Journal of Applied Nonlinear Dynamics 2(2) (2013) 141–150 Journal of Applied Nonlinear Dynamics Journal homepage: www.lhscientificpublishing.com/journals/JAND.html Self-Similar Property of Random Signals: Solution of Inverse Problem Raoul R. Nigmatullin 1† and J.A. Tenreiro Machado 2 1 Theoretical Physics Department, Institute of Physics, Kazan(Volga Region) Federal University, Kremlevskaya str., 18, 420008, Kazan, Tatarstan, Russian Federation 2 ISEP-Institute of Engineering, Polytechnic of Porto, Department of Electrical Engineering, Rua Dr. Antonio Bernardino de Almeida, 431 4200-072 Porto, Portugal Submission Info Communicated by Albert C.J. Luo Received 28 September 2012 Accepted 9 October 2012 Available online 1 July 2013 Keywords Scaling equation Inverse solution Self-similar properties of random signals Abstract Many random signals with clearly expressed trends can have self- similar properties. In order to see this self-similar property new presentation of signals is suggested. A novel algorithm for inverse solution of the scaling equation is developed. This original algo- rithm allows finding the scaling parameters, the corresponding power-law exponent and the unknown log-periodic function from the fitting procedure. The effectiveness of algorithm is tested in financial data revealing season fluctuations of annual, monthly and weekly prices. The general recommendations are given that allow the verification of this algorithm in general data series. ©2013 L&H Scientific Publishing, LLC. All rights reserved. 1 Introduction and formulation of the problem During three decades of intensive research it became clear that our world is presumably fractal and repeats itself on different scales, both in space and time. But nowadays it is not sufficient to say that the object/system studied has self-similar properties. It is necessary to find the fractal dimension, to determine the limits of applicability of the scaling properties and to prove the evi- dence/absence of log-periodic oscillations [1] that accompany any scaling process in time or space. The same phenomena were discovered in random economical and financial activities [2]. In general, research in this field simply supposes or postulates the existence of scaling properties of the system studied. Nevertheless, if the researcher did not make this supposition initially then he must find the justification evidences that the system has really scaling/self-similar properties. How to find the convincing arguments for the skeptical scholar if a scientist has only a set of numerical data † Corresponding author. Email address: renigmat@gmail.com, jtm@isep.ipp.pt ISSN 2164 - 6457, eISSN 2164 - 6473/$- see front materials © 2013 L&H Scientific Publishing, LLC. All rights reserved. DOI : 10.5890/JAND.2013.04.003