The Kohonen algorithm entails a topology conserving mapping of an input pattern space X subsetR(n) characterized by an a priori probability distribution P(x), xinX, onto a discrete lattice of neurons r with virtual positions w(r)inX. Extending results obtained by Ritter (1991) the authors show in the one-dimensional case for an arbitrary monotonously decreasing neighborhood function h(|r-r'|) that the point density D(W(r)) of the virtual net is a polynomial function of the probability density P(x) with D(w(r))~P(alpha)(w(r)). Here the distortion exponent is given by alpha=(1+12R)/3(1+6R) and is determined by the normalized second moment R of the neighborhood function. A Gaussian neighborhood interaction is discussed and the analytical results are checked by means of computer simulations.

Abstract-In this paper, we investigate the effectiveness of a financial time-series forecasting strategy which exploits the multiresolution property of the wavelet transform. A financial series is decomposed into an over complete, shift invariant scale-related representation. In transform space, each individual wavelet series is modeled by a separate multilayer perceptron (MLP). To better utilize the detailed information in the lower scales of wavelet coefficients (high frequencies) and general (trend) information in the higher scales of wavelet coefficients (low frequencies), we applied the Bayesian method of automatic relevance determination (ARD) to choose short past windows (short-term history) for the inputs to the MLPs at lower scales and long past windows (long-term history) at higher scales. To form the overall forecast, the individual forecasts are then recombined by the linear reconstruction property of the inverse transform with the chosen autocorrelation shell representation, or by another perceptron which learns the weight of each scale in the prediction of the original time series. The forecast results are then passed to a money management system to generate trades. Compared with previous work on combining wavelet techniques and neural networks to financial time-series, our contributions include 1) proposing a three-stage prediction scheme; 2) applying a multiresolution prediction which is strictly based on the autocorrelation shell representation, 3) incorporating the Bayesian technique ARD with MLP training for the selection of relevant inputs; and 4) using a realistic money management system and trading model to evaluate the forecasting performance. Using an accurate trading model, our system shows promising profitability performance. Results comparing the performance of the proposed architecture with an MLP without wavelet preprocessing on 10-year bond futures indicate a doubling in profit per trade ($AUD1753:$AUD819) and Sharpe ratio improvement of 0.732 versus 0.367, as well as significant improvements in the ratio of winning to loosing trades, thus indicating significant potential profitability for live trading.

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