The understanding of complex systems has become a central issue because such systems exist in a wide range of scientific disciplines. We here focus on financial markets as an example of a complex system. In particular we analyze financial data from the S&P 500 stocks in the 19-year period 1992–2010. We propose a definition of state for a financial market and use it to identify points of drastic change in the correlation structure. These points are mapped to occurrences of financial crises. We find that a wide variety of characteristic correlation structure patterns exist in the observation time window, and that these characteristic correlation structure patterns can be classified into several typical “market states”. Using this classification we recognize transitions between different market states. A similarity measure we develop thus affords means of understanding changes in states and of recognizing developments not previously seen.

The concept of fidelity decay is discussed from the point of view of the scattering matrix, and the scattering fidelity is introduced as the parametric cross-correlation of a given S-matrix element, taken in the time domain, normalized by the corresponding autocorrelation function. We show that for chaotic systems, this quantity represents the usual fidelity amplitude, if appropriate ensemble and/or energy averages are taken. We present a microwave experiment where the scattering fidelity is measured for an ensemble of chaotic systems. The results are in excellent agreement with random matrix theory for the standard fidelity amplitude. The only parameter, namely the perturbation strength could be determined independently from level dynamics of the system, thus providing a parameter free agreement between theory and experiment.

Financial markets are prominent examples for highly non-stationary systems. Sample averaged observables such as variances and correlation coefficients strongly depend on the time window in which they are evaluated. This implies severe limitations for approaches in the spirit of standard equilibrium statistical mechanics and thermodynamics. Nevertheless, we show that there are similar generic features which we uncover in the empirical return distributions for whole markets. We explain our findings by setting up a random matrix model. The great success of statistical mechanics and thermodynamics is borne out by their ability to characterize, in the equilibrium, large systems with many degrees of freedom in terms of a few state variables, for example temperature and pressure. Ergodicity (or quasi-ergodicity) is the prerequisite needed to introduce statistical ensembles. Systems out of equilibrium or, more generally, nonstationary systems still pose fundamental challenges [1][2][3][4]. Complex systems -the term "complex" is used in a broad sense -show a wealth of different aspects which can be traced back to non-stationarity [5,6]. Financial markets are presently in the focus, because they demonstrated their non-stationarity in a rather drastic way during the recent years. To assess a financial market as a whole, the correlations between the prices of the individual stocks are of crucial importance [7][8][9][10]. They fluctuate considerably in time, e.g., because the market expectations of the traders change, the business relations between the companies change, particularly in a state of crisis, and so on. The motion of the stock prices is in this respect reminiscent of that of particles in many-body systems such as heavier atomic nuclei. Depending on the excitation energy, the motion of the individual particles can be incoherent, i.e., uncorrelated in the above terminology, or coherent (collective), i.e., correlated, or even somewhere in-between [11][12][13]. This non-stationarity on the energy scale leads to very different spectral properties, [12,13]. Such an analogy can be helpful, but we do not want to overstretch it.Here, we want to show that the non-stationarity, namely the fluctuation of the correlations, induces generic features in financial time series. These become visible when looking at quantities which measure the stock price changes for the entire market. We have four goals. First, we carry out a detailed data analysis revealing the generic features. Second, we set up a random matrix model to explain them. Third, we demonstrate that the non-stationarity of the correlations leads to heavy tails. Fourth, we argue that our approach maps a noninvariant situation to an effectively invariant one. For an economic audience we discuss the consequences for portfolio management elsewhere [14].Consider K companies with stock prices S k (t), k = 1, . . . , K as functions of time t. The relative price changes over a fixed time interval ∆t, i.e., the returnsare well-known to have distributions with heavy tails, the ...

Abstract. We combine geometric data analysis and stochastic modeling to describe the collective dynamics of complex systems. As an example we apply this approach to financial data and focus on the non-stationarity of the market correlation structure. We identify the dominating variable and extract its explicit stochastic model. This allows us to establish a connection between its time evolution and known historical events on the market. We discuss the dynamics, the stability and the hierarchy of the recently proposed quasi-stationary market states.

Using supersymmetry techniques analytical expressions for the average of the fidelity amplitude f ǫ, and H 0 and H ǫ are taken from the Gaussian unitary ensemble (GUE) or the Gaussian orthogonal ensemble (GOE), respectively. As long as the perturbation strength is small compared to the mean level spacing, a Gaussian decay of the fidelity amplitude is observed, whereas for stronger perturbations a change to a single-exponential decay takes place, in accordance with results from literature. Close to the Heisenberg time τ = 1, however, a partial revival of the fidelity is found, which hitherto remained unnoticed. Random matrix simulations have been performed for the three Gaussian ensembles. For the case of the GOE and the GUE they are in perfect agreement with the analytical results.