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Statistical modeling of directional data using a robust hierarchical von mises distribution model: perspectives for wind energy

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Abstract

For describing wind direction, a variety of statistical distributions has been suggested that provides information about the wind regime at a particular location and aids the development of efficient wind energy generation. In this paper a systematic approach for data classification putting a special emphasis on the von Mises mixtures is presented. A von Mises mixture model is broad enough to cover, on one hand, symmetry and asymmetry, on the other hand, unimodality and multimodality of circular data. We developed an improved mathematical model of the classical von Mises mixture method, rests on number of principles which gives its internal coherence and originality. In principle, our hierarchical model of von Mises distributions is flexible to precisely modeled complex directional data sets. We define a new specific expectation–maximization (S-EM) algorithm for estimating the parameters of the model. The simulation showed that satisfactory fit of complex directional data could be obtained (error generally < 1%). Furthermore, the Bayesian Information Criterion is used to judge the goodness of fit and the suitability for this model versus common distributions found in the literature. The findings prove that our hierarchical model of von Mises distributions is relevant for modeling the complex directional data with several modes and/or prevailing data directions.

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References

  • Acoltzi TM, Martínez AT, Pérez JC (1989) 850 mb in Laguna Verde zone, Veracruz, Mexico. Atmósfera 2:181–187

    Google Scholar 

  • Akaike H (1974) A new look at the statistical model identification. IEEE Trans Autom Control 19(6):716–723

  • Bobée B (1999) Estimation des événements extrêmes de crue par l'analyse fréquentielle: une revue critique. La Houille Blanche 7/8:100–105

  • Bukal M, Marković I, Petrović I (2014) Composite distance based approach to von Mises mixture reduction. Inf Fusion 20:136–145

    Article  Google Scholar 

  • Calderara S, Prati A, Cucchiara R (2011) Mixtures of von mises distributions for people trajectory shape analysis. IEEE Trans Circuits Syst Video Technol 21(4):457–471

    Article  Google Scholar 

  • Carta JA, Bueno C, Ramírez P (2008a) Statistical modelling of directional wind speeds using mixtures of von Mises distributions: case study. Energy Convers Manag 49(5):897–907

    Article  Google Scholar 

  • Carta JA, Ramirez P, Bueno C (2008b) A joint probability density function of wind speed and direction for wind energy analysis. Energy Convers Manag 49(6):1309–1320

    Article  Google Scholar 

  • Dempster AP, Laird NM, Rubin DB (1977) Maximum likelihood from incomplete data via the EM algorithm. J R Stat Soc Ser B (methodol) 39(1):1–22

    MathSciNet  MATH  Google Scholar 

  • Di Marzio M, Panzera A, Taylor CC (2009) Local polynomial regression for circular predictors. Stat Probab Lett 79(19):2066–2075

    Article  MathSciNet  Google Scholar 

  • Dobigeon N, Tourneret JY (2007) Joint segmentation of wind speed and direction using a hierarchical model. Comput Stat Data Anal 51(12):5603–5621

    Article  MathSciNet  Google Scholar 

  • Gatto R (2009) Information theoretic results for circular distributions. Statistics 43(4):409–421

    Article  MathSciNet  Google Scholar 

  • Gatto R, Jammalamadaka SR (2003) Inference for wrapped symmetric α-stable circular models. Sankhyā Indian J Stat 65(2):333–355

    MathSciNet  MATH  Google Scholar 

  • Goldberger J, Gordon S, Greenspan H (2003) An efficient image similarity measure based on approximations of KL-divergence between two gaussian mixtures. In: ICCV, vol 3, pp 487–493‏

  • Goldberger J, Greenspan HK, Dreyfuss J (2008) Simplifying mixture models using the unscented transform. IEEE Trans Pattern Anal Mach Intell 30(8):1496–1502

    Article  Google Scholar 

  • Gupta MR, Chen Y (2011) Theory and use of the EM algorithm. Now Publishers Inc, Norwell

    Google Scholar 

  • Hasnat MA, Alata O, Trémeau A (2013) Hierarchical 3-d von mises-fisher mixture model. In: 1st Workshop on divergences and divergence learning (WDDL)

  • Hershey JR, Olsen PA (2007) Approximating the kullback leibler divergence between gaussian mixture models. In 2007 IEEE international conference on acoustics, speech and signal processing-ICASSP'07, vol 4. IEEE, pp IV–317

  • Hung WL, Chang-Chien SJ, Yang MS (2012) Self-updating clustering algorithm for estimating the parameters in mixtures of von Mises distributions. J Appl Stat 39(10):2259–2274

    Article  MathSciNet  Google Scholar 

  • Jammalamadaka SR, Sengupta A (2001) Topics in circular statistics, vol 5. world scientific

  • Jones M, Lotwick H (1984) Kernel density-estimation using the fast fourier-transform-a remark. J R Stat Soc Ser C-Appl Stat 33(1):120–122

    Google Scholar 

  • Jung C, Schindler D (2017) Global comparison of the goodness-of-fit of wind speed distributions. Energy Convers Manag 133:216–234

    Article  Google Scholar 

  • Kamisan NAB, Hussin AB, Zubairi YZ (2010) Finding the best circular distribution for southwesterly monsoon wind direction in Malaysia. Sains Malaysiana 39:387–393

    Google Scholar 

  • Karavasilis V, Nikou C, Likas A (2017) Real time visual tracking using a spatially weighted von mises mixture model. Pattern Recogn Lett 90:50–57

    Article  Google Scholar 

  • Lagona F (2016) Regression analysis of correlated circular data based on the multivariate von mises distribution. Environ Ecol Stat 23(1):89–113

    Article  MathSciNet  Google Scholar 

  • Maksimov VM (1967) Necessary and sufficient statistics for the family of shifts of probability distributions on continuous bicompact groups. Theoria Verojatna 12:307–321

    MathSciNet  MATH  Google Scholar 

  • Mardia KV, Jupp PE (1999) Directional statistics. Wiley, Chichester

    Book  Google Scholar 

  • Mardia KV, Jupp PE (2000) Directional statistics. Wiley series in probability and statistics. Wiley, Chichester

    MATH  Google Scholar 

  • Mardia KV, Voss J (2014) Some fundamental properties of a multivariate von mises distribution. Commun Stat-Theory Methods 43(6):1132–1144

    Article  MathSciNet  Google Scholar 

  • Marković I, Petrović I (2012) Bearing-only tracking with a mixture of von mises distributions. In 2012 IEEE/RSJ international conference on intelligent robots and systems. IEEE, pp 707–712

  • Masseran N, Razali AM, Ibrahim K, Latif MT (2013) Fitting a mixture of von mises distributions in order to model data on wind direction in Peninsular Malaysia. Energy Convers Manag 72:94–102

    Article  Google Scholar 

  • Merugu S, Ghosh J (2003) Privacy-preserving distributed clustering using generative models. In Third IEEE international conference on data mining. IEEE, pp 211–218

  • Nielsen F (2012) Closed-form information-theoretic divergences for statistical mixtures. In Proceedings of the 21st international conference on pattern recognition (ICPR2012). IEEE, pp 1723–1726 ‏

  • Oliveira M, Crujeiras RM, Rodríguez-Casal A (2012) A plug-in rule for bandwidth selection in circular density estimation. Comput Stat Data Anal 56(12):3898–3908

    Article  MathSciNet  Google Scholar 

  • Oliveira M, Crujeiras R, Rodríguez-Casal A (2014) CircSiZer: an exploratory tool for circular data. Environ Ecol Stat 21(1):143–159

    Article  MathSciNet  Google Scholar 

  • Oliveira-Brochado A, Martins FV (2005) Assessing the number of components in mixture models: a review.In: FEP Working Papers, (194)‏

  • Olsen PA, Dharanipragada S (2003) An efficient integrated gender detection scheme and time mediated averaging of gender dependent acoustic models. In: Eighth European conference on speech communication and technology.

  • Pfyffer S, Gatto R (2013) An efficient simulation algorithm for the generalized von Mises distribution of order two. Comput Stat 28(1):255–268

    Article  MathSciNet  Google Scholar 

  • Radhey SS (1987) Mise of kernel estimation of a density and its derivatives. Stat Probab Lett 5:153–159

    Article  Google Scholar 

  • Runnalls AR (2007) Kullback-Leibler approach to Gaussian mixture reduction. IEEE Trans Aerosp Electron Syst 43(3):989–999

    Article  Google Scholar 

  • Sain SR, Baggerly KA, Scott DW (1994) Cross-validation of multivariate densities. J Am Stat Assoc 89(427):807–817

    Article  MathSciNet  Google Scholar 

  • Schwarz G (1978) Estimating the dimension of a model. The annals of statistics, pp 461–464

  • Shaoping W, Ang L, Kuangyu W, Ximing W (2020) Robust kernels for kernel density estimation. Econ Lett. https://doi.org/10.1016/j.econlet:109-138

    Article  MathSciNet  MATH  Google Scholar 

  • Shieh GS, Johnson RA (2005) Inferences based on a bivariate distribution with von mises marginals. Ann Inst Stat Math 57(4):789–802

    Article  MathSciNet  Google Scholar 

  • Stone M (1974) Cross-validatory choice and assessment of statistical predictions. J R Stat Soc: Ser B (methodol) 36(2):111–133

    MathSciNet  MATH  Google Scholar 

  • Von Mises R (1918) Über die "Ganzzahligkeit" der Atomgewicht und verwandte Fragen. Physikalische Zeitschrift 19:490–500

  • Wang Y, Son S, Swartz SM, Goulbourne NC (2012) A mixed von mises distribution for modeling soft biological tissues with two distributed fiber properties. Int J Solids Struct 49(21):2914–2923

    Article  Google Scholar 

  • Williams B, Christensen WF, Reese CS (2011) Pollution source direction identification: embedding dispersion models to solve an inverse problem. Environmetrics 22(8):962–974

    Article  MathSciNet  Google Scholar 

  • Yin F, Fritsche C, Gustafsson F, Zoubir AM (2013) EM-and JMAP-ML based joint estimation algorithms for robust wireless geolocation in mixed LOS/NLOS environments. IEEE Trans Signal Process 62(1):168–182

    Article  MathSciNet  Google Scholar 

  • Žilinskas A (1982) Axiomatic approach to statistical models and their use in multimodal optimization theory. Math Program 22(1):104–116

    Article  MathSciNet  Google Scholar 

Download references

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Authors and Affiliations

  1. LRIT URAC 29, Faculty of Sciences, Mohammed V University in Rabat, Rabat, Morocco

    Said Benlakhdar & Mohammed Rziza

  2. RIITM, ENSIAS, Mohammed V University in Rabat, Rabat, Morocco

    Rachid Oulad Haj Thami

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  1. Said Benlakhdar
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  2. Mohammed Rziza
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  3. Rachid Oulad Haj Thami
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Correspondence to Said Benlakhdar.

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Benlakhdar, S., Rziza, M. & Thami, R.O.H. Statistical modeling of directional data using a robust hierarchical von mises distribution model: perspectives for wind energy. Comput Stat 37, 1599–1619 (2022). https://doi.org/10.1007/s00180-021-01173-5

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