PHYSICAL REVIEW E 88, 042130 (2013)
Spectral density of a Wishart model for nonsymmetric correlation matrices Vinayak* Instituto de Ciencias F´ısicas, Universidad Nacional Aut´onoma de M´exico, C.P. 62210 Cuernavaca, M´exico (Received 11 June 2013; revised manuscript received 20 August 2013; published 17 October 2013) The Wishart model for real symmetric correlation matrices is defined as W = AAt , where matrix A is usually a rectangular Gaussian random matrix and At is the transpose of A. Analogously, for nonsymmetric correlation matrices, a model may be defined for two statistically equivalent but different matrices A and B as ABt . The corresponding Wishart model, thus, is defined as C = ABt BAt . We study the spectral density of C for the case when A and B are not statistically independent. The ensemble average of such nonsymmetric matrices, therefore, does not simply vanishes to a null matrix. In this paper we derive a Pastur self-consistent equation which describes spectral density of C at large matrix dimension. We complement our analytic results with numerics. DOI: 10.1103/PhysRevE.88.042130
PACS number(s): 02.50.Sk, 05.45.Tp, 89.90.+n, 89.65.Gh
I. INTRODUCTION
Correlation matrices constitute a fundamental tool for multivariate time series analysis [1,2]. Wishart introduced random matrices of W = AAt /T type as a model for real symmetric correlation matrices [3]. In this model A is usually a rectangular matrix of dimension N × T , At is the transpose of A, and the matrix entries Aj ν are independent Gaussian variables with zero mean and a fixed variance. In due course this model gained much attention from various branches of science and now is applied to a vast domain including mathematical statistics [1,2], physics [4–7], communication engineering [8], econophysics [9–14], biological sciences [15,16], atmospheric science [17], etc. In random matrix theory (RMT) [4] the ensemble of such non-negative matrices is known as Wishart orthogonal ensemble (WOEs) or Laguerre orthogonal ensemble [18,19] where analytical results for spectral statistics are known in great detail. WOE characterizes a null hypothesis for symmetric correlation matrices where the spectral statistics set a benchmark or a reference against which any useful actual correlation must be viewed. For instance, the spectral density of WOE [20], has proved to be remarkably useful for identifying underlying actual correlations. Prime examples thereof are found in econophysics [9–11]. However, there have been important advances incorporating the actual correlations in random matrix ensembles. These ensembles are often referred to as the correlated Wishart ensembles [21–26]; the generalization for the WOE case is called the correlated Wishart orthogonal ensembles (CWOE). CWOE results are also important because these supply a reference against which correlations lying off the trend must be viewed [27]. In recent years there has been a growing interest in the analysis of nonsymmetric correlation matrices [28,29]. A nonsymmetric correlation matrix can be realized for timelagged correlations among the variables representing the same statistical system [30,31]. In a more general case it can be the matrix representing correlations among the variables of two different statistical systems [32]. The corresponding random matrix model which describes a null hypothesis for nonsymmetric correlation matrices is defined for two
*
[email protected]
1539-3755/2013/88(4)/042130(6)
statistically equivalent but independent matrices, e.g., AB = ABt /T , where A and B are rectangular matrices of dimensions N × T and M × T , respectively, and entries of both matrices are the Gaussian variables with mean zero and variance one. For rectangular AB , i.e., for N = M, a reference against which actual correlations have to be viewed is the statistics of singular values of AB [30,33,34] or equivalently the statistics of eigenvalues of a corresponding Wishart model of an N × N matrix, defined as C = ABt BAt /T 2 . On the other hand, for square AB , ample results are known for the statistics of complex eigenvalues [34–36], which could be useful for the spectral studies of correlation matrices such as the eigenvalue density used in Refs. [32,37]. Following the CWOE approach we generalize the Wishart model for nonsymmetric correlation matrices to the case where A and B are not statistically independent. In other words, the ensemble average ABt /T = η, where bar denotes the ensemble averaging and η is the N × M correlation matrix which defines the actual correlations between N input variables with M output variables. The joint probability density of the matrix elements of A and B can be described via −1 1 η A t t P (A,B) ∝ exp −tr (1) (A B ) , B ηt 1 where 1 is an identity matrix of dimensions N × N in the upper diagonal block and M × M in the lower diagonal block. In this paper we derive a self-consistent Pastur equation which describes the spectral density for C at large matrix dimension where η = 0. In the next section we define nonsymmetric correlation matrices from the CWOE approach, fix notations and describe some generalities of the work. In Sec. III, we state the main result of the paper. Details of the derivation of the result are given in Appendix B. In Sec. IV we present some numerical examples to complement analytics. In Sec. V we summarize our results. II. NONSYMMETRIC CORRELATION MATRICES: A CWOE PERSPECTIVE
CWOE is an ensemble of real symmetric matrices of type C = WW t /T , where W = ξ 1/2 W, ξ is a positive definite nonrandom matrix, and entries of the matrix W are independent 042130-1
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Gaussian variables with mean zero and variance one. Thus, C = ξ.
(2)
Spectral statistics of CWOE have been addressed by several authors. For example, the spectral density for large matrices has been derived in Refs. [20–24] and recently for finite dimensional matrices in Refs. [25,26]. Unlike WOE spectra, CWOE spectra may exhibit nonuniversal spectral statistics as noted in Ref. [24]. Suppose the matrix W constituted by two different random matrices A and B, as A W= , (3) B where A and B are of dimensions N × T and M × T , respectively. Then the matrix C is a partitioned matrix, defined in terms of A and B, as 1 AAt AB t , (4) C= T BAt BBt . Here the diagonal blocks, viz., AAt and BBt , and the offdiagonal blocks, viz., AB t and BAt , are, respectively, N × N , M × M, N × M, and M × N dimensional. Therefore, ξ is also partitioned, ξAA ξAB ξ= , (5) ξBA ξBB where the diagonal blocks ξAA and ξBB account for the correlations among the variables of A and of B, respectively. The off-diagonal blocks, e.g., AB t /T = ξAB , account for the correlations of A and B. By construction ξBA = [ξAB ]t . Without loss of generality we consider M N and T M. We consider the case where ξAB = 0 and wish to compare the spectral density with that of the null hypothesis, i.e., when ξAA = 1N×N , ξBB = 1M×M , and ξAB = 0. It is therefore important to remove cross correlations among the variables of individual matrices because only then will the diagonal blocks of (4) yield identity matrices on the ensemble averaging. Thus, we introduce decorrelated matrices [24] defined as −1/2
−1/2
A = ξAA A, B = ξBB B. −1/2
ABt BAt . T2 We further define the M × M symmetric matrix D, D=
BAt ABt , T2
ζ = ηηt .
−1/2
(7)
(8)
III. SPECTRAL DENSITY FOR LARGE MATRICES
To obtain the spectral density, ρ(λ), of C we closely follow the binary correlation method developed by French and his collaborators [5,39] and used in Ref. [24] to study the spectral statistics of CWOE. In this method we deal with the resolvent or the Stieltjes transform of the density, defined for a complex variable z as
1 G(z) = . (12) z1N×N − C N We use angular brackets to represent the spectral averaging, e.g., H k = k −1 tr H , and bars to denote averaging over the ensemble. Below we use a bar also to represent functions of the ensemble averaged scalar quantities. The ensemble averaged spectral density, can be determined via 1 ρ(λ) = ∓ ImG(λ ± i ), (13) π for infinitesimal > 0. For large z, the resolvent may be expressed in terms of moments, mp , of the density, as G(z) =
N , T M . = T
κM
∞ Cp N p=0
zp+1
=
∞ mp . p+1 z p=0
(14)
The remaining task now is to obtain a closed form of this summation. To obtain a closed form, in the above expansion we consider only those binary associations yielding leading order terms and avoid those resulting in terms of O(N −1 ). This method finally gives the so-called Pastur self-consistent equation for the resolvent [5,24,39], or the Pastur density [40], which holds for large matrices. In order to do the ensemble averaging we use the following exact results, valid for arbitrary fixed matrices and : 1 A At N = T N , (15) T A A N = t N ,
and the ratios, κN =
(11)
However, in the following we consider a large N limit, where N/T and M/T are finite so that the matrices C and D will never be deterministic. We consider only those cases where the spectrum of ζ does not exceed N . This is always valid for our model because the positive definiteness of ξ ensures an upper bound 1 for eigenvalues of ζ . For the completeness of the paper we prove this remark in Appendix A.
(6)
Note that we still have ABt /T = η, where η = ξAA ξAB ξBB and the null hypothesis is characterized for η = 0. This case has been studied by several authors [30,33,34,38]. In this paper we consider η = 0 and calculate the ensemble averaged spectral density of N × N symmetric matrices C, defined as C=
A few remarks are immediate. At first we note that the ensemble averages yield C = κM 1N×N + ηηt and D = κN 1M×M + ηt η. By construction it is obvious that the nonzero eigenvalues of D are identical to those of C. Next, for T → ∞ and N,M finite, since C = ξ , C = ηηt and D = ηt η. We finally define a symmetric matrix ζ as
(9) (10) 042130-2
1 N , N 1 A N A N = t N . N
A N At N =
(16) (17) (18)
SPECTRAL DENSITY OF A WISHART MODEL FOR . . .
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The dimensions of and are suitably adjusted in the above identities. Similar results can be written for the averaging over B. These are the same results as obtained for CWOE in Ref. [24]. However, here we have to take account of η, which gives two more important identities, viz., 1 A Bt N = T η N , (19) T 1 B At M = T ηt M . (20) T In this section we omit the detail computation of G(z), merely stating here the central result of the paper. We provide step by step details of the derivation in Appendix B. A compact result for the self-consistent Pastur equation can be written as −1
G(z) = [z − ζ Y 1 (z,G(z)) − Y 2 (z,G(z))] .
(21)
diagonal blocks are defined by equal-cross-correlation matrix model, e.g., [ξAA ]j k = δj k + (1 − δj k )a, for 1 j,k N , and the same for ξBB , where the correlation coefficient is b. In both examples we consider 0 < a,b,c < 1. This choice is necessary but not sufficient for the positive definiteness of ξ ; see Eq. (28). The spectrum of the equal-cross-correlation matrix is easy to calculate. For instance, the spectrum of ξAA is described by 1 − a and N a + 1 − a, where the former has degeneracy N − 1. The spectrum of ξBB is also described in the same way but for M and b. Moreover, the inverse of the square root of these matrices, those we need to define η, are also not difficult to calculate. For example, we simply have
−1/2 ξAA j k = δj k [(1 − a)−1/2 ] −
Here Y 1 (z,G(z)) =
{1 + κN [z G(z) − 1]} 1−
Y 2 (z,G(z)) =
2
κN G(z){1+κN [z G(z)−1]} 1−κN g(z,G(z))
Y (z,G(z)) 1 − κN g(z,G(z))
,
(22)
(23)
,
Y (z,G(z)) = κM + κN [z G(z) − 1] × {1 + κM + κN [z G(z) − 1]},
(a)
(24) 8
1 + κN [z G(z) − 1]
.
0.02
6
(25)
0
ρ(λ)
[z − Y 2 (z,G(z))]G(z) − 1
(26)
For the first case the spectrum of ζ can be determined analytically because of a trivial choice of ξAB . Yet eigenvalues, (ξ ) λj , of ξ may not be as trivial to obtain. However, in this case we find N − 1 eigenvalues 1 − a, M − 1 eigenvalues 1 − b,
and g(z,G(z)) =
1 [(1 − a)−1/2 − (N a + 1 − a)−1/2 ]. N
Equation (21), together with definitions (22)–(25), is the main result of this paper. This result is analogous to the result for CWOE which has been obtained first by Mar´cenko and Pastur [20] and then by others using different techniques [21,22,24]. For the uncorrelated case, i.e., for ζ = 0, Eq. (21) results in a cubic equation confirming thereby the result obtained in Ref. [34].
0.8
1
1.2
4
data Theory
2 0 6
IV. NUMERICAL EXAMPLES AND VERIFICATION OF THE RESULT (21)
(b)
0.04 0.02
A numerical technique has been developed in Ref. [24] for solving the Pastur equation which describes the density for CWOE. We use the same technique to solve our result (21). However, in our case the result is complicated and needs some treatment for meeting requirements of the numerical technique. We first note that G depends on Y 1 and Y 2 , while both the latter quantities depend on g and G. Since the g itself depends on G and Y 2 , at least one quantity G or g, has to be determined explicitly in terms of the other. It turns out that, using Eqs. (25) and (23), we can eliminate Y 2 from g. Therefore, for a given z, g, and in turn Y 1 and Y 2 , can be estimated using an initial guess of G. After resolving these we can use the numerical technique [24] to obtain the solution of (21). We demonstrate our result for two different correlation matrices. In the first example, we consider ξAB to be a rank 1 matrix, e.g., [ξAB ]j r = c for every integer 1 j N and 1 r M. In the second example we consider [ξAB ]j r = c δj r + (1 − δj r )c|j −r| . For simplicity, we consider that the
ρ(λ)
4 0
0.8
1
1.2
2
0
0
0.2
λ
0.4
FIG. 1. Spectral density, ρ(λ), where [ξAB ]j r = c for 1 j N and 1 r M, c = 0.8, and correlation coefficients of the equalcross-correlation matrices describing the diagonal blocks are a = b = 0.5. Symbols in the figure represent Monte Carlo simulations and solid lines are the theory obtained from the numerical solution of Eq. (21). In (a) we show results for N = 384 and in (b) we show results for N = 256. Dimension of the full matrix in both the figures is 1024 and T = 5120. In insets we show distribution of the separated eigenvalues where we have considered an ensemble of 10 000 matrices. Dashed lines in insets represent Gaussian distributions where the means and the variances have been calculated numerically.
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and the remaining two are given by (ξAA ) (ξBB ) (ξ ) (ξ ) 2 λ λNAA − λMBB + 4N Mc2 + λ ± N M (ξ ) , λ± = 2 (27) (ξ
)
(ξ
)
where λNAA = N a + 1 − a and λMBB = Mb + 1 − b. Note that the positive definiteness of ξ is ensured if (ξAA ) (ξBB ) λN λM > NMc2 .
(28)
The matrix ξAB is rank 1; so is η: c ηj r = . (ξAA ) (ξBB ) λN λM
(29) (ξ
) ξ
)
Using this we readily obtain ζj k = Mc2 /λNAA λM(BB and the ) (ξAA ) (ξBB ) 2 only nonzero eigenvalue, λ(ζ λM . It trivially N = N Mc /λN (ζ ) follows from the inequality (28) that λN < 1. In fact, due the positive definiteness of ξ it holds for more general cases as proved in Appendix A. To check our theoretical result (21) with numerics we begin with simulating C defined in the beginning of Sec. II. Next, we identify the off-diagonal block {AB} in C. Finally, we use −1/2 −1/2 the transformation ξAA AB t ξBB to obtain ABt , which we desire to calculate C. In numerical simulations we fix N + M = 1024, T = 5(N + M) and consider two values of N , (a)
8
Theory Data ζ=0
ρ(λ)
6 4 2 0 5
(b)
ρ(λ)
4 3 2 1 0
0
0.1
0.2
0.3
λ
0.4
0.5
FIG. 2. (Color online) Spectral density, ρ(λ), for the second example, where [ξAB ]j r = c δj r + (1 − δj r )c|j −r| , for 1 j N and 1 r M, c = 0.05, and the correlation coefficients which describe the diagonal blocks are a = b = 0.5. With an outlay similar to Fig. 1 we compare our theory with numerics. Color lines in this figure represent the uncorrelated case.
viz., N = 256 and 384. To compare numerics with theory we consider an ensemble of size 1000 of matrices C. In our first example, ζ has only one nonzero eigenvalue. For this spectrum our theory (21) yields the density for the bulk of the spectra. It suggests that the bulk should be described by density of the uncorrelated case. We verify this with numerics in Fig. 1, where a = b = 0.9 and c = 0.8. However, like the equal-cross-correlation matrix model of CWOE [24], in this case as well, we obtain one eigenvalue separated from the bulk [41]. Interestingly, here the bulk remains invariant with correlations as opposed to the CWOE case. Moreover, distribution of the separated eigenvalues is closely described by Gaussian distribution as shown in insets of Figs. 1(a) and 1(b). Our second example corresponds to a nontrivial spectrum of ζ . We consider the correlation matrix ξ as explained above with parameters a = b = 0.5 and c = 0.05. Note that the off-diagonal blocks have small contributions to the largest eigenvalues of ξ and therefore are difficult to trace in the analysis of separated eigenvalues of the corresponding CWOE. In Fig. 2 we compare our theory with numerics for N = 384 in Fig. 2(a) and for N = 256 in Fig. 2(b). As shown in the figure, even small correlations in ξAB cause notable changes in the density which are described well by our theory. V. CONCLUSION
In conclusion, we have studied a Wishart model for the nonsymmetric correlation matrices where the two constituting matrices are not statistically independent, incorporating thereby actual correlations in the theory. We have derived a Pastur self-consistent equation which describes the spectral density of this model. Our result is valid for large matrices. We have supplemented some numerical examples to demonstrate the result. A couple of interesting analytic problems for this model worth pursuing in the future are, viz., (i) to obtain results for spectral density of finite dimensional matrices and (ii) to obtain results for the two-point function and higher order spectral correlations. However, in both the problems calculation of the joint probability density for all the eigenvalues could be a starting point but it seems formidable because of some technicalities. On the other hand, for the unitary invariant ensembles the first problem could be solvable using the techniques of [42,43]. Besides, in the view of success of the supersymmetric method for CWOE [25,26] the first problem seems to be solvable. Moreover, the binary correlation method which has been used to obtain asymptotic result for the two-point function of CWOE [24] could be an effective tool to derive the same for this model. Finally, we believe that given the plenitude of the applications of RMT, these analytic result may not be confined only to time series analysis but in the other fields as well [35,36,44]. ACKNOWLEDGMENTS
The author of this paper is thankful to Thomas H. Seligman for discussions and encouragement. In particular, the author is thankful to F. Leyvraz for useful and illuminating discussions in the course of this work and for a careful reading
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of the manuscript. The author acknowledges referees for invaluable suggestions. Financial support from CONACyT through Projects No. 154586 and No. PAPIIT UNAM RR 113311 is acknowledged. The author was supported by DGAPA/UNAM through a postdoctoral fellowship. APPENDIX A: UPPER BOUND OF THE SINGULAR VALUES OF η
Since ξ is a positive definite matrix, the matrix X which results from the decorrelations, defined in Sec. II is also a positive definite matrix. In the following we show that the positive definiteness of X, and therefore of ξ , ensures an upper bound of the singular values of η. The matrix X is given by 1 η . (A1) X= ηt 1 Consider an (N + M) × (N + M) dimensional orthogonal matrix, O, composed of two orthogonal matrices O1 and O2 of dimensions N × N and M × M, respectively, defined as O1 0 , (A2) O= 0 O2 and = S, where S is a rectangular N × M dimensional diagonal matrix: Sj r = δj r sj and the sj ’s are the singular values of η. Then 1 S t OXO = . (A3) St 1 O1 η Ot2
Since X is a positive definite matrix, OXOt is also a positive definite matrix. We use Sylvester’s criterion [45], which states that a real symmetric matrix is positive definite if and only if all the leading principal minors of the matrix are positive. This criterion, for OXOt , leads to n number of inequalities where n = min{N,M}. For instance, for N < M we have N−j N inequalities, viz., k=1 (1 − sk2 ) > 0 for j = 0, . . . ,N − 1. These inequalities hold together if sk < 1, for all 1 k N , giving thereby an upper bound 1 due to the positive definiteness of OXOt and therefore due to the positive definiteness of X. APPENDIX B: DERIVATION OF THE RESULT (21)
We prefer to calculate a more general quantity GL , defined as
GL (z) = L
1 z1N×N − C
(B1)
.
Here L is an arbitrary but nonrandom N × N matrix. What follows from the identities (15)–(20) is that the binary associations of A only with At or with Bt give leading order terms; otherwise, O(N −1 ) or lower order terms. Therefore, in the expansion (14), we calculate terms arising only from the binary associations described below,
× L ABt BAt Cn ABt BAt Cp−n−2 + L ABt BAt Cn ABt BAt Cp−n−2 ,
where underbrackets are used to describe binary associations. Here we avoid terms due to binary associations of A with A or with B since the former are of O(N −1 ), because of the identity (16), and the latter vanish on ensemble averaging. The binary associations we consider in (B2) then yield a leading order equality. Finally, using identities (15) and (19) we get GL (z) =
L g L (z) + {1 + κN [z G(z) − 1]} z z ∞ κM Dn BBt + GL (z) + κM GL (z) , z T zn+1 n=1
L z−p−1 + L ABt BAt Cp−1 2 z T p=1 + L AB BA t
t
Cp−1
+
g L (z) =
∞ L ηBAt Cp p=0
T zp+1
(B4)
.
It should be mentioned that the angular brackets we are using for the spectral averaging are in accordance with the dimensionality of the matrices under trace operation. For instance the spectral average in the term, involving D, of Eq. (B3) is calculated over an M × M matrix. Binary associations across the traces, in intermediate steps from (B2) to (B3), are also ignored as they produce lower order terms; see identities (17) and (18). To calculate the summation in Eq. (B3) we consider binary associations similar to those in Eq. (B2), but for B. We obtain zκM
∞ Dn BBt n=1
T zn+1
= (κN [z G(z) − 1]{1 + κN [z G(z) − 1] + κM } + κN κM g(z))[1 − κN g(z)]−1 . (B5) In the derivation of the above equation we have used the resolvent G(z) defined for D as G(z) = (z1M×M − D)−1 . Since D and C have the same nonzero spectrum, G(z) can be easily given in terms of G(z), as κN G(z) − z−1 = [G(z) − z−1 ]. (B6) κM Finally, we have used the fact that for a square matrix the trace remains the same for its transpose. However, we still remain with g L (z). Considering again binary associations of B, in (B4), we find g L (z) = GLζ (z) {1 + κN [z G(z) − 1]} + κN g L (z)
∞ AAt Cn n=0
T zn+1
.
(B7)
Using binary associations of A, the leftover summation is computed to be
p−2 −p−1 ∞ z p=2 n=0
(B3)
where we have used definition (8) to write the last term in the right-hand side and
∞
GL (z) =
(B2)
∞ AAt Cn
T4
n=0
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T
zn+1
=
G(z){1 + κN [z G(z) − 1]} . 1 − κN g(z)
(B8)
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PHYSICAL REVIEW E 88, 042130 (2013)
It readily gives g L in terms of g(z) and G(z) as g L (z,G(z)) =
GLζ (z){1 + κN [z G(z) − 1]} 1−
κN G(z){1+κN [z G(z)−1]} 1−κN g(z,G(z))
.
(B9)
We can now rewrite Eq. (B3) in a closed form. For instance, we write
ζ Y 1 (z,G(z)) − Y 2 (z,G(z))]−1 , in Eq. (B10), we obtain
1 . (B11) GL (z) = L z − ζ Y 1 (z,G(z)) − Y 2 (z,G(z)) For L = 1N×N the above equation gives the central result (21) of the paper. Using Eq. (B9), for L = 1N×N , and definition (22), we write g(z,G(z)) as
zGL (z) = L + GLζ (z) Y 1 (z,G(z)) + GL (z) Y 2 (z,G(z)), (B10)
g(z,G(z)) =
Gζ Y 1 (z,G(z)) (z) 1 + κN [z G(z) − 1]
.
(B12)
where Y 1 (z,G(z)) and Y 2 (z,G(z)) are defined, respectively, in Eq. (22) and Eq. (23). Substituting L → L[z −
For L = ζ Y 1 (z,G(z)), it is straightforward to deduce definition (25) from Eq. (B11).
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[23] Z. Burda, J. Jurkiewicz, and B. Waclaw, Phys. Rev. E 71, 026111 (2005). [24] Vinayak and A. Pandey, Phys. Rev. E 81, 036202 (2010). [25] C. Recher, M. Kieburg, and T. Guhr, Phys. Rev. Lett. 105, 244101 (2010). [26] C. Recher, M. Kieburg, T. Guhr, and M. R. Zirnbauer, J. Stat. Phys. 148, 981 (2012). [27] M. Potters, J. P. Bouchaud, and L. Laloux, Acta Phys. Pol. B 36, 2767 (2005). [28] I. M. Johnstone, Ann. Stat. 29, 295 (2001). [29] J. P. Bouchaud and M. Potters, arXiv:0910.1205. [30] J.-P. Bouchaud, L. Laloux, M. A. Miceli, and M. Potters, Eur. Phys. J. 55, 201 (2007). [31] D. Wang, B. Podobnik, D. Horvati´c, and H. E. Stanley, Phys. Rev. E 83, 046121 (2011). [32] G. Livan and L. Rebecchi, Eur. Phys. J. B 85, 213 (2012). [33] R. R. M¨uller, Acta Phys. Pol. B 36, 1001 (2005). [34] Z. Burda, A. Jarosz, G. Livan, M. A. Nowak, and A. Swiech, Phys. Rev. E 82, 061114 (2010). [35] T. Kanazawa, T. Wettig, and N. Yamamoto, Phys. Rev. D 81, 081701 (2010). [36] G. Akemann, M. J. Phillips and H.-J. Sommers, J. Phys. A: Math. Theor. 43, 085211 (2010). [37] C. Biely and S. Thurner, Quantum Financ. 8, 705 (2008). [38] G. Akemann, J. R. Ipsen, and M. Kieburg, arXiv:1307.7560. [39] A. Pandey, Ann. Phys. (NY) 134, 110 (1981). [40] L. A. Pastur, Theoret. and Math. Phys. II 10, 67 (1972). [41] It is understood that in both cases the bulk density is normalized to 1 − N −1 . [42] S. H. Simon and A. L. Moustakas, Phys. Rev. E 69, 065101(R) (2004). [43] J. Baik et al., Ann. Prob. 33, 1643 (2005); S. P´ech´e, J. Multivariate Anal. 97, 874 (2006). ˇ [44] Vinayak and M. Znidariˇ c, J. Phys. A: Math. Theor. 45, 125204 (2012). [45] G. T. Gilbert, Am. Math. Mon. 98, 44 (1991).
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Spectral density of a Wishart model for nonsymmetric correlation matrices Vinayak* Instituto de Ciencias F´ısicas, Universidad Nacional Aut´onoma de M´exico, C.P. 62210 Cuernavaca, M´exico (Received 11 June 2013; revised manuscript received 20 August 2013; published 17 October 2013) The Wishart model for real symmetric correlation matrices is defined as W = AAt , where matrix A is usually a rectangular Gaussian random matrix and At is the transpose of A. Analogously, for nonsymmetric correlation matrices, a model may be defined for two statistically equivalent but different matrices A and B as ABt . The corresponding Wishart model, thus, is defined as C = ABt BAt . We study the spectral density of C for the case when A and B are not statistically independent. The ensemble average of such nonsymmetric matrices, therefore, does not simply vanishes to a null matrix. In this paper we derive a Pastur self-consistent equation which describes spectral density of C at large matrix dimension. We complement our analytic results with numerics. DOI: 10.1103/PhysRevE.88.042130
PACS number(s): 02.50.Sk, 05.45.Tp, 89.90.+n, 89.65.Gh
I. INTRODUCTION
Correlation matrices constitute a fundamental tool for multivariate time series analysis [1,2]. Wishart introduced random matrices of W = AAt /T type as a model for real symmetric correlation matrices [3]. In this model A is usually a rectangular matrix of dimension N × T , At is the transpose of A, and the matrix entries Aj ν are independent Gaussian variables with zero mean and a fixed variance. In due course this model gained much attention from various branches of science and now is applied to a vast domain including mathematical statistics [1,2], physics [4–7], communication engineering [8], econophysics [9–14], biological sciences [15,16], atmospheric science [17], etc. In random matrix theory (RMT) [4] the ensemble of such non-negative matrices is known as Wishart orthogonal ensemble (WOEs) or Laguerre orthogonal ensemble [18,19] where analytical results for spectral statistics are known in great detail. WOE characterizes a null hypothesis for symmetric correlation matrices where the spectral statistics set a benchmark or a reference against which any useful actual correlation must be viewed. For instance, the spectral density of WOE [20], has proved to be remarkably useful for identifying underlying actual correlations. Prime examples thereof are found in econophysics [9–11]. However, there have been important advances incorporating the actual correlations in random matrix ensembles. These ensembles are often referred to as the correlated Wishart ensembles [21–26]; the generalization for the WOE case is called the correlated Wishart orthogonal ensembles (CWOE). CWOE results are also important because these supply a reference against which correlations lying off the trend must be viewed [27]. In recent years there has been a growing interest in the analysis of nonsymmetric correlation matrices [28,29]. A nonsymmetric correlation matrix can be realized for timelagged correlations among the variables representing the same statistical system [30,31]. In a more general case it can be the matrix representing correlations among the variables of two different statistical systems [32]. The corresponding random matrix model which describes a null hypothesis for nonsymmetric correlation matrices is defined for two
*
[email protected]
1539-3755/2013/88(4)/042130(6)
statistically equivalent but independent matrices, e.g., AB = ABt /T , where A and B are rectangular matrices of dimensions N × T and M × T , respectively, and entries of both matrices are the Gaussian variables with mean zero and variance one. For rectangular AB , i.e., for N = M, a reference against which actual correlations have to be viewed is the statistics of singular values of AB [30,33,34] or equivalently the statistics of eigenvalues of a corresponding Wishart model of an N × N matrix, defined as C = ABt BAt /T 2 . On the other hand, for square AB , ample results are known for the statistics of complex eigenvalues [34–36], which could be useful for the spectral studies of correlation matrices such as the eigenvalue density used in Refs. [32,37]. Following the CWOE approach we generalize the Wishart model for nonsymmetric correlation matrices to the case where A and B are not statistically independent. In other words, the ensemble average ABt /T = η, where bar denotes the ensemble averaging and η is the N × M correlation matrix which defines the actual correlations between N input variables with M output variables. The joint probability density of the matrix elements of A and B can be described via −1 1 η A t t P (A,B) ∝ exp −tr (1) (A B ) , B ηt 1 where 1 is an identity matrix of dimensions N × N in the upper diagonal block and M × M in the lower diagonal block. In this paper we derive a self-consistent Pastur equation which describes the spectral density for C at large matrix dimension where η = 0. In the next section we define nonsymmetric correlation matrices from the CWOE approach, fix notations and describe some generalities of the work. In Sec. III, we state the main result of the paper. Details of the derivation of the result are given in Appendix B. In Sec. IV we present some numerical examples to complement analytics. In Sec. V we summarize our results. II. NONSYMMETRIC CORRELATION MATRICES: A CWOE PERSPECTIVE
CWOE is an ensemble of real symmetric matrices of type C = WW t /T , where W = ξ 1/2 W, ξ is a positive definite nonrandom matrix, and entries of the matrix W are independent 042130-1
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Gaussian variables with mean zero and variance one. Thus, C = ξ.
(2)
Spectral statistics of CWOE have been addressed by several authors. For example, the spectral density for large matrices has been derived in Refs. [20–24] and recently for finite dimensional matrices in Refs. [25,26]. Unlike WOE spectra, CWOE spectra may exhibit nonuniversal spectral statistics as noted in Ref. [24]. Suppose the matrix W constituted by two different random matrices A and B, as A W= , (3) B where A and B are of dimensions N × T and M × T , respectively. Then the matrix C is a partitioned matrix, defined in terms of A and B, as 1 AAt AB t , (4) C= T BAt BBt . Here the diagonal blocks, viz., AAt and BBt , and the offdiagonal blocks, viz., AB t and BAt , are, respectively, N × N , M × M, N × M, and M × N dimensional. Therefore, ξ is also partitioned, ξAA ξAB ξ= , (5) ξBA ξBB where the diagonal blocks ξAA and ξBB account for the correlations among the variables of A and of B, respectively. The off-diagonal blocks, e.g., AB t /T = ξAB , account for the correlations of A and B. By construction ξBA = [ξAB ]t . Without loss of generality we consider M N and T M. We consider the case where ξAB = 0 and wish to compare the spectral density with that of the null hypothesis, i.e., when ξAA = 1N×N , ξBB = 1M×M , and ξAB = 0. It is therefore important to remove cross correlations among the variables of individual matrices because only then will the diagonal blocks of (4) yield identity matrices on the ensemble averaging. Thus, we introduce decorrelated matrices [24] defined as −1/2
−1/2
A = ξAA A, B = ξBB B. −1/2
ABt BAt . T2 We further define the M × M symmetric matrix D, D=
BAt ABt , T2
ζ = ηηt .
−1/2
(7)
(8)
III. SPECTRAL DENSITY FOR LARGE MATRICES
To obtain the spectral density, ρ(λ), of C we closely follow the binary correlation method developed by French and his collaborators [5,39] and used in Ref. [24] to study the spectral statistics of CWOE. In this method we deal with the resolvent or the Stieltjes transform of the density, defined for a complex variable z as
1 G(z) = . (12) z1N×N − C N We use angular brackets to represent the spectral averaging, e.g., H k = k −1 tr H , and bars to denote averaging over the ensemble. Below we use a bar also to represent functions of the ensemble averaged scalar quantities. The ensemble averaged spectral density, can be determined via 1 ρ(λ) = ∓ ImG(λ ± i ), (13) π for infinitesimal > 0. For large z, the resolvent may be expressed in terms of moments, mp , of the density, as G(z) =
N , T M . = T
κM
∞ Cp N p=0
zp+1
=
∞ mp . p+1 z p=0
(14)
The remaining task now is to obtain a closed form of this summation. To obtain a closed form, in the above expansion we consider only those binary associations yielding leading order terms and avoid those resulting in terms of O(N −1 ). This method finally gives the so-called Pastur self-consistent equation for the resolvent [5,24,39], or the Pastur density [40], which holds for large matrices. In order to do the ensemble averaging we use the following exact results, valid for arbitrary fixed matrices and : 1 A At N = T N , (15) T A A N = t N ,
and the ratios, κN =
(11)
However, in the following we consider a large N limit, where N/T and M/T are finite so that the matrices C and D will never be deterministic. We consider only those cases where the spectrum of ζ does not exceed N . This is always valid for our model because the positive definiteness of ξ ensures an upper bound 1 for eigenvalues of ζ . For the completeness of the paper we prove this remark in Appendix A.
(6)
Note that we still have ABt /T = η, where η = ξAA ξAB ξBB and the null hypothesis is characterized for η = 0. This case has been studied by several authors [30,33,34,38]. In this paper we consider η = 0 and calculate the ensemble averaged spectral density of N × N symmetric matrices C, defined as C=
A few remarks are immediate. At first we note that the ensemble averages yield C = κM 1N×N + ηηt and D = κN 1M×M + ηt η. By construction it is obvious that the nonzero eigenvalues of D are identical to those of C. Next, for T → ∞ and N,M finite, since C = ξ , C = ηηt and D = ηt η. We finally define a symmetric matrix ζ as
(9) (10) 042130-2
1 N , N 1 A N A N = t N . N
A N At N =
(16) (17) (18)
SPECTRAL DENSITY OF A WISHART MODEL FOR . . .
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The dimensions of and are suitably adjusted in the above identities. Similar results can be written for the averaging over B. These are the same results as obtained for CWOE in Ref. [24]. However, here we have to take account of η, which gives two more important identities, viz., 1 A Bt N = T η N , (19) T 1 B At M = T ηt M . (20) T In this section we omit the detail computation of G(z), merely stating here the central result of the paper. We provide step by step details of the derivation in Appendix B. A compact result for the self-consistent Pastur equation can be written as −1
G(z) = [z − ζ Y 1 (z,G(z)) − Y 2 (z,G(z))] .
(21)
diagonal blocks are defined by equal-cross-correlation matrix model, e.g., [ξAA ]j k = δj k + (1 − δj k )a, for 1 j,k N , and the same for ξBB , where the correlation coefficient is b. In both examples we consider 0 < a,b,c < 1. This choice is necessary but not sufficient for the positive definiteness of ξ ; see Eq. (28). The spectrum of the equal-cross-correlation matrix is easy to calculate. For instance, the spectrum of ξAA is described by 1 − a and N a + 1 − a, where the former has degeneracy N − 1. The spectrum of ξBB is also described in the same way but for M and b. Moreover, the inverse of the square root of these matrices, those we need to define η, are also not difficult to calculate. For example, we simply have
−1/2 ξAA j k = δj k [(1 − a)−1/2 ] −
Here Y 1 (z,G(z)) =
{1 + κN [z G(z) − 1]} 1−
Y 2 (z,G(z)) =
2
κN G(z){1+κN [z G(z)−1]} 1−κN g(z,G(z))
Y (z,G(z)) 1 − κN g(z,G(z))
,
(22)
(23)
,
Y (z,G(z)) = κM + κN [z G(z) − 1] × {1 + κM + κN [z G(z) − 1]},
(a)
(24) 8
1 + κN [z G(z) − 1]
.
0.02
6
(25)
0
ρ(λ)
[z − Y 2 (z,G(z))]G(z) − 1
(26)
For the first case the spectrum of ζ can be determined analytically because of a trivial choice of ξAB . Yet eigenvalues, (ξ ) λj , of ξ may not be as trivial to obtain. However, in this case we find N − 1 eigenvalues 1 − a, M − 1 eigenvalues 1 − b,
and g(z,G(z)) =
1 [(1 − a)−1/2 − (N a + 1 − a)−1/2 ]. N
Equation (21), together with definitions (22)–(25), is the main result of this paper. This result is analogous to the result for CWOE which has been obtained first by Mar´cenko and Pastur [20] and then by others using different techniques [21,22,24]. For the uncorrelated case, i.e., for ζ = 0, Eq. (21) results in a cubic equation confirming thereby the result obtained in Ref. [34].
0.8
1
1.2
4
data Theory
2 0 6
IV. NUMERICAL EXAMPLES AND VERIFICATION OF THE RESULT (21)
(b)
0.04 0.02
A numerical technique has been developed in Ref. [24] for solving the Pastur equation which describes the density for CWOE. We use the same technique to solve our result (21). However, in our case the result is complicated and needs some treatment for meeting requirements of the numerical technique. We first note that G depends on Y 1 and Y 2 , while both the latter quantities depend on g and G. Since the g itself depends on G and Y 2 , at least one quantity G or g, has to be determined explicitly in terms of the other. It turns out that, using Eqs. (25) and (23), we can eliminate Y 2 from g. Therefore, for a given z, g, and in turn Y 1 and Y 2 , can be estimated using an initial guess of G. After resolving these we can use the numerical technique [24] to obtain the solution of (21). We demonstrate our result for two different correlation matrices. In the first example, we consider ξAB to be a rank 1 matrix, e.g., [ξAB ]j r = c for every integer 1 j N and 1 r M. In the second example we consider [ξAB ]j r = c δj r + (1 − δj r )c|j −r| . For simplicity, we consider that the
ρ(λ)
4 0
0.8
1
1.2
2
0
0
0.2
λ
0.4
FIG. 1. Spectral density, ρ(λ), where [ξAB ]j r = c for 1 j N and 1 r M, c = 0.8, and correlation coefficients of the equalcross-correlation matrices describing the diagonal blocks are a = b = 0.5. Symbols in the figure represent Monte Carlo simulations and solid lines are the theory obtained from the numerical solution of Eq. (21). In (a) we show results for N = 384 and in (b) we show results for N = 256. Dimension of the full matrix in both the figures is 1024 and T = 5120. In insets we show distribution of the separated eigenvalues where we have considered an ensemble of 10 000 matrices. Dashed lines in insets represent Gaussian distributions where the means and the variances have been calculated numerically.
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and the remaining two are given by (ξAA ) (ξBB ) (ξ ) (ξ ) 2 λ λNAA − λMBB + 4N Mc2 + λ ± N M (ξ ) , λ± = 2 (27) (ξ
)
(ξ
)
where λNAA = N a + 1 − a and λMBB = Mb + 1 − b. Note that the positive definiteness of ξ is ensured if (ξAA ) (ξBB ) λN λM > NMc2 .
(28)
The matrix ξAB is rank 1; so is η: c ηj r = . (ξAA ) (ξBB ) λN λM
(29) (ξ
) ξ
)
Using this we readily obtain ζj k = Mc2 /λNAA λM(BB and the ) (ξAA ) (ξBB ) 2 only nonzero eigenvalue, λ(ζ λM . It trivially N = N Mc /λN (ζ ) follows from the inequality (28) that λN < 1. In fact, due the positive definiteness of ξ it holds for more general cases as proved in Appendix A. To check our theoretical result (21) with numerics we begin with simulating C defined in the beginning of Sec. II. Next, we identify the off-diagonal block {AB} in C. Finally, we use −1/2 −1/2 the transformation ξAA AB t ξBB to obtain ABt , which we desire to calculate C. In numerical simulations we fix N + M = 1024, T = 5(N + M) and consider two values of N , (a)
8
Theory Data ζ=0
ρ(λ)
6 4 2 0 5
(b)
ρ(λ)
4 3 2 1 0
0
0.1
0.2
0.3
λ
0.4
0.5
FIG. 2. (Color online) Spectral density, ρ(λ), for the second example, where [ξAB ]j r = c δj r + (1 − δj r )c|j −r| , for 1 j N and 1 r M, c = 0.05, and the correlation coefficients which describe the diagonal blocks are a = b = 0.5. With an outlay similar to Fig. 1 we compare our theory with numerics. Color lines in this figure represent the uncorrelated case.
viz., N = 256 and 384. To compare numerics with theory we consider an ensemble of size 1000 of matrices C. In our first example, ζ has only one nonzero eigenvalue. For this spectrum our theory (21) yields the density for the bulk of the spectra. It suggests that the bulk should be described by density of the uncorrelated case. We verify this with numerics in Fig. 1, where a = b = 0.9 and c = 0.8. However, like the equal-cross-correlation matrix model of CWOE [24], in this case as well, we obtain one eigenvalue separated from the bulk [41]. Interestingly, here the bulk remains invariant with correlations as opposed to the CWOE case. Moreover, distribution of the separated eigenvalues is closely described by Gaussian distribution as shown in insets of Figs. 1(a) and 1(b). Our second example corresponds to a nontrivial spectrum of ζ . We consider the correlation matrix ξ as explained above with parameters a = b = 0.5 and c = 0.05. Note that the off-diagonal blocks have small contributions to the largest eigenvalues of ξ and therefore are difficult to trace in the analysis of separated eigenvalues of the corresponding CWOE. In Fig. 2 we compare our theory with numerics for N = 384 in Fig. 2(a) and for N = 256 in Fig. 2(b). As shown in the figure, even small correlations in ξAB cause notable changes in the density which are described well by our theory. V. CONCLUSION
In conclusion, we have studied a Wishart model for the nonsymmetric correlation matrices where the two constituting matrices are not statistically independent, incorporating thereby actual correlations in the theory. We have derived a Pastur self-consistent equation which describes the spectral density of this model. Our result is valid for large matrices. We have supplemented some numerical examples to demonstrate the result. A couple of interesting analytic problems for this model worth pursuing in the future are, viz., (i) to obtain results for spectral density of finite dimensional matrices and (ii) to obtain results for the two-point function and higher order spectral correlations. However, in both the problems calculation of the joint probability density for all the eigenvalues could be a starting point but it seems formidable because of some technicalities. On the other hand, for the unitary invariant ensembles the first problem could be solvable using the techniques of [42,43]. Besides, in the view of success of the supersymmetric method for CWOE [25,26] the first problem seems to be solvable. Moreover, the binary correlation method which has been used to obtain asymptotic result for the two-point function of CWOE [24] could be an effective tool to derive the same for this model. Finally, we believe that given the plenitude of the applications of RMT, these analytic result may not be confined only to time series analysis but in the other fields as well [35,36,44]. ACKNOWLEDGMENTS
The author of this paper is thankful to Thomas H. Seligman for discussions and encouragement. In particular, the author is thankful to F. Leyvraz for useful and illuminating discussions in the course of this work and for a careful reading
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of the manuscript. The author acknowledges referees for invaluable suggestions. Financial support from CONACyT through Projects No. 154586 and No. PAPIIT UNAM RR 113311 is acknowledged. The author was supported by DGAPA/UNAM through a postdoctoral fellowship. APPENDIX A: UPPER BOUND OF THE SINGULAR VALUES OF η
Since ξ is a positive definite matrix, the matrix X which results from the decorrelations, defined in Sec. II is also a positive definite matrix. In the following we show that the positive definiteness of X, and therefore of ξ , ensures an upper bound of the singular values of η. The matrix X is given by 1 η . (A1) X= ηt 1 Consider an (N + M) × (N + M) dimensional orthogonal matrix, O, composed of two orthogonal matrices O1 and O2 of dimensions N × N and M × M, respectively, defined as O1 0 , (A2) O= 0 O2 and = S, where S is a rectangular N × M dimensional diagonal matrix: Sj r = δj r sj and the sj ’s are the singular values of η. Then 1 S t OXO = . (A3) St 1 O1 η Ot2
Since X is a positive definite matrix, OXOt is also a positive definite matrix. We use Sylvester’s criterion [45], which states that a real symmetric matrix is positive definite if and only if all the leading principal minors of the matrix are positive. This criterion, for OXOt , leads to n number of inequalities where n = min{N,M}. For instance, for N < M we have N−j N inequalities, viz., k=1 (1 − sk2 ) > 0 for j = 0, . . . ,N − 1. These inequalities hold together if sk < 1, for all 1 k N , giving thereby an upper bound 1 due to the positive definiteness of OXOt and therefore due to the positive definiteness of X. APPENDIX B: DERIVATION OF THE RESULT (21)
We prefer to calculate a more general quantity GL , defined as
GL (z) = L
1 z1N×N − C
(B1)
.
Here L is an arbitrary but nonrandom N × N matrix. What follows from the identities (15)–(20) is that the binary associations of A only with At or with Bt give leading order terms; otherwise, O(N −1 ) or lower order terms. Therefore, in the expansion (14), we calculate terms arising only from the binary associations described below,
× L ABt BAt Cn ABt BAt Cp−n−2 + L ABt BAt Cn ABt BAt Cp−n−2 ,
where underbrackets are used to describe binary associations. Here we avoid terms due to binary associations of A with A or with B since the former are of O(N −1 ), because of the identity (16), and the latter vanish on ensemble averaging. The binary associations we consider in (B2) then yield a leading order equality. Finally, using identities (15) and (19) we get GL (z) =
L g L (z) + {1 + κN [z G(z) − 1]} z z ∞ κM Dn BBt + GL (z) + κM GL (z) , z T zn+1 n=1
L z−p−1 + L ABt BAt Cp−1 2 z T p=1 + L AB BA t
t
Cp−1
+
g L (z) =
∞ L ηBAt Cp p=0
T zp+1
(B4)
.
It should be mentioned that the angular brackets we are using for the spectral averaging are in accordance with the dimensionality of the matrices under trace operation. For instance the spectral average in the term, involving D, of Eq. (B3) is calculated over an M × M matrix. Binary associations across the traces, in intermediate steps from (B2) to (B3), are also ignored as they produce lower order terms; see identities (17) and (18). To calculate the summation in Eq. (B3) we consider binary associations similar to those in Eq. (B2), but for B. We obtain zκM
∞ Dn BBt n=1
T zn+1
= (κN [z G(z) − 1]{1 + κN [z G(z) − 1] + κM } + κN κM g(z))[1 − κN g(z)]−1 . (B5) In the derivation of the above equation we have used the resolvent G(z) defined for D as G(z) = (z1M×M − D)−1 . Since D and C have the same nonzero spectrum, G(z) can be easily given in terms of G(z), as κN G(z) − z−1 = [G(z) − z−1 ]. (B6) κM Finally, we have used the fact that for a square matrix the trace remains the same for its transpose. However, we still remain with g L (z). Considering again binary associations of B, in (B4), we find g L (z) = GLζ (z) {1 + κN [z G(z) − 1]} + κN g L (z)
∞ AAt Cn n=0
T zn+1
.
(B7)
Using binary associations of A, the leftover summation is computed to be
p−2 −p−1 ∞ z p=2 n=0
(B3)
where we have used definition (8) to write the last term in the right-hand side and
∞
GL (z) =
(B2)
∞ AAt Cn
T4
n=0
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T
zn+1
=
G(z){1 + κN [z G(z) − 1]} . 1 − κN g(z)
(B8)
VINAYAK
PHYSICAL REVIEW E 88, 042130 (2013)
It readily gives g L in terms of g(z) and G(z) as g L (z,G(z)) =
GLζ (z){1 + κN [z G(z) − 1]} 1−
κN G(z){1+κN [z G(z)−1]} 1−κN g(z,G(z))
.
(B9)
We can now rewrite Eq. (B3) in a closed form. For instance, we write
ζ Y 1 (z,G(z)) − Y 2 (z,G(z))]−1 , in Eq. (B10), we obtain
1 . (B11) GL (z) = L z − ζ Y 1 (z,G(z)) − Y 2 (z,G(z)) For L = 1N×N the above equation gives the central result (21) of the paper. Using Eq. (B9), for L = 1N×N , and definition (22), we write g(z,G(z)) as
zGL (z) = L + GLζ (z) Y 1 (z,G(z)) + GL (z) Y 2 (z,G(z)), (B10)
g(z,G(z)) =
Gζ Y 1 (z,G(z)) (z) 1 + κN [z G(z) − 1]
.
(B12)
where Y 1 (z,G(z)) and Y 2 (z,G(z)) are defined, respectively, in Eq. (22) and Eq. (23). Substituting L → L[z −
For L = ζ Y 1 (z,G(z)), it is straightforward to deduce definition (25) from Eq. (B11).
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