Random matrix approach in search for weak signals immersed in background noise

We present new, original and alternative method for searching signals coded in noisy data. The method is based on the properties of random matrix eigenvalue spectra. First, we describe general ideas and support them with results of numerical simulations for basic periodic signals immersed in artificial stochastic noise. Then, the main effort is put to examine the strength of a new method in investigation of data content taken from the real astrophysical NAUTILUS detector, searching for the presence of gravitational waves. Our method discovers some previously unknown problems with data aggregation in this experiment. We provide also the results of new method applied to the entire respond signal from ground based detectors in future experimental activities with reduced background noise level. We indicate good performance of our method what makes it a positive predictor for further applications in many areas.Comment: 15 pages, 16 figure

Dynamic asset trees and Black Monday

The minimum spanning tree, based on the concept of ultrametricity, is constructed from the correlation matrix of stock returns. The dynamics of this asset tree can be characterised by its normalised length and the mean occupation layer, as measured from an appropriately chosen centre called the `central node'. We show how the tree length shrinks during a stock market crisis, Black Monday in this case, and how a strong reconfiguration takes place, resulting in topological shrinking of the tree.Comment: 6 pages, 3 eps figues. Elsevier style. Will appear in Physica A as part of the Bali conference proceedings, in pres

Coexistence of solutions in dynamical mean-field theory of the Mott transition

In this paper, I discuss the finite-temperature metal-insulator transition of the paramagnetic Hubbard model within dynamical mean-field theory. I show that coexisting solutions, the hallmark of such a transition, can be obtained in a consistent way both from Quantum Monte Carlo (QMC) simulations and from the Exact Diagonalization method. I pay special attention to discretization errors within QMC. These errors explain why it is difficult to obtain the solutions by QMC close to the boundaries of the coexistence region.Comment: 3 pages, 2 figures, RevTe

Are Financial Crashes Predictable?

We critically review recent claims that financial crashes can be predicted using the idea of log-periodic oscillations or by other methods inspired by the physics of critical phenomena. In particular, the October 1997 `correction' does not appear to be the accumulation point of a geometric series of local minima.Comment: LaTeX, 5 pages + 1 postscript figur

Alternation of different fluctuation regimes in the stock market dynamics

Based on the tick-by-tick stock prices from the German and American stock markets, we study the statistical properties of the distribution of the individual stocks and the index returns in highly collective and noisy intervals of trading, separately. We show that periods characterized by the strong inter-stock couplings can be associated with the distributions of index fluctuations which reveal more pronounced tails than in the case of weaker couplings in the market. During periods of strong correlations in the German market these distributions can even reveal an apparent L\'evy-stable component.Comment: 19 page

Time scales involved in market emergence

In addressing the question of the time scales characteristic for the market formation, we analyze high frequency tick-by-tick data from the NYSE and from the German market. By using returns on various time scales ranging from seconds or minutes up to two days, we compare magnitude of the largest eigenvalue of the correlation matrix for the same set of securities but for different time scales. For various sets of stocks of different capitalization (and the average trading frequency), we observe a significant elevation of the largest eigenvalue with increasing time scale. Our results from the correlation matrix study go in parallel with the so-called Epps effect. There is no unique explanation of this effect and it seems that many different factors play a role here. One of such factors is randomness in transaction moments for different stocks. Another interesting conclusion to be drawn from our results is that in the contemporary markets the emergence of significant correlations occurs on time scales much smaller than in the more distant history.Comment: 13 page

Log-periodic self-similarity: an emerging financial law?

A hypothesis that the financial log-periodicity, cascading self-similarity through various time scales, carries signatures of a law is pursued. It is shown that the most significant historical financial events can be classified amazingly well using a single and unique value of the preferred scaling factor lambda=2, which indicates that its real value should be close to this number. This applies even to a declining decelerating log-periodic phase. Crucial in this connection is identification of a "super-bubble" (bubble on bubble) phenomenon. Identifying a potential "universal" preferred scaling factor, as undertaken here, may significantly improve the predictive power of the corresponding methodology. Several more specific related results include evidence that: (i) the real end of the high technology bubble on the stock market started (with a decelerating log-periodic draw down) in the begining of September 2000; (ii) a parallel 2000-2002 decline seen in the Standard & Poor's 500 from the log-periodic perspective is already of the same significance as the one of the early 1930s and of the late 1970s; (iii) all this points to a much more serious global crash in around 2025, of course from a level much higher (at least one order of magnitude) than in 2000.Comment: Talk given by S. Drozdz at International Econophysics Conference, Bali, August 28-31, 2002; typos correcte