Joanna Janczura

Universal Algorithm for Identification of Fractional Brownian Motion. A Case of Telomere Subdiffusion

We present a systematic statistical analysis of the recently measured individual trajectories of fluorescently labeled telomeres in the nucleus of living human cells. The experiments were performed in the U2OS cancer cell line. We propose an algorithm for identification of the telomere motion. By expanding the previously published data set, we are able to explore the dynamics in six time orders, a task not possible earlier. As a result, we establish a rigorous mathematical characterization of the stochastic process and identify the basic mathematical mechanisms behind the telomere motion. We find that the increments of the motion are stationary, Gaussian, ergodic, and even more chaotic--mixing. Moreover, the obtained memory parameter estimates, as well as the ensemble average mean square displacement reveal subdiffusive behavior at all time spans. All these findings statistically prove a fractional Brownian motion for the telomere trajectories, which is confirmed by a generalized p-variation test. Taking into account the biophysical nature of telomeres as monomers in the chromatin chain, we suggest polymer dynamics as a sufficient framework for their motion with no influence of other models. In addition, these results shed light on other studies of telomere motion and the alternative telomere lengthening mechanism. We hope that identification of these mechanisms will allow the development of a proper physical and biological model for telomere subdynamics. This array of tests can be easily implemented to other data sets to enable quick and accurate analysis of their statistical characteristics.

Regime-switching models for electricity spot prices: Introducing heteroskedastic base regime dynamics and shifted spike distributions

We calibrate Markov regime-switching (MRS) models to mean daily spot prices from the EEX market. Our empirical study shows that (i) models with shifted spike regime distributions lead to more realistic models of electricity spot prices and that (ii) introducing heteroskedasticity in the base regime leads to better spike identification and goodness-of-fit than in MRS models with the standard mean-reverting, constant volatility dynamics.

Impulsive Noise Cancellation Method for Copper Ore Crusher Vibration Signals Enhancement

In this paper, we deal with a problem of local damage detection in bearings in the presence of a high-energy impulsive noise. Such a problem was identified during diagnostics of bearings in raw materials crusher. Unfortunately, classical approaches cannot be applied due to the impulsive character of the noise. In this paper we propose, a procedure that cancels out impulsive noise rather than extracts signal of interest. The methodology is based on the regime switching model with two regimes: first corresponding to high-energy noncyclic impulses and second to the rest of the signal. We apply the proposed technique to a simulated signal as well as to the real one. Effectiveness of the method is presented graphically using time series, time-frequency spectrogram, and classical envelope analysis. The obtained results indicate efficiency of the method in impulsive noise cancellation and improve the ability to detect a damage.

Pricing electricity derivatives within a Markov regime-switching model: a risk premium approach

In this paper we derive analytic formulas for electricity derivatives under assumption that electricity spot prices follow a 3-regime Markov regime-switching model with independent spikes and drops and periodic transition matrix. Since the classical derivatives pricing methodology cannot be used in the case of non-storable commodities, we employ the concept of the risk premium. The obtained theoretical results are then used for the European Energy Exchange data analysis. We calculate the risk premium in the case of the calibrated 3-regime MRS model. We find a time varying structure of the risk premium and an evidence for a negative risk premium (or positive forward premium), especially at short times before delivery. Finally, we use the obtained risk premium to calculate prices of European options written on spot, as well as, forward prices.

Ergodicity testing for anomalous diffusion: Small sample statistics

The analysis of trajectories recorded in experiments often requires calculating time averages instead of ensemble averages. According to the Boltzmann hypothesis, they are equivalent only under the assumption of ergodicity. In this paper, we implement tools that allow to study ergodic properties. This analysis is conducted in two classes of anomalous diffusion processes: fractional Brownian motion and subordinated Ornstein-Uhlenbeck process. We show that only first of them is ergodic. We demonstrate this by applying rigorous statistical methods: mean square displacement, confidence intervals, and dynamical functional test. Our methodology is universal and can be implemented for analysis of many experimental data not only if a large sample is available but also when there are only few trajectories recorded.

A New Method For Automated Noise Cancellation In Electromagnetic Field Measurement

Abstract Electromagnetic field (EMF) measurements have limited accuracy, which is additionally impaired by meter self-noise influence. In this paper, a novel noise cancellation method is proposed, based on the Hidden Markov Model (HMM) methodology. It allows us to calculate the overall field intensity with a much higher accuracy than that obtained from other popular approaches, especially when EMF measurements are close to the noise level. The effectiveness of the new method is illustrated on two EMF datasets, one recorded in an urban and the other in a rural area. Its performance is further evaluated in a thorough simulation study using datasets representing the two distinct noisy environments.

Inference for Markov Regime-Switching Models of Electricity Spot Prices

In the last decade Markov regime-switching (MRS) models have been extensively used for modeling the unique behavior of spot prices in wholesale electricity markets. This popularity stems from the models’ relative parsimony and the ability to capture the stylized facts, in particular the mean-reverting character of electricity spot prices, the regime changes implied by fundamentals, and the resulting extreme price spikes. Due to the unobservable switching mechanism, the estimation of MRS models requires inferring model parameters and state process values at the same time. The situation becomes more complicated when the individual regimes are independent from each other and at least one of them is mean-reverting. Statistical validation of such models is also nontrivial. In this paper we review the available techniques and suggest efficient tools for statistical inference of MRS models.

Ergodicity testing using an analytical formula for a dynamical functional of alpha-stable autoregressive fractionally integrated moving average processes

In this paper we study asymptotic behavior of a dynamical functional for an α-stable autoregressive fractionally integrated moving average (ARFIMA) process. We find an analytical formula for this important statistics and show its usefulness as a diagnostic tool for ergodic properties. The obtained results point to the very fast convergence of the dynamical functional and show that even for short trajectories one may obtain reliable conclusions on the ergodic properties of the ARFIMA process. Moreover we use the obtained theoretical results to illustrate how the dynamical functional statistics can be used in the verification of the proper model for an analysis of some biophysical experimental data.

Identification and stochastic modelling of sources in copper ore crusher vibrations

A problem of rolling element bearings diagnostics for different machines is widely discussed in the literature. Most of the methods are based on the vibration signal analysis. However for some real signals the classical methods of damage detection are insufficient because of the specific nature of examined data. This specific nature is very often manifested through overlapping, mixing or interleaving of processes with different statistical properties and may be the result of different sources that have influence on the analysed signal. The problem of different sources identification and parametrisation of processes which are related to them is very challenging and requires advanced techniques. There are many methods which can be useful in this context however each signal should be analysed separately and there is no universal technique adequate to all possible time series. In this paper we propose a method of sources identification for vibration signal from the heavy duty crusher used in mineral processing plant. A crusher is a kind of machine which use a metal surface to crumble materials into small fractional pieces. During this process, as well as during entering material stream into the crusher, a lot of impacts appear. They are present in vibration signal acquired from bearings housing. Moreover, for some cases we also observe cyclic impulses which may be related to damage of rolling element bearings in the machine. The proposed sources identification method, especially useful for crushers vibrations, is based on the statistical analysis of examined data. Moreover by using advanced techniques of time series theory we propose a stochastic model that exhibits similar statistical properties as analysed signals. The introduced technique can be a starting point to damage detection of rolling element bearings of copper ore crushers.