Multifractals: Theory and Applications

Abstract

There is not an accepted definition of a multifractal measure, similar to the status of a fractal set. However, like the fractal set, there is an accepted notion that a multifractal probability measure is very irregular and does not have a density function. As such, we need other methods to describe such measures. The multifractal formalism describes the relationship between local power-law behaviour of a measure and global averaging (Rényi dimensions). The relationship between these two aspects takes the form of a Legendre transformation, and a measure satisfying such relationships may be termed a multifractal measure. Further, there may also be a requirement for a relationship with the Hausdorff dimension. The above ideas are reviewed in Part I of the thesis. The Legendre transformation relationship also appears in the theory of large deviations, relating a type of cumulant generating function to the power-law decay rate of a probability function. There is a direct relationship between certain multifractal constructions and the theory of large deviations. We appeal to this theory to determine the conditions under which a Borel probability measure conforms to the Legendre transformation relationships under two regimes: a lattice based construction, and a point centred construction. A parallel development using large deviations for each construction type is given. It is shown that each can be set up to follow an almost identical pattern. This construction can also be used for a particular random cascade construction, but will not work for cascades where the partitioning is irregular. These results form Part II of the thesis. The correlation dimension is a member of the family of Rényi dimensions, and lends itself to relatively easy estimation techniques. It is studied in Part III of the thesis. Various sources of bias are discussed, both those that are intrinsic to the process under investigation, and extrinsic or more a consequence of the (possibly deficient) experimental methodology. Properties of a modified Hill estimator combined with a bootstrap procedure are also investigated. This estimator is used to estimate the correlation dimensions of earthquake hypocentre locations tabulated in four earthquake catalogues. The purpose was to determine whether the fracturing process can be characterised by such an exponent, and if it is different in different regions. In this case study, hypocentre location error probably causes a considerable bias to the dimension estimates.