Multivariate embedding methods: forecasting high-frequency financial data in the first INFFC

Forecasting software is described, where each point to be forecast is embedded in an mdimensional library made from historic data. The approach is based on the well-known 'nearest neighbour' algorithm of Casdagli (1989) but there are important differences, including the facility for multivariate embedding, the use of predictor variables which may be different from the embedding variables, and the 'rolling library' which is of a constant size but is continuously updated as each successive point is forecast. In this way the univariate Casdagli algorithm has been developed into a more sophisticated 'pattern recognition' technique for short-term forecasting, whilst augmenting its original purpose of searching for evidence of chaos in time series. Because each point to be forecast has its own parameter estimates a certain amount of variability between successive forecasts is to be expected. However it was interesting to find that forecasts made on the training data were in fact exceptionally smooth over certain periods so that for some time (not usually longer than a few days) all points fell within similar close point groups. On the other hand there were other, shorter periods (typically a few hours long) where forecasts became 'chaotic', because adjacent points fell into totally different areas of the library. Hence a by-product of our work for the INFFC has been to provide empirical evidence of the local stability results of Yao and Tong (1994).