of the degree of correlation among the variables. When det R = 1, that is, R = I, x’s are uncorrelated. When rij = 1 for all i, j, then det R = 0 and x’s are maximally correlated. But det R can be equal to zero under other circumstances, namely when the rank of R is less than m. Thus det R does not completely clarify the degree of correlation among the variables. However, det R can serve as a useful general indicator of the degree of dependence among the variables. Corollary. det R ≤ det R
= (1 − r¯ )m−1 (1 + (m − 1)¯r ) . Proof. The function ϕ(x) = m 1
λi (R) ≤
m
(9)
xi is Schur-concave, and hence from (6):
λi ( R).
1
The eigenvalues of R are well-known to be (1 + (m − 1)¯r , 1 − r¯ , . . . , 1 − r¯ ).
90
I. Olkin / Statistics and Probability Letters 88 (2014) 88–90
Note that the Hadamard inequality yields the bound det R ≤ rii = 1 (see [MOB], pp. 4, 300). Because (1 − r¯ )m−1 (1 + (m − 1)¯r ) ≤ 1, the corollary provides an improved bound given knowledge of the mean r¯ . For example,
R=
R=
R=
1 .5 .2
.5 1
.8
−.3 .5 ,
1
.5 .1 1
.5
det R = 0.23, det R = 0.50;
1
.1
1 .1 −.3 1 .1 .3
.2 .8 ,
det R = 0.62, det R = 0.97;
1
.3 .5 ,
det R = 0.68, det R = 0.78.
1
In two cases, the bound provides a considerable improvement over the Hadamard inequality. The bound is particularly useful for larger-size correlation matrices because the computation of r¯ is relatively simple. We note that many likelihood ratio tests in multivariate analysis are based on determinants for which this result may prove useful. Comment. It is straightforward to show that the theorem holds for any Hermitian matrix. References Kendall, M.G., 1960. The evergreen correlation coefficient. In: Olkin, I., Ghurye, S., Hoeffding, W., Madow, W., Mann, H. (Eds.), Contributions to Probability and Statistics, Essays in honor of Harold Hotelling. Stanford Univ. Press, Stanford, Calif., pp. 274–277. Marshall, A.W., Olkin, I., Arnold, B.C., 2011. Inequalities: Theory of Majorization and its Applications, second edition. In: Springer Series in Statistics, Springer, New York.