Multiresolution Embedding for Time-Series Analysis

Vance Bjorn
MIT Center for Biological and Computational Learning
vcb@ai.mit.edu

Andreas Weigend
Institute of Cognitive Science, University of Colorado at Boulder

Abstract

The standard method of embedding time-series data for analysis to use a moving window. However, by the inverse relationship between time and frequency localization, all information of the signal contained in frequencies with period less than twice the window size is lost when using that embedding. Increasing the window size may lessen the problem, but at the cost of adding more degrees of freedom (the curse of dimensionality). Both of these problems are of particular concern for financial time series, which contain information and processes at many scales. This work solves the challenge of embedding all frequencies/processes that comprise a point with a minimal dimensionality.

We propose the construction of a ``Multiresolution Embedding Matrix", for each time t of the time-series. Each row of this matrix are the three last coefficients of a scale of an edge wavelet transform (Average Interpolation construction method). This matrix contains frequency data of the time-series up to points. Since we are using an orthogonal wavelet transform, the last four points of the time series can be perfectly reconstructed. As we go back in time past four points the reconstruction gradually starts to remove the details of the underlying series. The end result is a representation of the state of the time-series that is very precise for recent points, but increasingly generalized further back in time.

This matrix, M(t), can reconstruct the value of the point t. A learning mechanism such as a neural network can be used to learn the mapping between M(t) and M(t+1). However, we can exploit the fact that the rows of the matrix represent frequencies or processes on different scales, which is perhaps related to the continum of time horizons in financial markets. You do not have to worry about a huge dimensional mapping since we believe structure lies on individual time scales.

We demonstrate the use of this method to answer one of the key issues in time-series analysis - what is the ``best" time scale for predictions. Until recently most instruments were modeled with daily data, but now, high-frequency data is commonly available. We use the multiresolution embedding matrix to model the dynamics on each scale separately. In three separate examples we demonstrate how we can localize the predictable dynamics: 1) on a filtered logistic map (in the time domain), 2) on a series obtained through the evolution of wavelet coefficients, and 3) on the real-world problem of exchange rate prediction.

The following plots show some of the results we have obtained so far using 30 minute DMark data that has been generously provided by Olsen and Associates. The first plot shows an example of reconstruction resulting from the use of edge wavelets It can be clearly seen that the last few points, the ones that contain the most important information for prediction, have been perfectly reconstructed, while points further back gradually lose detail. In this example, the reconstruction continues to closely track the signal for 1024 points while only requiring 30 coefficients. The following two plots show predictability versus scale for four years of the 30-minute DMark data. The results were obtained by using linear prediction to predict the edge wavelet coefficients needed for reconstruction of the next point in each scale at time t+1 from the edge wavelet coefficients at time t. We see that the daily scale has the most predictability.