Krzysztof Urbanowicz

Chaos control in economical model by time-delayed feedback method

A two-dimensional map describing chaotic behaviour of an economic model has been stabilized on various periodic orbits by the use of Pyragas time-delayed feedback control. The method avoids fancy data processing used in the Ott–Grebogi–Yorke approach and is based solely on the plain measurement and time lag of a scalar signal which in our case is a value of sales of a firm following an active investment strategy (Behrens–Feichtinger model). We show that the application of this control method is very straightforward and one can easily switch from a chaotic trajectory to a regular periodic orbit and simultaneously improve the systems economic properties.

Observations of deterministic chaos in financial time series by recurrence plots, can one control chaotic economy?

Abstract:Several economical time series such as exchange rates US

Noise-level estimation of time series using coarse-grained entropy.

/British Pound, USA Treasure Bonds rates and Warsaw Stock Index WIG have been investigated using the method of recurrence plots. The percentage of recurrence REC and the percentage of determinism DET have been calculated for the original and for shuffled data. We have found that in some cases the values of REC and DET parameters are about 20% lower for the surrogate data which indicates the presence of unstable periodical orbits in the considered data. A similar result has been obtained for the chaotic Lorenz model contaminated by noise. Our investigations suggest that real economical dynamics is a mixture of deterministic and stochastic chaos. We show how a simple chaotic economic model can be controlled by appropriate influence of time-delayed feedback.

Combustion process in a spark ignition engine: Dynamics and noise level estimation

We present a method of noise-level estimation that is valid even for high noise levels. The method makes use of the functional dependence of coarse-grained correlation entropy K2(epsilon ) on the threshold parameter epsilon. We show that the function K2(epsilon ) depends, in a characteristic way, on the noise standard deviation sigma. It follows that observing K2 (epsilon ) one can estimate the noise level sigma. Although the theory has been developed for the Gaussian noise added to the observed variable we have checked numerically that the method is also valid for the uniform noise distribution and for the case of Langevin equation corresponding to the dynamical noise. We have verified the validity of our method by applying it to estimate the noise level in several chaotic systems and in the Chua electronic circuit contaminated by noise.

Estimation of a noise level using coarse-grained entropy of experimental time series of internal pressure in a combustion engine

We analyze the experimental time series of internal pressure in a four cylinder spark ignition engine. In our experiment, performed for different spark advance angles, apart from the usual cyclic changes of engine pressure we observed additional oscillations. These oscillations are with longer time scales ranging from one to several hundred engine cycles depending on engine working conditions. Based on the pressure time dependence we have calculated the heat released per combustion cycle. Using the time series of heat release to calculate the correlation coarse-grained entropy we estimated the noise level for internal combustion process. Our results show that for a larger spark advance angle the system is more deterministic.

How random is your heart beat

We report our results on non-periodic experimental time series of pressure in a single cylinder spark ignition engine. The experiments were performed for different levels of loading. We estimate the noise level in internal pressure calculating the coarse-grained entropy from variations of maximal pressures in successive cycles. The results show that the dynamics of the combustion is a non-linear multidimensional process mediated by noise. Our results show that so defined level of noise in internal pressure is not monotonous function of loading.

Investment strategy due to the minimization of portfolio noise level by observations of coarse-grained entropy

We measure the content of random uncorrelated noise in heart rate variability using a general method of noise level estimation using a coarse-grained entropy. We show that usually, except for atrial fibrillation, the level of such noise is within 5–15% of the variance of the data and that the variability due to the linearly correlated processes is dominant in all cases analyzed but atrial fibrillation. The nonlinear deterministic content of heart rate variability remains significant and may not be ignored.

Risk evaluation with enhanced covariance matrix

Using a recently developed method of noise level estimation that makes use of properties of the coarse-grained entropy, we have analyzed the noise level for the Dow Jones index and a few stocks from the New York Stock Exchange. We have found that the noise level ranges from 40% to 80% of the signal variance. The condition of a minimal noise level has been applied to construct optimal portfolios from selected shares. We show that the implementation of a corresponding threshold investment strategy leads to positive returns for historical data.

Noise estimation by use of neighboring distances in takens space and its applications to stock market data

We propose a route for the evaluation of risk based on a transformation of the covariance matrix. The approach uses a ‘potential’ or ‘objective’ function. This allows us to rescale data from different assets (or sources) such that each data set then has similar statistical properties in terms of their probability distributions. The method is tested using historical data from both the New York and Warsaw stock exchanges.

Anti-deterministic behaviour of discrete systems that are less predictable than noise

We present a method that uses distances between nearest neighbors in Takens space to evaluate a level of noise. The method is valid even for high noise levels. The method has been verified by estimation of noise levels in several chaotic systems. We have analyzed the noise level for Dow Jones and DAX indexes and we have found that the noise level ranges from 25 to 80% of the signal variance.