Financial time series prediction using least squares support vector machines within the evidence framework

For financial time series, the generation of error bars on the point predic tion is important in order to estimate the corresponding risk. The Bayesian evidence framework, already successfully applied to design of multilayer p erceptrons, is applied in this paper to least squares support vector machin e (LS-SVM) regression in order to infer nonlinear models for predicting a t ime series and the related volatility. On the first level of inference, a s tatistical framework is related to the LS-SVM formulation which allows to i nclude the time-varying volatility of the market by an appropriate choice o f several hyperparameters. By the use of equality constraints and a 2-norm, the model parameters of the LS-SVM are obtained from a linear Karush-Kuhn- Tucker system in the dual space, Error bars on the model predictions are ob tained by marginalizing over the model parameters. The hyperparameters of t he model are inferred on the second level of inference. The inferred hyperp arameters, related to the volatility, are used to construct a volatility mo del within the evidence framework. Model comparison is performed on the thi rd level of inference in order to automatically tune the parameters of the kernel function and to select the relevant inputs. The LS-SVM formulation a llows to derive analytic expressions in the feature space and practical exp ressions are obtained in the dual space replacing the inner product by the related kernel function using Mercer's theorem. The one step ahead predicti on performances obtained on the prediction of the weekly 90-day T-bill rate and the daily DAX30 closing prices show that significant out of sample sig n predictions can be made with respect to the Pesaran-Timmerman test statis tic.