# Multifractals: The scalability of random multiplicative processes

Google Tech Talks

May 15, 2007

ABSTRACT

Many interesting data are of sufficient size and complexity that they can only be studied statistically. Broad distributions, such as those produced by a multiplicative process, do not scale according to conventional laws. For these non-Gaussian distributions, a different exponent is needed to describe how each moment of the distribution scales. Multifractal analysis provides a means of determining these scaling exponents for a given data distribution. In this talk, we will introduce the concept of multifractals by first extending the more familiar concept of fractals, and then by deriving the theory of multifractals from statistical mechanics. We then provide an introduction to multiplicative random processes as a universal mechanism underlying the generation of many data. Then, after a brief review of an algorithm often used for calculating the multifractal spectrum (the Wavelet Transform Modulus Maxima), we will present a few applications of multifractals that we hope to be of interest: volatility of financial time series, characterization of (internet) traffic, and probabilistic measures on suffix trees.

May 15, 2007

ABSTRACT

Many interesting data are of sufficient size and complexity that they can only be studied statistically. Broad distributions, such as those produced by a multiplicative process, do not scale according to conventional laws. For these non-Gaussian distributions, a different exponent is needed to describe how each moment of the distribution scales. Multifractal analysis provides a means of determining these scaling exponents for a given data distribution. In this talk, we will introduce the concept of multifractals by first extending the more familiar concept of fractals, and then by deriving the theory of multifractals from statistical mechanics. We then provide an introduction to multiplicative random processes as a universal mechanism underlying the generation of many data. Then, after a brief review of an algorithm often used for calculating the multifractal spectrum (the Wavelet Transform Modulus Maxima), we will present a few applications of multifractals that we hope to be of interest: volatility of financial time series, characterization of (internet) traffic, and probabilistic measures on suffix trees.