Time series analysis and fractal models with applications  Page description

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Details of project

  
Identifier
47067
Type K
Principal investigator Terdik, György
Title in Hungarian Idősorok analízise és sztochasztikus fraktál modellek tanulmányozása alkalmazásokkal
Title in English Time series analysis and fractal models with applications
Panel Mathematics and Computing Science
Department or equivalent Department of Information Technology (University of Debrecen)
Participants Fazekas, István
Iglói, Endre
Ispány, Márton
Kormos, János
Starting date 2004-01-01
Closing date 2008-12-31
Funding (in million HUF) 7.400
FTE (full time equivalent) 0.00
state closed project





 

Final report

  
Results in Hungarian
Frakcionális Ornstein—Uhlenbeck lepedő tulajdonságait vizsgáltuk abból a szempontból, hogy olyan modellt konstruáljunk, aminek spektruma az origó környezetében nem izotróp módon viselkedik. A Lévy-Flight-ok (közel stabilis Lévy-folyamtok) fontos szerepet játszanak a nem Gauss jelenségek vizsgálatában. Egzakt eredményeket bizonyítottunk a Lévy-Flight-ok aszimptotikus egész és tört rendű momentumaira, ezzel összefüggésben sikerült kimutatni ezek multi-fraktál tulajdonságát. A nemlineáris vektor értékű regresszió problémájával foglalkoztunk, amikor a megfigyelések hibája stacionárius eloszlású. Egzakt formulát adtunk meg a paraméter becslések aszimptotikus szórásmátrixára, és alkalmaztuk eredményünket valódi adatokra is. A magfüggvényes sűrűségfüggvény becslés aszimptotikus normalitását bizonyítottuk úgy, hogy a mezőt egyre nagyobb tartományon figyeljük meg, de közben megfigyelési helyeket is sűrítjük. Kiderül, hogy az aszimptotikus kovariancia függ a sávszélesség és az osztópontok távolságának arányától.
Results in English
The Fractional Ornstein—Uhlenbeck sheet is investigated, non-isotropic stationary model is constructed and applied for real data. We showed that Lévy-Flights are fractals and proved asymptotical formulae for moments and cumulants.with integer and fractal order. Functional limit theorems are proved for a sequence of Galton-Watson processes with immigration, where the offspring mean tends to its critical value 1 under weak conditions for the variances of offspring and immigration processes. Int he limit theorems the norming factors depend on these variances. We proved the asymptotic normality of the kernel density estimates in 2D. The asymptotic covariance is shown to be dependent of the ratio of the window size and distance between point on the lattice. The nonlinear multiple regression with stationary errors is investigated. A clear formula is given for the asymptotic variance of the parameter estimator and it is applied for the identification of fitting models to real data.
Full text http://real.mtak.hu/1686/
Decision
Yes





 

List of publications

  
Subba Rao, T. and Terdik, Gy: On the estimation and tests associated with multivariate multiple regression- with applications to environmental time series, HYDSTAT Int. Conf on the FUTURE OF STATISTICAL THEORY, PRACTICE AND EDUCATION December 29, 2004-- January 1, Indian School of Business, HYDERABAD, ANDHRA PRADESH, INDIA, 2004
Fazekas, I. and Chuprunov, A.: A central limit theorem for random fields., Acta Mathematica Academiae Paedagogicae Nyíregyháziensis, 20, no. 1, 93--104, 2004
Fazekas, I., Kukush, A. and Zwanzig, S.: Correction of nonlinear orthogonal regression estimator., Ukrainian Mathematical Journal, 56, no. 8, 1101--1118, 2004
Terdik, György; Woyczynski, Wojbor A.; Piryatinska, Alexandra,: Are Lévy-Flights Multifractals?, 5th Joint Conference on Mathematics and Computer Science, June 9-12, Debrecen, Hungary, 2004
Terdik, György; Woyczynski, Wojbor A.; Piryatinska, Alexandra,: Are Lévy-Flights Multifractals?, Pure Mathematics and Applications, 15, 323-333, 2005
Terdik, Gy,: Multivariate Non-Linear Regression with Dependent Residuals, Workshop STATDEP 2005, STATISTICS FOR DEPENDENT DATA, C R E S T (Centre de Recherche en Economie et Statistique) January 26-29, PARIS/MALAKOFF, 2005
Fazekas, I. and Chuprunov, A.: Integral analogues of almost sure limit theorems, Periodica Mathematica Hungarica, 50, 1-2, 61-78, 2005
Fazekas, I., Kukush, A.: On covariance-matching constrained kriging., Perspectives in Modern Statistical Inference III, Mikulov, Czech Rebublic, July 18-22, 2005, 2005
Fazekas, I., Kukush, A.: Kriging and prediction of nonlinear functionals., MicroCAD, Miskolc, Hungary, 2005, 2005
E. Iglói: A rate-optimal trigonometric series expansion of the fractional Brownian motion, Electronic J. Probability (EJP), 10, 1381-1397, 2005
Fazekas, I. and Chuprunov, A.: Almost sure limit theorems for random allocations, Studia Sci. Math. Hung. 42(2), 173--194, 2005
Chuprunov, A. and Fazekas, I.: Inequalities and strong laws of large numbers for random allocations., Acta Math. Hung., 109(1-2), 163--182, 2005
Subba Rao, T. and Terdik, Gy: Non-Linear Regression with Applications, Springer Verlag, New York, Lect. Notes in Statistics No.187, Dependence in Probability and Statistics, P. Bertail, P. Doukhan, P. Soulier, eds., 2006
Terdik, Gy. , W. A. Woyczynski,: Rosiński Measures for Tempered Stable and Related Ornstein-Uhlenbeck Processes, PMS, Urbanik Volume, Vol. 26, Fasc. 2, pp213--243, 2006
Ispány, M.: Statistical inference for multitype branching processes with immigration., 9th Vilnius Conference on Probab. Theory, June 25-30, 2006, Vilnius, Lithuania, p. 164., 2006
Fazekas, I. and Rychlik, Z.: Almost sure limit theorems for semi-selfsimilar processes., Probability and Mathematical Statistics. 25(2), 241--255, 2005
Fazekas, I.: Burkholder's inequality for multiindex martingales., Annales Mathematicae et Informaticae. 32, 45--51, 2005
Fazekas, I. and Chuprunov, A.: Asymptotic normality of kernel type density estimators for random fields., Stat. Inference Stoch. Process. 9(2), 161--178, 2006
Fazekas, I. and Filzmoser, P.: A functional central limit theorem for kernel type density estimators., MicroCAD, Miskolc, Hungary, 2006., 2006
Ispány, M.: A new martingale limit theorem with applications for critical branching processes with immigration, 56th Session of the ISI 22 - 29 AUG, Lisboa 2007, 2007
Fazekas, I.: Almost sure limit theorems for semi-selfsimilar processes., Levy Processes: Theory and Applications, Copenhagen, August 13-17, 2007
Terdik, Gy.: Lévy flights and tempered stable distributions, XXVI. Seminar on Stability Problems for Stochastic Models, August 27 - September 2, Sovata-Bai, Romania, 2006
Terdik, Gy.: Non-Gaussian long range dependence, International Conference "Modern Stochastics:Theory and Applications, INTERNATIONAL CONFERENCE, MODERN STOCHASTICS: THEORY AND APPLICATIONS, Kyiv, Ukraine, 2006
Terdik, Gy. Subba Rao, T. and Rao Jammalamadaka,: On multivariate nonlinear regression models with stationary correlated errors, Journal of Statistical Planning and Inference Volume 137, Issue 11, 1 November 2007, 3793-3814, 2007
Fazekas, I. and Chuprunov, A.: An almost sure functional limit theorem for the domain of geometric partial attraction of semistable laws., J. Theoretical Probability, 20 (2), 339--353, 2007
Terdik, György; Woyczynski, Wojbor A.; Piryatinska, Alexandra,: Fractional- and integer-order moments, and multiscaling for smoothly truncated Lévy flights, Physics Letters A, 348, 94–109, 2006
Fazekas, I., Kukush, A.: Kriging and prediction of nonlinear functionals., Austrian Journal of Statistics, 34(2), 175-184, 2005
Gyorgy Terdik and Wojbor A. Woyczynski: Notes on fractional Ornstein--Uhlenbeck random sheets, Publ. Math. Debrecen, 66/1-2, 153-181, 2005
E. Iglói: Renormalization group of and convergence to the LISDLG process, ESAIM: Probability and Statistics 8, 102-114, 2004
Rao Jammalamadaka, Subba Rao, T. and Terdik, Gy: Higher order cumulants of random vectors and applications to statistical inference and time series, Sankhya (A, Methodology), The Indian Journal of Statistics, Volume 68, Part 2, pp. 326-356, 2006
Ispány, M.: Limit theorems for normalized nearly critical branching processes with immigration., Publicationes Mathematicae Debrecen 72, 17-34., 2008
Terdik, Gy. , Gyires, T.: Levy Flights and Fractal Modeling of Internet Traffic, IEEE/ACM Transactions on Networking, nyomdában, 2008
Ispány, M., Krasznahorkay, I: On speed of stochastic CART model search, Austrian Journal of Statistics 36, Nr. 1., 27-40, 2007



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