Spectral density of a Wishart model for nonsymmetric Correlation Matrices
aa r X i v : . [ m a t h - ph ] A ug 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 (Dated: September 20, 2018)The Wishart model for real symmetric correlation matrices is defined as W = AA t , where matrix A is usually a rectangular Gaussian random matrix and A t is the transpose of A . Analogously,for nonsymmetric correlation matrices, a model may be defined for two statistically equivalentbut different matrices A and B as AB t . The corresponding Wishart model, thus, is defined as C = AB t BA t . We study the spectral density of C for the case when A and B are not statisticallyindependent. The ensemble average of such nonsymmetric matrices, therefore, does not simplyvanishes to a null matrix. In this paper we derive a Pastur self-consistent equation which describesspectral density of large C . We complement our analytic results with numerics. PACS numbers: 02.50.Sk, 05.45.Tp, 89.90.+n
I. INTRODUCTION
Correlation matrices are a fundamental tool for mul-tivariate time series analysis [1, 2]. Wishart introducedrandom matrices of W = AA t /T type as a model forreal symmetric correlation matrices [3]. In this model A is usually a rectangular matrix of dimension N × T , A t is the transpose of A and the matrix entries A jν areindependent Gaussian variables with zero mean and afixed variance. In due course this model gained muchattention from various branches of science, and now isapplied to a vast domain including mathematical statis-tics [1, 2], physics [4–7], communication engineering [8],econophysics[9–14], biological sciences [15, 16], atmo-spheric science [17], etc. In random matrix theory (RMT)[4] ensembles of such nonnegative matrices are known asWishart orthogonal ensembles (WOE) or Laguerre or-thogonal ensembles [18, 19] where analytical results forspectral statistics are known in great detail.WOE characterizes a null hypothesis for symmetriccorrelation matrices where the spectral statistics set abenchmark or a reference against which any useful actual correlation must be viewed. For instance, the spectraldensity of WOE [20], has proved to be remarkably use-ful for identifying underlying actual correlations. Primeexamples thereof are found in econophysics [9–11]. How-ever, there have been important advances incorporatingactual correlations in random matrix ensembles. Theseensembles are often referred to as the correlated Wishartensembles [22–26]; the generalization for the WOE caseis the correlated Wishart orthogonal ensemble (CWOE).CWOE results are also important because these supplya reference against which the correlations lying off thetrend must be viewed [27]In recent years there has been a growing interest in theanalysis of nonsymmetric correlation matrices [28, 29].A nonsymmetric correlation matrix can be realized for ∗ Electronic address: [email protected] time-lagged correlations among the variables represent-ing the same statistical system [30, 31]. In a more generalcase it can be the matrix representing correlations amongthe variables of two different statistical systems [32]. Thecorresponding random matrix model which describes anull hypothesis for nonsymmetric correlation matrices isdefined for two statistically equivalent but independentmatrices, e.g., Σ AB = AB t /T where A and B are rectan-gular matrices of dimensions N × T and M × T respec-tively, and entries of both the matrices are the Gaussianvariables with mean zero and variance one. For rectan-gular Σ AB , i.e., for N = M , a reference against whichthe actual correlations have to be viewed is the statis-tics of singular values of Σ AB [30, 33, 34] or equivalentlythe statistics of eigenvalues of a corresponding Wishartmodel of N × N matrices, defined as C = AB t BA t /T .On the other hand, for square Σ AB , ample amount of re-sults are known for the statistics of complex eigenvalues[34–36] which could be useful for the spectral studies ofcorrelation matrices such as the eigenvalue density usedin [32].Following the CWOE approach we generalize theWishart model for nonsymmetric correlation matrices tothe case where A and B are not statistically indepen-dent. In other words, the ensemble average AB t /T = η where bar denotes the ensemble averaging and η is an N × M correlation matrix which defines the nonrandomcorrelations between N input variables with M outputvariables. The joint probability density of the matrixelements A and B can be described as P ( A , B ) ∝ exp " − tr ((cid:18) ηη t (cid:19) − (cid:18) AB (cid:19) (cid:0) A t B t (cid:1)) , (1)where is an identity matrix of dimensions N × N in theupper diagonal block and M × M in the lower diagonalblock. In this paper we derive a self-consistent Pasturequation which describes the spectral density for large C where η = 0.In the next section we define the nonsymmetric corre-lation matrices from CWOE approach, fix notations anddescribe some generalities of the work. In section III, westate the main result of the paper. Details of the deriva-tion of the result is given in Appendix B. In Sec. IVwe present some numerical examples to complement ourresult. This is followed by conclusion. II. NONSYMMETRIC CORRELATIONMATRICES: A CWOE PERSPECTIVE
CWOE is an ensemble of real symmetric matrices oftype C = WW t /T where W = ξ / W , ξ is a positivedefinite nonrandom matrix and entries of the matrix W are independent Gaussian variables with mean zero andvariance once. Thus C = ξ. (2)Spectral statistics of CWOE have been addressed by sev-eral authors. For example, the spectral density for largematrices has been derived by [20–24] and recently for fi-nite dimensional matrices by [25, 26]. Unlike WOE spec-tra, CWOE spectra may exhibit nonuniversal spectralstatistics as noted in [24].Suppose the matrix W constitutes of two different ran-dom matrices A and B , as W = (cid:18) AB (cid:19) , (3)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 C = 1 T (cid:18) AA t AB t BA t BB t . (cid:19) , (4)Here the diagonal blocks viz., AA t and BB t , and theoff-diagonal blocks viz., AB t and BA t , are respectively N × N , M × M , N × M and M × N dimensional. Therefore ξ is also partitioned: ξ = (cid:18) ξ AA ξ AB ξ BA ξ BB (cid:19) , (5)where diagonal blocks ξ AA and ξ BB account for the cor-relations among the variables of A and of B , respectively.Off-diagonal blocks, e.g., AB t /T = ξ AB , account for thecorrelations of A and B . By construction ξ BA = [ ξ AB ] t .Without loss of generality we consider M ≥ N and T ≥ M .We consider the case where ξ AB = 0 and wish to com-pare the spectral density with that of the null hypothesis,i.e., when ξ AA = N × N , ξ BB = M × M and ξ AB = 0. Itis therefore important to remove the cross-correlationsamong the variables of individual matrices because onlythen the diagonal blocks of (4) will yield identity matriceson the ensemble averaging. Thus we introduce decorre-lated matrices [24] defined as A = ξ − / A , B = ξ − / B . (6) Note that we still have AB t /T = η where η = ξ − / ξ AB ξ − / and the null hypothesis is characterizedfor η = 0. This case has been studied by several authors[30, 33, 34]. In this paper we consider η = 0 and calculateensemble averaged spectral density of N × N symmetricmatrices C , defined as C = AB t BA t T . (7)We further define M × M symmetric matrix D , D = BA t AB t T , (8)and the ratios, κ N = NT , (9) κ M = MT . (10)A few remarks are immediate. At first we note thatthe ensemble averages yield, C = κ M N × N + ηη t and D = κ N M × M + η t η . By construction it is obvious thatthe nonzero eigenvalues of D are identical to those of C .Next, for T → ∞ , since C = ξ , C = ηη t and D = η t η .We finally define a symmetric matrix ζ , as ζ = ηη t . (11)However, in the following we consider a large N limitwhere N/T and
M/T are finite so that matrices C and D will never be deterministic. We consider only thosecases where the spectrum of ζ does not exceed N . Thisis always valid for our model because the positive defi-niteness of ξ ensures an upper bound 1 for eigenvaluesof ζ . For the completeness of the paper we prove thisremark in Appendix A. III. SPECTRAL DENSITY FOR LARGEMATRICES
To obtain the spectral density, ρ ( λ ), of C we closelyfollow the binary correlation method developed by Frenchand his collaborators [5, 37] and used in [24] to study thespectral statistics of CWOE. In this method we deal withthe resolvent or the Stieltjes transform of the density,defined for a complex variable z as G ( z ) = (cid:28) z N × N − C (cid:29) N . (12)We use the angular brackets to represent the spectral av-eraging, e.g., h H i k = k − tr H , and bar to denote averag-ing over the ensemble. Below we use bar also to representfunctions of ensemble averaged scalar quantities.The ensemble averaged spectral density, can be deter-mined via ρ ( λ ) = ∓ π ℑ G ( λ ± i ǫ ) , (13)for infinitesimal ǫ >
0. For large z , the resolvent may beexpressed in terms of moments, m p , of the density, as G ( z ) = ∞ X p =0 (cid:10) C p (cid:11) N z p +1 = ∞ X p =0 m p z p +1 . (14)The remaining task now is to obtain a closed form of thissummation. To obtain a closed form, in the above expan-sion we consider only those binary associations yieldingleading order terms and avoid those resulting in terms of O ( N − ). This method finally gives the so called Pasturself-consistent equation for the resolvent [5, 24, 37], orthe Pastur density [38], which holds for large matrices.In order to do the ensemble averaging we use the fol-lowing exact results, valid for arbitrary fixed matrices Φand Ψ: 1 T h A Φ A t Ψ i N = h Φ i T h Ψ i N , (15) h A Φ A Ψ i N = (cid:10) Φ t Ψ (cid:11) N , (16) h A Φ i N h Ψ A t i N = 1 N h ΨΦ i N , (17) h A Φ i N h A Ψ i N = 1 N (cid:10) Ψ t Φ (cid:11) N . (18)The dimensions of Φ and Ψ are suitably adjusted in theabove identities. Similar results can be written for theaveraging over B . These are the same results as obtainedfor CWOE in [24]. However, here we have to take accountof η which gives further two important identities, viz.,1 T h A Φ B t Ψ i N = h Φ i T h η Ψ i N , (19)1 T h B Φ A t Ψ i M = h Φ i T (cid:10) η t Ψ (cid:11) M . (20)In this section we omit detail computation of G ( z ),merely stating here the central result of the paper. Weprovide step by step details of the derivation in AppendixB. A compact result for the self-consistent Pastur equa-tion can be written as G ( z ) = D(cid:0) z − ζY ( z, G ( z )) − Y ( z, G ( z )) (cid:1) − E . (21)Here Y ( z, G ( z )) = (cid:2) κ N (cid:0) z G ( z ) − (cid:1)(cid:3) − κ N G ( z ) (cid:2) κ N (cid:0) z G ( z ) − (cid:1)(cid:3) − κ N g ( z, G ( z )) , (22) Y ( z, G ( z )) = Y ( z, G ( z ))1 − κ N g ( z, G ( z )) , (23) Y ( z, G ( z )) = κ M + κ N (cid:0) z G ( z ) − (cid:1) × (cid:2) κ M + κ N (cid:0) z G ( z ) − (cid:1)(cid:3) , (24) and g ( z, G ( z )) = [ z − Y ( z, G ( z ))] G ( z ) −
11 + κ N (cid:0) z G ( z ) − (cid:1) . (25)Eq. (21) together with definitions (22-25) is the main re-sult of this paper. This result is analogous to the resultfor CWOE which has been obtained first by Mar´cenkoand Pastur [20] and then by others using different tech-niques [21, 22, 24]. For the uncorrelated case, i.e., for ζ = 0, Eq. (21) results in a cubic equation confirmingthereby the result obtained in [34]. IV. NUMERICAL EXAMPLES ANDVERIFICATION OF THE RESULT (21)
A numerical technique has been developed in [24] forsolving the Pastur equation which describes the densityfor CWOE. We use the same technique to solve our result(21). However, in our case the result is complicated andneeds some treatments for meeting requirements of thenumerical technique. We first note that G depends on Y and Y while both the latter quantities depend on g and G . Since g itself depends on G and Y , at leastone quantity G or g , has to be determined explicitly interms of other. It turns out that, using Eqs. (25) and(23), we can eliminate Y from g . Therefore, for a given z , g , and in turn Y and Y , can be estimated using aninitial guess of G . After resolving these we can use thenumerical technique [24] to obtain the solution of (21).We demonstrate our result for two different correlationmatrices. In a first example we consider ξ AB to be a rankone matrix, e.g., [ ξ AB ] jr = c for every integer 1 ≤ j ≤ N and 1 ≤ r ≤ M . In a second example we consider[ ξ AB ] jr = c δ jr + (1 − δ jr ) c | j − r | . For simplicity, we con-sider that the diagonal blocks are defined by equal-cross-correlation matrix model, e.g., [ ξ AA ] jk = δ jk + (1 − δ jk ) a ,for 1 ≤ j, k ≤ N and the same for ξ BB where thecorrelation coefficient is b . In both examples we con-sider 0 < a, b, c <
1. The spectrum of an equal-cross-correlation matrix is easy to calculate. For instance, thespectrum of ξ AA is described by 1 − a and N a + 1 − a where the former has degeneracy N −
1. The spectrumof ξ BB is also described in the same way but for M and b . The inverse of the square root of these matrices, thosewe need to define η , are also not difficult to calculate.For example, we simply have[ ξ − / ] jk = δ jk [(1 − a ) − / ] − N h (1 − a ) − / − ( N a + 1 − a ) − / i . (26)For the first case the spectrum of ζ can be determinedanalytically because of a trivial choice of ξ AB . Yet eigen-values, λ ( ξ ) j , of ξ may not be as trivial to obtain. How-ever, in this case we find N − − a , M − λ ρ ( λ ) ρ ( λ ) dataTheory (a)(b) FIG. 1: Spectral density, ρ ( λ ), where [ ξ AB ] jr = c for 1 ≤ j ≤ N and 1 ≤ r ≤ M , c = 0 .
8, and correlation coefficientsof the equal cross-correlation matrices describing the diago-nal blocks are a = b = 0 .
5. Symbols in the figure representMonte Carlo simulations and solid lines are the theory ob-tained from the numerical solution of Eq. (21). In Fig. 1(a)we show results for N = 384 and in Fig. 1(b) we show resultsfor N = 256. The dimension of the full matrix in both thefigures is 1024 and T = 5120. In the inset we show distribu-tion of the separated eigenvalues where we have consideredensemble of 10000 matrices. Dashed lines in the inset repre-sent Gaussian distribution where the mean and the variancehave been calculated numerically. eigenvalues 1 − b and the remaining two are given by λ ( ξ ) ± = λ ( ξ AA ) N + λ ( ξ BB ) M ± q [ λ ( ξ AA ) N − λ ( ξ BB ) M ] + 4 N M c , (27)where λ ( ξ AA ) N = N a + 1 − a and λ ( ξ BB ) M = M b + 1 − b . Notethat the positive definiteness of ξ is ensured if λ ( ξ AA ) N λ ( ξ BB ) M > N M c . (28)The matrix ξ AB is rank one, so is η : η jr = c q λ ( ξ AA ) N λ ( ξ BB ) M (29)Using this we readily obtain ζ jk = M c /λ ( ξ AA ) N λ ξ (BB ) M andthe only nonzero eigenvalue, λ ( ζ ) N = N M c /λ ( ξ AA ) N λ ( ξ BB ) M . ρ ( λ ) TheoryData ζ=0 λ ρ ( λ ) (a)(b) FIG. 2: Spectral density, ρ ( λ ), for the second example where[ ξ AB ] jr = c δ jr +(1 − δ jr ) c | j − r | , for 1 ≤ j ≤ N and 1 ≤ r ≤ M , c = 0 .
05 and the correlation coefficients which describe thediagonal blocks are a = b = 0 .
5. With an outlay similar toFig. 1 we compare our theory with numerics. Solid red linesin this figure represent the uncorrelated case.
It trivially follows from the inequality (28) that λ ( ζ ) N < C defined in the beginning of theSec. II. Next, we identify the off-diagonal block { AB } in C . Finally we use the transformation ξ − / AB t ξ − / toobtain AB t that we desire to calculate C . In numericalsimulations we fix N + M = 1024, T = 5( N + M ) andconsider two values of N , viz., N = 256 and 384. Tocompare the numerics with the theory we consider anensemble 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 forthe bulk of the spectra. It suggests that the bulk shouldbe described by density of the uncorrelated case. We ver-ify this with numerics in Fig. 1, where a = b = 0 . c = 0 .
8. However, like the equal-cross-correlation matrixmodel of CWE [24], in this case as well, we obtain oneeigenvalue separated from the bulk [39]. Interestingly,here the bulk remains invariant with correlations as op-posed to the CWE case. Moreover, the distribution of theseparated eigenvalues is closely described by a Gaussiandistribution as shown in insets of Figs. 1(a) and 1(b).Our second example corresponds to a non-trivial spec-trum of ζ . We consider the correlation matrix ξ as ex-plained above with parameters a = b = 0 . c = 0 . ξ and therefore are dif-ficult to be traced in the analysis of separated eigenvaluesof the corresponding CWOE. In Fig. 2 we compare ourtheory with numerics for N = 384 in Fig. 2(a) and for N = 256 in Fig. 2(b). As shown in the figure, even smallcorrelations in ξ AB render notable changes in the densitywhich are described well by our theory. V. CONCLUSION
In conclusion, we have studied a Wishart model for thenonsymmetric correlation matrices where the two consti-tuting matrices are not statistically independent, incor-porating thereby actual correlations in the theory. Wehave derived a Pastur self-consistent equation which de-scribes the spectral density of this model. Our result isvalid for large matrices. We have supplemented somenumerical examples to demonstrate the result.A couple of interesting analytic problems for this modelworth persuing in future, viz., ( i ) to obtain result forspectral density of finite dimensional matrices, and ( ii )to obtain results for the two-point function and higherorder spectral correlations. However, in both the prob-lems calculation of the joint probability density for allthe eigenvalues could be a starting point but it seemsformidable because of some technicalities. On the otherhand, for the unitary invariant ensembles the first prob-lem could be solvable using the techniques of [40, 41].Besides, in the view of success of the supersymmetricmethod for CWOE [25, 26] the first problem seems tobe solvable. Moreover, the binary correlation methodwhich has been used to obtain asymptotic result for thetwo-point function of CWOE [24] could be an effectivetool to derive the same for this model. Finally, we be-lieve that given the plenitude of the applications of RMT,these analytic results may not be confined only to timeseries analysis but in other fields as well [35, 36, 42]. VI. ACKNOWLEDGMENTS
The author of this paper is thankful to Thomas H.Seligman for discussions and encouragement. In partic-ular, the author is thankful to F. Leyvraz for useful andilluminating discussions in the course of this work. Theauthor acknowledges referees for invaluable suggestions.Financial support from the project 44020 by CONA-CyT, Mexico, and project PAPIIT UNAM RR 11311,Mexico, in the course of this work is acknowledged.The author is a postdoctoral fellow supported byDGAPA/UNAM.
Appendix A: Upper bound of the singular values of η Since ξ is a positive definite matrix, the matrix X whichresults from the decorrelations, defined in Sec. II, is alsoa positive definite matrix. In the following we show thatthe positive definiteness of X , and therefore of ξ , ensuresan upper bound of the singular of η . The matrix X isgiven by X = (cid:18) ηη t (cid:19) . (A1)Consider an ( N + M ) × ( N + M ) dimensional orthogonalmatrix, O , composed of two orthogonal matrices O and O of dimensions N × N and M × M , respectively, definedas O = (cid:18) O O (cid:19) , (A2)and O η O t = S where S is a rectangular N × M dimen-sional diagonal matrix: S jr = δ jr s j and the s j ’s are thesingular values of η . Then OXO t = (cid:18) SS t (cid:19) . (A3)Since X is a positive definite matrix, therefore OXO t isalso a positive definite matrix. We use the Sylvester’scriterion [43] which states that a real symmetric matrixis positive definite iff all the leading principal minors ofthe matrix are positive. This criterion, for OXO t , leadsto n number of inequalities where n = min { N, M } . Forinstance, for N < M we have N inequalities: Q N − jk (1 − s k ) >
0, for j = 0 , ..., N −
1. These inequalities holdtogether if s j <
1, for all the j ’s, giving thereby an upperbound 1 due to the positive definiteness of OXO t andtherefore due to the positive definiteness of X . Appendix B: Derivation of the result (21)
We prefer to calculate a more general quantity G L ,defined as G L ( z ) = (cid:28) L z N × N − C (cid:29) . (B1)Here L is an arbitrary but nonrandom N × N matrix.What follows from the identities (15-20) is that the bi-nary associations of A only with A t or with B t give lead-ing order terms, otherwise O ( N − ) or lower order terms.Therefore, in the expansion (14), we calculate only thebinary associations described below. G L ( z ) = h L i z + ∞ X p =1 z − p − T n h L AB t BA t C p − i + h L AB t BA t C p − i o + ∞ X p =2 p − X n =0 z − p − T × n h L AB t BA t C n AB t BA t C p − n − i + h L AB t BA t C n AB t BA t C p − n − i o . (B2)Here we avoid terms due to binary associations of A with A or with B since the former are O ( N − ), because ofthe identity (16), and the latter vanish on ensemble aver-aging. The binary associations we consider in (B2) thenyield a leading order equality. Using the identities (15,19)we get G L ( z ) = h L i z + g L ( z ) z (cid:8) κ N (cid:0) z G ( z ) − (cid:1)(cid:9) + κ M z G L ( z ) + κ M G L ( z ) ∞ X n =1 h D n BB t i T z n +1 , (B3)where we have used definition (8) to write the last termin the right hand side and g L ( z ) = ∞ X p =0 h L η BA t C p i T z p +1 . (B4)It should be mentioned that the angular brackets we areusing for the spectral averaging are in accordance withthe 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. Bi-nary associations across the traces, in intermediate stepsfrom (B2) to (B3), are also ignored as they produce lowerorder terms; see identities (17,18).To calculate the summation in Eq. (B3) we considerthe binary associations similar to those in Eq. (B2), butfor B . We obtain zκ M ∞ X n =1 h D n BB t i T z n +1 = h κ N (cid:0) z G ( z ) − (cid:1) × (cid:2) κ N (cid:0) z G ( z ) − (cid:1) + κ M (cid:3) + κ N κ M g ( z ) i × [1 − κ N g ( z )] − . (B5)In the derivation of the above equation we haveused the resolvent G ( z ) defined for D as G ( z ) = (cid:10) ( z M × M − D ) − (cid:11) . Since D and C have the samenonzero spectrum, G ( z ) can be easily given in terms of G ( z ), as G ( z ) − z − = κ N κ M (cid:0) G ( z ) − z − (cid:1) . (B6)Finally, we have used the fact that for a square matrixthe trace remains the same for its transpose.However, we still remain with g L ( z ). Consideringagain the binary associations of B , in (B4), we find g L ( z ) = G Lζ ( z ) (cid:8) κ N (cid:0) z G ( z ) − (cid:1)(cid:9) + κ N g L ( z ) ∞ X n =0 h AA t C n i T z n +1 . (B7)Using the binary associations of A , the left over summa-tion is computed to be ∞ X n =0 h AA t C n i T z n +1 = G ( z ) (cid:2) κ N (cid:0) z G ( z ) − (cid:1)(cid:3) − κ N g ( z ) . (B8)It readily gives g L in terms of g ( z ) and G ( z ) as g L ( z, G ( z )) = G Lζ ( z ) (cid:2) κ N (cid:0) z G ( z ) − (cid:1)(cid:3) − κ N G ( z ) (cid:2) κ N (cid:0) z G ( z ) − (cid:1)(cid:3) − κ N g ( z, G ( z )) . (B9)We can now rewrite the equation (B3) in a closed form.For instance, we write zG L ( z ) = h L i + G Lζ ( z ) Y ( z, G ( z )) + G L ( z ) Y ( z, G ( z )) , (B10)where Y ( z, G ( z )) and Y ( z, G ( z )) are defined respec-tively in Eq. (22) and Eq. (23). Substituting L → L [ z − ζ Y ( z, G ( z )) − Y ( z, G ( z ))] − , in Eq. (B10), weobtain G L ( z ) = (cid:28) L z − ζY ( z, G ( z )) − Y ( z, G ( z )) (cid:29) . (B11)For L = N × N the above equation gives the central result(21) of the paper.Using Eq. (B9), for L = N × N , and definition (22),we write g ( z, G ( z )) as g ( z, G ( z )) = G ζ Y ( z,G ( z )) ( z )1 + κ N (cid:0) z G ( z ) − (cid:1) . (B12)For L = ζ Y ( z, G ( z )), it is straightforward to deduce thedefinition (25) from Eq. (B11). [1] S. S. Wilks, Mathematical Statistics (Wiley, New York,1962). [2] R. J. Muirhead,
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