Preprint of the EMBS International Conference 95
A WAVELET BASED NEURAL NETWORK FOR PREDICTION OF ICP SIGNAL Fu-Chiang Tsui1 2
Mingui Sun1 3
;
;
Robert J. Sclabassi1 2 3
Ching-Chung Li2
; ;
1
2
Laboratory for Computational Neuroscience Departments of Electrical Engineering and 3 Neurological Surgery University of Pittsburgh, Pittsburgh, PA 15261, U.S.A
[email protected]
ABSTRACT We present a wavelet-based neural network for multi-step prediction of the intracranial pressure(ICP) signal. A multiresolution dynamic predictor(MDP) is proposed, which utilizes the discrete wavelet transform computing wavelet coefficients from coarse scale to fine scale and recurrent neural networks(RNNs) forming dynamic nonlinear models for prediction. It has the ability to predict the ICP in both long-term with coarse resolution and short-term with fine resolution. Computational results up to three scale levels have demonstrated the effectiveness of the MDP for multistep prediction as compared with the the raw data. 1. INTRODUCTION In this paper, we discuss intracranial pressure(ICP) signal prediction for patients in the intensive care unit(ICU)[1][2] by applying both recurrent neural networks(RNNs)[2] and the discrete wavelet transform(DWT)[3]. Traditional methods [4][5] for time-series prediction using neural networks are not effective for the ICP prediction due to the high sampling rate(200Hz) and the desired long-term prediction. To improve performance and reduce the computational complexity, a set of RNNs operating in wavelet domain is utilized where both the input and output to each network are wavelet coefficients(x(0), x(1), x(2), : : :, and x(n)) at a corresponding scale. Our computational results validate this predictor. In Section 2, we briefly explain the concept of the multiresolution dynamic predictor(MDP). In section 3, we apply our algorithms to predict ICP signals based on the predicted wavelet coefficients. Finally, experimental results are discussed in Section 4. 2. MULTIRESOLUTION DYNAMIC PREDICTOR(MDP) Our MDP applies wavelet coefficients, obtained by the DWT of Wang[3], instead of raw data in RNNs. Unlike other DWTs, the DWT of Wang can be computed from coarse to fine scale levels, with a comparatively short computation This work was supported by the Ben Franklin Foundation of Pennsylvania, Computational Diagnostics, Inc. and NIH(NS30318)
time. This DWT is constructed by both scaling( (x)) and wavelet( (x)) functions[3] defined as,
(x)
1X 6 4
=
j
(x)
=
=0
4
j
?1) (x ? j )3+
(
j
(1)
? 37 (2x) + 127 (2x ? 1) ? 37 (2x ? 2)(2)
where the subscript “+” of x+ denotes x > 0. A recurrent neural network(RNN)[2] with only one feedback path from output to input layer in Fig-1 is a nonlinear dynamic model used to recursively predict wavelet coefficients at a given scale. During the training phase, switches S 1, S 2, and S 3 are closed and S 4 is open, forming simply a feedforward network for training connection weights. In the prediction phase, switches S 1, S 2, and S 3 are open and S 4 is closed for a specified duration. After obtaining predicted coefficients from multiscale RNNs, an interpolation function can be computed by multiplying predicted coefficients with scaling and wavelet functions at each scale. We find that the error for the predicted signal at the coarsest level (projected function in V0 space) is insignificant if the error of the predicted coefficients is small. The algorithm for ICP signal prediction by using the wavelet based RNNs includes the following computational steps. 1. Subsampling raw ICP signal. 2. Computing N levels of DWT on the ICP signal. 3. Training N Networks for wavelet coefficients at N scales. 4. Predicting coefficients at multiple scales from RNNs. 5. Constructing the projected functions based on the predicted coefficients. 3. EXPERIMENTAL RESULTS ICP data from the headtrauma ICU at University of Pittsburgh medical center was collected via an X-window based real-time data collection package developed by the authors[1] [2]. The ICP data is subsampled at every 128 raw samples in V0 , corresponding to samples at integer indices defined in [2][3]. The first 352-seconds of ICP data is used for training. After taking the DWT on ICP signal up to the W1 space, we
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Figure 1: A recurrent neural network trained by back propagation.
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4. DISCUSSION By using the MDP, we have obtained both short-term and long-term prediction with fine resolution and coarse resolution, respectively. For a certain time interval, the multi-step prediction for each RNN can be combined together to obtain a fine resolution prediction. Moreover, this resolution grows geometrically. For example, in the interval [500; 599] corresponding to 31:36 seconds, the projected function(PF) in V2 applies 203 wavelet coefficients in contrast to the PF in V0 which uses only 50 coefficients. In addition, we can distribute each RNN to a separate processor for parallel processing and compute the finest resolution from all RNNs. Our comparison to the RNN operating in the raw data space shows that the MDP has much improved performance in computational efficiency and prediction accuracy. 5. ACKNOWLEDGMENT We acknowledge valuable discussions with Dr. Jianzhong Wang and Dr. Xiaopu Yan on the Discrete Wavelet Trans-
24 22 20 18 550
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used N40;4;2;1, N30;4;2;1, and N41;4;2;1 RNNs to model coefficients in V0 , W0 , and W1 respectively. The projected functions in V0 , V1 , and V2 were obtained by combining the predicted interpolation functions in each space. Fig-2(a) shows the predicted function from the wavelet coefficients in V0 for 64 seconds. Fig-2(b,c) displays the predicted functions from the wavelet coefficients in V1 and V2 within 32 seconds. From Fig-2(a,b,c), it is observed that the predicted projected function at the finest scale produces a better approximation of the raw ICP waveforms. However, more wavelet coefficients are employed to construct interpolation functions at finer scales. The forecasted MAPs (mean absolute percentage errors) for the projected functions at three levels are 4:51%, 3:55%, 3:96% respectively in terms of 100 points in each unit interval. On the other hand, a RNN, N41;4;2;1, operating in the raw data could not capture the natural ICP dynamics; its prediction diverged after a 2-second interval and it also took approximately one day to train a 5-minute raw ICP data. In comparison, our MDP used only 20 minutes to train the three RNNs and predicted wavelet coefficients successively for 32 seconds at finer resolution and 64 seconds at coarser resolution.
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Figure 2: Predicted(dash) and real(solid) projected functions in (a) V0 , (b) V1 = V0 W0 , (c)V2 = V1 W2 spaces. form. Thanks also go to Dr. Donald W. Marion for his valuable discussions on the patients studied. 6. REFERENCES [1] M. Sun, F.-C. Tsui, D. W. Marion and R. J. Sclabassi, “The Wavelets and Their Applications to the ICU Monitoring,” in Intelligent Engineering Systems Through Artificial Neural Networks, Vol. 4, C.H. Dagli, B.R. Fernandez, J. Ghosh and R.T.S. Kumara ed., AMSE Press, 1994, pp. 541-546. [2] F.-C. Tsui, M. Sun, C.C. Li, and R.J. Sclabassi, “Recurrent Neural Networks and Discrete Wavelet Transform for Time Series Modeling and Prediction,” in Proceedings ICASSP-95, Detroit, Michigan, May8-12, 1995, pp 3359-3362. [3] W. Cai and J.-Z. Wang. “Adaptive Wavelet Collocation Methods for Initial Value Boundary Problems of Nonlinear PDE’s,” Submitted to SIAM Numer. Anal., 1994. [4] A.S. Weigend, B.A. Huberman, abd D.E. Rumelhart, “Predicting The Future: A Connectionist Approach,” in Int. Journal of Neural Systems, Vol. 1, No. 3, 1990, pp. 193-209. [5] A. Lapedes, R. Farber. “Nonlinear Signal Processing Using Neural Networks: Prediction and System Modeling,” Technical Report LA-UR-87-2662, Los Alamos National Laboratory, Los Alamos, New Mexico 87545, 1987.
