The problem of three-dimensional (3-D) wind profile prediction is addressed based a trinion wind model, which inherently reckons the coupling of the three perpendicular components of a wind field. The augmented trinion statistics are developed and employed to enhance the prediction performance due to its full exploitation of the second-order statistics. The proposed trinion domain processing can be regarded as a more compact version of the existing quaternion-valued approach, with a lower computational complexity. Simulations based on recorded wind data are provided to demonstrate the effectiveness of the proposed methods

Gou, X.

Liu, W.

Liu, Z.

Xu, Y.

arXiv

White Rose Research Online

This is a repository copy of Three-Dimensional Wind Profile Prediction with Trinion-Valued Adaptive Algorithms.White Rose Research Online URL for this paper:http://eprints.whiterose.ac.uk/93737/Version: Accepted VersionProceedings Paper:Gou, X., Liu, Z., Liu, W. et al. (1 more author) (2015) Three-Dimensional Wind Profile Prediction with Trinion-Valued Adaptive Algorithms. In: 2015 IEEE International Conference on Digital Signal Processing (DSP). , 21-24 July 2015, Singapore. IEEE , pp. 566-569. https://doi.org/10.1109/ICDSP.2015.7251937eprints@whiterose.ac.ukhttps://eprints.whiterose.ac.uk/Reuse Unless indicated otherwise, fulltext items are protected by copyright with all rights reserved. The copyright exception in section 29 of the Copyright, Designs and Patents Act 1988 allows the making of a single copy solely for the purpose of non-commercial research or private study within the limits of fair dealing. The publisher or other rights-holder may allow further reproduction and re-use of this version - refer to the White Rose Research Online record for this item. Where records identify the publisher as the copyright holder, users can verify any specific terms of use on the publisher’s website. Takedown If you consider content in White Rose Research Online to be in breach of UK law, please notify us by emailing eprints@whiterose.ac.uk including the URL of the record and the reason for the withdrawal request. arXiv:1506.02952v1  [math.OC]  9 Jun 2015Three-Dimensional Wind Profile Prediction withTrinion-Valued Adaptive AlgorithmsXiaoming Gou∗†, Zhiwen Liu∗, Wei Liu† and Yougen Xu∗∗School of Information and Electronics, Beijing Institute of Technology, Beijing 100081, China†Communications Research Group, Department of Electronic and Electrical Engineering,University of Sheffield, Sheffield S1 3JD, United KingdomEmail: w.liu@sheffield.ac.ukAbstract—The problem of three-dimensional (3-D) wind profileprediction is addressed based a trinion wind model, whichinherently reckons the coupling of the three perpendicularcomponents of a wind field. The augmented trinion statisticsaredeveloped and employed to enhance the prediction performancedue to its full exploitation of the second-order statistics. Theproposed trinion domain processing can be regarded as a morecompact version of the existing quaternion-valued approach, witha lower computational complexity. Simulations based on recordedwind data are provided to demonstrate the effectiveness of theproposed methods.Index Terms—wind profile prediction, trinion-valued repre-sentation, least mean squares, adaptive filtering, widely linearprocessingI. I NTRODUCTIONHypercomlex numbers are high-dimensional extensions ofreal numbers [1], [2], and they have been introduced to solvemultivariate signal processing problems, such as colour imageprocessing [3]–[6], vector-sensor array signal processing [7]–[12], human gesture spotting [13], [14], wind profile predic-tion [15]–[17], [22], and wireless communications [23]. Inthe last application, anemometer readings are modeled withhypercomplex numbers and the wind profile is predicted byadaptive filtering algorithms. In particular, pure quaternionshave been widely used to model three-dimensional (3D) windspeed. When external atmospheric parameters are available, afull quaternion-valued model can be considered [15].The quaternion-valued model leads to improved perfor-mance over real-valued models, since it accounts for thecoupling of the wind measurements and can be extended toexploiting the augmented quaternion statistics [15]. However,pure quaternions do not form a mathematical ring [18], sincethe product of two pure quaternions is not a pure quaternionin general. As a result, the related adaptive algorithms for3-Dwind profile prediction initialised in the pure quaternion willhave to work in the full quaternion domain. The predictionresults have to be truncated from full quaternions to purequaternions, implying redundant computations in the updateprocess. For example, it takes 16 real-valued multiplicationsand 12 real-valued additions to implement a full-quaternion--imag axis ı-imag axisreal axis3D wind speedvavcvbFig. 1. Trinion-valued three-dimensional wind speed model.valued multiplication. By comparison, it only takes 9 real-valued multiplications and 6 real-valued additions to calculatethe multiplication of any two 3-D numbers.In this work, we aim to develop a more compact 3-D windspeed model based on a 3-D mathematical ring called trinions,termed by an anonymous author who provided all possibledefinitions of 3-D numbers [19]. We first define the gradientoperation in the trinion domain and then derive the least meansquares (LMS) adaptive algorithm, which is further extendedto augmented statistics.The rest of this paper is organised as follows. A briefintroduction to trinions is provided in Section II. The trinio -valued LMS (TLMS) algorithm is developed in Section III,and the augmented TLMS (ATLMS) algorithm in Section IV.Simulation results are presented in Section V and conclusionsare drawn in Section VI.II. T RINIONSAs shown in Fig. 1, the 3-D wind speed is a tri-variatesignal composed of three perpendicular components, and canbe modeled with 3-D hypercomplex numbers (see Fig. 1).There are various definitions of a 3-D numberv composedof one real part (va) and two imaginary parts (vb andvc),v = va + ıvb + vc, (1)according to definitions about the relationships among thethree base elements 1,ı, and.To form an Abelian group of these three, the following rulesapply [20]:ı2 = , ı = ı = −1, 2 = −ı . (2)Trinions subject to the above rules form a commutative ring,namely, for two trinionsv andw, we havevw = wv.The modulus ofv is given by [20]|v| =√v2a + v2b + v2c . (3)We define the following conjugate ofv,v∗ = va − vb − ıvc, (4)so that|v|2 is equal to the real part ofvv∗, i.e. |v|2 = ℜ(vv∗).To develop an LMS-like algorithm in the trinion domain,the trinion-valued gradient needs to be defined. In the complexdomain, a variablez and its conjugate can be viewed as twoindependent variables, based on which the complex-valuedgradient can be defined [21]. Similarly, in the quaterniondomain, a variableq and its three involutions can be viewedas four independent variables, based on which the quaternion-valued gradient can be derived [22], [24]. However, to our bestknowledge, the trinion involution is not available in general. Inthis paper, we define the following gradients of a functionf(x)with respect to the trinion-valued variablex and its conjugate,∇xf =13(∇xaf − ∇xbf − ı∇xcf),∇x∗f =13(∇xaf + ı∇xbf + ∇xcf),(5)respectively, wherex = xa + ıxb + xc.Unlike the quaternion-valued gradient, the trinion-valuedgradient remains the same, no matter which side of the sub-gradients the imaginary unitsı and are on, since trinions arecommutative. We can then calculate the derivatives of somesimple functions, for instance,∂x∂x=∂x∗∂x∗= 1,∂x∂x∗=∂x∗∂x=1− ı+ 3.(6)III. T RINION-VALUED LMS FILTERINGSimilar to the LMS algorithm in the complex-valued do-main, the trinion-valued errore(n) is given bye(n) = d(n)−wT(n)x(n), (7)whered(n) is the reference signal,w(n) is the weight vector,x(n) = [x(n−P );x(n−P − 1); · · · ;x(n−L−P +1)]T isthe filter input,L is the filter length, andP is the predictionstep. The cost function is expressed asJ(n) = |e(n)|2. (8)Using the steepest descent method, the following gradient needto be calculated∇w∗J(n) =13[∇waJ(n) + ı∇wbJ(n) + ∇wcJ(n)]. (9)For details of the calculation, please refer to Appendix at theend. Using results there, the update of the weight vector canbe obtained asw(n+ 1) = w(n) + µe(n)x∗(n), (10)whereµ is the step size. The algorithm formulated above istermed as the trinion-valued LMS (TLMS) algorithm.IV. A UGMENTED TRINION STATISTICSA zero-mean trinion-valued vectorv is composed of threezero-mean real-valued random vectorsva, vb, and vc, andtheir complete second-order statistics can be found in thefollowing six real-valued covariance matrices:Cvava = E{vavTa }, Cvbvb = E{vbvTb },Cvcvc = E{vcvTc }, Cvavb = E{vavTb },Cvbvc = E{vbvTc }, Cvcva = E{vcvTa }.(11)And they can be more efficiently represented by three trinion-valued covariance matrices:Cvv = E{vvH},Cvvı = E{vvıH},Cvv = E{vvH},(12)where superscriptH denotes the Hermitian transpose and thetwo additional mappings ofv are defined asvı = vb − ıva − vc,v = vc − ıvb − va.(13)Neither of these two mappings is an involution, and they aredefined as shorthand notations only. Then we can obtainCvava =12ℜ(Cvv + Cvvı),Cvbvb =12ℜ(Cvvı − Cvvı),Cvcvc =12ℜ(Cvv − ıCvv),Cvavb =12ℜ(Cvvı + Cvv),Cvbvc =12ℜ(ıCvv − Cvv),Cvcva =12ℜ(Cvvı − ıCvv).(14)To take the complete second-order statistics into consideration,we need to make use of the augmented input vector composedof the original input vectorx(n) and its two mappingsxı(n),x(n), namely,xaug(n) =x(n)xı(n)x(n) , (15)and the predicted estimatey(n) will be given byy(n) = waugT(n)xaug(n)= wT1 (n)x(n) +wT2 (n)xı(n) +wT3 (n)x(n),(16)Fig. 2. Predicted result from the TLMS algorithm.wherewaug = [w1;w2;w3]. Analogous to the derivation ofthe TLMS algorithm, the update equation of the augmentedTLMS (ATLMS) algorithm is given bywaug(n+ 1) = waug(n) + ρeaug(n)xaug∗(n), (17)whereρ is the step size.The computational complexity for each update of the weightvector of the trinion-based and quaternion-based filteringalgo-rithms are summarised in Table I. It can be seen that the trinionmodel can effectively reduce the computational complexitycompared to the quaternion model.TABLE ICOMPUTIONAL COMPLEXITY PER UPDATE OF THE WEIGHT VECTORAlgorithm Real multiplications Real additionsQLMS [17] 16L+ 4 16LAugmented QLMS 64L+ 4 64LTLMS 9L+ 3 9LAugmented TLMS 27L+ 3 27LV. SIMULATIONSIn this section, both proposed algorithms (TLMS andATLMS) are applied to anemometer readings provided byGoogle’s RE<C Initiative [25]. The wind speed measured onMay 25, 2011 is used for demonstration. The step size is setto be6× 10−5. The filter length is 8, and the prediction stepis 1. All algorithms are initialised with zeros. The predictedresults provided by TLMS and ATLMS algorithms are shownin Figs. 2 and 3, respectively. From the results, we can seethat both algorithms can track the wind data effectively.The learning curves averaged over 200 trials of the proposedalgorithms are shown in Fig. 4, compared with the quaternion-based QLMS and AQLMS algorithms. It can be observed thatboth augmented algorithms (AQLMS and ATLMS) have aFig. 3. Predicted result from the ATLMS algorithm.Fig. 4. Averaged learning curves.faster convergence rate than the original ones (QLMS andTLMS), due to their full exploitation of the second-orderstatistics. Meanwhile, the proposed TLMS algorithm has asimilar performance with the QLMS algorithm, while theATLMS algorithm is comparable with the AQLMS algorithm.However, as shown in Table I, the proposed trinion-basedalgorithms have a much lower computational complexity.VI. CONCLUSIONA compact model for three-dimensional wind profile pre-diction based on trinion algebra has been proposed, with twoLMS-type adaptive filtering algorithms derived using partiland full trinion-valued second-order statistics, respectiv ly.Numerical simulations using recorded wind data have shownthat the two algorithms have a similar performance to theirquaternion-valued counterparts, but with a much lower com-putational complexity.APPENDIXThe cost functionJ(n) as a function of real-valued variablesis given byJ = (da −wTaxa +wTb xc +wTc xb)2+ (db −wTaxb −wTb xa +wTc xc)2+ (dc −wTaxc −wTb xb −wTc xa)2,(18)where the time index ‘n’ has been dropped for the sake ofcompact notation. Then the three component-wise gradientscan be computed as∇waJ = 2[(xaxTa + xbxTb + xcxTc )wa+ (xbxTa + xcxTb − xaxTc )wb+ (xcxTa − xaxTb − xbxTc )wc− (daxa + dbxb + dcxc)],(19)∇wbJ = 2[(xaxTb + xbxTc − xcxTa )wa+ (xcxTc + xaxTa + xbxTb )wb+ (xcxTb − xaxTc + xbxTa )wc+ (daxc − dbxa − dcxb)],(20)∇wcJ = 2[(xaxTc − xbxTa − xcxTb )wa+ (xbxTc − xcxTa + xaxTb )wb+ (xaxTa + xcxTc + xbxTb )wc+ (daxb + dbxc − dcxa)].(21)Finally, we can obtain the expression for the gradient ofJ(n)by substituting (19)–(21) into (9),∇w∗J(n) =23e(n)x∗(n), (22)which yields the update equation in (10), as the term23canbe absorbed into the step size.ACKNOWLEDGMENTThis work was supported in part by the National Natu-ral Science Foundation of China (61331019, 61490690), theChina Scholarship Council Postgraduate Scholarship Program(2014), and the UK National Grid.REFERENCES[1] I. L. Kantor and A. S. Solodovnikov,Hypercomplex numbers: Anelementary introduction to algebras. Springer-Verlag, 1989.[2] J. P. Ward,Quaternions and Cayley numbers: Algebra and applications.Dordrecht: Kluwer Academic Publisher, 1997.[3] N. Le Bihan and S. J. Sangwine, “Quaternion principle comp nentanalysis of color images,” inProceedings of the IEEE InternationalConference on Image Processing, Barcelona, Spain, pp. 809–812, Sep.2003.[4] S. C. Pei, J. H. Chang, and J. J. Ding, “Commutative reducebi-quaternions and their Fourier transform for signal and image processingapplications,”IEEE Transactions on Signal Processing, vol. 52, no. 7,pp. 2012–2031, Jul. 2004.[5] T. A. Ell, “Multi-vector color-image filters,” inProceedings of the IEEEInternational Conference on Image Processing, San Antonio, TX, USA,pp. 245–248, Sep. 2007.[6] L. Q. Guo, M. Zhu, and X. H. Ge, “Reduced biquaternion canonicaltransform, convolution and correlation,”Signal Processing, vol. 91, no.8, pp. 2147–2153, Aug. 2011.[7] S. Miron, N. Le Bihan, and J. I. Mars, “Quaternion-MUSIC for vector-sensor array processing,”IEEE Transactions Signal Processing, vol. 54,no. 4, pp. 1218–1229, Apr. 2006.[8] N. Le Bihan, S. Miron, and J. I. Mars, “MUSIC algorithm forvector-sensors array processing using biquaternions,”IEEE Transactions onSignal Processing, vol. 55, no. 9, pp. 4523–4533, Sep. 2007.[9] J. F. Jiang and J. Q. Zhang, “Geometric algebra of Euclidean 3-spacefor electromagnetic vector-sensor array processing, partI: Modeling,”IEEE Transactions on Antennas and Propagation, vol. 58, no. 12, pp.3961–3973, Dec. 2010.[10] X. F. Gong, Z. W. Liu, and Y. G. Xu, “Coherent source localiz tion:Bicomplex polarimetric smoothing with electromagnetic vetor-sensors,”IEEE Transactions on Aerospace and Electronic Systems, vol. 47, no.3, pp. 2268–2285, Jul. 2011.[11] X. R. Zhang, W. Liu, Y. G. Xu, and Z. W. Liu, “Quaternion-valuedrobust adaptive beamformer for electromagnetic vector-sensor arrayswith worst-case constraint,”Signal Processing, vol. 104, pp. 274–283,Nov. 2014.[12] M. B. Hawes and W. Liu, “Design of fixed beamformers basedon vector-sensor arrays,”International Journal of Antennas and Propagation, vol.2015, 2015.[13] B. Che Ujang, C. Jahanchahi, C. C. Took, and D. P. Mandic,“Adaptiveconvex combination approach for the identification of improper quater-nion processes,”IEEE Transactions on Neural Networks and LearningSystems, vol. 25, no. 1, pp. 172–182, Jan. 2014.[14] F. A. Tobar and D. P. Mandic, “Quaternion reproducing kernel Hilbertspaces: Existence and uniqueness conditions,”IEEE Transactions onInformation Theory, vol. 60, no. 9, pp. 5736–5749, Sep. 2014.[15] C. C. Took, G. Strbac, K. Aihara, and D. P. Mandic, “Quaternion-valuedshort term joint forecasting of three-dimensional wind andtmosphericparameters,”Renewable Energy, vol. 36, no. 6, pp. 1754–1760, Jun.2011.[16] J. Navarro-Moreno, R. M. Fernandez-Alcalá, C. C. Took, and D. P.Mandic, “Prediction of wide-sense stationary quaternion ra dom sig-nals,” Signal Processing, vol. 93, no. 9, pp. 2573–2580, Sep. 2013.[17] Q. Barthélemy, A. Larue, and J. I. Mars, “About QLMS derivations,”IEEE Signal Processing Letters, vol. 21, no. 2, pp. 240–243, Feb. 2014.[18] R. B. Allenby. Rings, fields and groups: An introduction to abstractalgebra. Butterworth-Heinemann, 1991.[19] Async, OMIC’s N-nion’s site [Online]. Available:http://asyncbrain.baf.cz/m/nt/index.htm.[20] D. Assefa, L. Mansinha, K. F. Tiampo, H. Rasmussen, and K. Abdella,“The trinion Fourier transform of color images,”Signal Processing, vol.91, no. 8, pp. 1887–1900, Aug. 2011.[21] A. van den Bos, “Complex gradient and Hessian,”IEE Proceedings -Vision Image Signal Process, vol. 141, no. 6, pp. 380–383, Dec. 1994.[22] M. D. Jiang, W. Liu, and Y. Li, “A general quaternion-valued gradientoperator and its applications to computational fluid dynamics and adap-tive beamforming,” inProceedings of the 19th International Conferenceon Digital Signal Processing, Hong Kong, China, pp. 821–826, Aug.2014.[23] W. Liu, “Antenna array signal processing for a quaternion-valued wire-less communication system,” inProc. the Benjamin Franklin Symposiumon Microwave and Antenna Sub-systems (BenMAS), Philadelphia, US,September 2014.[24] M. D. Jiang, Y. Li, and W. Liu, “Properties and applications of arestricted HR gradient operator,”arXiv:1407.5178 [math.OC], July2014.[25] Google, “RE<C: Surface level wind data collection,”Google Code 2011[Online]. Available: http://code.google.com/p/google-rec-csp/

Three-Dimensional Wind Profile Prediction with Trinion-Valued Adaptive Algorithms

Xiaoming Gou

Zhiwen Liu

Wei Liu

Yougen Xu

Crossref

Three-dimensional wind profile prediction with trinion-valued adaptive algorithms

Three-Dimensional Wind Profile Prediction with Trinion-Valued Adaptive Algorithms

Abstract

Similar works

Full text

Available Versions

White Rose Research Online

Crossref