Multivariate analysis of short time series in terms of ensembles of correlation matrices

Abstract

When dealing with non-stationary systems, for which many time series are available, it is common to divide time in epochs, i.e. smaller time intervals and deal with short time series in the hope to have some form of approximate stationarity on that time scale. We can then study time evolution by looking at properties as a function of the epochs. This leads to singular correlation matrices and thus poor statistics. In the present paper, we propose an ensemble technique to deal with a large set of short time series without any consideration of non-stationarity. Given a singular data matrix, we randomly select subsets of time series and thus create an ensemble of non-singular correlation matrices. As the selection possibilities are binomially large, we will obtain good statistics for eigenvalues of correlation matrices, which are typically not independent. Once we defined the ensemble, we analyze its behavior for constant and block-diagonal correlations and compare numerics with analytic results for the corresponding correlated Wishart ensembles. We discuss differences resulting from spurious correlations due to repetitive use of time-series. The usefulness of this technique should extend beyond the stationary case if, on the time scale of the epochs, we have quasi-stationarity at least for most epochs.

Complex Market Dynamics in the Light of Random Matrix Theory

“…As usual, the market mode captures the mean market correlation corresponding to the maximum eigenvalue, which is separate from rest of the eigenvalues (see Ref. [36] for the comparison of the behavior of maximum eigenvalues in correlated Wishart ensembles). The group modes, which tell about the sectoral behavior of the market, largely coincide with the random modes and correspond to the random behavior of the stocks.…”

Non-linear correlation analysis in financial markets using hierarchical clustering

“…Next, we analyze the time evolution of distribution of eigenvalues of correlation matrices as shown in figure 3; note that the plot is logarithmic. All the correlation matrices are singular and thus, we have a delta peak at zero eigenvalues in addition to bulk distribution (which follows random matrix theory predictions) and outliers that represent correlations [9,[15][16][17][18][19]. The largest eigenvalue, which is linearly correlated with average correlations, attains very large values in crisis periods as seen from distribution of eigenvalues for both PCC and DCC.…”

Instability of networks: effects of sampling frequency and extreme fluctuations in financial data

“…Real-world dynamic networks display time-varying degrees of stability [ 31 ]. Often, stability of the network is directly influenced by external perturbations [ 36 ]. However, the origin of network instability may also stem from the inherent dynamical properties of the underlying system.…”