In this paper, we introduce a novel algorithm for calculating arbitrary order cumulants of multidimensional data. Since the $d{th}$ order cumulant can be presented in the form of a $d$-dimensional tensor, the algorithm is presented using tensor operations. The algorithm provided in the paper takes advantage of supersymmetry of cumulant and moment tensors. We show that the proposed algorithm considerably reduces the computational complexity and the computational memory requirement of cumulant calculation as compared with existing algorithms. For the sizes of interest, the reduction is of the order of $d!$ compared to the naive algorithm.


  1. high-order cumulants
  2. nonnormally distributed data
  3. numerical algorithms

MSC codes

  1. 65Y05
  2. 15A69
  3. 65C60

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G. S. Amin and H. M. Kat, Hedge fund performance 1990--2000: Do the “money machines" really add value?, J. Financ. Quant. Anal., 38 (2003), pp. 251--274.
J. C. Arismendi and H. Kimura, Monte Carlo Approximate Tensor Moment Simulations, available at SSRN 2491639, 2014.
N. Balakrishnan, N. L. Johnson, and S. Kotz, A note on relationships between moments, central moments and cumulants from multivariate distributions, Stat. Probab. Lett., 39 (1998), pp. 49--54.
O. E. Barndorff-Nielsen and D. R. Cox, Asymptotic Techniques for Use in Statistics, Chapman & Hall, London, 1989.
H. Becker, L. Albera, P. Comon, M. Haardt, G. Birot, F. Wendling, M. Gavaret, C.-G. Bénar, and I. Merlet, EEG extended source localization: Tensor-based vs. conventional methods, NeuroImage, 96 (2014), pp. 143--157.
J. Bezanson, J. Chen, S. Karpinski, V. Shah, and A. Edelman, Array operators using multiple dispatch: A design methodology for array implementations in dynamic languages, in Proceedings of ACM SIGPLAN International Workshop on Libraries, Languages, and Compilers for Array Programming, Edinburgh, UK, ACM, 2014, p. 56, https://dl.acm.org/citation.cfm?doid=2627373.2627383.
J. Bezanson, A. Edelman, S. Karpinski, and V. B. Shah, Julia: A fresh approach to numerical computing, SIAM Rev., 59 (2017), pp. 65--98.
J. Bezanson, S. Karpinski, V. B. Shah, and A. Edelman, Julia: A Fast Dynamic Language for Technical Computing, arXiv:1209.5145, (2012).
G. Birot, L. Albera, F. Wendling, and I. Merlet, Localization of extended brain sources from EEG/MEG: The ExSo-MUSIC approach, NeuroImage, 56 (2011), pp. 102--113.
J.-F. Cardoso and E. Moulines, Asymptotic performance analysis of direction-finding algorithms based on fourth-order cumulants, IEEE Trans. Signal Process., 43 (1995), pp. 214--224.
P. Chevalier, L. Albera, A. Ferréol, and P. Comon, On the virtual array concept for higher order array processing, IEEE Trans. Signal Process., 53 (2005), pp. 1254--1271.
P. Chevalier, A. Ferréol, and L. Albera, High-resolution direction finding from higher order statistics: The 2q-MUSIC algorithm, IEEE Trans. Signal Process., 54 (2006), pp. 2986--2997.
P. Chevalier, A. Ferréol, L. Albera, and G. Birot, Higher order direction finding with polarization diversity: THE PD-2q-music algorithms, in 15th European Signal Processing Conference, IEEE, New York, 2007, pp. 257--261.
L. Comtet, Advanced Combinatorics, Reidel, Boston, 1974.
R.-G. Cong and M. Brady, The interdependence between rainfall and temperature: Copula analyses, Sci. World J., 2012 (2012), Article ID 405675.
E. Di Nardo, G. Guarino, and D. Senato, A unifying framework for k-statistics, polykays and their multivariate generalizations, Bernoulli, 14 (2008), pp. 440--468, https://projecteuclid.org/euclid.bj/1208872113.
K. Domino, The use of the multi-cumulant tensor analysis for the algorithmic optimisation of investment portfolios, Phys. A, 467 (2017), pp. 267--276.
K. Domino, T. Błachowicz, and M. Ciupak, The use of copula functions for predictive analysis of correlations between extreme storm tides, Phys. A, 413 (2014), p. 489–497.
K. Domino, P. Gawron, and Ł. Pawela, Cummulants.jl, https://doi.org/10.5281/zenodo.1185137, 2017.
K. Domino, P. Gawron, and Ł. Pawela, SymmetricTensors.jl, https://doi.org/10.5281/zenodo.996222, 2017.
E. Eban, G. Rothschild, A. Mizrahi, I. Nelken, and G. Elidan, Dynamic copula networks for modeling real-valued time series, in Proceedings of the Sixteenth International Conference on Artificial Intelligence and Statistics, Proc. Mach. Learn. Res. 31, C. M. Carvalho and P. Ravikumar, eds., PMLR, Scottsdale, AZ, 2013, pp. 247--255, http://proceedings.mlr.press/v31/eban13a.html.
B. Friman, F. Karsch, K. Redlich, and V. Skokov, Fluctuations as probe of the QCD phase transition and freeze-out in heavy ion collisions at LHC and RHIC, Euro. Phys. J. C-Particles Fields, 71 (2011), pp. 1--11.
J. Gabelli and B. Reulet, High frequency dynamics and the third cumulant of quantum noise, J. Stat. Mech. Theory Exp., 2009 (2009), p. P01049.
M. Geng, H. Liang, and J. Wang, Research on methods of higher-order statistics for phase difference detection and frequency estimation, in 2011 4th International Congress on Image and Signal Processing (CISP), Vol. 4, IEEE, New York, 2011, pp. 2189--2193.
X. Geng, K. Sun, L. Ji, H. Tang, and Y. Zhao, Joint skewness and its application in unsupervised band selection for small target detection, Sci. Rep., 5 (2015), 9915, https://doi.org/10.1038/srep09915.
R. L. Graham, D. E. Knuth, and O. Patashnik, Concrete Mathematics: A Foundation for Computer Science, Addison-Wesley, Boston, 1989.
A. Guizot, The Hedge Fund Compliance and Risk Management Guide, Vol. 371, John Wiley & Sons, New York, 2006.
E. Jondeau, E. Jurczenko, and M. Rockinger, Moment Component Analysis: An Illustration with International Stock Markets, Swiss Finance Institute Research Paper, 2015.
M. G. Kendall et al., The Advanced Theory of Statistics, Charles Griffin, London, 1946.
D. E. Knuth, The Art of Computer Programming: Sorting and Searching, Vol. 3B, Pearson Education, Boston, 1998.
J. R. Latimer and N. Namazi, Cumulant filters---A recursive estimation method for systems with non-Gaussian process and measurement noise, in Proceedings of the 35th Southeastern Symposium on System Theory, IEEE, New York, 2003, pp. 445--449.
J. Liang, Joint azimuth and elevation direction finding using cumulant, IEEE Sens. J., 9 (2009), pp. 390--398.
J. Liu, Z. Huang, and Y. Zhou, Extended 2q-MUSIC algorithm for noncircular signals, Signal Process., 88 (2008), pp. 1327--1339.
E. Lukacs, Characteristics Functions, Griffin, London, 1970.
X. Luo, Error estimation for moment analysis in heavy-ion collision experiment, J. Phys. G, 39 (2012), p. 025008.
E. S. Manolakos and H. M. Stellakis, Systematic synthesis of parallel architectures for the computation of higher order cumulants, Parallel Comput., 26 (2000), pp. 655--676.
I. W. Martin, Consumption-based asset pricing with higher cumulants, Rev. Econo. Stud., 80 (2013), pp. 745--773.
P. McCullagh, Tensor Methods in Statistics, Vol. 161, Chapman & Hall, London, 1987.
P. McCullagh and J. Kolassa, Cumulants, Scholarpedia, 4 (2009), p. 4699.
E. Moulines and J.-F. Cardoso, Second-order versus fourth-order MUSIC algorithms: An asymptotical statistical analysis, in Proceedings of the IEEE Signal Processing Workshop on Higher-Order Statistics, Chamrousse, France, 1991.
J.-F. Muzy, D. Sornette, J. Delour, and A. Arneodo, Multifractal returns and hierarchical portfolio theory, Quant. Finance, 1 (2001), pp. 131--148.
B. Ozga-Zielinski, M. Ciupak, J. Adamowski, B. Khalil, and J. Malard, Snow-melt flood frequency analysis by means of copula based 2D probability distributions for the Narew River in Poland, J. Hydrol. Region. Stud., 6 (2016), pp. 26--51.
P. Pal and P. Vaidyanathan, Multiple level nested array: An efficient geometry for 2q-th order cumulant based array processing, IEEE Trans. Signal Process., 60 (2012), pp. 1253--1269.
B. Porat and B. Friedlander, Direction finding algorithms based on high-order statistics, IEEE Trans. Signal Process., 39 (1991), pp. 2016--2024.
D.-B. Pougaza, A. Mohammad-Djafari, and J.-F. Bercher, Using the Notion of Copula in Tomography, arXiv:0812.1316, 2008.
G.-C. Rota and B. D. Taylor, The classical umbral calculus, SIAM J. Math. Anal., 25 (1994), pp. 694--711.
M. Rubinstein, E. Jurczenko, and B. Maillet, Multi-moment Asset Allocation and Pricing Models, Vol. 399, John Wiley & Sons, New York, 2006.
M. D. Schatz, T. M. Low, R. A. van de Geijn, and T. G. Kolda, Exploiting symmetry in tensors for high performance: Multiplication with symmetric tensors, SIAM J. Sci. Comput., 36 (2014), pp. C453--C479.
R. J. Scherrer, A. A. Berlind, Q. Mao, and C. K. McBride, From finance to cosmology: The copula of large-scale structure, Astrophys. J. Lett., 708 (2009), p. L9.

Information & Authors


Published In

cover image SIAM Journal on Scientific Computing
SIAM Journal on Scientific Computing
Pages: A1590 - A1610
ISSN (online): 1095-7197


Submitted: 26 September 2017
Accepted: 9 April 2018
Published online: 5 June 2018


  1. high-order cumulants
  2. nonnormally distributed data
  3. numerical algorithms

MSC codes

  1. 65Y05
  2. 15A69
  3. 65C60



Funding Information

Narodowe Centrum Nauki https://doi.org/10.13039/501100004281 : 2014/15/B/ST6/05204

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