Atomic Decomposition of Financial Data

When looking at a time series, it is often instructive to consider the data as observations sampled from a noisy version of some underlying data generating process. This data generating process may be considered to be a function from a function space. We can specify very simple functions, known as atoms, which may be taken in linear combinations to represent any function within a particular function space. The atoms are described as members of a family of functions indexed by parameters. Quite commonly used for functions underlying time series data are the parameters location and frequency. This type of atom is known as a time-frequency atom. After we have specified the family of atoms that we wish to use to represent our underlying data generating process, the difficult problem of choosing the most effective, parsimonious representation from this family remains to be solved. Several techniques, such as Matching Pursuit and Basis Pursuit, have been suggested to solve this problem. In the current study, we investigate the use of several families of atoms, both individually and in combination, to decompose exchange rate data in search of structure that has been overlooked in more traditional approaches.