Emerging spectra of singular correlation matrices under small power-map deformations

Correlation matrices are a standard tool in the analysis of the time evolution of complex systems in general and financial markets in particular. Yet most analysis assume stationarity of the underlying time series. This tends to be an assumption of varying and often dubious validity. The validity of the assumption improves as shorter time series are used. If many time series are used, this implies an analysis of highly singular correlation matrices. We attack this problem by using the so-called power map, which was introduced to reduce noise. Its nonlinearity breaks the degeneracy of the zero eigenvalues and we analyze the sensitivity of the so-emerging spectra to correlations. This sensitivity will be demonstrated for uncorrelated and correlated Wishart ensembles.

Figure 1

Portfolio variance ${\Omega }^2$ normalized to the minimal portfolio variance ${\Omega }_0^2$. The length $T$ of the time series was varied, and the number of companies was fixed at $N=100$. Results are shown for sample correlations without noise reduction (black dashed-dotted line) and for power-mapped correlations (blue solid line).The red dashed line at ${\Omega }^2/{\Omega }_0^2=4.37$ corresponds to a homogeneous portfolio.

Figure 2

Density of eigenvalues of ${\mathsf{C}}^{\left(q\right)}$ where $q=1.001$, $N=1024$, and $\kappa =1/2$ for the WOE case. In Fig. 2a we show the density of emerging spectra while in Fig. 2b we show the density of the nonzero eigenvalues which is closely described by the Marćenko-Pastur law (6), shown by a solid line. Both densities are shown on different scales. The density in Fig. 2a is normalized to $1-\kappa$ while the density in Fig. 2b is normalized to $\kappa$. Note that the density in Fig. 2a is not quite symmetric.

Figure 3

Comparison of theory with simulations for WOE where $T$ is the variable. Solid lines are used to represent theoretical results and symbols are used to represent numerics. In (a) and (b) we compare first two moments $\overline{\delta m_1}$ and $\overline{\delta m_2}$ obtained from the numerical simulations with the theory (21). In Figs. 3c and 3d we compare numerical results of scaling and shifting parameters $s$ and $r$, obtained from Eqs. (25) and (26) with theory (27). In (e) and (f) we compare numerical results of $\overline{\delta m_1^{\text{mp}}}$ and $\overline{\delta m_2^{\text{mp}}}$ with theory (28). Finally, in (g) and (h) we compare $\overline{\delta m_1^{\left(0\right)}}$ and $\overline{\delta m_2^{\left(0\right)}}$ with the theory (29).

Figure 4

The spectral density of the originally nonzero bulk spectra and the well separated emerging spectra of ${\mathsf{C}}^{\left(\alpha \right)}$. In this figure we show results for $N=1024$, $T=512$. We consider the simplest model for a nonrandom correlation matrix with elements ${\xi }_{jk}={\delta }_{jk}+c(1-{\delta }_{jk})$. In the top figures [(a), (b), and (c)] we show the bulk densities $\overline{\rho }\left(\lambda \right)$, ${\overline{\rho }}_1\left(\Delta \lambda \right)$, and ${\overline{\rho }}_0\left(\lambda \right)$, respectively, for nonzero spectra, nonzero spectral corrections, and for emerging spectra, where $c=0.5$. In insets of these figures, numerical densities of the corresponding separated individual eigenvalues are shown. In (a) we plot the CWOE theory (30). In the inset of the same figure we plot the theoretical density using results for the mean and the variance given in Ref. [20]. In (b) we use the ansatz (22) by estimating scaling and shifting parameters from (31) and (34), with a redefined variance $(1-c)$, to plot the theory. Distribution of separated individual eigenvalues in insets of (b) and (c) both are not Gaussian. Averaged mean positions as a function of the correlation coefficient $c$ are shown in the bottom figures [(d), (e), and (f)], respectively, for the separated individual eigenvalues in $\overline{\rho }\left(\lambda \right)$, ${\overline{\rho }}_1\left(\Delta \lambda \right)$, and ${\overline{\rho }}_0\left(\lambda \right)$. In this figure we use solid lines for the theory and open circles for the numerical results.

Figure 5

Comparison of theoretical and numerical results for CWOE where the nonrandom matrix elements are ${\xi }_{jk}={\delta }_{jk}+c(1-{\delta }_{jk})$. We choose $c=0.5$ and varied $T$ for fixed $N=1024$. This figure repeats the pattern of Fig. 3. In this figure we compare the moments only for the bulk densities. The null hypothesis, i.e., the corresponding WOE theory is shown by dashed lines.

Figure 6

The spectral density of the originally nonzero bulk spectra and the well-separated emerging spectra are shown respectively in (a) and (b). The power-mapped correlation matrices ${\mathsf{C}}^{\left(q\right)}$ used result from a CWOE where the averaged correlation matrix $\xi$ has three-diagonal blocks. The parameters are chosen as $N=N_1+N_2+N_3=1024$, where $N_1=N/2$, $N_2=N_3=N/4$ and the corresponding correlation coefficients are, respectively, $c_1=0.9$, $c_2=0.45$, and $c_3=0.225$. In the top figures we consider $T=512$ while it is varied in the bottom figures. In the inset of (a) the density of the separated individual eigenvalues are shown in good agreement with the theory. The first two moments, ${\overline{\delta m}}_1^{\left(0\right)},{\overline{\delta m}}_2^{\left(0\right)}$ are shown as function of $T$, respectively, in (c) and in (d). We use solid lines to represent theoretical curves and open circles for the numerical results. We use black dashed lines to represent WOE results.

Figure 7

The spectral density of the originally nonzero bulk spectra and the well-separated emerging spectra are shown respectively in (a) and (b). The power-mapped correlation matrices ${\mathsf{C}}^{\left(q\right)}$ used result from a CWOE where the averaged correlation matrix $\xi$ has an exponentially decaying band structure ${\xi }_{jk}=c^{|j-k|}$. With an outlay similar to that used in Fig. 6 we display spectral densities and the moments of emerging spectra for the CWOE. We consider here $N=1024$, $\kappa =1/2$ and the correlation coefficient $c=0.8$ (shown by open circles) and $c=0.5$ (shown by open squares). In the inset of (a), tails of the densities are amplified; separated individual eigenvalues are not seen. Black dashed lines are used to represent the corresponding WOE results.