Multidimensional data are frequently encountered in numerous industrial systems, standard techniques regarding statistical multichannel prognostic methods naturally cannot fully exploit and capture the coupled nature and spatiotemporal relationship of the available information among channels. To alleviate a bottleneck problem, in this paper, a new approach named sparse-based fractional Levy quaternion extended Kalman filter (SFLQEKF) is formulated for the prognostic of stochastic time series. Initially, the stochastic time series is decomposed into the static component and the dynamic component resorting to the piecewise representation through a new improved sparse low-rank matrix (SLRM) approximation algorithm namely singular value penalty resonance sparse signal decomposition (SVP-RSSD). Afterward, both static component and dynamic component are predicted individually through the proposed fractional Levy quaternion extended Kalman filter (FLQEKF) method, the final time series of interest can be obtained by integrating the predicted static- and dynamic components correspondingly, in which a new predictive derivation of the proposed FLQEKF model is formulated for the first time by considering fractional-order characteristic and Levy flight process damping term, hence the issue of oscillating indeterminacy induced from classical extended Kalman filter (EFK) can be addressed. Eventually, the prognostic behavior of the proposed approach is investigated via the stochastic time series conducted from two experimental cases, the predicted results illustrate the availability and applicability of the proposed approach compared with three state-of-the-art benchmarks.
FengGLZhangLYYangJH, et al. Long-term prediction of time series using fuzzy cognitive maps. Eng Appl Artif Intel2021; 102: 104274.
2.
FangXLPaynabarKGebraeelN. Multistream sensor fusion-based prognostics model for systems with single failure modes. Reliab Eng Syst Saf2017; 159: 322–331.
3.
LiQZuoMJLiangSY. Prognosis of bearing degeneration using adaptive quaternion least mean biquadrate under framework of hypercomplex data. IEEE Sens J2020; 20(5): 2659–2670.
4.
CastellanoJMLJiménezIMMozosRS, et al. Weaning outcome prediction from heterogeneous time series using normalized compression distance and multidimensional scaling. Expert Syst Appl2013; 40(5): 1737–1747.
5.
BrandiGMatteoTD. Predicting multidimensional data via tensor learning. J Comput Sci2021; 53: 101372.
6.
WangTLuGLYanP. Multi-sensors based condition monitoring of rotary machines: an approach of multidimensional time-series analysis. Measurement2019; 134: 326–335.
7.
HuJHKimCHalaszP, et al. Artificial intelligence for performance prediction of organic solvent nanofiltration membranes. J Memb Sci2021; 619: 118513.
8.
ZhouYPHuangMH. Lithium-ion batteries remaining useful life prediction based on a mixture of empirical mode decomposition and ARIMA model. Microelectron Reliab2016; 65: 265–273
9.
WangZPQuJFFangXY, et al. Prediction of early stabilization time of electrolytic capacitor based on ARIMA-Bi-LSTM hybrid model. Neurocomputing2020; 403(25): 63–79.
10.
LiQLiangSYYangJG, et al. Long range dependence prognostics for bearing vibration intensity chaotic time series. Entropy2016; 18(1): 23.
11.
AnhVLunneyKPeiriscS. Stochastic models for characterisation and prediction of time series with long-range dependence and fractality. Environ Model Softw1997; 12(1): 67–73.
12.
ZhangHMoZWangJ, et al. Nonlinear drifted fractional Brownian motion with multiple hidden state variables for remaining useful life prediction of lithium-ion batteries. IEEE Trans Reliab2020; 69(2): 768–780.
13.
LiQLiangSY. Degradation trend prognostics for rolling bearing using improved R/S statistic model and fractional Brownian motion approach. IEEE Access2017; 6: 21103–21114.
14.
QianYYanR. Remaining useful life prediction of rolling bearings using an enhanced particle filter. IEEE Trans Instrum Meas2015; 64(10): 2696–2707.
15.
DongGChenZWeiJ, et al. Battery health prognosis using Brownian motion modeling and particle filtering. IEEE Trans Ind. Electron2018; 65(11); 8646–8655.
16.
WangLWuYXXuJP, et al. Nonlinear dynamic numerical analysis and prediction of complex system based on bivariate cycling time stochastic differential equation. Alexandria Eng J2020; 59(4): 2065–2082.
17.
QuJQQianZPeiY. Day-ahead hourly photovoltaic power forecasting using attention-based CNN-LSTM neural network embedded with multiple relevant and target variables prediction pattern. Energy2021; 232; 120996.
18.
LiJMYaoXFWangXD, et al. Multiscale local features learning based on BP neural network for rolling bearing intelligent fault diagnosis. Measurement2020; 153: 107419.
19.
BhattacharyyaSBhattacharjeeSMondalNK. A quantum backpropagation multilayer perceptron (QBMLP) for predicting iron adsorption capacity of calcareous soil from aqueous solution. Appl Soft Comput2015; 27: 299–312.
20.
KumarAZhouYQGandhiCP, et al. Bearing defect size assessment using wavelet transform based deep convolutional neural network (DCNN). Alexandria Eng J2020; 59(2): 999–1012.
21.
ChenDLQinYWangY, et al. Health indicator construction by quadratic function-based deep convolutional auto-encoder and its application into bearing residual useful lifetime prediction. ISA Trans2021; 114: 44–56.
22.
NittaTFuruyaT. A 3D vector version of the backpropagation algorithm. In: Proceedings of international joint conference on neural networks (IJCNN), Baltimore, MD, USA, 1992, vol. 2, pp. 511–516.
23.
ChengYJLuCLiTY, et al. Residual lifetime prediction for lithium-ion battery based on functional principal component analysis and Bayesian approach. Energy2015; 90(2): 1983–1993.
24.
WenSCYangCH. Time series analysis and prediction of nonlinear systems with ensemble learning framework applied to deep learning neural networks. Inf Sci2021; 572; 167–181.
25.
SilvaCCMartinsRD. Polar and axial vectors versus quaternions. Am J Phys2002; 70(9): 958–963.
26.
MaclachlanCRiedAW. The arithmetic of hyperbolic 3-manifolds. New York: Springer Verlag, 2003.
27.
AltmannSL. Hamilton, Rodrigues, and the Quaternion scandal. Math Mag1989; 62(5): 291–308.
28.
ZhangFZ. Quaternions and matrices of quaternions. Linear Algebra Appl1997; 251: 21–57.
29.
ArenaPFortunaLMuscatoG, et al. Lecture notes in control and information sciences: neural networks in multidimensional domains. New York: Springer Verlag, 1998.
30.
BihanNLMarsJ. Singular value decomposition of quaternion matrices: a new tool for vector-sensor signal processing. Signal Process2004; 84(7): 1177–1199.
31.
YangTFMaJFMiaoYB, et al. Quaternion weighted spherical Bessel-Fourier moment and its invariant for color image reconstruction and object recognition. Inf Sci2019; 505: 388–405.
32.
LiuW. Channel equalization and beamforming for quaternion-valued wireless communication systems. J Franklin Inst2017; 354(18): 8721–8733.
YiCCLvYDangZ, et al. Quaternion singular spectrum analysis using convex optimization and its application to fault diagnosis of rolling bearing. Measurement2017; 103: 321–332.
35.
MaYLChengJSHuNQ, et al. Symplectic quaternion singular mode decomposition with application in gear fault diagnosis. Mech Mach Theory2021; 160: 104266.
36.
ArenaPBaglioSFortunaL, et al. Chaotic time series prediction via quaternionic multilayer perceptrons. In: IEEE international conference on systems, man and cybernetics. Intelligent systems for the 21st century, Vancouver, BC, Canada, 1995, pp. 1790–1794. New York: IEEE.
37.
TookCCMandicDP. The Quaternion LMS algorithm for adaptive filtering of hypercomplex processes. IEEE Trans Signal Process2009; 57(4): 1316–1327.
38.
TookCCMandicDBenestyJ. Study of the quaternion LMS and four-channel LMS algorithms. In: 2009 IEEE international conference on acoustics, speech and signal processing, Taipei, Taiwan, 2009, pp. 3109–3112. New York: IEEE.
39.
TookCCMandicDPAiharaK. Quaternion-valued short term forecasting of wind profile. In: The 2010 international joint conference on neural networks (IJCNN), Barcelona, Spain, Barcelona, Spain, 2010, pp. 1–6. New York: IEEE.
40.
UjangCABCTookCCKavcicA, et al. A split quaternion nonlinear adaptive filter. In: 2009 IEEE international conference on acoustics, speech and signal processing, Taipei, Taiwan, 2009, pp. 1745–1748. New York: IEEE.
41.
HermesCWohlerCSchenkK, et al. Long-term vehicle motion prediction. In: IEEE intelligent vehicles symposium, 2009, pp. 652–657. New York: IEEE.
42.
ZhouWDingGHaoY, et al. Quaternion central divided difference Kalman filtering algorithm and its applications to initial alignment of SINS. In: IEEE international conference on information and automation, Harbin, China, 2010, pp. 1846–1851. New York: IEEE.
43.
TookCCMandicDP. Quaternion-valued stochastic gradient-based adaptive IIR filtering. IEEE Trans Signal Process2010; 58(7): 3895–3901.
44.
TookCCMandicDP. A quaternion widely linear adaptive filter. IEEE Trans Signal Process2010; 58(8): 4427–4431.
45.
TookCCStrbacGAiharaK, et al. Quaternion-valued short-term joint forecasting of three-dimensional wind and atmospheric parameters. Renew Energ2011; 36: 1754–1760.
46.
JiangMLiuWLiY, et al. Frequency-domain quaternion-valued adaptive filtering and its application to wind profile prediction. In: 2013 IEEE international conference of IEEE region 10 (TENCON 2013), Xi’an, China, 2013, pp. 1–5. New York: IEEE.
47.
JunCLuT. Gyro fault prediction algorithm based on UKF. In: 2012 International conference on image analysis and signal processing, Huangzhou, China, 2012, pp. 1–4. New York: IEEE.
48.
JahanchahiCMandicDP. A class of uaternion Kalman filters. IEEE Trans Neural Netw Learn Syst2014; 25(3): 533–544.
49.
XiaYJahanchahiCNittaT, et al. Performance bounds of quaternion estimators. IEEE Trans Neural Netw Learn Syst2015; 26(12): 3287–3292.
50.
Navarro-MorenoJRuiz-MolinaJC. Semi-widely linear estimation of Cη-proper quaternion random signal vectors under Gaussian and stationary conditions. Signal Process2016; 119: 56–66.
51.
Jiménez-LópezJDFernández-AlcaláRMNavarro-MorenoJ, et al. Widely linear estimation of quaternion signals with intermittent observations. Signal Process2017; 136: 92–101.
52.
Navarro-MorenoJFernández-AlcaláRMJiménez-LópezJD, et al. Widely linear estimation for multisensor quaternion systems with mixed uncertainties in the observations. J Franklin Inst2019; 356: 3115–3138.
53.
Fernández-AlcaláRMNavarro-MorenoJJiménez-LópezJD, et al. Semi-widely linear estimation algorithms of quaternion signals with missing observations and correlated noises. J Franklin Inst2020; 357: 3075–3096.
54.
HuangWXieHShenC, et al. A robust strong tracking cubature Kalman filter for spacecraft attitude estimation with quaternion constraint. Acta Astronaut2016; 121: 153–163.
55.
HikosakaMDingTHiroseA. Proposal of polarization state prediction using quaternion neural networks for fading channel prediction in mobile communications. In: 2016 international joint conference on neural networks (IJCNN), Vancouver, BC, Canada, 2016, pp. 4105–4111. New York: IEEE.
56.
ChenTChenBMaW, et al. Quaternion least mean kurtosis algorithm for adaptive filtering of 3D and 4D signal processes. In: 20th international conference on information fusion (Fusion), Xi’an, China, 2017, pp. 1–5. New York: IEEE.
57.
XiangMTookCCMandicDP. Cost-effective quaternion minimum mean square error estimation: from widely linear to four-channel processing. Signal Process2017; 136: 81–91.
58.
XiangMScalzo DeesBMandicDP. Multiple-model adaptive estimation for 3-D and 4-D Signals: a widely linear quaternion approach. IEEE Trans Neural Netw Learn Syst2019; 30(1): 72–84.
59.
SveierAEgelandO. Pose estimation using dual quaternions and moving horizon estimation. IFAC PapersOnLine2018; 51(13): 186–191.
60.
MengüçECAcırNMandicDP. Widely linear quaternion-valued least-mean kurtosis algorithm. IEEE Trans Signal Process2020; 68: 5914–5922.
61.
MengüçEC. Design of quaternion-valued second-order Volterra adaptive filters for nonlinear 3-D and 4-D signals. Signal Process2020; 174: 107619.
62.
LiuXZhangMJChenJW, et al. Trajectory tracking with quaternion-based attitude representation for autonomous underwater vehicle based on terminal sliding mode control. Appl Ocean Res2020; 104: 102342.
63.
LinJTLiuZChenCLP, et al. Quaternion broad learning system: a novel multi-dimensional filter for estimation and elimination tremor in teleoperation. Neurocomputing2020; 380: 78–86.
64.
XiaoJYWenSPYangXJ, et al. New approach to global Mittag-Leffler synchronization problem of fractional-order quaternion-valued BAM neural networks based on a new inequality. Neural Networks2020; 122: 320–337.
65.
ZhangWWShaCLCaoJD, et al. Adaptive quaternion projective synchronization of fractional order delayed neural networks in quaternion field. Appl Math Comput2021; 400: 126045.
66.
SongQKChenSHZhaoZJ, et al. Passive filter design for fractional-order quaternion-valued neural networks with neutral delays and external disturbance. Neural Networks2021; 137: 18–30.
67.
LiHLHuCZhangL, et al. Non-separation method-based robust finite-time synchronization of uncertain fractional-order quaternion-valued neural networks. Appl Math Comput2021; 409: 126377.
68.
WuZW. Multiple asymptotic stability of fractional-order quaternion-valued neural networks with time-varying delays. Neurocomputing2021; 448: 301–312.
69.
WangGQiaoJCXueR, et al. Quaternion kernel recursive least-squares algorithm. Signal Process2021; 178: 107810.
70.
FreskE. Quaternion based extended Kalman filter as an attitude and heading reference system. Lulea University of Technology, 2012.
71.
FreskENikolakopoulosG. Full quaternion based attitude control for a quadrotor. In: 2013 European control conference (ECC), Zurich, Switzerland, 2013, pp. 3864–3869. New York: IEEE.
72.
FreskENikolakopoulosGGustafssonT. A generalized reduced-complexity inertial navigation system for unmanned aerial vehicles. IEEE Trans Control Syst Technol2017; 25(1): 192–207.
FreskENikolakopoulosG. Experimental evaluation of a full quaternion based attitude quadrotor controller. In: 20th IEEE conference on emerging technologies & factory automation (ETFA), Luxembourg, Luxembourg, 2015, pp. 1–4. New York: IEEE.
75.
SelesnickIW. Resonance-based signal decomposition: a new sparsity-enabled signal analysis method. Signal Process2011; 91: 2793–2809.
76.
SelesnickIWParekhABayramI. Convex 1-D total variation denoising with non-convex regularization. IEEE Signal Process Lett2015; 22(2): 141–144.
77.
ParekhASelesnickIW. Convex fused LASSO denoising with non-convex regularization and its use for pulse detection. In: 2015 IEEE signal processing in medicine and biology symposium (SPMB), Philadelphia, PA, USA, 2015. New York: IEEE.
78.
ParekhASelesnickIW. Improved sparse low-rank matrix estimation. Signal Process2017; 139: 62–69.
79.
AfonsoMVDiasJMBFigueiredoMA. Fast image recovery using variable splitting and constrained optimization. IEEE Trans Image Process2019; 19: 2345–2356.
80.
AfonsoMVDiasJMBFigueiredoMAT. An augmented Lagrangian approach to the constrained optimization formulation of imaging inverse problems. IEEE Trans Image Process2011; 20: 681–695.
81.
TibshiraniR. Regression shrinkage and selection via the lasso. J R Stat Soc Series B Stat Methodol1996; 58(1): 267–288.
82.
TibshiraniR. The lasso method for variable selection in the cox model. Stat Med1997; 16(4): 385–395.
83.
HollandJ. Adaptation in natural and artificial systems. Cambridge, MA: MIT Press, 1992.
84.
MitchellM. An introduction to genetic algorithms. Cambridge, MA: MIT Press, 1996.
85.
HamiltonWR. On quaternions, or on a new system of imaginaries in algebra. Dublin Philos Mag J Sci1844; 25(163): 10–13.
86.
HamiltonW. Elements of quaternions. 2nd ed.London, UK: Longmans, Green, & Company, 1899.
87.
KuipersJB. Quaternions and rotation sequences. Princeton, NJ: Princeton University Press, Ed., 1998.
88.
WardJP. Quaternions and Cayley numbers: algebra and applications. London, UK: Kluwer Academic Publisher, 1997.
89.
SierociukDDzielinskiA. Fractional Kalman filter algorithm for the states, parameters and order of fractional system estimation. Int J Appl Math Comput Sci2006; 16(1): 129–140.
90.
SierociukDZiubinskiP. Fractional order estimation schemes for fractional and integer order systems with constant and variable fractional order colored noise. Circuits Syst Signal Process2014; 33(12): 3861–3882.
91.
SierociukD. Fractional Kalman filter algorithms for correlated system and measurement noises. Control Cybern2013; 42(2): 471–490.
92.
SierociukD.ZiubinskiP. Variable order fractional Kalman filters for estimation over lossy network. Lect Notes Electr Eng2015; 320: 285–294.
93.
ViswanathanGMRaposoEPda LuzMGE. Lévy flights and super diffusion in the context of biological encounters and random searches. Phys Life Rev2008; 5(3): 133–150.
94.
Barndorff-NielsenOEMikoschTResnickSI. Lévy processes: theory and applications. Birkhäuser Basel: Springer, 2001.
95.
BrockmannDHufnagelLGeiselT. The scaling laws of human travel. Nature2006; 439(7075): 462–465.
96.
HirschMSmaleSDevaneyRL. Differential equations, dynamical systems and linear algebra. New York: Academic Press, 1974.
97.
HochreiterSSchmidhuberJ. Long short-term memory. Neural Comput1997; 9(8): 1735–1780.
98.
ParkKChoiYChoiWJ, et al. LSTM-based battery remaining useful life prediction with multi-channel charging profiles. IEEE Access2020; 8: 20786–20798.
HeJGuanXPengT, et al. A multi-feature integration method for fatigue crack detection and crack length estimation in riveted lap joints using Lamb waves. Smart Mater Struct2013; 22(10): 105007.
101.
PengTHeJXiangY, et al. Probabilistic fatigue damage prognosis of lap joint using Bayesian updating. J Intell Mater Syst Struct2015; 26(8): 965–979.
102.
YangJHeJGuanX, et al. A probabilistic crack size quantification method using in-situ Lamb wave test and Bayesian updating. Mech Syst Signal Process2016; 78: 118–133.
103.
NectouxPGouriveauRMedjaherK, et al. PRONOSTIA: an experimental platform for bearings accelerated degradation tests. In: IEEE 2012 IEEE conference on prognostics and health management (PHM), Denver, CO, USA, 2012, pp. 1–8.
