Fractal dimensions for rainfall time series

Abstract

Fractals are objects which have a similar appearance when viewed at different scales. Such objects have detail at arbitrarily small scales, making them too complex to be represented by Euclidean space. They are assigned a dimension which is non-integer. Some natural phenomena have been modelled as fractals with success; examples include geologic deposits, topographical surfaces and seismic activity. In particular, time series data has been represented as a curve with dimension between one and two. There are many different ways of defining fractal dimension. Most are equivalent in the continuous domain, but when applied in practice to discrete data sets lead to different results. Three methods for estimating fractal dimension are evaluated for accuracy. Two standard algorithms, Hurst's rescaled range analysis and the box-counting method, are compared with a recently introduced method which has not yet been widely used. It will be seen that this last method offers superior efficiency and accuracy, and it is recommended for fractal dimension calculations for time series data. We have applied these fractal analysis techniques to rainfall time series data from a number of gauge locations in Queensland. The suitability of fractal analysis for rainfall time series data is discussed, as is the question of how the theory might aid our interpretation of rainfall data.