Calibration of the subdiffusive arithmetic Brownian motion with tempered stable waiting-times

In the classical analysis many models used to real data description are based on the standard Brownian diffusion-type processes. However, some real data exhibit characteristic periods of constant values. In such cases the popular systems seem not to be applicable. Therefore we propose an alternative approach, based on the combination of the popular arithmetic Brownian motion and tempered stable subordinator. The probability density function of the proposed model can be described by a Fokker-Planck type equation and therefore it has many similar properties as the popular arithmetic Brownian motion. In this paper we propose the estimation procedure for the considered tempered stable subdiffusive arithmetic Brownian motion and calibrate the analyzed process to the real financial data.Subdiffusion, Tempered stable distribution, Calibration

Regime variance testing - a quantile approach

This paper is devoted to testing time series that exhibit behavior related to two or more regimes with different statistical properties. Motivation of our study are two real data sets from plasma physics with observable two-regimes structure. In this paper we develop estimation procedure for critical point of division the structure change of a time series. Moreover we propose three tests for recognition such specific behavior. The presented methodology is based on the empirical second moment and its main advantage is lack of the distribution assumption. Moreover, the examined statistical properties we express in the language of empirical quantiles of the squared data therefore the methodology is an extension of the approach known from the literature. The theoretical results we confirm by simulations and analysis of real data of turbulent laboratory plasma

Estimation of coefficients for periodic autoregressive model with additive noise -- a finite-variance case

Periodic autoregressive (PAR) time series is considered as one of the most common models of second-order cyclostationary processes. In real applications, the signals with periodic characteristics may be disturbed by additional noise related to measurement device disturbances or to other external sources. The known estimation techniques for PAR models assume noise-free model, thus may be inefficient for such cases. In this paper, we propose four estimation techniques for the noise-corrupted finite-variance PAR models. The methodology is based on Yule-Walker equations utilizing the autocovariance function. Thus, it can be used for any type of the finite-variance additive noise. The presented simulation study clearly indicates the efficiency of the proposed techniques, also for extreme case, when the additive noise is a sum of the Gaussian additive noise and additive outliers. This situation corresponds to the real applications related to condition monitoring area which is a main motivation for the presented research

Application of alpha-stable distribution approach for local damage detection in rotating machines

In this paper a novel method for informative frequency band selection for local damage detection is presented. Local damage in bearings/gearbox provides specific vibration signature, i.e. train of impulses with cycle related to fault frequency. The proposed approach is based on the α-stable distribution, which is an extension of the Gaussian one. The choice of this distribution is motivated by its superiority towards other distributions when modeling impulsive data. We introduce here the new selector (to select informative frequency band) which is based on the stability parameter α. Moreover we propose also the new time-frequency maps based on the measures of dependence adequate for α-stable distribution, namely autocodifference and autocovariation maps. The introduced methodology is illustrated by analysis of simulated and real vibration signals from heavy-duty rotating machinery. The results prove that proposed approach allows detection of multiple damages in signal and location of informative frequency band related to these damages. Moreover the analyzed examples indicate the α-stable distribution approach for some cases can give better results in contrast to the classical methodology based on the spectral kurtosis