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The normal distribution is used as a unified probability distribution, however, our researcher found that it is not good agreed with the real-life dynamical systems data. We collected and analyzed representative naturally occurring data series (e.g., the earth environment, sunspots, brain waves, electrocardiograms, some cases are classic chaos systems and social activities). It is found that the probability density functions (PDFs) of first or higher order differences for these datasets are consistently fat-tailed bell-shaped curves, and their associated cumulative distribution functions (CDFs) are consistently S-shaped when compared to the near-straight line of the normal distribution CDF. It is proved that this profile is not because of numerical or measure error, and the t-distribution is a good approximation. This kind of PDF/CDF is a universal phenomenon for independent time and space series data, which will make researchers to reconsider some hypotheses about stochastic dynamical models such as Wiener process, and therefore merits investigation.
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Electric signals have been recently recorded at the Earths surface with amplitudes appreciably larger than those hitherto reported. Their entropy in natural time is smaller than that, $S_u$, of a ``uniform distribution. The same holds for their entropy upon time-reversal. This behavior, as supported by numerical simulations in fBm time series and in an on-off intermittency model, stems from infinitely ranged long range temporal correlations and hence these signals are probably Seismic Electric Signals (critical dynamics). The entropy fluctuations are found to increase upon approaching bursting, which reminds the behavior identifying sudden cardiac death individuals when analysing their electrocardiograms.
We consider an independently identically distributed random dynamical system generated by finitely many, non-uniformly expanding Markov interval maps with a finite number of branches. Assuming a topologically mixing condition and the uniqueness of the equilibrium state of product form, we establish an almost-sure weighted equidistribution of cycles with respect to a natural stationary measure, as the periods of the cycles tend to infinity. This result is an analogue of Bowens theorem on periodic orbits of topologically mixing Axiom A diffeomorphisms in random setup. We also prove averaging results over all samples, as well as another samplewise result. We apply our result to the random $beta$-expansion of real numbers, and obtain a new formula for the mean relative frequencies of digits in the series expansion.
In high frequency financial data not only returns but also waiting times between trades are random variables. In this work, we analyze the spectra of the waiting-time processes for tick-by-tick trades. The numerical problem, strictly related with the real inversion of Laplace transforms, is analyzed by using Tikhonovs regularization method. We also analyze these spectra by a rough method using a comb of Diracs delta functions.
When both the difference between two quantities and their individual values can be measured or computational predicted, multiple quantities can be determined from the measurements or predictions of select individual quantities and select pairwise differences. These measurements and predictions form a network connecting the quantities through their differences. Here, I analyze the optimization of such networks, where the trace ($A$-optimal), the largest eigenvalue ($E$-optimal), or the determinant ($D$-optimal) of the covariance matrix associated with the estimated quantities are minimized with respect to the allocation of the measurement (or computational) cost to different measurements (or predictions). My statistical analysis of the performance of such optimal measurement networks -- based on large sets of simulated data -- suggests that they substantially accelerate the determination of the quantities, and that they may be useful in applications such as the computational prediction of binding free energies of candidate drug molecules.
In this paper we present the concept of description of random processes in complex systems with the discrete time. It involves the description of kinetics of discrete processes by means of the chain of finite-difference non-Markov equations for time correlation functions (TCF). We have introduced the dynamic (time dependent) information Shannon entropy $S_i(t)$ where i=0,1,2,3,... as an information measure of stochastic dynamics of time correlation $(i=0)$ and time memory (i=1,2,3,...). The set of functions $S_i(t)$ constitute the quantitative measure of time correlation disorder $(i=0)$ and time memory disorder (i=1,2,3,...) in complex system. Harnessing the infinite set of orthogonal dynamic random variables on a basis of Gram-Shmidt orthogonalization procedure tends to creation of infinite chain of finite-difference non-Markov kinetic equations for discrete TCF and memory function.The solution of the equations above thereof brings to the recurrence relations between the TCF and MF of senior and junior orders. The results obtained offer considerable scope for attack on stochastic dynamics of discrete random processes in a complex systems. Application of this technique on the analysis of stochastic dynamics of RR-intervals from human ECGs shows convincing evidence for a non-Markovian phenomemena associated with a peculiarities in short and long-range scaling. This method may be of use in distinguishing healthy from pathologic data sets based in differences in these non-Markovian properties.
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