The Analysis of Foreign Exchange Data Using Waveform Dictionaries

This paper uses waveform dictionaries to decompose the signals contained within three foreign exchange rates using tick-by-tick observations obtained world wide. The three exchange rates examined are the Japanese Yen and the German Deutsche Mark against the U.S. dollar and the Deutsche Mark against the Yen. The data were provided by Olsen Associates. A waveform dictionary is a class of transforms that generalizes both windowed Fourier transforms and wavelets. Each wave form is parameterized by location, frequency, and scale. Such transforms can analyze signals that have highly localized structures in either time or frequency space as well as broad band structures; that is, waveforms can, in principle, detect everything from shocks represented by Dirac Delta functions, to \chirps", short bursts of energy within a narrow band of frequencies, to the presence of frequencies that occur sporadically, and nally to the presence of frequencies that hold over the entire observed period. Waveform dictionaries are most useful in analyzing data that are not stationary and non-stationarity up to second order is well recognized in the context of foreign exchange rates.