Complex systems are usually non-stationary and their dynamics is often dominated by collective effects. The collective motion of the whole system or of some of its parts, manifests itself in the time-dependent structures of covariance and correlation matrices. In the case of the collective motion of the entire system, one speaks of the market motion, which is always associated with the largest eigenvalue of the covariance and correlation matrices. The other large, isolated, eigenvalues indicate collectivity in parts of the system. In the case of finance, these correspond to industrial sectors.

The measured correlations of financial time series in subsequent epochs change considerably as a function of time. When studying the whole correlation matrices, quasi-stationary patterns, referred to as market states, are seen by applying clustering methods. They emerge, disappear or reemerge, but they are dominated by the collective motion of all stocks. Thus, the question arises, if one can extract more refined information on the system by subtracting the dominating market motion in a proper way. To this end we introduce a new approach by clustering reduced-rank correlation matrices that are obtained by subtracting the dyadic matrix belonging to the largest eigenvalue from the standard correlation matrices. The resulting dynamics is remarkably different and the corresponding market states are quasi-stationary over a long period of time. Our approach adds to attempts to separate endogenous from exogenous effects in the literature.

The identification of indicators for large events is highly desirable in terms of assessing systemic risks. We collect pieces of evidence across a variety of observables. We use the quasi-stationary market states mentioned previously. We study their precursor properties in the US stock markets, including two endogenous crises, the dot-com bubble burst and the pre-phase of the Lehman Brothers crash. We identify certain interesting features and critically discuss their suitability as indicators.

We measure a remaining collectivity in reduced-rank covariance matrices, which we refer to as average sector collectivity. We plot the average sector collectivity versus the collectivity corresponding to the largest eigenvalue to study the whole market trajectory in a two dimensional space spanned by both collectivities. We do so to capture the average sector collectivity in a much more precise way. Additionally, we observe that larger values in the average sector collectivity are often accompanied by trend shifts in the mean covariances and mean correlations. As of 2015/2016 the collectivity in the US stock markets changed fundamentally.