Extracting meaningful information from financial data

Physica A 287 (2000) 383–395

www.elsevier.com/locate/physa

Extracting meaningful information from nancial data Milan Rajkovic Institute of Nuclear Sciences VinÄca, P.O. Box 522, Belgrade 11001, Yugoslavia Received 30 April 2000; received in revised form 9 June 2000

Abstract A method for extracting information carrying eigenvalues of the correlation matrix is presented based on the topological transformation of the manifold de ned by the data matrix itself. The transformation, performed with the use of the minimum spanning tree and the barycentric transformation, linearizes the topological manifold and the singular value decomposition is performed on the nal data matrix corresponding to the linearized hypersurface. It is shown that the results of this procedure are superior to the results of the random matrix theory as applied to the nancial data. The method may be used independently or in conjunction with the random c 2000 Elsevier Science B.V. matrix theory. Other possible uses of the method are mentioned. All rights reserved. PACS: 05.40; 02.40.−k; 02.10.s Keywords: Financial markets data; Random matrices; Minimum spanning tree; Barycentric transformation

1. Introduction Recently, a number of papers appeared in the literature pointing out the importance of random matrix theory in analyzing the empirical correlation matrices emerging in the study of multivariate nancial time series (e.g. Refs. [1,2]). In general, the results of the random matrix theory have shown to be of great value in extracting the information carrying part from the signals perturbed by an unknown amount of random noise. The problem of identifying the meaningful part of the signal is important in various other areas of statistical analysis applied to signal processing as well as to the study of nonlinear dynamical systems. In case of nancial assets, the study of correlation matrices is of particular importance due to the impact of the correlations between price E-mail address: [email protected] (M. Rajkovic). c 2000 Elsevier Science B.V. All rights reserved. 0378-4371/00/$ - see front matter PII: S 0 3 7 8 - 4 3 7 1 ( 0 0 ) 0 0 3 7 7 - 0

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changes of dierent stocks on risk management of a given stock portfolio. In particular, in the theory of the optimal portfolios, the composition of the least risky portfolio has a considerable weight on the eigenvectors of the price changes correlation matrix corresponding to the smallest eigenvalues [3]. Actually, the eigenvectors that determine the least risky portfolios are most susceptible to the in uence of noise, hence the need for a reliable and ecient methods for extracting risk controllable information from the irrelevant noise. For that purpose a random matrix method, whose development was greatly in uenced by physicists, has been recently advocated (e.g. Refs. [1,2]) and its main features are as follows. The properties of the empirical correlation matrix are compared to the purely random matrix (null hypothesis), and deviations from the random matrix case are interpreted as signs of information carrying components. Since each eigenvalue corresponds to the variance of the elements of the data matrix, including one eigenvalue at a time in the meaningful information only part lowers the nominal value of the variance making it an adjustable parameter. The best t to the smoothed density of eigenvalues of the purely random correlation matrix determines the optimal variance and hence the part of the spectrum corresponding to the signal and to the noise. Therefore, in the rst approximation, the location of the spectrum edge is determined by tting the part of the eigenvalue density which contains most of the eigenvalues, a procedure that could be re ned by including the eects of the niteness of the correlation matrix and the eects of variability in the variance for dierent assets [1]. It is clear from the foregoing that in spite of the usefulness in applying the results of the random matrix theory the method requires subjective control of the tting procedure and of the degree of re nement necessary for dierent assets. Hence, due to the speci c needs of the nancial analysis, it is desirable to obtain an ecient and reliable study of price movements and the corresponding correlation matrices. The alternative procedure proposed in this paper is based on the topological characteristics of the signal (price changes) locus in the N -dimensional vector space, and in applying the singular value decomposition to the data matrix corresponding to the linearized topological hypersurface. The linearization enables easily detectable separation of information carrying component of the signal from noise, and since the performance of the algorithm is very ecient even on personal computers, the algorithm may be used for the on-line analysis of the nancial data. The method may be used globally, i.e., on the whole data set, or it may be used locally on clusters of points de ned by speci c clustering criteria. It can be eectively applied to the singular value decomposition (SVD)-based noise reduction procedures in the analysis of nonlinear dynamic systems, a subject to be treated elsewhere [4]. Moreover, the method is applicable both to deterministic and to stochastic data, making it useful in a wide area of applications. A presentation of our method as well as its applications to nancial time series is the topic of the present paper. It is organized in the following manner. In Section 2 we brie y sketch the mathematical and topological basis of the method followed by description of the minimum spanning tree in Section 3 and barycentric transformation in Section 4, the major ingredients of the procedure of unfolding the linearized manifold. In Section 5 we present the procedure for separating dominant from nondominant

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(noise) singular values (or eigenvalues) and nally we compare the results of our method to the results obtained by applying random matrix theory.

2. Topological characteristics of the data point set An attribute of a collection of observations or a collection of signals that is often taken as a measure of information carrying capacity is its dimensionality. If it is assumed that the observations or signals are realizations from a class of observations, the dimensionality D is the least number of functionally independent parameters needed to identify any observation or a signal from a given class. Since every observation (signal) can be represented by a point in a N -dimensional metric (vector) space, the signal locus de nes a K-dimensional topological hypersurface. Then the minimum number of parameters needed to generate the data vectors (or data points) is the topological dimensionality that corresponds to the measure of information carrying capacity. Hence, the problem of estimating intrinsic dimensionality is equivalent to the estimating topological dimensionality of the hypersurface de ned by the data vectors. The dimension of the information carrying capacity also represents the number of informationally signi cant eigenvalues of the covariance data matrix so the method of estimating the topological dimensionality of the data hypersurface is directly related to the analysis of the cross-correlation data matrices, and in particular to the nancial data matrices, a primary target of our study. At this point, we mention an important and well-known property that the singular values of an AM ×N matrix are equal to the square roots of the corresponding correlation matrix CN ×N = 1=M AAT eigenvalues. It is important to stress that the decomposition of the correlation matrix yields an eigenvalue spectrum for which it is impossible to perform the separation of the dominant from nondominant (i.e., noise) eigenvalues without applying a specially designed criterion. Naturally, in the case of estimating the topological dimension of data-generated hypersurface the data matrix itself is used in the calculations instead of the correlation matrix. It is interesting to mention that the concept of topological dimension, whose value is an integer, has been completely overlooked in the recent physics literature since its application to chaotic dynamics, an area of great recent popularity, remained unclear and to a large extent overshadowed by the various fractal dimensions. It is important to stress that all statistical tools used to nd the eigenvalues of the data matrix or the data-correlation matrix, such as the principal component analysis, singular value decomposition and Karhunen–Loeve decomposition, are intrinsically linear methods. However, relations in dynamical data are often nonlinear which is equivalent to the curvature of the relevant manifold generated by the data vectors. Hence, an application of linear methods to nonlinear data may more often than not give misleading results. One way to overcome this problem may be followed along these lines: (i) Application of linear methods to the data corresponding to the locally linear region of the manifold. The search for the linear region of the manifold is performed iteratively by making smaller and smaller local regions until the curvature and torsion

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tensors become equal to zero. This method may be particularly useful for studying the highly nonlinear spatio-temporal dynamics of dynamical systems [5,6]. (ii) Application of linear methods to the data corresponding to globally linearized manifolds. In this approach, the high-dimensional data manifold is deformed (unfolded) until it is absorbed by a linear subspace. It is clear that the rst method preserves the topological features of the original manifold while the second one, used in this paper, distorts the manifold so it is a challenge to preserve as many as possible topological features while performing the linearization. In the present framework, the whole manifold is initially globally linearized, after which the dimensionality is determined from the number of dominant singular values of the corresponding con guration data matrix.

3. Minimum spanning tree of the data points conÿguration The problem of reconstructing the surface or the hypersurface from the data points is one of the most challenging problems of combinatorial topology nowadays. The procedure for three-dimensional manifolds, for example, consists in triangulating the data set and it may be possible to atten such manifolds into two-dimensional surfaces for the purpose of noise reduction. Since the surface reconstruction has its application in numerous scienti c, engineering and medical procedures related to computer software, the problem is con ned to two or three dimensions and in general the procedures do not generalize easily to manifolds of higher dimensions, if at all feasible [7] Hence, since we are generally dealing with high-dimensional data, we chose not to perform reconstruction of the manifold followed by the linearization procedure and chose instead to work with points and clusters of points. The method represents a modi cation of the method originally proposed in Ref. [8], designed to study the structure in multivariate data point clusters. Data vectors (or points) de ne the con guration which can be uniquely determined by specifying all interpoint distances. For a given con guration of points the number of interpoint distances is a function of the embedding dimension and the idea is to nd the underlying information repository of the con guration which can be unfolded into a linearized topological object as well as reconstructed in the opposite direction, i.e., from a linearized object back to the original con guration. The choice for the underlying deformable object, the structure invariant, is the minimum spanning tree which has been used successfully in various clustering criteria and dynamic system identi cation algorithms as well as in the determination of the fractal dimension of the data point manifolds and attractors [9]. Formally stated, the data set is a graph G, consisting of a vertex set V (the points) and edge set E, each having a weight corresponding to the Euclidean distance between the two vertices. The minimum spanning tree (MST) of M points in a metric space is a tree, ( graph G = (V; E) with no closed-loop paths) spanning all M points and whose total length is a minimum. Some of the most important MST characteristics are that it minimizes (maximizes) all increasing (decreasing) symmetric functions of the interpoint distances. The examples

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are the sum or the product of the interpoint distances. The MST is invariant under similarity transformations such as translation, rotation and change in sign, as well as all transformations that preserve the ordering of the interpoint distances. For an M point set the MST consists of (M − 1) connected pairs of points. The number of these M − 1 connectivities is less than the minimum number of interpoint distances necessary to uniquely determine the con guration which implies that the initial con guration, once described by the MST, is no longer uniquely determined and that, in general, it is possible, while preserving the MST structure to reduce the dimensionality of the embedding space by a suitable transformation. 4. The barycentric transformation The barycentric transformation uses the MST structure to iteratively linearize the original con guration (manifold). Since the barycentric transformation involves several topological concepts they will be presented in a rather formal way. Deÿnition. Let  k = (a0 ; : : : ; ak ) be a k-dimensional simplex (a convex set of linearly independent points). A point with barycentric coordinates 1=(k + 1); : : : ; 1=(k + 1) is called the barycenter of the simplex  k : Speci cally, if coordinates of the point i are xij and the weights (lengths) are gi , then for each coordinate j of the barycenter b b(j) =

P i=1 gi · xij P i=1 gi

(1)

Due to prior normalization in our procedure, all weights (lengths) are equal to 1. Here it should be pointed out that a k-dimensional simplex is usually de ned as a set which consists of k + 1 linearly independent points x0 ; x1 ; : : : ; xk of a Euclidean space of dimension of at least k together with all the points given by x = 0 x0 + 1 x1 + · · · + k xk ; where 0 + 1 + · · · + k = 1 and 0 ; 1 ; : : : ; k ¿0 : Given a con guration of M points and the corresponding MST structure the barycentric transformation has the following features. For each point of the original con guration, the transformed con guration consists of n points, where n − 1 equals the connectivity of the reference point (number of points connected by an MST branch to the point considered), plus the reference point itself, hence (n−1)+1=n: Therefore, the barycentric transformation is obtained by replacing each (reference) point of the con guration by the barycenter of the cluster formed by all points connected to it, including the reference point itself. The barycentric con guration is a “con ned” mapping since each

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Fig. 1. First iteration of the barycentric transformation which linearizes the points con guration de ning a square wave-type pattern. The second iteration results in a straight line (not shown). Primed numbers indicate transformed points. Shaded triangle de ned by points 1, 2 and 3 determines the barycentric transformation of point 2 (since points 1 and 3 are points connected to it).

new con guration remains either on or within the boundaries of the previous con guration. This property is presented in Fig. 1 for a simple two-dimensional case. Another aspect of the barycentric transformation is easily noticed in this gure. Namely, the transformation represents a topological moving average operator since the subset size determining the barycenter depends on the connectivity of each point. This moving operator smoothens the original MST branches, replacing each point of the con guration by the barycenter of its corresponding subset. Both the “con nement” and the linearizing property of the mapping is clearly seen in this gure. Once the whole con guration has undergone the barycentric transformation there is a need to check the degree of linearization. However, the con nement property of the transformation implies that each iteration of the mapping shortens the MST length reducing the size of the con guration, converging eventually toward a centroid, i.e., to one point of the con guration. Thus, it is necessary to restore the original MST length, the con guration invariant, after each iteration, and since the characteristic feature of the con guration is conserved, the iterative procedure is stabilized when the linearization of the tree branches (edges) is no longer possible. As a check of the linearization we have used two criteria. The rst one uses the results of an important theorem [10] that if points are uniformly distributed within the hypersphere of radius r in a K-dimensional space and if RK = ||x1 − x2 ||=2r ; where x1 and x2 are random variables representing points in this hypersphere and RK is their normalized Euclidean interpoint distance, then the variance of RK is a decreasing function of k, that is K var(RK ) ≈ const: ;

(2)

where var(RK ) is the variance of RK : Thus, increasing the variance of the interpoint distances has the eect of decreasing the dimensionality, or linearizing the original con guration. Hence, the linearization may be detected when the variance of the interpoint distances levels o, which indicates a smoothening of the original hypersurface. Since the assumption of uniform distribution of points inside the hypersphere introduces a bias particularly in case of data generated by nonlinear dynamical systems

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and since the variance oset between two iterations enters as a heuristic parameter we have decided to use the curvature tensor as the criterion. In general, the necessary and sucient condition for the hypersurface to be “ at” is that the curvature and torsion tensors be equal to zero [11]. Hence, using the numerical method to compute these two quantities [12], we were able to determine the exact termination of the smoothing procedure, and compare the two criteria. In the case of nancial data matrices, quite good results are obtained for the values of variance tolerance level between 0.02 and 0.04. Once the linearization procedure is terminated the singular values and the eigenvectors are computed from the nal con guration, and the determination of the dimensionality is much simpler than in the case of original con guration. In most of the cases the irrelevant singular values (eigenvalues) are easily detected as being very close to zero. An important feature of the linearization procedure is tendency of the singular values (eigenvalues) of a random submatrix to approach one another as they tend to zero, hence enabling easy detection of information carrying singular values (eigenvalues) from noise. Good separation of dominant singular values and the irrelevant (noise) ones is obtained for a signal-to-noise ratio (SNR) of up to ∼ −8 dB, where the SNR level is de ned as S SNR = 10 log 10 dB : N As a benchmark to test the in uence of noise we have used, among others, the well-known chaotic Lorentz and Rossler attractors. For lower SNR ratios we have devised a method for separating signal from noise based on the angle between the corresponding subspaces, and the details are presented in the next section. 5. Separation of signal and noise subspaces Singular value decompostion of the linearized data may be represented in a matrix form as ! ! ∗ r X  V 0 a a H i ui viH ; (3) = S = UV = (Ua Us ) Vb∗ 0 s i=1 where ui and vi are the orthonormal characteristic vectors of the matrix SS H (or S H S), and {i } are the corresponding characteristic values. Note that the following expression is a general form of the decomposition since it is also valid for the complex data. In all our cases we have dealt with the real data. The index r represents the rank of the matrix S. Starting with the rst n terms in the expansion given in Eq. (3), the matrix S˜ is formed and compared with the original matrix S. Determining the number of dominant, i.e., relevant modes corresponds to estimating the reduced matrix having the same rank as the original one. For evaluating the quality of the reduced matrix, and hence the eectiveness of the rank reduction procedure, the concept of angle between subspaces proves to be very useful tool. If the matrix S˜ represents the acute perturbation

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of S, in the sense that the angle between the column space of S (denoted by R(S)) ˜ is acute, then there is and the corresponding column space of S˜ (denoted by R(S)) ˜ no vector in R(S) orthogonal to R(S) and vice versa. Conceptually, the procedure of rank reduction is performed by decreasing the column space and hence the number ˜ and R(S) is close to, or of singular values iteratively until the angle between R(S) equal to 0, so that the no vector from the column space of S (R(S)) is orthogonal to ˜ and vice versa. Therefore, the idea is to get an operationally useful expression R (S) for testing the angle between the two subspaces, taking into account that noise as well truncation errors introduce the perturbation. Speci cally, the angle between two subspaces M and N is de ned as [13] kcos Â(M; N )k = kPM PN k2 ;

(4)

where PM and PN are orthogonal projection operators onto the subspaces M and N , respectively, and the 2-norm is de ned as the largest singular value of the matrix, i.e., kAk2 = max(kAxk2 =kxk2 ) =  max : An alternative, more useful, formulation of the angle between two subspaces involves sin  rather than cos Â: In general, for any two subspaces sin  ∼

1 : k(PM − PN )−1 k2

Furthermore, PM = MM † , where M † denotes the Moore–Penrose pseudoinverse, represents the orthogonal projection of the matrix M on R(M ). In general, when kPM − PN k2 ¡ 1; then rank(M ) = rank(N ) and there is no vector in R(M ) orthogonal to R(N ) and vice versa [14], so that the rank equivalence represents the necessary, but not sucient, condition for matrix N to be the acute perturbation of M . The matrix S de ned in Eq. (3) may be written as S = A + E, where matrix E represents the perturbation due to the algorithm, round-o and truncation errors. Moreover, assuming that matrix elements are real from now on, S can be written as S = S˜ + S0 = U1 1 V1T + U2 2 V2T ;

(5)

where matrix S˜ is, as before, obtained by retaining the rst n terms in the expansion given by Eq. (2). Note that in this case PM =MM T represents the orthogonal projection of the matrix M on R(M ): Assuming that all matrix norms are spectral norms from now on, the error in approximating matrix S is kS˜ − Sk = n+1 6kS − Ak = kEk; while kS˜ − Ak = kS˜ − S + S − Ak6kS˜ − Sk + kS − Ak = n+1 + kEk62kEk : Now, with all norms assumed spectral ones, kPS˜ − PA k = kPS˜ PA⊥ k = kPA⊥ PS˜ k T T = kPA⊥ S˜S˜ k = kPA⊥ (A + E − S0 )S˜ k

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T T = kPA⊥ (E − S0 )S˜ k = kPA⊥ EPS˜T S˜ k T 6 kPA⊥ EPS˜T k · kS˜ k : T

6 kEk kS˜ k = kEkn−1 ; ˜ Hence, the where n is the smallest singular value of the n rank approximate matrix S: ˜ condition kPS˜ − PA k ¡ 1 implies kEk ¡ n ; and in that case the matrix S is the acute perturbation of A. In the opposite case, when n ∼ kEk or n ¿ kEk; the subspaces ˜ and R(A) are orthogonal to each other and the rank approximate order n is the R(S) ˜ Furthermore, the following inequalities hold [15]: dimension of the R(S). √ √ max|aij |6maxkaij k6kAk ¡ n maxkaij k6 mn max|aij | : Now, the elements eij of the matrix E satisfy −6eij 6; where  may be the round-o or truncation error. Since max|eij | =  ; and denoting the jth column of E as ej ; we get √ maxkej k = max[|eij |2 + · · · + |emj |2 ]1=2 6 m : Combining the above expressions we obtain √ 6kEk = 1 6 mn : Hence, taking for kEk the maximum value, becomes √ mn : kPS˜ − PA k6 n

√ mn, the expression for kPS˜ − PA k (6)

Therefore, we get a very useful expression containing the known smallest eigenvalue ˜ the dimensions of the local data matrix m and n of the n rank approximate matrix S, and the error  that can be estimated from the precision mode used. 6. Overview of the algorithm The method of determining the dominant, i.e., information carrying eigenvalues of the data correlation matrix, corresponds to calculating the topological dimension of the manifold corresponding to the data matrix. The original con guration is initially centered at 0 and the corresponding matrix is normalized. As a next step in the algorithm the singular values of the original data matrix (or eigenvalues of the correlation matrix) are determined in order to compare the results with the nal, unfolded values. The minimum spanning tree is formed as the major invariant of the initial con guration, whose length remains constant during the linearization procedure. The next step

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in the procedure is the actual linearization mapping, the barycentric transformation after which the MST length is restored. After each iteration the linearization is checked using the numerically computed curvature and torsion tensors and the procedure is continued until these two values are equal (or as close as possible) to zero. Finally, the singular values are computed for the nal con guration and the number of dominant singular values is determined using the angle between subspaces concept, in the case the orthogonality between the two subspaces is not obviously re ected in the close to zero values of the nondominant singular values. 7. Results In order to compare the results of the linearization procedure with the predictions of the random matrix theory (RMT) in the analysis of correlation matrices of nancial data, we have studied numerically the matrix of 400 assets (N = 400) of the NYSE, based on daily variations during the period 1980 –1989, for a total of 1000 days (M = 1000). Applying results of the RMT, the density of eigenvalues was rst calculated, de ned as 1 dn() () = N d where n() is the number of eigenvalues of the N × N correlation matrix less than . The empirically obtained density of eigenvalues is compared to the predictions of the RMT in the limit of in nite size data matrix, given by Ref. [1] and references therein p ( max − )( −  min ) Q ; (7) () = 22  where Q = M=N , and 2 is equal to the variance of the elements of the data matrix (normalized to 1 prior to forming the correlation matrix). The maximum and minimum values of the eigenvalues are expressed as s ! 1 1 2 : 1+ ±2 max min =  Q Q The partial empirical density of eigenvalues is presented in Fig. 2. Under the assumption that the correlation matrix is purely random, expression given by Eq. (7), being clearly a function of variance 2 ; may suggest cuto values indicated by indices 1 and 2 in Fig. 2. Dierent cuto values may be obtained using more re ned tting procedure as well as correcting the expression in Eq. (7) to include the eects of the nite matrix size. However, it is important to know whether an eigenvalue or a group of eigenvalues belongs to an information subspace or to a noise subspace, since this may be of crucial importance for the risk control. Clearly, the results of RMT give only an operationally rough range of eigenvalues which should be included for risk control. As a contrast, we have performed the linearization algortihm on the data matrix itself, and the singular value decomposition of the matrix corresponding to the nal,

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Fig. 2. Density of eigenvalues of the correlation matrix based on data comprising 400 assets of NYSE during the period 1980 –1989. Assuming that the matrix is purely random except for its highest eigenvalue, and adjusting the variance in Eq. (7), the suggested separation of signal and noise according to two best ts is indicated by indices 1 and 2. Index 3 refers to the cuto value determined by the linearization procedure.

linearized, con guration of points, followed by determination of the angle between the complementary subspaces, suggests that 23 eigenvalues belong to the information carrying part of the spectrum (index 3 in Fig. 2). More importantly, results of SVD on the nal con guration, partially presented in Fig. 3, clearly shows the characteristic smooth behavior of singular values belonging to purely random part of the data as they tend to zero. Also, the dominant singular values are clearly separated from the ones belonging to the noise subspace. Similar results, clearly suggesting the number of information carrying eigenvalues, were obtained for several sets of NYSE data for dierent time periods.

8. Conclusion The algorithm for performing global linearization of arbitrary manifolds is presented along with its use in the nancial time series analysis. The core of the method is the procedure of unfolding and linearizing manifolds de ned by point clusters. The algorithm uses the minimum spanning tree as an information invariant representing the data as well as a preserved graph structure connecting the data points. A barycentric transformation linearizes the con guration de ned by the MST and a singular value decomposition is performed on the matrix representing the nal, linearized con guration. The number of eectively information carrying singular values is determined by determining the orthogonality between signal and noise subspaces based on the angle

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Fig. 3. First 80 singular values of the matrix corresponding to the nal, linearized con guration of data points. Twenty-three singular values contain meaningful information while the rest of them correspond to the purely random data. Note the characteristic smooth approach to zero of the SVs corresponding to noise.

between the corresponding subspaces. A global linearization method presented in this paper emerges as an eective and ecient numerical tool to analyze nancial data sets either as a self-contained analysis program or in conjunction with the random matrix theory, which it actually complements as far as study of correlation matrices of nancial data is concerned. The behavior of eigenvalues (singular values) belonging to random part of the data exhibit a characteristic and very useful feature of monotonically approaching the zero value, thus enabling an easy identi cation of noise. Moreover, the method is robust with respect to noise level, since good results are obtained for relatively low signal-to-noise levels, mainly due to the method of detecting the nearly orthogonal subspaces in matrix decomposition. References [1] L. Laloux, P. Cizeau, J.P. Bouchaud, M. Potters, Noise dressing of nancial correlation matrices, Phys. Rev. Lett. 83 (1999) 1467.

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[2] V. Plerou, P. Gopikrishnan, B. Rosenow, L.A. Nunes Amaral, H. Eugene Stanley, Universal and nonuniversal properties of cross correlations in nancial time series, Phys. Rev. Lett. 83 (1999) 1471. [3] J.P. Bouchaud, M. Potters, Theory of Financial Risk, Alea-Saclay, Eyrolles, Paris, 1997. [4] M. Rajkovic, SVD noise reduction methods in nonlinear dynamical systems, in preparation. [5] M.M. Skoric, M.S. Jovanovic, M.R. Rajkovic, Route to Turbulence in Stimulated Raman Backscattering, Phys. Rev. E 53 (1996) 4056. [6] M. Rajkovic, Multiresolution Local Adaptive Method for the Analysis of Spatially Extended Systems, in: J.N. SHrensen, N. Aubry, E. Hop nger (Eds.), Simulation and Identi cation of Organized Structures in Flows, Kluwer Publications, Amsterdam, 1999, pp. 489–499. [7] H. Hoppe, Smoothening of Topological Surfaces, Ph.D. Thesis, Department of Computer Science and Engineering, University of Washington, 1997. [8] D. Schwartzman, J. Vidal, An algorithm for determining the topological dimensionality of point clusters, IEEE Trans. Comput. 24 (1975) 1175. [9] R. van de Weygaert, B.J.T. Jones, V.J. Martinez, The minimal spanning tree as an estimator for generalized dimensions, Phys. Lett. A 169 (1992) 145. [10] R.D. Lord, The distribution of distances in an hypersphere, Stud. Math. Stat. 25 (1954) 794. [11] R.L. Bishop, R.J. Crittenden, Geometry of Manifolds, Academic Press, New York, 1964. [12] Patrick J. Rabier, Werner C. Rheinboldt, On a computational method for the second fundamental tensor and its application to bifurcation problems, Numer. Math. 57 (1990) 681. [13] I.C.F. Ipsen, C.D. Meyer, The Angle Between Complementary Subspaces, NCSU Technical Report #NA-019501, January 1995. [14] G.W. Stewart, Error and perturbation bounds for subspaces associated with certain eigenvalue problems, SIAM Rev. 15 (1973) 727. [15] G.H. Golub, C.F. Van Loan, Matrix Computations, 2nd Edition, The Johns Hopkins Press, Baltimore, 1989.