Application of Hölder Function to Expansion Intensity of Spatial Phenomena Analysis

Adrianna Damiana Mastalerz-Kodzis; Ewa Katarzyna Pośpiech

doi:10.18778/0208-6018.335.04

Authors

Adrianna Damiana Mastalerz-Kodzis University of Economics in Katowice, Faculty of Management, Department of Statistics, Econometrics and Mathematics
Ewa Katarzyna Pośpiech University of Economics in Katowice, Faculty of Management, Department of Statistics, Econometrics and Mathematics

DOI:

https://doi.org/10.18778/0208-6018.335.04

Keywords:

stochastic process, Hurst exponent, Hölder function, spatial modelling

Abstract

The development of methods describing time series using stochastic processes took place in the 20th century. Among others, stationary processes were modelled with Hurst exponent, whereas non‑stationary processes with Hölder function. The characteristic feature of this type of processes is the analysis of the memory present in the time series. At the turn of the 21st century interest in statistics and spatial econometrics, as well as analyses carried out within the new economic geography arose. In this article, we have proposed the implementation of methods taken from the analysis of time series in the modelling of spatial data and the application of selected measures in studying the intensity of expansion in spatial phenomena. As the intensity measure we use Hölder point exponents. The article is composed of two parts. The first one contains the description of study methodology, the second – examples of application.

Downloads

Download data is not yet available.

References

Ayache A., Lévy‑Véhel J. (1999), Generalized Multifractional Brownian Motion: Definition and Preliminary Results, [in:] M. Dekking, J. Lévy‑Véhel, E. Lutton, C. Tricot (eds.), Fractals: Theory and Applications in Engineering, Springer‑Verlag, New York.
Google Scholar

Ayache A., Taqqu M.S. (2004), Multifractional processes with random exponent, “Stochastic Processes and their Applications”, no. 111(1), pp. 119–156.
Google Scholar

Baltagi B.H. (2005), Econometric Analysis of Panel Data, John Wiley & Sons, New York.
Google Scholar

Barrière O. (2007), Synthèse et estimation de mouvements browniens multifractionnaires et autres processus à régularité prescrite, Définition du processus autorégulé multifractionnaire et applications. PhD thesis, IRCCyN.
Google Scholar

Bass F. (1969), A New product growth for model consumer durables, “Managment Science”, no. 15(5), pp. 215–227.
Google Scholar

Box G.E.P., Jenkins G.M. (1976), Time series analysis forecasting and control, Holden‑Day, San Francisco.
Google Scholar

Daoudi K., Lévy‑Véhel J., Meyer Y. (1998), Construction of continuous functions with prescribed local regularity, “Journal of Constructive Approximations”, no. 014(03), pp. 349–385.
Google Scholar

Domański R. (2002), Gospodarka przestrzenna, Wydawnictwo Naukowe PWN, Warszawa.
Google Scholar

Echelard A., Barrière O., Lévy‑Véhel J. (2010), Terrain modelling with multifractional Brownian motion and self‑regulating processe, “ICCVG”, no. 6374, pp. 342–351.
Google Scholar

Falconer K.J., Lévy‑Véhel J. (2008), Multifractional, multistable and other processes with prescribed local form, “Journal of Theoretical Probability”, https://link.springer.com/article/10.1007/s10959–008–0147–9 [accessed: .....].
Google Scholar

Fuller W.A. (1996), Introduction to Statistical Time Series, Wiley, New York.
Google Scholar

Getis A., Mur J., Zoller H. (2004), Spatial Econometrics and Spatial Statistics, Palgrave Macmillan, New York.
Google Scholar

Granger C.W.J., Mizon G.E. (1994), Nonstationary Time Series Analysis and Cointegration, Oxford University Press, New York.
Google Scholar

Hagerstrand T. (1952), The propagation and innovation waves, “Lund Studies in Geography”, no. 4, Lund, Gleerup.
Google Scholar

Hsiao C. (2003), Analysis of Panel Data, Cambridge University Press, Cambridge.
Google Scholar

Kopczewska K. (2007), Ekonometria i statystyka przestrzenna, Wydawnictwo CeDeWu, Warszawa.
Google Scholar

Krugman P.R. (1991), Geography and Trade, The MIT Press, Cambridge.
Google Scholar

Lévy‑Véhel J., Mendivil F. (2011), Multifractal and higher dimensional zeta functions, “Nonlinearity”, no. 24(1), pp. 259–276.
Google Scholar

Lévy‑Véhel J., Seuret S. (2004), The 2‑microlocal Formalism, Fractal Geometry and Applications, A Jubilee of Benoit Mandelbrot, “ Proceedings of Symposia in Pure Mathematics”, no. 72(2), pp. 153–215.
Google Scholar

Mandelbrot B.B. (1982), The Fractal Geometry of Nature, WH Freeman & Co, New York.
Google Scholar

Mastalerz‑Kodzis A. (2003), Modelowanie procesów na rynku kapitałowym za pomocą multifraktali, “Prace Naukowe”, Akademia Ekonomiczna im. Karola Adamieckiego w Katowicach, Katowice.
Google Scholar

Mastalerz‑Kodzis A. (2016), Algorytm modelowania danych przestrzennych o zadanej lokalnej regularności, [in:] J. Mika, M. Miśkiewicz‑Nawrocka (eds.), Metody i modele analiz ilościowych w ekonomii i zarządzaniu, Wydawnictwo Uniwersytetu Ekonomicznego w Katowicach, Katowice.
Google Scholar

Matyas L., Sevestre P. (eds.) (2006), The Econometrics of Panel Data, Kluwer Academic Publishers, Dordrecht.
Google Scholar

Paelinck J.H.P., Klaassen L.H. (1983), Ekonometria przestrzenna, PWN, Warszawa.
Google Scholar

Peltier R.F., Lévy‑Véhel J. (1995), Multifractional Brownian Motion: Definition and Preliminary Results, INRIA Recquencourt, Rapport de recherche no. 2645.
Google Scholar

Perfect E., Tarquis A.M., Bird N.R.A. (2009), Accuracy of generalized dimensions estimated from grayscale images using the method of moments, “Fractals”, vol. 17, no. 3, pp. 351–363.
Google Scholar

Peters E.E. (1994), Fractal Market Analysis, John Wiley and Sons, New York.
Google Scholar

Suchecki B. (2010), Ekonometria przestrzenna, Wydawnictwo C.H. Beck, Warszawa.
Google Scholar

Zeliaś A. (ed.) (1991), Ekonometria przestrzenna, PWE, Warszawa.
Google Scholar