Fractal modeling of time series data

Fractals are a relatively new area of mathematics and have seen great use in producing synthetic scenes that look realistic. One way to produce fractals is with Iterated Function Systems (IFS's), and in this thesis we explore four IFS models. The first IFS model is the self-affine fractal model where data are modeled with a fractal that is composed of affine transformations of the fractal in {dollar}Rsp2{dollar}. We present background materials about this model and show how to use it to produce fractal functions. Then we present an inverse algorithm so that this model may be used to produce a given function. An application to a mountain profile is presented.; The second IFS model explored is the piece-wise self-affine fractal model where a fractal is produced that is composed of affine transformations of pieces of itself in {dollar}Rsp2{dollar} and but not necessarily the entire function. We first show how to use this model to generate fractal functions and then give an inverse algorithm so that this model may be applied for the representation of a variety of data types. We illustrate the utility of this model with applications to well-logging data, seismograms, electrocardiograms and speech data. Performance of this model is examined in terms of quantization of the model parameters and with a comparison to the classical modeling technique of autoregressive-moving-average models.; The third model we explore is hidden-variable fractal interpolation where the IFS's operate in three dimensional space, {dollar}Rsp3{dollar}, and the fractal in {dollar}Rsp3{dollar} is composed of affine transformations of itself. We present background material on this model and then give an inverse algorithm for identification of the model parameters so that this model may be applied to arbitrary data. Applications are presented where the model is used to represent sunspot data and electrocardiograms.; The fourth model is the piece-wise hidden-variable model where this model operates in three dimensional space yet the IFS is such that pieces of the fractal in {dollar}Rsp3{dollar} are mapped to other pieces of the fractal. An inverse algorithm is provided for this model and applications are given to seismograms and speech data.