Entropy-Based Wavelet De-noising Method for Time Series Analysis

Abstract

Abstract:The existence of noise has great influence on the real features of observed time series, thus noise reduction in time series data is a necessary and significant task in many practical applications. When using traditional de-noising methods, the results often cannot meet the practical needs due to their inherent shortcomings. In the present paper, first a set of key but difficult wavelet de-noising problems are discussed, and then by applying information entropy theories to the wavelet de-noising process, i.e., using the principle of maximum entropy (POME) to describe the random character of the noise and using wavelet energy entropy to describe the degrees of complexity of the main series in original series data, a new entropy-based wavelet de-noising method is proposed. Analysis results of both several different synthetic series and typical observed time series data have verified the performance of the new method. A comprehensive discussion of the results indicates that compared with traditional wavelet de-noising methods, the new proposed method is more effective and universal. Furthermore, because it uses information entropy theories to describe the obviously different characteristics of noises and the main series in the series data is observed first and then de-noised, the analysis process has a more reliable physical basis, and the results of the new proposed method are more reasonable and are the global optimum. Besides, the analysis process of the new proposed method is simple and is easy OPEN ACCESSEntropy 2009, 11 1124 to implement, so it would be more applicable and useful in applied sciences and practical engineering works.

Entropy-Based Method of Choosing the Decomposition Level in Wavelet Threshold De-noising

“…All of them are also analyzed by the white noise testing method for comparison. Other wavelet-based de-noising methods are not included here, considering that they have been discussed and compared with the entropy-based WTD method in [13]. Moreover, three quantitative indexes, SNR (signal-to-noise ratio), MSE (mean square error) and R xy (lag-0 cross-correlation coefficient) in Equation (7), are used to judge the accuracy of de-noising results obtained by using different DLs, mainly to ensure the soundness of conclusions.…”

“…Compared with them, the wavelet threshold de-noising (WTD) method is more effective and is especially applicable in various engineering activities, because it can elucidate the localized characteristics of non-stationary time series both in the temporal and frequency domains [9][10][11][12]. Although being theoretically powerful, in practice the WTD method is influenced by four basic but key issues, namely the choice of wavelet, the choice of decomposition level (DL), threshold estimation and choice of thresholding rules, respectively [13]. Many studies have been conducted in various fields to develop the WTD method.…”

“…The authors have reviewed traditional de-noising methods and the wavelet-based de-noising methods in [13]. Three of their main defects were summarized; they are the probability description of noise, the accuracy of threshold estimation methods and the validity of thresholding rules, respectively.…”

“…Finally, analytic results of both synthetic and observed series verified the performance of the improved WTD method. To the authors' knowledge, the issue of choice of decomposition level in WTD is little discussed, whether in [13] or in other studies, and no effective and operable approaches could be followed presently.…”

“…Moreover, the proposed method is also based on the wavelet energy entropy to keep in step with the analytic process of WTD in [13]. To achieve this purpose, in Section 2, the WTD method is briefly introduced, and then the energy distributions of noises following various probability distributions are described using WEE, based on which the method of choosing DL is proposed.…”