An entropy based in wavelet leaders to quantify the local regularity of a signal and its application to analyze the Dow Jones index

Abstract

Local regularity analysis is useful in many fields, such as financial analysis, fluid mechanics,PDE theory, signal and image processing. Different quantifiers have been proposedto measure the local regularity of a function. In this paper we present a new quantifier of local regularity of a signal: the pointwise wavelet leaders entropy. We define this new measure of regularity by combining the concept of entropy, coming from the information theory and statistical mechanics, with the wavelet leaders coefficients. Also we establish its inverse relation with one of the well-known regularity exponents, the pointwise H¨older exponent. Finally, we apply this methodology to the financial data series of the Dow Jones Industrial Average Index, registered in the period 1928?2011, in order to compare the temporal evolution of the pointwise H¨older exponent and the pointwise wavelet leaders entropy. The analysis reveals that temporal variation of these quantifiers reflects the evolution of the Dow Jones Industrial Average Index and identifies historical crisis events.