Fu-Chiang Tsui'I2 Ching-ChungLiz
Robert .Sclabassi''2 ' I .
Laboratory for Computational Neuroscience, Departments of Neurological Surgery' and Electrical Engineering' University of Pittsburgh, Pittsburgh, PA 15213-2582, USA, tsui@neuronet . p i t t .edu Phone: 412-648-9230 FAX: 41.2-692-5921
ABSTRACTAn adaptive recurrent neural network (ARNN) in wavelet coefficient space computed from the discrete wavelet transform (DWT) is presented in this paper for generating an adaptive, long-term, coarse resolution prediction of a time series. The weights inside the ARNN are updated by the incoming data, i.e., the network modifies itself with time. With the aid of the newly developed DWT of Cai-Wang,this ARNN is efficient and takes less time to train than a NN in data space since it deals only with wavelet coefficients instead of raw data. Results are demonstrated by applying this method to the long-term prediction of intracranial pressure (ICP) data recorded from head-trauma patients.
Appendix. The propositi'on states that the projection function at the coarsest scale level is continuous inits coefficients; this assures that the smaller the error in the predicted coefficients is, the less the deviation in the predicted projection function will be.
The DWT of Cai-Wang[1J computes wavelet coefficients from coarse to fine scales by applying the 4-th order €3-spline function to interpolate data in the time domain. It allows a description of the data at coarse scalelevels without information at the fine scale levels. In this way, the number of computed wavelet coefficients necessary is fewer than from other DWTs when only coarse scalelevels are considered. However, the DWTs which compute wavelet coefficients from fine to coarse scale levels have the advantage of providing de-correlation of a stationary signal in wavelet coefficient spaces whenorthonormal or compactly supported biorthogonal wavelet bases are used. We have applied the DWT of Cai-Wang in our previous work; however, the continuous prediction of wavelet coefficients is considered in this paper. The DWT is designed for a finite interval [0, L ] ,and thus needs basis functions defined at the left boundary, interior and the right boundary. The interior (d(2)) and boundary (d b ( 2 ) ) scaling functions in space &)corresponding to the coarsest level under consideration are[ 11
1. INTRODUCTIONAND MOTIVATION
In many time series applications, one-step prediction schemes are used to predict the next sample of the data based on the previous samples. However, one-step prediction may not provide enough information, especially in situations where a longer waming ofimpending changes would be useful. Thus, a long-term prediction scheme is desirable. However, most of the time-series observed in the real world are non-stationary, making long-term prediction very difficult, thus adaptive long-term prediction is necessary and appears to be suitable for quasi-stationary signals. In this paper, a method is presented for continuous,on-line, adaptive, nonlinear, andlong-term prediction in a coarse resolution of a quasi-stationary time-series by applying both an adaptive recurrent neural network (ARNN) and the discrete wavelet transform (DWT) on an interval[ 11. The predictor can provide not only the long-term prediction in coarse resolution but can also accommodate changes in dynamics of the analyzed system. In Section 2, we describe our approach, the scalingfunctions we used and the prediction procedures. In Section 3, we demonstrate our long-term adaptive predictor by applying it to intracranial pressure (ICP) data acquired from a head-trauma patient in the neuro intensive care unit (NICU) at the University of Pittsburgh Medical Center. This predictor may be used as an aid in assessing physiological status of head-trauma patients, allowing the early...