Correlation Matrix Spectra: A Tool for Detecting Non-apparent Correlations?

It has been shown that, if a model displays long-range (power-law) spatial correlations, its equal-time correlation matrix of this model will also have a power law tail in the distribution of its high-lying eigenvalues. The purpose of this letter is to show that the converse is generally incorrect: a power-law tail in the high-lying eigenvalues of the correlation matrix may exist even in the absence of equal-time power law correlations in the original model. We may therefore view the study of the eigenvalue distribution of the correlation matrix as a more powerful tool than the study of correlations, one which may in fact uncover structure, that would otherwise not be apparent. Specifically, we show that in the Totally Asymmetric Simple Exclusion Process, whereas there are no clearly visible correlations in the steady state, the eigenvalues of its correlation matrix exhibit a rich structure which we describe in detail.

It has been shown that, if a model displays long-range (power-law) spatial correlations, its equaltime correlation matrix of this model will also have a power law tail in the distribution of its high-lying eigenvalues. The purpose of this letter is to show that the converse is generally incorrect : a power-law tail in the high-lying eigenvalues of the correlation matrix may exist even in the absence of equal-time power law correlations in the original model. We may therefore view the study of the eigenvalue distribution of the correlation matrix as a more powerful tool than the study of correlations, one which may in fact uncover structure, that would otherwise not be apparent. Specifically, we show that in the Totally Asymmetric Simple Exclusion Process, whereas there are no clearly visible correlations in the steady state, the eigenvalues of its correlation matrix exhibit a rich structure which we describe in detail.
The analysis of correlation matrices has attracted considerable attention almost for a hundred years starting with multivariate analysis in finance [1]. In two pioneering papers Laloux et al. and Plerou et al. analysed a complex time signal-a time series of stock pricesand successfully disentangled the part due to chance and the systematic part via an analysis of the eigenvalues of the correlation matrix [2][3][4]. The same tools of correlation matrix analysis have recently gained attention from physicists in the discussion of critical phenomena and phase transitions: If we consider an extended system undergoing some kind of dynamics, the equal time correlations between the various components of the system yield a correlation matrix, the eigenvalues of which can be analysed. In this context, it has recently been shown [5], that a power-law decay of correlations in space leads to a power-law behaviour for the large eigenvalues of the correlation matrix . We are thus led to ask whether the opposite is true. That is, does the observation of such power-law behaviour in the eigenvalues imply a powerlaw in spatial correlations?
In a trivial sense, it is possible to find systems for which no correlations are apparent, and yet the power-law behaviour of the eigenvalues remains: We simply take a system which does display spatial power-law correlations, and "scramble" the components by randomly permuting them. Such an operation leaves the eigenvalues invariant, so that their power-law behaviour testifies to the existence of the spatial correlations, even though the latter have been masked by the random permutation.
However, we may ask whether there exist less trivial counterexamples. In the following, we shall suggest that there probably are: we shall analyse the so-called Totally Asymmetric Simple Exclusion Process, (TASEP) which shows little or no apparent spatial correlation, which additionally surely does not have a power-law decay, and yet, as we report here, displays a marked power-law feature in the spectra of its correlation matrix on one critical line of its phase diagram as well as further anomalous structure in the rest of the phase diagram. Of course, it is difficult rigorously to exclude the possibility that non-trivial spatial correlations for this system have, in fact, been hidden by a "scrambling" process similar to that described in the last paragraph, but, in view of the system's simplicity, this does not seem likely. TASEP is a model consisting of a many-particle hopping system where particles are located on a discrete lattice that evolves in continuous time. Particles can hop to the next lattice site, in only one direction (say to the right-hand side), on a one-dimensional lattice at a random time, with rate one, provided that the target site is empty. We here consider the problem with open boundary conditions where both sides of the lattice are coupled with particle reservoirs. If the first site of the lattice is empty then a particle can hop from the reservoir into the system with a transition rate α and the particles leave the system from the last site of the lattice with a transition rate β. TASEP has been used to describe directed transport in 1D, such as arises, for instance, in unidirectionally moving vehicular traffic along roads [6,7]. There are several reasons to choose TASEP for the present study: The equal-time correlation functions for this stationary non-equilibrium system are known exactly [8,9] and the phase diagram (see Fig. 1) of the exact and the mean field solutions coincide. In phase I, there is a high density of occupied sites which fluctuates little in time, in phase II there is a corresponding low-density phase, in which the density is also approximately timeindependent, whereas in III the density is equal to 1/2, independent of α and β. Finally, in the transition line between I and II, a phase exists in which the density oscillates between a high value corresponding to a nearby point of I, and a low value corresponding to a nearby point of II.
We are interested in the density-density two point correlation C i,j on the lattice. This is defined as the probability to find a particle in a lattice site j, given there is a particle at lattice site i. The analytical expression for this correlation function C(r) (average correlation between any two points at distance r) is given in [9]. On the line α = β < 0.5, where both the high density and low density phase coexist, the two point correlation function do not decay in a power law fashion. Rather, the spatial correlations appear to decay at a scale of the order of the system size, as shown in the inset of Figure 2.
In the present letter we analyse the spectrum of the correlation matrix and particularly, its behaviour as α and β vary. For α = β < 0.5 we find a power-law ( Fig.  2) in the Zipf plot (see below) and thus an example, where we find a power-law even though the obvious two-point function in space does not show such a behaviour. Let us now compare the eigenvalue density of the correlation matrix with the null-hypothesis, that is, the eigenvalue spectrum of the correlation matrix of a completely uncorrelated signal. These correlations remain different from zero if they are taken on the scale of the fluctuations, that is, on the scale of the square root of the duration of the signal. The eigenvalue distribution may be calculated exactly in this case, and the eigenvalue density was determined analytically by Marčenko and Pastur [10]. This eigenvalue density has the remarkable feature that it vanishes outside a finite interval. We may thus meaningfully speak of deviations from the Marčenko-Pastur (MP) result whenever eigenvalues appear significantly outside this interval.
We thus compare our eigenvalue spectra with the MP distribution in order to see to what extent our eigenvalues differ from an uncorrelated signal. This is very much in the line of [2,3]. Our key result can now be stated as follows: we find agreement in the high and low density regions, that is, in the interior of regions I and II, so that these are indeed well described by a random process. On the other hand, in all other parts of the phase diagram, characteristic differences are observed: in the constant current phase corresponding to III, we find a significant deviation from the MP prediction in that a significant number of eigenvalues below the MP threshold are observed. We find similar deviations on the I/III and II/III lines, and different ones at the triple point α = β = 1/2.
To construct the correlation matrices, we have generated the times series by Monte-Carlo simulation. The random update rule was been used to generate the time series. The lattice size is N , with 10 3 ≤ N ≤ 10 4 . For each parameter value we have considered the length of the time series as T = 20 × N . Obviously the correlation matrix C is an N × N dimensional matrix. The results are averaged over an ensemble of 100 configurations.
We shall analyse the eigenvalues using the so called Zipf plot, also known as "scree diagram" or ranking-ofeigenvalues plot, in which the eigenvalues in decreasing order λ n are plotted against their rank n, typically on a doubly logarithmic plot. Such a plot makes an initial power-law very prominent.
Let us first look at the structure of eigenvalues on the I/II coexistence line, that is, for α = β < 0.5. For any value of α and β on this low density-high density coexistence line, though the spatial two point correlation function C(r) decay with distance r, but do not decay in . So the density of eigenvalues of the correlation matrix will obviously be different from the MP distribution [ Fig. 3]. We do find an initial power-law decay on the Zipf plot for the eigenvalues on this coexistence line. The power we find (λ n ∼ n −θ with θ ≈ 2) obtained is same for any value of α and β, as long as α = β < 0.5 [ Fig. 2]. There are other differences between the observed distribution on the I/II separation line and the MP distribution: first, the range over which the power-law is observed, varies with the parameter value. It is higher for the lower values of α = β. As a result, the density of high-lying eigenvalues differ for different values of α and β on this line. Second, we observe a shift of the bulk to lower eigenvalues in compared with the MP distribution [ Fig. 3]. This shift actually compensates the contribution from the higher eigenvalues, since the sum of the eigenvalues remains constant and equal to the dimension of the matrix. However for lower values of α, the density profile for the eigenvalues are deformed and the deformation becomes more prominent as the value of α = β decreases [ Fig. 3]. Finally, when α 0.25, a second peak appears in the eigenvalue density, in sharp contrast to the MP result, in which only one peak appears. This phenomenon may perhaps be explained as follows: the density of particles inside the lattice, in the low density region is lower for the lower values of α. Similarly it is higher for the lower values of β, in the high density region. Hence for α = β < 0.5, that is on the I/II coexistence line, for the lower values of α and β there will be larger strings of particles in the lattice followed by a string of empty lattice sites of similar length. As a result correlation length inside the lattice increases as α decreases (for α = β < 0.5). This correlation length is not enough to show a power law decay in case of two point correlation function, but it may well be related to the presence of a larger number eigenvalues above the MP threshold. These then display the power law behaviour observed in the Zipf plot.
We now proceed to consider the new peak in the eigenvalue density that appears for α = β 0.25. In particular, it is natural to ask whether, inside the two peaks of eigenvalue density which arise for lower values of α and β, the correlations of the unfolded eigenvalues (say ξ) [12], are identical. We test two independent statistical properties of unfolded eigenvalues ξ: the distribution of nearest-neighbour spacing s = ξ i+1 − ξ i and the statistics of number variance Σ 2 (x). The distributions of nearest-neighbour spacing of unfolded eigenvalues, which are obtained from the first and the second peak of the density of eigenvalues [ Fig. 3] appear to be universal. That means the behaviour of the nearest-neighbour (nn) spacing distributions are not distinguishable from that of the Wishart ensemble for both the peaks.
Number variance, the variance of the number of unfolded eigenvalues in the intervals of length x, is defined as is the number of unfolded eigenvalues in the interval [ξ−x/2, ξ+x/2]. The average is made along ξ. If the eigenvalues are uncorrelated, Σ 2 (x) = x; whereas if all unfolded eigenvalues are equidistant, Σ 2 (x) = 0. We found the unfolded eigenvalues obtained from the first peak of the distribution [ Fig  3] does not follow the universal behavior for the number variance statistics, while for the second peak it appears to be universal [ Fig. 4]. In the maximal or constant current region, we also find significant deviations from MP in the probability distribution of eigenvalues [see Fig. 5]. This consists in the appearance of eigenvalues below the lower threshold of MP, and is quite pronounced. One or two high eigenvalues (outside the limits of MP) are also observed. The deviation of the probability distribution of eigenvalues of the correlation matrix is also present on the transition lines of I/III and II/III. But there the number of belowthreshold eigenvalues distribution is smaller than in the constant current phase. The plot for the eigenvalue density for the maximal current and for the triple point is shown in Fig 5. Transition from the MC phase to the triple point is continuous, as the number of lower eigenvalues decreases slowly as the triple point is approached.
In different parts of the phase diagram, the probability densities of the eigenvalues have a deviation from the Marčenko-Pastur but also the distribution of the lowest eigenvalues, over the configuration space is significantly different from that of the low density or high density regions [ Fig. 6].
We have also checked whether the effects can be accounted for by edge effects: we did not notice any significant such effect for any value of α and β for the entire phase diagram. This is also true for the higher eigenvalues when α = β < 0.5. If there is any edge effect at all in the spectrum of eigenvalues, it is not detectable with the present computational accuracy.
On the I/II phase coexistence line, we have taken different parts of the lattice and repeated the correlation matrix analysis. We indeed observed the power law in the Zipf plot (with the same value of θ) for all the parts of the lattice which will be discussed in detail at [17]. On this coexistence line the motion of the domain wall is nonlocalised over the lattice [18]. Whether or not the motion of the domain wall is responsible for the observed power law will be studied as a future problem [17]. We will also attempt to connect the formula of two point function given in [9] to the exact solutions derived recently [19] for arbitrary correlations at least in an approximate fashion.
In conclusion, we have shown that the analysis of the density of eigenvalues of the correlation matrix of a signal is sensitive to non-trivial correlations, which cannot otherwise be reliably characterised by direct numerical observation. Such is the case of TASEP, in which twobody correlations are weak, though they extend over the whole system at the phase coexistence line. Comparison of the correlation matrix spectrum with those generated by a random signal provide clear evidence that the signal produced by TASEP has significant correlation in some parts of the phase diagram. In particular at the α = β < 1/2 line long-range weak correlations in space induce a power law in the spectrum.