György Terdik

Automated detection of neonate EEG sleep stages

The paper integrates and adapts a range of advanced computational, mathematical and statistical tools for the purpose of analysis of neonate sleep stages based on extensive electroencephalogram (EEG) recordings. The level of brain dysmaturity of a neonate is difficult to assess by direct physical or cognitive examination, but dysmaturity is known to be directly related to the structure of neonatal sleep as reflected in the nonstationary time series produced by EEG signals which, importantly, can be collected trough a noninvasive procedure. In the past, the assessment of sleep EEG structure has often been done manually by experienced clinicians. The goal of this paper is to develop rigorous algorithmic tools for the same purpose by providing a formal scheme to separate different sleep stages corresponding to different stationary segments of the EEG signal based on statistical analysis of the spectral and nonlinear characteristics of the sleep EEG recordings. The methods developed in this paper can, potentially, be translated to other areas of biomedical research.

Lévy flights and fractal modeling of internet traffic

The relation between burstiness and self-similarity of network traffic was identified in numerous papers in the past decade. These papers suggested that the widely used Poisson based models were not suitable for modeling bursty, local-area and wide-area network traffic. Poisson models were abandoned as unrealistic and simplistic characterizations of network traffic. Recent papers have challenged the accuracy of these results in todays networks. Authors of these papers believe that it is time to reexamine the Poisson traffic assumption. The explanation is that as the amount of Internet traffic grows dramatically, any irregularity of the network traffic, such as burstiness, might cancel out because of the huge number of different multiplexed flows. Some of these results are based on analyses of particular OC48 Internet backbone connections and other historical traffic traces. We analyzed the same traffic traces and applied new methods to characterize them in terms of packet interarrival times and packet lengths. The major contribution of the paper is the application of two new analytical methods. We apply the theory of smoothly truncated Levy flights and the linear fractal model in examining the variability of Internet traffic from self-similar to Poisson. The paper demonstrates that the series of interarrival times is still close to a self-similar process, but the burstiness of the packet lengths decreases significantly compared to earlier traces.

Does the Internet Still Demonstrate Fractal Nature

The self-similar nature of bursty Internet traffic has been investigated for the last decade. A first generation of papers, approximately from 1994 to 2004, argued that the traditionally used Poisson models oversimplified the characteristics of network traffic and were not appropriate for modeling bursty, local-area, and wide-area network traffic. Since 2004, a second generation of papers has challenged the suitability of these results in networks of the new century and has claimed that the traditional Poisson-based and other models are still more appropriate for characterizing today’s Internet traffic. A possible explanation was that as the speed and amount of Internet traffic grow spectacularly, any irregularity of the network traffic, such as self-similarity, might cancel out as a consequence of high-speed optical connections, new communications protocols, and the vast number of multiplexed flows. These papers analyzed traffic traces of Internet backbone collected in 2003. In one of our previous papers we applied the theory of smoothly truncated Levy flights and the linear fractal model in examining the variability of Internet traffic from self-similar to Poisson. We demonstrated that the series of interarrival times was still close to a self-similar process, but the burstiness of the packet lengths decreased significantly compared to earlier traces. Since then, new traffic traces have been made public, including ones captured from the Internet backbone in 2008. In this paper we analyze these traffic traces and apply our new analytical methods to illustrate the tendency of Internet traffic burstiness. Ultimately, we attempt to answer the question: Does the Internet still demonstrate fractal nature?

Angular Spectra for non-Gaussian Isotropic Fields

Cosmic Microwave Background (CMB) Anisotropies is a subject of intensive research in several fields of sciences. In this paper we start a systematic development of basic notions and theory in statistics according to the application for CMB. The main result of this paper is the necessary and sufficient condition for isotropy of a non-Gaussian field in terms of spectra. Clear formulae for bi-, tri- and polyspectra and bi-, tri-, and higher order covariances are also given. Keywords: Bispectrum, Trispectrum, Angular poly-Spectra, Cosmic microwave background radiation; Gaussianity; spherical random fields

Multifractal study of Internet traffic

The self-similar/multifractal nature of Internet traffic has been observed by several measurements and statistical studies. It has not been decided yet whether the traces are following either self-similar or multifractal flows. In this paper we analyzed Internet data traffic, both the classical Bellcore-data and data measured at our MAN by a local protocol analyzer. A new multifractal stochastic model was applied and showed its relevance in these cases.

Internet Traffic Modeling with Lévy Flights

Measurements of local and wide-area network traffic in the 90s established the relation between burstiness and self-similarity of network traffic. Several papers demonstrated that the widely used Poisson based models could not be applied for the past decades network traffic. If the traffic had been a Poisson process, the traffics burst lengths would have been smoothed by averaging over a long time scale contradicting with the observations of the past decades traffic characteristics. Poisson models were abandoned as unsuitable characterizations of network traffic. Recent papers have questioned the direct applicability of these results in networks of the new century. Some authors of these papers demand the revision of previous assumptions on the Poisson traffic models. They argue that as newer and newer network technologies are implemented and the amount of Internet traffic grows exponentially, the burstiness of network traffic might cancel out due to the huge number of aggregated traffic flows. Some results are based on analyses of high-speed Internet backbone links and other traffic traces. We analyzed the same traffic traces and applied novel methods to characterize them in terms of packet interarrival time. We demonstrate that the series of interarrival times is still close to a self-similar process.

On the Frequency Variogram and on Frequency Domain Methods for the Analysis of Spatio‐Temporal Data

The covariance function and the variogram play very important roles in modelling and in prediction of spatial and spatio-temporal data. The assumption of second order stationarity, in space and time, is often made in the analysis of spatial data and the spatio-temporal data. Several times the assumption of stationarity is considered to be very restrictive, and therefore, a weaker assumption that the data is Intrinsically stationary both in space and time is often made and used, mainly by the geo-statisticians and other environmental scientists. In this paper we consider the data to be intrinsically stationary. Because of the inclusion of time dimension,the estimation and derivation of the sampling properties of various estimators related to spatio-temporal data become complicated. In this paper our object is to present an alternative way, based on Frequency Domain methods for modelling the data. Here we consider Discrete Fourier Transforms (DFT) defined for the (Intrinsic) time series data observed at several locations as our data, and then consider the estimation of the parameters of spatio-temporal covariance function, estimation of Frequency Variogram, tests of independence etc. We use the well known property that the Discrete Fourier Transforms of stationary time series evaluated at distinct Fourier Frequencies are asymptotically independent and distributed as complex normal in deriving many results considered in this paper.

Statistical Analysis of Next Generation Network Traffics Based on Wavelets and Transformation ON/(ON+OFF)

The significant increase of trunk channel bandwidth makes much easier to integrate different types of traffics on the tier links without activating high processing power consuming QoS (Quality of Service) mechanisms in the intermediate nodes. Self-similarity, long range dependence and fractal characteristics of packet flows are strongly influenced by the QoS parameters in congested network environment. Several models are proposed for the qualitative and quantitative evaluation of physical phenomenon supervened on different OSI layers at the routers and switches. Most of these claims relatively long traces for evaluating both scale independence and fractal characteristics. The highlights of common usage of wavelet and ON/(ON+OFF) transformations in network traffic analysis are evaluated in this chapter. We take into consideration the channel load and the channel intensity as complex time series for evaluation the statistical characteristics of changes in time of the flows nature in packet switched networks. UDP and TCP traffics in tier and LAN networks are considered and statistically analyzed based on MRA (Multi Resolution Analysis) wavelets method. A fast detection algorithm of data and real time traffic burstiness is presented for a QoS based packet switched network environment with congestion.

Wavelet analysis of QoS based network traffic

Integration of different network services into a converged infrastructure is one of the most intensive challenges in the converged network application development area. Huge differences exist regarding several characteristics of data, voice and video content traffics. While data traffic requires error free services, real time applications like interactive video transfer and VoIP distress of the high delay and jitter values. DiffServ becomes more and more popular Quality of Service (QoS) mechanism not only in WAN but LAN environment as well. Coloring the IP packets of different traffic streams based on QoS traffic classes generate increased diversification of the statistical characteristics detected at the measuring points of the production network. Self similarity, long range dependence and fractal characteristics of these packet flows are strongly influenced by the QoS parameters in congested network environment. Several models are proposed for the qualitative and quantitative evaluation of physical phenomenon supervened on different OSI layers at the intermediate nodes. Most of these claims relatively long traces to evaluate the scale independence and fractal characteristics. The wavelets based Multi Resolution Analysis (MRA) proposes fast pyramidal algorithm requiring ∼O(n) computation steps for determining the self similarity measure of the traces with n samples, consisting serious interest in the low delay aspect of the burst detection. In this paper several UDP and TCP traffics are considered and statistically analyzed based on MRA method. Fast detection algorithm of real time traffic burstiness is presented for QoS based packet switched WAN environment with congestion.

Long-range dependence in third order and bispectrum singularity

In this paper the third order long-range dependence (LRD) is defined in terms of the bispectrum and third order cumulants (bicovariances). Two particular non-Gaussian processes with second order LRD are considered together with their bispectra and bicovariances.