Construction of a Simple Multifractal Spectrum as an Alternative to the Legendre Spectra in Multifractal Formalisms


  • Roberto N. Padua
  • Joel G. Adanza Negros Oriental State University
  • Efren O. Barabat University of San Jose-Recoletos
  • Dionisel Y. Regalado



monofractal, multifractal statistical observations, power-law


It is important to realize at the beginning of a statistical analysis whether the data are from a monofractal or multifractal distribution because the methods of analysis are different for each. In seismic sequence analysis, for instance, the monofractal method uses the R/S and DFA (range-scale and detrended fluctuation analysis, respectively) techniques while the multifractal formalism uses the partition function technique (PFT) and the Legendre spectra outputting three parameters: maximum  of the spectrum, asymmetry B and width W of the curve (Lapenna et al. (2003)). In this paper, we introduce a simple test of mono or multifractality of data sets. The test is based on fitting a power-law distribution to a random sample obtained from some unknown distribution G(.). For each quantile, a fractal dimension  is obtained. This corresponds to the Legendre spectra or multifractal spectra. A regression function is fitted to the points (tk,  ) and the slope b of this line is tested. If b = 0, then the observations are deduced to have come from a monofractal distribution f(x). The paper proposed a test for monofractality which, in effect, also tests for multifractality or non-fractality of a set of observations. For monofractal observations, the proposed new multifractal spectral analysis revealed a single point (singularity at a point) while for multifractal observations, a single-humped continuous quadratic function is observed. The parameters of the quadratic function are interpreted as the measure of asymmetry (B), ruggedness (C) and width (W). The new proposed multifractal spectrum function is easier to calculate and is consistent with the more complicated Legendre spectrum proposed in the literature.

Author Biographies

Roberto N. Padua

Dr. Roberto Natividad Padua, scientist, received his PhD in Mathematical Statistics from Clemson University, South Carolina, USA under the Fullbright-Hays scholarship grant. He is an accomplished author, a multi-awarded researcher and an internationally acclaimed lectured. Dr. Padua ia a former Commissioner of CHED and currently a consultant to several State and Private Universities, conducting lectures on research and Fractal Statistics. Dr. Padua is a Summa cum Laude, BS in Mathematics Teaching graduate from the Philippine Normal College under the National Science Development Board (NSDB) program. He obtained his MS in Mathematics Education Degree from the Centro Escolar University as a Presidential Scholar.

Joel G. Adanza, Negros Oriental State University

graduated from Mapua Institute of Technology with the degree of Bachelor of Science in Industrial Engineering in 1977 and finished his MA. Ed.(major in Mathematics) at the University of Perpetual Help System. He has co-authored three books in Statistics and Logic. He has also presented research papers in graph theory and mathematics education in the annual and regional convention of the MSP. At present he is an assistant professor II teaching mathematics at the Negros Oriental State University. He is also pursuing his Ph.D. in Mathematics.

Efren O. Barabat, University of San Jose-Recoletos

an Electronics Engineer, graduated from the University of San Jose-Recoletos in 2010, Cum Laude honors. He ranked as top 9 examinee in the April 2011 ECE Licensure Examination. He worked as Field Engineer in SMART Communications, Inc. from 2011 to 2012. Currently, a full-time faculty member of the Electronics Engineering Department of USJ-R College of Engineering, handling Mathematics and Major Subjects of ECE.

Dionisel Y. Regalado

research assistant, is a graduate from Mindanao University of Science and Technology with a degree in Bachelor of Science in Mathematics Teaching.


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How to Cite

Padua, R. N., Adanza, J. G., Barabat, E. O., & Regalado, D. Y. (2013). Construction of a Simple Multifractal Spectrum as an Alternative to the Legendre Spectra in Multifractal Formalisms. Recoletos Multidisciplinary Research Journal, 1(1).




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