Recent manuscripts

ˇ        Gy. Terdik, W.A. Woyczynski, A. Piryatinska, (2006) Fractional- and integer-order moments, and multiscaling for smoothly truncated Lévy flights, Physics Letters A 348  94–109
We study multiscale properties of smoothly truncated Lévy flights. The behavior of both fractional- and integer-order moments 〈|X_{a}(t)|^{ρ}〉, for both small and large values of the scaling parameter a is investigated. In the former case we obtain the behavior close to that of the α-stable flight, and for the latter, close to that of the Brownian motion.

ˇ        Gy. Terdik, AW.A. Woyczynski, and A. Piryatinska, (2005) Are Levy flights multiscale? Pure Mathematics and Applications 15, 323-333.

ˇ        György TERDIK  and Wojbor A. WOYCZYŃSKI, Rosiński Measures for Tempered Stable and Related Ornstein-Uhlenbeck Processes
Several concrete parametric classes of tempered stable distributions are discussed in terms of explicit calculations of their Rosiński measures. The hope that they will provide a family of concrete models useful in applied areas and for which the fitting can be done by parametric methods. Related Ornstein-Uhlenbeck processes are analyzed. The emphasis throughout the paper is on obtaining exact analytic formulas.

ˇ        Gy. Terdik, T. Subba Rao and S. Rao Jammalamadaka, On Multivariate Nonlinear Regression Models with Stationary Correlated Errors. To appear: JSPIRoy Volume, 2006
In this paper we consider the statistical analysis of multivariate multiple nonlinear regression models with correlated errors, using Finite Fourier Transforms. Consistency and asymptotic normality of the weighted least squares estimates are established under various conditions on the regressor variables. These conditions involve different types of scalings, and the scaling factors are obtained explicitly for various types of nonlinear regression models including an interesting model which requires the estimation of unknown frequencies. The estimation of frequencies is a classical problem occurring in many areas like signal processing, environmental time series, astronomy and other areas of physical sciences. We illustrate our methodology using two real data sets taken from geophysics and environmental sciences. The data we consider from geophysics  is polar motion (which is now widely known as "Chandlers Wobble") where one has to estimate the drift parameters, the offset parameters and the two periodicities associated with elliptical motion. The data was first analyzed by Arato, Kolmogorov and Sinai who treat it as a bivariate time series satisfying a finite order time series model. They estimate the periodicities using the coefficients of the fitted models. Our analysis shows that the two dominant frequencies are 12 hours and 410 days. The second example, we consider is the minimum/maximum monthly temperatures observed at the Antarctic Peninsula (Faraday/Vernadsky station). It is now widely believed that over the past 50 years there is a steady warming in this region, and if this is true, the warming has serious consequences on ecology, marine life etc. as it can result in melting of ice shelves and glaciers. Our objective here is to estimate any existing temperature trend in the data and we use the nonlinear regression methodology developed here to achieve that goal.

ˇ        S. Rao Jammalamadaka, Gyorgy Terdik and T. Subba Rao (2005), Higher oder Cumulants of Random Vector, Differential Operators, and Applications to Statistial Inferences and Time series
This paper provides a unified and comprehensive approach that is useful in deriving expressions for higher order cumulants of random vectors. The use of this methodology is then illustrated in three diverse and novel contexts, namely: (i) in obtaining a lower bound (Bhattacharya bound) for the variance-covariance matrix of a vector of unbiased estimators where the density depends on several parameters, (ii) in studying the asymptotic theory of multivariable statistics when the population is not necessarily Gaussian and finally, (iii) in the study of multivariate nonlinear time series models and in obtaining higher order cumulant spectra. The approach depends on expanding the characteristic functions and cumulant generating functions in terms of the Kronecker products of differential operators. Our objective here is to derive such expressions using only elementary calculus of several variables and also to highlight some important applications in statistics.

 

Computer network traffic has recently been the subject of various types of statistical studies including fractal analysis, and in particular, measuring and modeling Long-Range Dependence (LRD), investigating self-similarity, and showing multifractal properties . The common agreement among several empirical findings about the general properties of traces is summarized in as follows.

    1. Many signals show significant LRD, but behavior inconsistent with strict self-similarity.
    2. For many signals, the scaling behavior of moments as the signal is aggregated is a nontrivial function of the moment order.
    3. Many signals have increments that are inherently positive, skewed and hence non-Gaussian.

    There are some additional properties motivated by our experimental study of ATM traces, providing strong evidence of the presence of Γ distribution and real-valued bispectrum.  ∙  The marginal distribution of signals of ATM traces is close to Γ distribution.

    1. Signals of ATM traces have a real-valued bispectrum.

    Having these properties in mind we will study in the present paper a certain nonlinear diffusion process, superposition of such processes with random coefficients, the limit of the centralized integral processes of the superposition processes and its increment process. We will consider a possible application too. The main objective is to find a multifractal model which has an analytically and statistically tractable higher order cumulant structure.

 In Section 1 we introduce the notions of multifractality, self-similarity and dilative stability. In Section 2 the basic process of superposition is considered. We call it Diffusion with Linear differential Generator (DLG). It is pointed out that the finite-dimensional distribution of the DLG process is multivariate Γ.

    In Section 3 we will consider a triangular array of random coefficient DLG processes. The covariance structure of the LISDLG process is the same as that of the Fractional Brownian Motion (FBM). Consequently, the spectrum of the ΔLISDLG process is the same as that of the discrete time Fractional Gaussian Noise. The bispectrum of the ΔLISDLG process is also given, and it has the extraordinary property that it is real-valued. If the bispectrum of a linearly regular long-range dependent discrete time process is real (non-zero), then the process must be nonlinear. The logarithm of the m^{th} order cumulants of the LISDLG process scales linearly with the logarithm of time. A similar scaling behavior also holds for the ΔLISDLG process. In particular, below means that the logarithm of the m^{th} order cumulants of the series averaged at level n depends linearly on log n with coefficient 2(H-1) being independent of m, where H is the Hurst parameter. This fact is more special than the multifractal property, nevertheless it holds under more general circumstances than our case, e.g. for the OU type models of , see Subsection 3.5. Moreover, the LISDLG process has an interesting property, which the authors call dilative stability, which implies the above mentioned scaling behaviour

    In Section 4 we will apply our ΔLISDLG model to real data. The time series of ATM traces measured in SUNET fits our model very well. The feasibility of carrying out parameter estimation utilizing the dilative stability is also discussed to some extent.

    An appendix giving some basic facts of multivariable Γ distribution closes the paper.

 

  • Endre Iglói, György Terdik(1999) LONG-RANGE DEPENDENCE THROUGH GAMMA-MIXED ORNSTEIN--UHLENBECK PROCESS  The limit process of aggregational models (i) sum of random coefficient AR(1) processes with independent Brownian motion (BM) inputs and (ii) sum of AR(1) processes with random coefficients of Gamma distribution and with input of common BM's, proves to be Gaussian and stationary and its transfer function is the mixture of transfer functions of Ornstein-Uhlenbeck (OU) processes by Gamma distribution. It is called Gamma-mixed Ornstein -Uhlenbeck process (GMOU). For independent Poisson alternating 0{1 reward processes with proper random intensity it is shown that the standardized sum of the processes converges to the standardized GMOU process. The GMOU process has various interesting properties and it is a new candidate for the successful modeling of several Gaussian stationary data with long-range dependence. Possible applications and problems are also considered Keywords Stationarity, Long-range dependence, Spectral representation, Ornstein-Uhlenbeck process, Aggregational model, Stochastic differential equation, Fractional Brownian motion input, Heart rate variability.
  • Endre Iglói, György Terdik(1997), Bilinear Stochastic Systems with Fractional Brownian Motion Input.Technical report No. 97/12, Kossuth University of Debrecen, Department of Mathematics and Informatics
    The partial derivatives according to the time and the fractional Brownian motion of some particular class of stationary processes are defined. Althought the fractional Brownian motion is not semimartingale the bilinear SDE with fractional Brownian motion input is considered and solved. The solution is explicitly given in both frequency and time domain in case when the coefficient of the bilinear term is pure imaginary. The stationary Stratonovich solution of the bilinear SDE with white noise input is also considered.
  • Endre Iglói, György Terdik(1996), Bilinear Stochastic Systems with Long Range Dependence in Continuous Time.
        Application of long range dependent model includes several fields of science and economics as geophysics, hydrology, turbulence, weather and so on. Recently it has been successfully used for modelling network traffic data. The basic stochastic process of this kind is the fractional Brownian motion defined in . The fractional Brownian motion is given as a particular fractional operator on the standard Brownian motion. The linear or Gaussian parametric models of long range dependent phenomena are both the linear stochastic differential equations with fractional Brownian motion input and the fractional operator on the solution of a linear stochastic differential equation. Actually these two types of processes are equivalent. Because of most of the observations are not Gaussian there is a need of nonlinear modelling for long range dependence. One of the possibility to get rid of Gaussianity is the bilinear model started by Subba Rao in discrete time case. The easy way to get a long range nonGaussian process is putting the fractional operator on the solution of the bilinear SDE. It is more painful to consider a bilinear SDE with fractional Brownian motion input.
    In this paper we start with the bilinear SDE with white noise input and list the basic ideas leading to the stationary solution given in both time domain and chaotic frequency domain forms. The fractional integral operator on this stationary solution is applied and its basic properties are pointed out. In section three the bilinear SDE with fractional Brownian motion input is considered. The problem of the stochastic integration by the fractional Brownian motion is solved and the stationary solution of the SDE is explicitly given in case when the coefficient of the bilinear term is pure imaginary.
  • György Terdik(1999), Bilinear Stochastic Models and Related Problems of Nonlinear Time Series Analysis
  • Gy. Terdik and J. Máth(1998), A new test of linearity for time series based on the bispectrum. J. of Time Series, vol. 19, No 6, pp. 737-753.
        The linearity of a time series is checked by the help of its bispectrum. The linearity is meant by the most general definition, i.e. the time series is linear if the best predictor is linear. The bispectrum is estimated via stretching the data and smoothing by Rao-Gabr optimal window. The test is worked out for linearity versus quadratic predictability. It turns out that the test statistics is asymptotically χ˛ -distributed under the hypotheses that the time series is linear. The results are demonstrated by simulation and real data.
       Key words: Linearity Test, Bispectrum, Quadratic Prediction, Bilinear Model