{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,6,19]],"date-time":"2026-06-19T12:02:41Z","timestamp":1781870561354,"version":"3.54.5"},"reference-count":25,"publisher":"MDPI AG","issue":"6","license":[{"start":{"date-parts":[[2010,6,10]],"date-time":"2010-06-10T00:00:00Z","timestamp":1276128000000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/3.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Entropy"],"abstract":"<jats:p>In this paper, the energy distributions of various noises following normal, log-normal and Pearson-III distributions are first described quantitatively using the wavelet energy entropy (WEE), and the results are compared and discussed. Then, on the basis of these analytic results, a method for use in choosing the decomposition level (DL) in wavelet threshold de-noising (WTD) is put forward. Finally, the performance of the proposed method is verified by analysis of both synthetic and observed series. Analytic results indicate that the proposed method is easy to operate and suitable for various signals. Moreover, contrary to traditional white noise testing which depends on \u201cautocorrelations\u201d, the proposed method uses energy distributions to distinguish real signals and noise in noisy series, therefore the chosen DL is reliable, and the WTD results of time series can be improved.<\/jats:p>","DOI":"10.3390\/e12061499","type":"journal-article","created":{"date-parts":[[2010,6,10]],"date-time":"2010-06-10T12:52:37Z","timestamp":1276174357000},"page":"1499-1513","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":43,"title":["Entropy-Based Method of Choosing the Decomposition Level in Wavelet Threshold De-noising"],"prefix":"10.3390","volume":"12","author":[{"given":"Yan-Fang","family":"Sang","sequence":"first","affiliation":[{"name":"State Key Laboratory of Pollution Control and Resource Reuse, Department of Hydrosciences, School of Earth Sciences and Engineering, Nanjing University, Nanjing 210093, China"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Dong","family":"Wang","sequence":"additional","affiliation":[{"name":"State Key Laboratory of Pollution Control and Resource Reuse, Department of Hydrosciences, School of Earth Sciences and Engineering, Nanjing University, Nanjing 210093, China"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Ji-Chun","family":"Wu","sequence":"additional","affiliation":[{"name":"State Key Laboratory of Pollution Control and Resource Reuse, Department of Hydrosciences, School of Earth Sciences and Engineering, Nanjing University, Nanjing 210093, China"}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"1968","published-online":{"date-parts":[[2010,6,10]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"183","DOI":"10.1029\/91WR02269","article-title":"Uncorrelated measurement error in flood frequency inference","volume":"28","author":"Kuczera","year":"1992","journal-title":"Water Resour. Res."},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"237","DOI":"10.1016\/j.jhydrol.2009.01.001","article-title":"Using long-term data sets to understand transit times in contrasting headwater catchments","volume":"367","author":"Hrachowitz","year":"2009","journal-title":"J. Hydrol."},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"1526","DOI":"10.1016\/j.advwatres.2009.07.004","article-title":"Stochastic observation error and uncertainty in water quality evaluation","volume":"32","author":"Wang","year":"2009","journal-title":"Adv. Water Resour."},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"165","DOI":"10.1016\/j.jhydrol.2009.01.042","article-title":"The relation between periods\u2019 identification and noises in hydrologic series data","volume":"368","author":"Sang","year":"2009","journal-title":"J. Hydrol."},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"613","DOI":"10.1109\/18.382009","article-title":"De-noising by soft-thresholding","volume":"41","author":"Donoho","year":"1995","journal-title":"IEEE Trans. Inform. Theory"},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"2595","DOI":"10.1109\/78.482110","article-title":"Filtering random noise from deterministic signals via data compression","volume":"43","author":"Natarajan","year":"1995","journal-title":"IEEE Trans. Signal Process."},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"41","DOI":"10.1006\/jsvi.2000.3275","article-title":"Estimation of speech components by AFC analysis in a noisy environment","volume":"241","author":"Kazama","year":"2001","journal-title":"J. Sound Vib."},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"147","DOI":"10.1016\/S0022-1694(01)00534-0","article-title":"Noise reduction in chaotic hydrologic time series: facts and doubts","volume":"256","author":"Elshorbagy","year":"2002","journal-title":"J. Hydrol."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"61","DOI":"10.1175\/1520-0477(1998)079<0061:APGTWA>2.0.CO;2","article-title":"A practical guide to wavelet analysis","volume":"79","author":"Torrence","year":"1998","journal-title":"Bull. Amer. Meteorol. Soc."},{"key":"ref_10","doi-asserted-by":"crossref","unstructured":"Percival, D.B., and Walden, A.T. (2000). Wavelet Methods for Time Series Analysis, Cambridge University Press.","DOI":"10.1017\/CBO9780511841040"},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"1113","DOI":"10.1109\/78.923292","article-title":"Asymptotic behavior of the minimum mean squared error threshold for noisy wavelet coefficients of piecewise smooth signals","volume":"49","author":"Jansen","year":"2001","journal-title":"IEEE Trans. Signal Process."},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"296","DOI":"10.1109\/LSP.2006.870355","article-title":"Minimum risk thresholds for data with heavy noise","volume":"13","author":"Jansen","year":"2006","journal-title":"IEEE Signal Process. Lett."},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"1123","DOI":"10.3390\/e11041123","article-title":"Entropy-based wavelet de-noising method for time series analysis","volume":"1","author":"Sang","year":"2009","journal-title":"Entropy"},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"713","DOI":"10.1109\/18.119732","article-title":"Entropy based algorithms for best basis selection","volume":"38","author":"Coifman","year":"1992","journal-title":"IEEE Trans. Inform. Theor."},{"key":"ref_15","first-page":"808","article-title":"Removing noise from music using local trigonometric bases and wavelet packets","volume":"42","author":"Berger","year":"1994","journal-title":"J. Audio Eng. Soc."},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"717","DOI":"10.1016\/S0022-460X(02)01556-0","article-title":"An approach based on simplified KLT and wavelet transform for enhancing speech degraded by non-stationary wideband noise","volume":"268","author":"Lou","year":"2003","journal-title":"J. Sound Vib."},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"177","DOI":"10.1016\/j.bspc.2006.08.004","article-title":"Novel wavelet domain Wiener filtering de-noising techniques: application to bowel sounds captured by means of abdominal surface vibrations","volume":"1","author":"Dimoulas","year":"2006","journal-title":"Biomed. Signal Process. Contr."},{"key":"ref_18","doi-asserted-by":"crossref","unstructured":"Chui, C.K. (1992). An Introduction to Wavelets, Vol. 1 (Wavelet Analysis and Its Applications), Academic Press.","DOI":"10.1016\/B978-0-12-174590-5.50029-0"},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"859","DOI":"10.1016\/j.sigpro.2005.06.017","article-title":"Wavelet-based signal de-noising via simple singularities approximation","volume":"86","author":"Bruni","year":"2006","journal-title":"Signal Process."},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"1679","DOI":"10.1016\/j.compstruc.2007.02.025","article-title":"Correcting data from an unknown accelerometer using recursive least squares and wavelet de-noising","volume":"85","author":"Chanerley","year":"2007","journal-title":"Comput. Struct."},{"key":"ref_21","unstructured":"Box, G.E.P., Jenkins, G.M., and Reinsel, G.C. (1994). Time Series Analysis, Forecasting and Control, Prentice-Hall."},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"620","DOI":"10.1103\/PhysRev.106.620","article-title":"Information theory and statistical mechanics","volume":"106","author":"Jaynes","year":"1957","journal-title":"Phys. Rev."},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"275","DOI":"10.1016\/j.jhydrol.2005.04.003","article-title":"Recent advances in wavelet analyses: Part 1. A review of concepts","volume":"314","author":"Labat","year":"2005","journal-title":"J. Hydrol."},{"key":"ref_24","unstructured":"Wang, W.S., Ding, J., and Li, Y.Q. (2005). Hydrology Wavelet Analysis (in Chinese), Chemical Industry Press."},{"key":"ref_25","unstructured":"Sang, Y.F., Wu, J.C., Wang, D., and Ling, C.P. (, January September). New Model of Groundwater Simulation and Prediction Based on Wavelet De-noising. Proceedings of 7th International Conference on Calibration and Reliability in Groundwater Modeling, Wuhan, China."}],"container-title":["Entropy"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/1099-4300\/12\/6\/1499\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T22:02:40Z","timestamp":1760220160000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/1099-4300\/12\/6\/1499"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2010,6,10]]},"references-count":25,"journal-issue":{"issue":"6","published-online":{"date-parts":[[2010,6]]}},"alternative-id":["e12061499"],"URL":"https:\/\/doi.org\/10.3390\/e12061499","relation":{},"ISSN":["1099-4300"],"issn-type":[{"value":"1099-4300","type":"electronic"}],"subject":[],"published":{"date-parts":[[2010,6,10]]}}}