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Energy-based wavelet de-noising of hydrologic time series.

Sang YF, Liu C, Wang Z, Wen J, Shang L - PLoS ONE (2014)

Bottom Line: It can distinguish noise from deterministic components in series, and uncertainty of de-noising result can be quantitatively estimated using proper confidence interval, but WTD method cannot do this.The results also indicate the influences of three key factors (wavelet choice, decomposition level choice and noise content) on wavelet de-noising.If too much noise is included in a series, accurate de-noising result cannot be obtained by the proposed method or WTD, but the series would show pure random but not autocorrelation characters, so de-noising is no longer needed.

View Article: PubMed Central - PubMed

Affiliation: Key Laboratory of Water Cycle & Related Land Surface Processes, Institute of Geographic Sciences and Natural Resources Research, Chinese Academy of Sciences, Beijing, China; Key Laboratory of Land Surface Process and Climate Change in Cold and Arid Regions, Chinese Academy of Sciences, Lanzhou, China.

ABSTRACT
De-noising is a substantial issue in hydrologic time series analysis, but it is a difficult task due to the defect of methods. In this paper an energy-based wavelet de-noising method was proposed. It is to remove noise by comparing energy distribution of series with the background energy distribution, which is established from Monte-Carlo test. Differing from wavelet threshold de-noising (WTD) method with the basis of wavelet coefficient thresholding, the proposed method is based on energy distribution of series. It can distinguish noise from deterministic components in series, and uncertainty of de-noising result can be quantitatively estimated using proper confidence interval, but WTD method cannot do this. Analysis of both synthetic and observed series verified the comparable power of the proposed method and WTD, but de-noising process by the former is more easily operable. The results also indicate the influences of three key factors (wavelet choice, decomposition level choice and noise content) on wavelet de-noising. Wavelet should be carefully chosen when using the proposed method. The suitable decomposition level for wavelet de-noising should correspond to series' deterministic sub-signal which has the smallest temporal scale. If too much noise is included in a series, accurate de-noising result cannot be obtained by the proposed method or WTD, but the series would show pure random but not autocorrelation characters, so de-noising is no longer needed.

Show MeSH
Energy distributions of the gauss (G) (a), 2-parameter lognormal (LN2) (b), and Pearson-III (P) (c) distributed noise with 95% confidence interval.
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pone-0110733-g001: Energy distributions of the gauss (G) (a), 2-parameter lognormal (LN2) (b), and Pearson-III (P) (c) distributed noise with 95% confidence interval.

Mentions: The above steps are applied to three noise types, and the results are presented in Fig. 1. According to the study in the literature [25], “db5” wavelet is used to analyze these noise data with the same length of 1000. The sampling number of Monte-Carlo test is 50,000 to obtain stable sampling result. The results indicate that arithmetic mean of energy of noise's sub-signal exponentially decreases with level, with the base of 2, and it is due to the grid of dyadic DWT [26] – [27]. Uncertainty can be estimated by determining 95% confidence interval. Arithmetic mean and mode values of energy of sub-signal under each level are the same, so the upper and lower limits of 95% confidence interval are symmetrical to arithmetic mean value. Consequently, energy distributions of three noise types (G, LN2 and P) are similar to each other. They strictly follow the exponentially decreasing rule with the base of 2. Therefore, it is thought that the shape of energy distribution of noise has no relation with noise type.


Energy-based wavelet de-noising of hydrologic time series.

Sang YF, Liu C, Wang Z, Wen J, Shang L - PLoS ONE (2014)

Energy distributions of the gauss (G) (a), 2-parameter lognormal (LN2) (b), and Pearson-III (P) (c) distributed noise with 95% confidence interval.
© Copyright Policy
Related In: Results  -  Collection

License
Show All Figures
getmorefigures.php?uid=PMC4215914&req=5

pone-0110733-g001: Energy distributions of the gauss (G) (a), 2-parameter lognormal (LN2) (b), and Pearson-III (P) (c) distributed noise with 95% confidence interval.
Mentions: The above steps are applied to three noise types, and the results are presented in Fig. 1. According to the study in the literature [25], “db5” wavelet is used to analyze these noise data with the same length of 1000. The sampling number of Monte-Carlo test is 50,000 to obtain stable sampling result. The results indicate that arithmetic mean of energy of noise's sub-signal exponentially decreases with level, with the base of 2, and it is due to the grid of dyadic DWT [26] – [27]. Uncertainty can be estimated by determining 95% confidence interval. Arithmetic mean and mode values of energy of sub-signal under each level are the same, so the upper and lower limits of 95% confidence interval are symmetrical to arithmetic mean value. Consequently, energy distributions of three noise types (G, LN2 and P) are similar to each other. They strictly follow the exponentially decreasing rule with the base of 2. Therefore, it is thought that the shape of energy distribution of noise has no relation with noise type.

Bottom Line: It can distinguish noise from deterministic components in series, and uncertainty of de-noising result can be quantitatively estimated using proper confidence interval, but WTD method cannot do this.The results also indicate the influences of three key factors (wavelet choice, decomposition level choice and noise content) on wavelet de-noising.If too much noise is included in a series, accurate de-noising result cannot be obtained by the proposed method or WTD, but the series would show pure random but not autocorrelation characters, so de-noising is no longer needed.

View Article: PubMed Central - PubMed

Affiliation: Key Laboratory of Water Cycle & Related Land Surface Processes, Institute of Geographic Sciences and Natural Resources Research, Chinese Academy of Sciences, Beijing, China; Key Laboratory of Land Surface Process and Climate Change in Cold and Arid Regions, Chinese Academy of Sciences, Lanzhou, China.

ABSTRACT
De-noising is a substantial issue in hydrologic time series analysis, but it is a difficult task due to the defect of methods. In this paper an energy-based wavelet de-noising method was proposed. It is to remove noise by comparing energy distribution of series with the background energy distribution, which is established from Monte-Carlo test. Differing from wavelet threshold de-noising (WTD) method with the basis of wavelet coefficient thresholding, the proposed method is based on energy distribution of series. It can distinguish noise from deterministic components in series, and uncertainty of de-noising result can be quantitatively estimated using proper confidence interval, but WTD method cannot do this. Analysis of both synthetic and observed series verified the comparable power of the proposed method and WTD, but de-noising process by the former is more easily operable. The results also indicate the influences of three key factors (wavelet choice, decomposition level choice and noise content) on wavelet de-noising. Wavelet should be carefully chosen when using the proposed method. The suitable decomposition level for wavelet de-noising should correspond to series' deterministic sub-signal which has the smallest temporal scale. If too much noise is included in a series, accurate de-noising result cannot be obtained by the proposed method or WTD, but the series would show pure random but not autocorrelation characters, so de-noising is no longer needed.

Show MeSH