{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,14]],"date-time":"2026-04-14T20:46:48Z","timestamp":1776199608837,"version":"3.50.1"},"reference-count":24,"publisher":"MDPI AG","issue":"2","license":[{"start":{"date-parts":[[2025,1,23]],"date-time":"2025-01-23T00:00:00Z","timestamp":1737590400000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"National Science Foundation of China","award":["42304043"],"award-info":[{"award-number":["42304043"]}]},{"name":"National Science Foundation of China","award":["KF2023YB01-11"],"award-info":[{"award-number":["KF2023YB01-11"]}]},{"name":"National Science Foundation of China","award":["SYS-ZX01-2024-01"],"award-info":[{"award-number":["SYS-ZX01-2024-01"]}]},{"name":"Key Laboratory of Smart Earth","award":["42304043"],"award-info":[{"award-number":["42304043"]}]},{"name":"Key Laboratory of Smart Earth","award":["KF2023YB01-11"],"award-info":[{"award-number":["KF2023YB01-11"]}]},{"name":"Key Laboratory of Smart Earth","award":["SYS-ZX01-2024-01"],"award-info":[{"award-number":["SYS-ZX01-2024-01"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Entropy"],"abstract":"<jats:p>To overcome the limitations of traditional evaluation indicators in determining the optimal wavelet decomposition level, this paper proposes an adaptive method for selecting the best decomposition level by combining the Jarque\u2013Bera test and a composite weighting approach. Firstly, in the noise extraction stage, the Jarque\u2013Bera test is employed to ensure that the extracted noise follows Gaussian white noise characteristics, thereby avoiding issues of insufficient denoising or signal distortion. Secondly, in the evaluation stage of the denoised signal, a comprehensive consideration of the geometric and physical meanings of various evaluation metrics, as well as the Pearson correlation coefficients between them, is undertaken. The RMSE and smoothness are selected as evaluation indicators for the denoising performance. Since these two metrics describe signal characteristics from different dimensions, a weighted combination approach is used to generate a single composite evaluation index. Additionally, to overcome the limitations of using a single weighting method, a composite weighting strategy is proposed by combining the entropy weight method and the coefficient of variation method. The composite coefficient between these two weighting methods is calculated using the variance coefficient method, yielding a new composite evaluation metric. A smaller value of this metric indicates better denoising performance, and the corresponding optimal decomposition level is more accurately determined. The simulation results demonstrate that the proposed comprehensive evaluation method can accurately determine the optimal wavelet decomposition level in both known and unknown truth-value cases, exhibiting a high accuracy and good applicability. Furthermore, the experimental results show that using the optimal decomposition level determined by the proposed method for wavelet denoising leads to smoother peak regions, more stable waveforms and significantly improved denoising performance.<\/jats:p>","DOI":"10.3390\/e27020108","type":"journal-article","created":{"date-parts":[[2025,1,23]],"date-time":"2025-01-23T04:54:31Z","timestamp":1737608071000},"page":"108","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":4,"title":["Determination Method of Optimal Decomposition Level of Discrete Wavelet Based on Joint Jarque\u2013Bera Test and Combination Weighting Method"],"prefix":"10.3390","volume":"27","author":[{"ORCID":"https:\/\/orcid.org\/0009-0004-1574-3015","authenticated-orcid":false,"given":"Zhanpeng","family":"Zhang","sequence":"first","affiliation":[{"name":"Key Laboratory of Smart Earth, Beijing 100080, China"},{"name":"Institute of Geospatial Information, Information Engineering University, Zhengzhou 450001, China"}]},{"given":"Changjian","family":"Liu","sequence":"additional","affiliation":[{"name":"Key Laboratory of Smart Earth, Beijing 100080, China"},{"name":"Institute of Geospatial Information, Information Engineering University, Zhengzhou 450001, China"}]},{"given":"Min","family":"Wang","sequence":"additional","affiliation":[{"name":"Key Laboratory of Smart Earth, Beijing 100080, China"},{"name":"Institute of Geospatial Information, Information Engineering University, Zhengzhou 450001, China"}]},{"given":"Shuang","family":"Sun","sequence":"additional","affiliation":[{"name":"Key Laboratory of Smart Earth, Beijing 100080, China"},{"name":"Institute of Geospatial Information, Information Engineering University, Zhengzhou 450001, China"}]},{"given":"Zhao","family":"Zhan","sequence":"additional","affiliation":[{"name":"Key Laboratory of Smart Earth, Beijing 100080, China"},{"name":"Institute of Geospatial Information, Information Engineering University, Zhengzhou 450001, China"}]}],"member":"1968","published-online":{"date-parts":[[2025,1,23]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","unstructured":"Durrett, R. (2019). Probability: Theory and Examples, Cambridge University Press. Cambridge Series in Statistical and Probabilistic Mathematics.","DOI":"10.1017\/9781108591034"},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"1117","DOI":"10.1111\/anae.16420","article-title":"The Central Limit Theorem: The Remarkable Theory That Explains All of Statistics","volume":"79","author":"Sidebotham","year":"2024","journal-title":"Anaesthesia"},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"439","DOI":"10.1016\/j.ins.2018.07.015","article-title":"Wavelet-Denoising Multiple Echo State Networks for Multivariate Time Series Prediction","volume":"465","author":"Xu","year":"2018","journal-title":"Inf. Sci."},{"key":"ref_4","first-page":"5110875","article-title":"A Cycle Slip Detection and Repair Method Using BDS Triple-Frequency Optimization Combination with Wavelet Denoising","volume":"2022","author":"Li","year":"2022","journal-title":"Int. J. Aerosp. Eng."},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"163","DOI":"10.1515\/bmt-2018-0001","article-title":"Optimal Level and Order Detection in Wavelet Decomposition for PCG Signal Denoising","volume":"64","author":"Rouis","year":"2019","journal-title":"Biomed. Eng. Biomed. Tech."},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"3963","DOI":"10.1007\/s00477-024-02788-y","article-title":"Improving Deep Learning-Based Streamflow Forecasting under Trend Varying Conditions through Evaluation of New Wavelet Preprocessing Technique","volume":"38","author":"Behbahani","year":"2024","journal-title":"Stoch. Environ. Res. Risk Assess."},{"key":"ref_7","first-page":"749","article-title":"A Hybrid Indicator for Determining the Best Decomposition Scale of Wavelet Denoising","volume":"41","author":"Tao","year":"2012","journal-title":"Acta Geod. Cartogr. Sin."},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"2220","DOI":"10.4028\/www.scientific.net\/AMM.644-650.2220","article-title":"Adaptive Algorithm for Determination of Optimal Wavelet Decomposition Level Based on Jarque-Bera Test","volume":"644\u2013650","author":"Liu","year":"2014","journal-title":"Appl. Mech. Mater."},{"key":"ref_9","doi-asserted-by":"crossref","unstructured":"Li, X., Liao, K., He, G., and Zhao, J. (2023). Research on Improved Wavelet Threshold Denoising Method for Non-Contact Force and Magnetic Signals. Electronics, 12.","DOI":"10.3390\/electronics12051244"},{"key":"ref_10","doi-asserted-by":"crossref","unstructured":"Guo, Y., Zhou, X., Li, J., Ba, R., Xu, Z., Tu, S., and Chai, L. (2023). A Novel and Optimized Sine\u2013Cosine Transform Wavelet Threshold Denoising Method Based on the Sym4 Basis Function and Adaptive Threshold Related to Noise Intensity. Appl. Sci., 13.","DOI":"10.3390\/app131910789"},{"key":"ref_11","first-page":"688","article-title":"A reliable evaluation indicator of wavelet de-noising","volume":"40","author":"Zhu","year":"2015","journal-title":"Geomat. Inf. Sci. Wuhan Univ."},{"key":"ref_12","first-page":"698","article-title":"A Kind of Wavelet De-Noising Composite Evaluation Index Based on Entropy Method","volume":"38","author":"Wang","year":"2018","journal-title":"J. Geod. Geodyn."},{"key":"ref_13","first-page":"718","article-title":"A quality evaluation method for wavelet denoising based on combinatorial weighting method","volume":"49","author":"Li","year":"2023","journal-title":"J. Beijing Univ. Aeronaut. Astronaut."},{"key":"ref_14","unstructured":"Mallat, S. (2008). A Wavelet Tour of Signal Processing, Third Edition: The Sparse Way, Academic Press Inc.. [3rd ed.]."},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"112021","DOI":"10.1016\/j.ymssp.2024.112021","article-title":"Full-Field Extraction of Subtle Displacement Components via Phase-Projection Wavelet Denoising for Vision-Based Vibration Measurement","volume":"224","author":"Li","year":"2025","journal-title":"Mech. Syst. Signal Process."},{"key":"ref_16","unstructured":"Tian, Y. (2012). Tests for Normality Based on Skewness and Kurtosis. [Master\u2019s Thesis, Shanghai JiaoTong University]."},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"433","DOI":"10.1177\/00236772241276808","article-title":"Testing for Normality: A User\u2019s (Cautionary) Guide","volume":"58","author":"Gosselin","year":"2024","journal-title":"Lab. Anim."},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"480","DOI":"10.1007\/s44199-024-00098-4","article-title":"Testing Normality of Data for Uncertain Level of Significance","volume":"23","author":"Aslam","year":"2024","journal-title":"J. Stat. Theory Appl."},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"112026","DOI":"10.1016\/j.ymssp.2024.112026","article-title":"Probability Distributions and Typical Sparsity Measures of Hilbert Transform-Based Generalized Envelopes and Their Application to Machine Condition Monitoring","volume":"224","author":"Chen","year":"2025","journal-title":"Mech. Syst. Signal Process."},{"key":"ref_20","first-page":"165","article-title":"Parameters selection of stationary wavelet denoising algorithm","volume":"41","author":"Zhang","year":"2019","journal-title":"J. Natl. Univ. Def. Technol."},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"131475","DOI":"10.1016\/j.energy.2024.131475","article-title":"A Novel Battery Abnormality Diagnosis Method Using Multi-Scale Normalized Coefficient of Variation in Real-World Vehicles","volume":"299","author":"Hong","year":"2024","journal-title":"Energy"},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"012004","DOI":"10.1088\/1742-6596\/2846\/1\/012004","article-title":"Energy Efficiency Evaluation of Wind Turbines Based on Entropy Weight Method and Stacked Autoencoder","volume":"2846","author":"Ma","year":"2024","journal-title":"J. Phys. Conf. Ser."},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"37","DOI":"10.1007\/s10291-023-01576-5","article-title":"Real-Time Multipath Mitigation Based on Spatiotemporal Correlations in BDS Precise Point Positioning","volume":"28","author":"Wang","year":"2024","journal-title":"GPS Solut."},{"key":"ref_24","doi-asserted-by":"crossref","unstructured":"Li, L., Shen, Y., and Li, X. (2023). Mitigating Satellite-Induced Code Pseudorange Variations at GLONASS G3 Frequency Using Periodical Model. Remote Sens., 15.","DOI":"10.3390\/rs15020431"}],"container-title":["Entropy"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/1099-4300\/27\/2\/108\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,8]],"date-time":"2025-10-08T10:34:03Z","timestamp":1759919643000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/1099-4300\/27\/2\/108"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2025,1,23]]},"references-count":24,"journal-issue":{"issue":"2","published-online":{"date-parts":[[2025,2]]}},"alternative-id":["e27020108"],"URL":"https:\/\/doi.org\/10.3390\/e27020108","relation":{},"ISSN":["1099-4300"],"issn-type":[{"value":"1099-4300","type":"electronic"}],"subject":[],"published":{"date-parts":[[2025,1,23]]}}}