{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,29]],"date-time":"2026-04-29T06:50:20Z","timestamp":1777445420963,"version":"3.51.4"},"reference-count":77,"publisher":"MDPI AG","issue":"2","license":[{"start":{"date-parts":[[2021,2,11]],"date-time":"2021-02-11T00:00:00Z","timestamp":1613001600000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"DOI":"10.13039\/501100012166","name":"National Key Research and Development Program of China","doi-asserted-by":"publisher","award":["2020YFC1606801"],"award-info":[{"award-number":["2020YFC1606801"]}],"id":[{"id":"10.13039\/501100012166","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Entropy"],"abstract":"<jats:p>Trend prediction based on sensor data in a multi-sensor system is an important topic. As the number of sensors increases, we can measure and store more and more data. However, the increase in data has not effectively improved prediction performance. This paper focuses on this problem and presents a distributed predictor that can overcome unrelated data and sensor noise: First, we define the causality entropy to calculate the measurement\u2019s causality. Then, the series causality coefficient (SCC) is proposed to select the high causal measurement as the input data. To overcome the traditional deep learning network\u2019s over-fitting to the sensor noise, the Bayesian method is used to obtain the weight distribution characteristics of the sub-predictor network. A multi-layer perceptron (MLP) is constructed as the fusion layer to fuse the results from different sub-predictors. The experiments were implemented to verify the effectiveness of the proposed method by meteorological data from Beijing. The results show that the proposed predictor can effectively model the multi-sensor system\u2019s big measurement data to improve prediction performance.<\/jats:p>","DOI":"10.3390\/e23020219","type":"journal-article","created":{"date-parts":[[2021,2,12]],"date-time":"2021-02-12T16:12:10Z","timestamp":1613146330000},"page":"219","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":39,"title":["Distributed Deep Fusion Predictor for a Multi-Sensor System Based on Causality Entropy"],"prefix":"10.3390","volume":"23","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-2230-0077","authenticated-orcid":false,"given":"Xue-Bo","family":"Jin","sequence":"first","affiliation":[{"name":"Artificial Intelligence College, Beijing Technology and Business University, Beijing 10048, China"},{"name":"China Light Industry Key Laboratory of Industrial Internet and Big Data Beijing Technology and Business University, Beijing 10048, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Xing-Hong","family":"Yu","sequence":"additional","affiliation":[{"name":"Artificial Intelligence College, Beijing Technology and Business University, Beijing 10048, China"},{"name":"China Light Industry Key Laboratory of Industrial Internet and Big Data Beijing Technology and Business University, Beijing 10048, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Ting-Li","family":"Su","sequence":"additional","affiliation":[{"name":"Artificial Intelligence College, Beijing Technology and Business University, Beijing 10048, China"},{"name":"China Light Industry Key Laboratory of Industrial Internet and Big Data Beijing Technology and Business University, Beijing 10048, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Dan-Ni","family":"Yang","sequence":"additional","affiliation":[{"name":"Electrical and Information Engineering College, Tianjin University, Tianjin 300072, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-8047-1010","authenticated-orcid":false,"given":"Yu-Ting","family":"Bai","sequence":"additional","affiliation":[{"name":"Artificial Intelligence College, Beijing Technology and Business University, Beijing 10048, China"},{"name":"China Light Industry Key Laboratory of Industrial Internet and Big Data Beijing Technology and Business University, Beijing 10048, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Jian-Lei","family":"Kong","sequence":"additional","affiliation":[{"name":"Artificial Intelligence College, Beijing Technology and Business University, Beijing 10048, China"},{"name":"China Light Industry Key Laboratory of Industrial Internet and Big Data Beijing Technology and Business University, Beijing 10048, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Li","family":"Wang","sequence":"additional","affiliation":[{"name":"Artificial Intelligence College, Beijing Technology and Business University, Beijing 10048, China"},{"name":"China Light Industry Key Laboratory of Industrial Internet and Big Data Beijing Technology and Business University, Beijing 10048, China"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2021,2,11]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1177\/1729881419839596","article-title":"A review of mobile robots: Concepts, methods, theoretical framework, and applications","volume":"16","author":"Rubio","year":"2019","journal-title":"Int. J. Adv. Robot. Syst."},{"key":"ref_2","doi-asserted-by":"crossref","unstructured":"Zhao, Z., Wang, X., Yao, P., and Bai, Y. (2020). A health performance evaluation method of multirotors under wind turbulence. Nonlinear Dyn., 102.","DOI":"10.1007\/s11071-020-06041-3"},{"key":"ref_3","doi-asserted-by":"crossref","unstructured":"Zhang, X., Zhao, Z., Wang, Z., and Wang, X. (2021). Fault detection and identification method for quadcopter based on airframe vibration signals. Sensors, 21.","DOI":"10.3390\/s21020581"},{"key":"ref_4","doi-asserted-by":"crossref","unstructured":"Jin, X., Yu, X., and Wang, X. (2020). Deep Learning Predictor for Sustainable Precision Agriculture Based on Internet of Things System. Sustainability, 12.","DOI":"10.3390\/su12041433"},{"key":"ref_5","doi-asserted-by":"crossref","unstructured":"Jin, X., Yang, N., and Wang, X. (2020). Hybrid Deep Learning Predictor for Smart Agriculture Sensing Based on Empirical Mode Decomposition and Gated Recurrent Unit Group Model. Sensors, 20.","DOI":"10.3390\/s20051334"},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"103172","DOI":"10.1016\/j.micpro.2020.103172","article-title":"Intelligent based novel embedded system based IoT enabled air pollution monitoring system","volume":"77","author":"Senthilkumar","year":"2020","journal-title":"Microprocess. Microsyst."},{"key":"ref_7","doi-asserted-by":"crossref","unstructured":"Jin, X., Yang, N., and Wang, X. (2019). Integrated Predictor Based on Decomposition Mechanism for PM2.5 Long-Term Prediction. Appl. Sci., 9.","DOI":"10.3390\/app9214533"},{"key":"ref_8","doi-asserted-by":"crossref","unstructured":"Jin, X., Sun, S., Wei, H., and Yang, F. (2018). Advances in Multi-Sensor Information Fusion: Theory and Applications 2017. Sensors, 18.","DOI":"10.3390\/s18041162"},{"key":"ref_9","doi-asserted-by":"crossref","unstructured":"Lee, D., and Kim, K. (2019). Recurrent neural network-based hourly prediction of photovoltaic power output using meteorological information. Energies, 12.","DOI":"10.3390\/en12020215"},{"key":"ref_10","doi-asserted-by":"crossref","unstructured":"Jin, X., Yang, N., and Wang, X. (2020). Deep hybrid model based on EMD with classification by frequency characteristics for long-term air quality prediction. Mathematics, 8.","DOI":"10.3390\/math8020214"},{"key":"ref_11","doi-asserted-by":"crossref","unstructured":"Huang, C.J., and Kuo, P.H. (2018). A deep CNN-LSTM model for particulate matter (PM2.5) forecasting in smart cities. Sensors, 18.","DOI":"10.3390\/s18072220"},{"key":"ref_12","doi-asserted-by":"crossref","unstructured":"Bai, Y., Wang, X., and Sun, Q. (2019). Spatio-temporal prediction for the monitoring-blind area of industrial atmosphere based on the fusion network. Int. J. Environ. Res. Public Health, 16.","DOI":"10.3390\/ijerph16203788"},{"key":"ref_13","doi-asserted-by":"crossref","unstructured":"Bai, Y., Jin, X., Wang, X., Wang, X., and Xu, J. (2020). Dynamic correlation analysis method of air pollutants in spatio-temporal analysis. Int. J. Environ. Res. Public Health, 17.","DOI":"10.3390\/ijerph17010360"},{"key":"ref_14","unstructured":"Ziemann, T., Peri, H., and Singh, A. (2020). System and method for enhancing trust for person-related data sources. (10,542,043), U.S. Patent."},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"eaau4996","DOI":"10.1126\/sciadv.aau4996","article-title":"Detecting and quantifying causal associations in large nonlinear time series datasets","volume":"5","author":"Runge","year":"2019","journal-title":"Sci. Adv."},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"9107167","DOI":"10.1155\/2019\/9107167","article-title":"Compound autoregressive network for prediction of multivariate time series","volume":"2019","author":"Bai","year":"2019","journal-title":"Complexity"},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"916","DOI":"10.1016\/j.ejor.2018.12.013","article-title":"Structural combination of seasonal exponential smoothing forecasts applied to load forecasting","volume":"275","year":"2019","journal-title":"Eur. J. Oper. Res."},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"532","DOI":"10.1007\/s40313-019-00467-w","article-title":"A novel moving average forecasting approach using fuzzy time series data set","volume":"30","author":"Gautam","year":"2019","journal-title":"J. Control. Autom. Electr. Syst."},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"1485","DOI":"10.1016\/j.ijforecast.2018.02.001","article-title":"Online adaptive lasso estimation in vector autoregressive models for high dimensional wind power forecasting","volume":"35","author":"Messner","year":"2019","journal-title":"Int. J. Forecast."},{"key":"ref_20","doi-asserted-by":"crossref","unstructured":"Alsharif, M., Younes, M., and Kim, J. (2019). Time series ARIMA model for prediction of daily and monthly average global solar radiation: The case study of Seoul, South Korea. Symmetry, 11.","DOI":"10.3390\/sym11020240"},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"899","DOI":"10.1016\/j.eneco.2019.05.026","article-title":"A hybrid short-term electricity price forecasting framework: Cuckoo search-based feature selection with singular spectrum analysis and SVM","volume":"81","author":"Zhang","year":"2019","journal-title":"Energy Econ."},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"879","DOI":"10.1016\/j.renene.2018.07.060","article-title":"Wind speed and wind direction forecasting using echo state network with nonlinear functions","volume":"131","author":"Chitsazan","year":"2019","journal-title":"Renew. Energy"},{"key":"ref_23","first-page":"79","article-title":"A novel dbn model for time series forecasting","volume":"44","author":"Ren","year":"2017","journal-title":"IAENG Int. J. Comput. Sci."},{"key":"ref_24","doi-asserted-by":"crossref","unstructured":"Sulaiman, J., and Wahab, S.H. (2018). Heavy rainfall forecasting model using artificial neural network for flood prone area. IT Convergence and Security 2017, Springer.","DOI":"10.1007\/978-981-10-6451-7_9"},{"key":"ref_25","doi-asserted-by":"crossref","unstructured":"Izonin, I., Tkachenko, R., Verhun, V., and Zub, K. (2020). An approach towards missing data management using improved GRNN-SGTM ensemble method-ScienceDirect. Eng. Sci. Technol. Int. J., in press.","DOI":"10.1016\/j.jestch.2020.10.005"},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"10252","DOI":"10.1109\/TVT.2019.2933232","article-title":"RNN-based path prediction of obstacle vehicles with deep ensemble","volume":"10","author":"Min","year":"2019","journal-title":"IEEE Trans. Veh. Technol."},{"key":"ref_27","doi-asserted-by":"crossref","unstructured":"Sundermeyer, M., Schl\u00fcter, R., and Ney, H. (2012, January 9\u201313). LSTM neural networks for language modeling. Proceedings of the Thirteenth Annual Conference of the International Speech Communication Association, Portland, OR, USA.","DOI":"10.21437\/Interspeech.2012-65"},{"key":"ref_28","doi-asserted-by":"crossref","unstructured":"Wang, Y., Liao, W., and Chang, Y. (2018). Gated recurrent unit network-based short-term photovoltaic forecasting. Energies, 11.","DOI":"10.3390\/en11082163"},{"key":"ref_29","doi-asserted-by":"crossref","first-page":"3847","DOI":"10.1049\/iet-gtd.2018.6687","article-title":"Short-term power load forecasting based on multi-layer bidirectional recurrent neural network","volume":"13","author":"Tang","year":"2019","journal-title":"IET Gener. Transm. Distrib."},{"key":"ref_30","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1016\/j.scitotenv.2019.01.333","article-title":"A hybrid model for spatiotemporal forecasting of PM2.5 based on graph convolutional neural network and long short-term memory","volume":"664","author":"Qi","year":"2019","journal-title":"Sci. Total. Environ."},{"key":"ref_31","doi-asserted-by":"crossref","unstructured":"Tian, C., Ma, J., and Zhang, C. (2018). A deep neural network model for short-term load forecast based on long short-term memory network and convolutional neural network. Energies, 11.","DOI":"10.3390\/en11123493"},{"key":"ref_32","doi-asserted-by":"crossref","first-page":"3046","DOI":"10.1007\/s00034-017-0705-4","article-title":"Iterative parameter estimation for signal models based on measured data","volume":"37","author":"Xu","year":"2018","journal-title":"Circuits Syst. Signal Process."},{"key":"ref_33","doi-asserted-by":"crossref","first-page":"1687814017730003","DOI":"10.1177\/1687814017730003","article-title":"The parameter estimation algorithms based on the dynamical response measurement data","volume":"9","author":"Xu","year":"2017","journal-title":"Adv. Mech. Eng."},{"key":"ref_34","doi-asserted-by":"crossref","first-page":"211","DOI":"10.1016\/j.dsp.2016.11.010","article-title":"Joint state and multi-innovation parameter estimation for time-delay linear systems and its convergence based on the Kalman filtering","volume":"62","author":"Ding","year":"2017","journal-title":"Dig. Signal Proc."},{"key":"ref_35","doi-asserted-by":"crossref","first-page":"2506","DOI":"10.1049\/iet-cta.2016.0202","article-title":"Performance analysis of the generalised projection identification for time-varying systems","volume":"10","author":"Ding","year":"2016","journal-title":"IET Control. Theory Appl."},{"key":"ref_36","doi-asserted-by":"crossref","first-page":"2178","DOI":"10.1007\/s00034-019-01261-4","article-title":"Weighted parameter estimation for Hammerstein nonlinear ARX systems","volume":"39","author":"Ding","year":"2020","journal-title":"Circuits Syst. Signal Proc."},{"key":"ref_37","doi-asserted-by":"crossref","first-page":"112575","DOI":"10.1016\/j.cam.2019.112575","article-title":"Gradient estimation algorithms for the parameter identification of bilinear systems using the auxiliary model","volume":"369","author":"Ding","year":"2020","journal-title":"J. Comput. Appl. Math."},{"key":"ref_38","doi-asserted-by":"crossref","first-page":"1189","DOI":"10.1007\/s12555-016-0081-z","article-title":"A filtering based multi-innovation extended stochastic gradient algorithm for multivariable control systems","volume":"15","author":"Pan","year":"2017","journal-title":"Int. J. Control Autom. Syst."},{"key":"ref_39","doi-asserted-by":"crossref","first-page":"677","DOI":"10.1049\/iet-cta.2019.0413","article-title":"Recursive parameter estimation and its convergence for bilinear systems","volume":"14","author":"Zhang","year":"2020","journal-title":"IET Control. Theory Appl."},{"key":"ref_40","doi-asserted-by":"crossref","first-page":"23","DOI":"10.1016\/j.sigpro.2018.01.012","article-title":"The least squares based iterative algorithms for parameter estimation of a bilinear system with autoregressive noise using the data filtering technique","volume":"147","author":"Li","year":"2018","journal-title":"Signal Process."},{"key":"ref_41","doi-asserted-by":"crossref","first-page":"726","DOI":"10.1016\/j.jfranklin.2019.11.003","article-title":"Recursive identification of bilinear time-delay systems through the redundant rule","volume":"357","author":"Zhang","year":"2020","journal-title":"J. Frankl. Inst."},{"key":"ref_42","doi-asserted-by":"crossref","first-page":"228","DOI":"10.1049\/iet-spr.2016.0220","article-title":"Parameter estimation algorithms for dynamical response signals based on the multi-innovation theory and the hierarchical principle","volume":"11","author":"Xu","year":"2017","journal-title":"IET Signal Process."},{"key":"ref_43","doi-asserted-by":"crossref","first-page":"899","DOI":"10.3389\/fgene.2019.00899","article-title":"On the use of the pearson correlation coefficient for model evaluation in genome-wide prediction","volume":"10","author":"Waldmann","year":"2019","journal-title":"Front. Genet."},{"key":"ref_44","doi-asserted-by":"crossref","unstructured":"Amarkhil, Q., Elwakil, E., and Hubbard, B. (2020). A meta-analysis of critical causes of project delay using spearman\u2019s rank and relative importance index integrated approach. Can. J. Civ. Eng., Just-IN.","DOI":"10.1139\/cjce-2020-0527"},{"key":"ref_45","doi-asserted-by":"crossref","unstructured":"Duan, S., Yang, W., and Wang, X. (2019, January 11\u201313). Grain pile temperature forecasting from weather factors: A support vector regression approach. Proceedings of the 2019 IEEE\/CIC International Conference on Communications in China (ICCC), Changchun, China.","DOI":"10.1109\/ICCChina.2019.8855910"},{"key":"ref_46","doi-asserted-by":"crossref","first-page":"113500","DOI":"10.1016\/j.apenergy.2019.113500","article-title":"Building energy performance forecasting: A multiple linear regression approach","volume":"253","author":"Ciulla","year":"2019","journal-title":"Appl. Energy"},{"key":"ref_47","first-page":"2564","article-title":"Ultra short-term PV power forecasting based on ELM segmentation model","volume":"2017","author":"Jing","year":"2017","journal-title":"J. Eng."},{"key":"ref_48","doi-asserted-by":"crossref","unstructured":"Lin, C.Y., Chang, Y.S., and Chiao, H.T. (2019, January 6\u20139). Design a Hybrid Framework for Air Pollution Forecasting. Proceedings of the 2019 IEEE International Conference on Systems, Man and Cybernetics (SMC), Bari, Italy.","DOI":"10.1109\/SMC.2019.8914257"},{"key":"ref_49","doi-asserted-by":"crossref","unstructured":"Gao, D., Zhou, Y., Wang, T., and Wang, Y. (2020). A Method for predicting the remaining useful life of lithium-ion batteries based on particle filter using Kendall rank correlation coefficient. Energies, 13.","DOI":"10.3390\/en13164183"},{"key":"ref_50","doi-asserted-by":"crossref","unstructured":"Contreras-Reyes, J.E., and Hern\u00e1ndez-Santoro, C. (2020). Assessing granger-causality in the southern humboldt current ecosystem using cross-spectral methods. Entropy, 22.","DOI":"10.3390\/e22101071"},{"key":"ref_51","doi-asserted-by":"crossref","first-page":"084102","DOI":"10.1103\/PhysRevLett.100.084102","article-title":"Detrended cross-correlation analysis: A new method for analyzing two nonstationary time series","volume":"100","author":"Podobnik","year":"2008","journal-title":"Phys. Rev. Lett."},{"key":"ref_52","doi-asserted-by":"crossref","unstructured":"Jin, X., Zhang, J., Su, T., Bai, Y., Kong, J., and Wang, X. (2021). Wavelet-deep optimized model for nonlinear multi-component data forecasting. Comput. Intell. Neurosci., accept.","DOI":"10.1155\/2021\/8810046"},{"key":"ref_53","doi-asserted-by":"crossref","first-page":"4346803","DOI":"10.1155\/2020\/4346803","article-title":"Deep-learning prediction model with serial two-level decomposition based on bayesian optimization","volume":"2020","author":"Jin","year":"2020","journal-title":"Complexity"},{"key":"ref_54","first-page":"1","article-title":"A survey on Bayesian deep learning","volume":"53","author":"Wang","year":"2020","journal-title":"ACM Comput. Surv. (CSUR)"},{"key":"ref_55","doi-asserted-by":"crossref","unstructured":"Mukhopadhyay, P., and Mallick, S. (2019). Bayesian deep learning for seismic facies classification and its uncertainty estimation. Soc. Explor. Geophys., 2488\u20132492.","DOI":"10.1190\/segam2019-3216870.1"},{"key":"ref_56","unstructured":"Zhang, R., Li, C., and Zhang, J. (2019). Cyclical stochastic gradient MCMC for Bayesian deep learning. arXiv."},{"key":"ref_57","first-page":"315908","article-title":"Closed-loop estimation for randomly sampled measurements in target tracking system","volume":"2014","author":"Jin","year":"2014","journal-title":"Math. Probl. Eng."},{"key":"ref_58","doi-asserted-by":"crossref","first-page":"219","DOI":"10.5772\/54471","article-title":"Target tracking of a linear time invariant system under irregular sampling","volume":"9","author":"Jin","year":"2012","journal-title":"Int. J. Adv. Robot. Syst."},{"key":"ref_59","doi-asserted-by":"crossref","unstructured":"Li, G., Yang, L., Lee, C.G., Wang, X., and Rong, M. (2020). A Bayesian deep learning RUL framework integrating epistemic and aleatoric uncertainties. IEEE Trans. Ind. Electron., 1.","DOI":"10.1109\/TIE.2020.3009593"},{"key":"ref_60","unstructured":"Harper, R., and Southern, J. (2020). A Bayesian deep learning framework for end-to-end prediction of emotion from heartbeat. IEEE Trans. Affect. Comput."},{"key":"ref_61","unstructured":"Abdi, H. (2007). The Kendall rank correlation coefficient. Encyclopedia of Measurement and Statistics, Salkind, N.J., Ed, SAGE Publications Inc."},{"key":"ref_62","first-page":"13","article-title":"Flexible Bayesian analysis of the von bertalanffy growth function with the use of a log-skew-t distribution","volume":"115","author":"Wiff","year":"2017","journal-title":"Fish. Bull."},{"key":"ref_63","doi-asserted-by":"crossref","first-page":"1021","DOI":"10.1109\/LSP.2019.2915000","article-title":"Kullback\u2013Leibler divergence between multivariate generalized gaussian distributions","volume":"26","author":"Bouhlel","year":"2019","journal-title":"IEEE Signal Process. Lett."},{"key":"ref_64","doi-asserted-by":"crossref","first-page":"1756","DOI":"10.1007\/s12555-017-0482-7","article-title":"Hierarchical parameter estimation for the frequency response based on the dynamical window data","volume":"16","author":"Xu","year":"2018","journal-title":"Int. J. Control Autom. Syst."},{"key":"ref_65","doi-asserted-by":"crossref","first-page":"1623","DOI":"10.1016\/j.jfranklin.2018.08.030","article-title":"State space model identification of multirate processes with time-delay using the expectation maximization","volume":"356","author":"Gu","year":"2019","journal-title":"J. Frankl. Inst."},{"key":"ref_66","doi-asserted-by":"crossref","first-page":"141","DOI":"10.1080\/00207721.2018.1544303","article-title":"Hierarchical Newton and least squares iterative estimation algorithm for dynamic systems by transfer functions based on the impulse responses","volume":"50","author":"Xu","year":"2019","journal-title":"Int. J. Syst. Sci."},{"key":"ref_67","doi-asserted-by":"crossref","first-page":"1276","DOI":"10.1049\/iet-cta.2019.0731","article-title":"Hierarchical multi-innovation generalised extended stochastic gradient methods for multivariable equation-error autoregressive moving average systems","volume":"14","author":"Xu","year":"2020","journal-title":"IET Control. Theory Appl."},{"key":"ref_68","doi-asserted-by":"crossref","first-page":"455","DOI":"10.1049\/iet-spr.2019.0481","article-title":"Recursive coupled projection algorithms for multivariable output-error-like systems with coloured noises","volume":"14","author":"Pan","year":"2020","journal-title":"IET Signal Process."},{"key":"ref_69","doi-asserted-by":"crossref","first-page":"937","DOI":"10.1002\/acs.3113","article-title":"Separable multi-innovation stochastic gradient estimation algorithm for the nonlinear dynamic responses of systems","volume":"34","author":"Xu","year":"2020","journal-title":"Int. J. Adapt. Control Signal Process."},{"key":"ref_70","doi-asserted-by":"crossref","first-page":"2415","DOI":"10.1007\/s11071-017-3594-y","article-title":"Recursive parameter identification of the dynamical models for bilinear state space systems","volume":"89","author":"Zhang","year":"2017","journal-title":"Nonlinear Dyn."},{"key":"ref_71","doi-asserted-by":"crossref","first-page":"3079","DOI":"10.1016\/j.jfranklin.2018.01.011","article-title":"Combined state and parameter estimation for a bilinear state space system with moving average noise","volume":"355","author":"Zhang","year":"2018","journal-title":"J. Frankl. Inst."},{"key":"ref_72","doi-asserted-by":"crossref","first-page":"2176","DOI":"10.1049\/iet-cta.2020.0104","article-title":"Bias compensation-based parameter and state estimation for a class of time-delay nonlinear state-space models","volume":"14","author":"Gu","year":"2020","journal-title":"IET Control. Theory Appl."},{"key":"ref_73","doi-asserted-by":"crossref","first-page":"1704","DOI":"10.1049\/iet-cta.2018.0156","article-title":"State filtering-based least squares parameter estimation for bilinear systems using the hierarchical identification principle","volume":"12","author":"Zhang","year":"2018","journal-title":"IET Control. Theory Appl."},{"key":"ref_74","doi-asserted-by":"crossref","first-page":"10102","DOI":"10.1016\/j.jfranklin.2019.06.032","article-title":"Hierarchical recursive generalized extended least squares estimation algorithms for a class of nonlinear stochastic systems with colored noise","volume":"356","author":"Wang","year":"2019","journal-title":"J. Frankl. Inst."},{"key":"ref_75","doi-asserted-by":"crossref","first-page":"875","DOI":"10.1002\/acs.2995","article-title":"Highly computationally efficient state filter based on the delta operator","volume":"33","author":"Zhang","year":"2019","journal-title":"Int. J. Adapt. Control Signal Process."},{"key":"ref_76","doi-asserted-by":"crossref","first-page":"5492","DOI":"10.1002\/rnc.5084","article-title":"Two-stage auxiliary model gradient-based iterative algorithm for the input nonlinear controlled autoregressive system with variable-gain nonlinearity","volume":"30","author":"Fan","year":"2020","journal-title":"Int. J. Robust Nonlinear Control"},{"key":"ref_77","doi-asserted-by":"crossref","first-page":"1157","DOI":"10.1002\/acs.3027","article-title":"State estimation for bilinear systems through minimizing the covariance matrix of the state estimation errors","volume":"33","author":"Zhang","year":"2019","journal-title":"Int. J. Adapt. Control Signal Process."}],"container-title":["Entropy"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/1099-4300\/23\/2\/219\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T05:22:44Z","timestamp":1760160164000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/1099-4300\/23\/2\/219"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2021,2,11]]},"references-count":77,"journal-issue":{"issue":"2","published-online":{"date-parts":[[2021,2]]}},"alternative-id":["e23020219"],"URL":"https:\/\/doi.org\/10.3390\/e23020219","relation":{},"ISSN":["1099-4300"],"issn-type":[{"value":"1099-4300","type":"electronic"}],"subject":[],"published":{"date-parts":[[2021,2,11]]}}}