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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.

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Energy distributions of SS1 (a) and SS2 (b) series obtained by different wavelets.
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pone-0110733-g003: Energy distributions of SS1 (a) and SS2 (b) series obtained by different wavelets.

Mentions: The decomposition level 10 (log21500) is used, and five wavelets (“db2”, “db16”, “sym5”, “coif4” and “bior3.9”) are used to remove noise in SS1 series by the proposed method. Energy distributions of SS1 series obtained by five wavelets are depicted in Fig. 3(a), in which sub-signals under ten levels (Ls) are reconstructed by detail wavelet coefficients. When applying different wavelets to SS1 series, energies of those sub-signals after L6 overstep 95% confidence interval, although their values vary with the wavelet used. Therefore, these sub-signals are regarded as deterministic components, and their sum is the de-noised SS1 series. The same analyses are conducted to SS2 series using the decomposition level 8 (log2500) and five wavelets. As shown in Fig. 3(b), sub-signals of SS2 series after L5 obtained by any wavelet have the energies overstepping 95% confidence interval, so they are regarded as deterministic components, and their sum is the de-noised SS2 series.


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 SS1 (a) and SS2 (b) series obtained by different wavelets.
© Copyright Policy
Related In: Results  -  Collection

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

pone-0110733-g003: Energy distributions of SS1 (a) and SS2 (b) series obtained by different wavelets.
Mentions: The decomposition level 10 (log21500) is used, and five wavelets (“db2”, “db16”, “sym5”, “coif4” and “bior3.9”) are used to remove noise in SS1 series by the proposed method. Energy distributions of SS1 series obtained by five wavelets are depicted in Fig. 3(a), in which sub-signals under ten levels (Ls) are reconstructed by detail wavelet coefficients. When applying different wavelets to SS1 series, energies of those sub-signals after L6 overstep 95% confidence interval, although their values vary with the wavelet used. Therefore, these sub-signals are regarded as deterministic components, and their sum is the de-noised SS1 series. The same analyses are conducted to SS2 series using the decomposition level 8 (log2500) and five wavelets. As shown in Fig. 3(b), sub-signals of SS2 series after L5 obtained by any wavelet have the energies overstepping 95% confidence interval, so they are regarded as deterministic components, and their sum is the de-noised SS2 series.

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