Smooth representation of trends by a wavelet-based technique

Abstract

A methodology is presented, to generate a smooth function for a set of discrete, finite data. The objective function is expressed as the weighted summation of some wavelet-based basis functions. The required wavelet filters are provided by a lifting-scheme wavelet method. Using this technique, there is no need to assume a trend for the data points beyond the boundaries. To determine the basis functions and the coefficients, the wavelet packet approach is coupled with the lifting scheme. The proposed technique is fast, robust and easily understandable. Presence of noises in the data degrades the performance of the function approximation algorithm. An iterative technique is developed to remove the worthless noise elements from the data set. In each iteration, some parts of the true data points are recovered so that by a few passes, most of the true function is obtained. To show the capability of the techniques some sets of data obtained from simulated and laboratory experimental systems are investigated. A comparison with the classical wavelet-based function approximation methods, indicates that the proposed algorithm for trend representation is significantly better. In addition, the proposed technique for trend denoising compares satisfactorily with a wavelet packet-based noise removal algorithm and the Hamming FIR digital filter.