|Table of Contents|

Citation:
　Dongxing Xu,Jianchuan Yin.Probabilistic Interval Prediction of Ship Roll Motion Using Multi-Resolution Decomposition and Non-Parametric Kernel Density Estimation[J].Journal of Marine Science and Application,2026,(4):1030-1041.[doi:10.1007/s11804-025-00722-4]
Click and Copy

Probabilistic Interval Prediction of Ship Roll Motion Using Multi-Resolution Decomposition and Non-Parametric Kernel Density Estimation

Info

Title:
Probabilistic Interval Prediction of Ship Roll Motion Using Multi-Resolution Decomposition and Non-Parametric Kernel Density Estimation
Author(s):
Dongxing Xu1; 2; 3;  Jianchuan Yin1; 2; 3
Affilations:
Author(s):
Dongxing Xu1; 2; 3;  Jianchuan Yin1; 2; 3
1. Naval Architecture and Shipping College, Guangdong Ocean University, Zhanjiang 524005, China;
2. Guangdong Provincial Engineering Research Center for Ship Intelligence and Safety, Zhanjiang 524005, China;
3. Guangdong Provincial Key Laboratory of Intelligent Equipment for South China Sea Marine Ranching, Zhanjiang 524088, China
Keywords:
Ship roll motion probabilistic prediction; Data preprocessing strategy; Neural network; Non-parametric kernel density estimation
分类号:
-
DOI:
10.1007/s11804-025-00722-4
Abstract:
The prediction of ship roll motion is important for the safety of marine vessels and the efficiency of offshore operations. This study aims to address the issues of low prediction accuracy and incapability to illustrate the uncertainty of ship roll motion in traditional deterministic prediction models. Thus, a probabilistic interval prediction scheme for ship roll motion is established utilizing the novel data preprocessing strategy and non-parametric kernel density estimation (KDE). The novel data preprocessing method based on multi-resolution decomposition and reconstruction strategy is used to reduce the non-stationarity of the time series and number of mode components. This method is also utilized to extract trends from the mode components for preserving more effective information. Accordingly, the dynamic recurrent neural network is adopted to enable the identification and deterministic prediction of reconstruction data for ship roll motion. The deterministic prediction values of ship motion status can be obtained by reconstructing the predicted outcomes of the mode components. Non-parametric KDE is applied to predict the fluctuation range of ship roll status by combining deterministic prediction results. The feasibility and effectiveness of the probability prediction scheme are exhibited through the measured data of M.V. YuKun. Results show that the deterministic prediction accuracy of the proposed scheme is superior to those of traditional deterministic prediction models, with maximum improvements of 22.7% and 27.5% for single-and multi-step prediction accuracies, respectively. The minimum prediction interval coverage probability at the 85% confidence level is 98%. The reliable fluctuation range of ship roll can be realized through the non-parametric KDE, which can offer more effective reference information for ship operators. The proposed model not only satisfies the requirements of deterministic prediction accuracy but also produces reliable uncertainty estimates of ship roll motion status.

References:

[1] Afshar K, Bigdeli N (2011) Data analysis and short term load forecasting in Iran electricity market using singular spectral analysis (SSA). Renew Energ 36(5): 2620-2627. https://doi.org/10.1016/j.energy.2011.02.003
[2] An XL , Jiang DX , Zhao MH, Liu C (2012) Short-term prediction of wind power using EMD and chaotic theory. Commun Nonlinear Sci Numer Simulat 17(2): 1036-1042. https://doi.org/10.1016/j.cnsns. 2011.06.003
[3] Chung JC, Bien Z, Kim YS (1990) A note on ship-motion prediction based on wave-excitation input estimation. IEEE J Oceanic Eng 15(3): 244-250. DOI: 10.1109/48.107153
[4] Ding SF, Ma Q, Zhou L, Han S, Dong WB (2023) Multipoint heave motion prediction method for ships based on the PSO-TGCN model. China Ocean Eng 37(6): 1022-1031. DOI: 10.1007/s13344-023-0085-4
[5] Duan WY, Huang LM, Han Y, Zhang YH, Huang S (2015) A hybrid AR-EMD-SVR model for the short-term prediction of nonlinear and non-stationary ship motion. J Zhejiang Univ-Sc A (Appl Phys Eng) 16(7): 562-576. DOI:10.1631/jzus.A1500040
[6] Elman JL (1990) Finding structure in time. Cognitive Sci 14(2): 179-211. https://doi.org/10.1207/s15516709cog1402_1
[7] Elman JL (1993) Learning and development in neural networks: The importance of starting small. Cognition 48(1): 71-99. https://doi.org/10.1016/0010-0277(93)90058-4
[8] Fossen TI (2011) Handbook of marine craft hydrodynamics and motion control. John Wiley & Sons Ltd, Chichester, West Sussex, United Kingdom
[9] Fu HX, Gu ZQ, Wang YC (2022) Ship pitch prediction based on Bi-ConvLSTM-CA model. J Mar Sci Eng 10(7): 840. https://doi.org/10.3390/jmse10070840
[10] Gao N, Hu AK, Hou LX, Chang X (2023) Real-time ship motion prediction based on adaptive wavelet transform and dynamic neural network. Ocean Eng 280: 114466. https://doi.org/10.1016/j.oceaneng.2023.114466
[11] Geng XY, Li YB, Sun Q (2023) A novel short-term ship motion prediction algorithm based on EMD and adaptive PSO-LSTM with the sliding window approach. J Mar Sci Eng 11: 466. https://doi.org/10.3390/jmse11030466
[12] Gramacki A (2018) Nonparametric kernel density estimation and its computational aspects. Springer International Publishing, Cham, Switzerland. https://doi.org/10.1007/978-3-319-71688-6
[13] Han LQ, Shi Y (2023) Artificial neural network theory and application. China Machine Press, Beijing, China
[14] Harvey A, Oryshchenko V (2012) Kernel density estimation for time series data. Int J Forecasting 28(1): 3-14. https://doi.org/10.1016/j.ijforecast.2011.02.016
[15] Huang LM, Duan WY, Han Y, Chen YS (2014) A review of short-term prediction techniques for ship motions in seaway. J Ship Mech 18(12): 1534-1542. DOI: 10.3969/j.issn.1007-7294.2014.12.013
[16] Huang NE, Shen Z, Long, SR, Wu MC, Shih HH, Zheng QA, Yen NC, Tung CC, Liu HH (1998) The empirical mode decomposition and the Hilbert spectrum for nonlinear and non-stationary time series analysis. Proc R Soc London Ser A Math Phys Eng Sci 454(1971): 903-995. https://doi.org/10.1098/rspa.1998.0193
[17] Jiang H, Duan SL, Huang LM, Han Y, Yang H, Ma QW (2020) Scale effects in AR model real-time ship motion prediction. Ocean Eng 203: 107202. https://doi.org/10.1016/j.oceaneng.2020.107202
[18] Kwiatkowski D, Phillips PCB, Schmidt P, Shin Y (1992) Testing the null hypothesis of stationarity against the alternative of a unit root. J Econometrics 54(1): 159-178. https://doi.org/10.1016/0304-4076(92)90104-Y
[19] Li GY, Kawan B, Wang H, Zhang HX (2017) Neural-network-based modelling and analysis for time series prediction of ship motion. Ship Technol Res 64(1): 30-39. http://dx.doi.org/10.1080/09377255. 2017.1309786
[20] Li L, Wang TT, Yin HR, Niu J, Liang YB, He JN, Miao SH (2022a) Point-interval joint prediction method of dissolved gas volume fraction in transformer oil based on GWO-LSTM and NKDE. High Volt Appar 58(11): 88-97. (in Chinese) DOI:10.13296/j.1001-1609.hva.2022.11.012
[21] Li MW, Geng J, Han DF, Zheng TJ (2016) Ship motion prediction using dynamic seasonal RvSVR with phase space reconstruction and the chaos adaptive efficient FOA. Neurocomputing 174: 661-680. https://doi.org/10.1016/j.neucom.2015.09.089
[22] Li MW, Xu DY, Geng J, Hong WC (2022b) A ship motion forecasting approach based on empirical mode decomposition method hybrid deep learning network and quantum butterfly optimization algorithm. Nonlinear Dyn 107: 2447-2467. https://doi.org/10.1007/s11071-021-07139-y
[23] Li SQ, Liu XM, Lin AJ (2020) Fractional frequency hybrid model based on EEMD for financial time series forecasting. Commun Nonlinear Sci Numer Simulat 89: 105281. https://doi.org/10.1016/j.cnsns.2020.105281
[24] Li XM, Han ZW, Zhao TY, Zhang JL, Xue D (2021) Modeling for indoor temperature prediction based on time-delay and Elman neural network in air conditioning system. J Build Eng 33: 101854. https://doi.org/10.1016/j.jobe.2020.101854
[25] Lin Z, Yang Q, Guo ZQ, Li J (2011) An improved Autoregressive method with Kalman Filtering Theory for vessel motion predication. Int J Intell Eng Syst 4(4): 11-18. DOI:10.22266/ijies2011.1231.02
[26] Luo QF, Li J, Zhou YQ, Liao L (2020) Using spotted hyena optimizer for training feedforward neural networks. Cogn Syst Res 65: 1-16. https://doi.org/10.1016/j.cogsys.2020.09.001
[27] Mao ZY, Zhao Y, Zhao JX, Shao LH, Zuo H (2025) A dual-channel ultra-short-term ship motion prediction model using temporal convolutional network, frequency-enhanced channel attention and informer. Phy Fluids 37(2): 022145. https://doi.org/10.1063/5.0257465
[28] Mi XW, Liu H, Li YF (2019) Wind speed prediction model using singular spectrum analysis, empirical mode decomposition and convolutional support vector machine. Energ Convers Manage 180: 196-205. https://doi.org/10.1016/j.enconman.2018.11.006
[29] Moreno SR, dos Santos Coelho L (2018) Wind speed forecasting approach based on singular spectrum analysis and adaptive neuro fuzzy inference system. Renew Energ 126: 736-754. https://doi.org/10.1016/j.renene.2017.11.089
[30] Peng XY, Zhang B, Zhou HG (2019) An improved particle swarm optimization algorithm applied to long short-term memory neural network for ship motion attitude prediction. T I Meas Control 41(15): 4462-4471. DOI: 10.1177/0142331219860731
[31] Peng XY, Zhao XR, Wei NX, Xie N (2001) AR algorithm for extremely short-term prediction of large ship’s motion. Ship Eng (5): 5-7, 10. (in Chinese)
[32] Qiu XP (2020) Neural networks and deep learning. China Machine Press, Beijing, China
[33] Reis J, Batista P, Oliveira P, Silvestre C (2023) Discrete-time Kalman filter for heave motion estimation. Ocean Eng 277: 114240. https://doi.org/10.1016/j.oceaneng.2023.114240
[34] Song CY, Zhang XK, Zhang GQ (2022) Attitude prediction of ship coupled heave-pitch motions using nonlinear innovation via full-scale test data. Ocean Eng 264: 112524. https://doi.org/10.1016/j.oceaneng.2022.112524
[35] Sun Q, Tang Z, Gao JP, Zhang GC (2022) Short-term ship motion attitude prediction based on LSTM and GPR. Appl Ocean Res 118: 102927. https://doi.org/10.1016/j.apor.2021.102927
[36] Suhermi N, Suhartono, Prastyo DD, Ali B (2018) Roll motion prediction using a hybrid deep learning and ARIMA model. Proced Comput Sci 144: 251-258. https://doi.org/10.1016/j.procs.2018. 10.526
[37] Tarkhaneh O, Shen HF (2019) Training of feedforward neural networks for data classification using hybrid particle swarm optimization, Mantegna Lévy flight and neighborhood search. Heliyon 5(4): e01275. https://doi.org/10.1016/j.heliyon.2019.e01275
[38] Wang YC, Dou YB, Yang ZQ, Fu HX (2024) Ship pitch prediction method based on LSTMC and multi-head attention. Ocean Eng 309: 118236. https://doi.org/10.1016/j.oceaneng.2024.118236
[39] Wang YC, Wang H, Zou DX, Fu HX (2021) Ship roll prediction algorithm based on Bi-LSTM-TPA combined model. J Mar Sci Eng 9(4): 387. https://doi.org/10.3390/jmse9040387
[40] Wei YY, Chen ZZ, Zhao C, Chen X (2023a) Deterministic ship roll forecasting model based on multi-objective data fusion and multi-layer error correction. Appl Soft Comput 132: 109915. https://doi.org/10.1016/j.asoc.2022.109915
[41] Wei YY, Chen ZZ, Zhao C, Tu YH, Chen X, Yang R (2021) A BiLSTM hybrid model for ship roll multi-step forecasting based on decomposition and hyperparameter optimization. Ocean Eng 242: 110138. https://doi.org/10.1016/j.oceaneng.2021.110138
[42] Wei YY, Chen ZZ, Zhao C, Chen X, He JH, Zhang CY (2023b) A three-stage multi-objective heterogeneous integrated model with decomposition-reconstruction mechanism and adaptive segmentation error correction method for ship motion multi-step prediction. Adv Eng Inform 56: 101954. https://doi.org/10.1016/j.aei.2023.101954
[43] Wu ZH, Huang NE (2009) Ensemble empirical model decomposition: A noise assisted data analysis method. Adv Adapt Data Anal 1(1): 1-41. DOI: 10.1142/S1793536909000047
[44] Xia JD, Zheng WL, Wang ZH, Li J (2022) Short-term prediction of ship motion attitude based on EMD-LSTM. Comput Digit Eng 50(7): 1434-1438. DOI: 10.3969/j.issn.1672-9722.2022.07.007
[45] Yang Y, Yu DJ, Cheng JS (2006) A roller bearing fault diagnosis method based on EMD energy entropy and ANN. Journal of Sound and Vibration 294(1-2): 269-277. https://doi.org/10.1016/j.jsv.2005.11.002
[46] Yin JC, Zou ZJ, Xu F (2013) On-line prediction of ship roll motion during maneuvering using sequential learning RBF neural networks. Ocean Eng 61: 139-147. https://doi.org/10.1016/j.oceaneng.2013. 01.005
[47] Yin JC, Perakis NA, Wang N (2018) A real-time ship roll motion prediction using wavelet transform and variable RBF network. Ocean Eng 160: 10-19. https://doi.org/10.1016/j.oceaneng.2018. 04.058
[48] Yin JC, Zou ZJ, Xu, F, Wang NN (2014) Online ship roll motion prediction based on grey sequential extreme learning machine. Neurocomputing 129: 168-174. https://doi.org/10.1016/j.neucom. 2013.09.043
[49] Yu CJ, Li YL, Zhang MJ (2017) An improved wavelet transform using singular spectrum analysis for wind speed forecasting based on Elman Neural Network. Energ Convers Manage 148: 895-904. https://doi.org/10.1016/j.enconman.2017.05.063
[50] Yuan PY, Zhao WG, Zhao Y (2025) Intelligent prediction algorithm of ship roll and pitch motion based on SSA-optimized BiLSTM network. Ocean Eng 320: 120331. https://doi.org/10.1016/j.oceaneng.2025.120331
[51] Zambom AZ, Dias R (2013) A review of kernel density estimation with applications to econometrics. Int Economet Rev 5(1): 20-42. https://doi.org/10.48550/arXiv.1212.2812
[52] Zhang B, Wang S, Deng LW, Jia MQ, Xu JZ (2023) Ship motion attitude prediction model based on IWOA-TCN-Attention. Ocean Eng 272: 113911. https://doi.org/10.1016/j.oceaneng.2023.113911
[53] Zhang B, Peng XY, Gao J (2020a) Ship motion attitude prediction based on ELM-EMD-LSTM integrated model. J Ship Mech 24(11): 1413-1421. DOI: 10.3969/j.issn.1007-7294.2020.11.005
[54] Zhang GY, Tan F, Wu YX (2020b) Ship motion attitude prediction based on an adaptive dynamic particle swarm optimization algorithm and bidirectional LSTM Neural Network. IEEE Access 8: 90087-90098. DOI: 10.1109/ACCESS.2020.2993909
[55] Zhang T, Zheng XQ, Liu MX (2021) Multiscale attention-based LSTM for ship motion prediction. Ocean Eng 230(13): 109066. https://doi.org/10.1016/j.oceaneng.2021.109066
[56] Zhang XL, Lu XM, Jia RS, Kan ST (2018) Micro-seismic signal denoising method based on variational mode decomposition and energy entropy. J China Coal Soc 43(2): 356-363. (in Chinese) DOI: 10.13225/j.cnki.jccs.2017.4153
[57] Zhou BW, Ma XJ, Luo YH, Yang DS (2019) Wind power prediction based on LSTM networks and nonparametric kernel density estimation. IEEE Access 7: 165279-165292
[58] Zhou L, Duan YX, Ren ZR, An S (2021) Active heave compensation control system and motion prediction research. J Huazhong Univ of Sci & Tech (Natural Science Edition) 49(3): 98-104. (in Chinese) DOI: 10.13245/j.hust.210318

Memo

Memo:
Received date:2025-3-7;Accepted date:2025-4-17。<br>Foundation item:Funding Supported by the National Natural Science Foundation of China (under Grants of 52271361 and 52231014), the Natural Science Foundation of Guangdong Province of China (under Grant 2023A1515010684) and the Special Projects of Key Areas for Colleges and Universities of Guangdong Province (under Grant 2021ZDZX1008).<br>Corresponding author:Jianchuan Yin，E-mail:yinjianchuan@gdou.edu.cn
Last Update: 2026-10-16