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Vitaliy Babak, Volodymyr Eremenko, Artur Zaporozhets


In this paper, it was proposed to carry out a preliminary normalization of diagnostic parameters using the Johnson distribution, which with three basic distribution groups (SL, SB, SU), covers a wide class of empirical distributions. The mathematical description of the family allows us to find the approximating probability density function in an explicit form, to determine the distribution parameters for obtaining the corresponding function (curve), as well as the inverse function for finding the quantiles of the specified levels. To assess the accuracy of the obtained normalized data, they were compared with the data obtained by replacing the resulting law with a Gaussian one. Percentages of values were compared in the implementation under study, which concentrated in the limits of estimated quantiles. Implementations were obtained using the simulation method. By the same method, the correctness (relative systematic error) of determining the quantile values of the specified levels was evaluated. The error value δ was estimated between the conditionally true quantile value calculated from the generated pseudo-general complex and the value estimated using the methods considered in the paper. Obtained data show that the relative error in the calculation of quantiles using the Johnson distribution does not exceed 0.07% and decreases in two orders of magnitude than the currently accepted procedure for replacing sample laws with Gaussian.


Johnson distribution; Gaussian distribution; uniform distribution; uncertainty; approximation; quantiles; statistical diagnostics.

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S. Gholizadeh, “A review of non-destructive testing methods of composite materials,” Procedia Structural Integrity, vol. 1, pp. 50-57, 2016

B. Salski, W. Gwarek, P. Korpas, S. Reszewicz, A.Y.B. Chong, P. Theodorakeas, I. Hatziioannidis, V. Kappatos, C. Selcuk, T.-H. Gan, M. Koui, M. Iwanowski, B. Zielinski, “Non-destructive testing of carbon-fibre-reinforced polymer materials with a radio-frequency inductive sensor,” Composite Structures, vol. 122, pp. 104-112, 2015

J. Dong, B. Kim, A. Locquet, P. McKeon, N. Declercq, D.S. Citrin, “Nondestructive evaluation of forced delamination in glass fiber-reinforced composites by terahertz and ultrasonic waves,” Composites Part B: Engineering, vol. 79, pp. 667-675, 2015.

A. Zaporozhets, V. Eremenko, R. Serhiienko, S. Ivanov, “Methods and hardware for diagnosing thermal power equipment based on Smart Grid Technology”, in: N. Shakhovska, M.O. Medykovskyy (Eds.), Advances in Intelligent Systems and Computing III, Springer, Cham, 2019, pp. 476-492.

A.A. Zaporozhets, V.S. Eremenko, R.V. Serhiienko and S.A. Ivanov, “Development of an intelligent system for diagnosing the technical condition of the heat power equipment,” Proceedings of the 2018 IEEE 13th International Scientific and Technical Conference on Computer Sciences and Information Technologies (CSIT), Lviv, Ukraine, 11-14 September, 2018, pp. 48-51.

M.-Y. Moon, K.K. Choi, N. Gaul, D. Lamb, “Treating Epistemic Uncertainty Using Bootstrapping Selection of Input Distribution Model for Confidence-Based Reliability Assessment,” Journal of Mechanical Design, vol. 141, issue 3, 031402, 2019

Z.Q. John Lu, “The elements of statistical learning: data mining, inference, and prediction,” Journal of the Royal Statistical Society: Series A (Statistics in Society), vol. 173, issue 3, pp. 693-694, 2010.

S. Nunes, M. Pimentel, F. Riberio, P. Milheiro-Oliveira, A. Carvalho, “Estimation of the tensile strength of UHPFRC layers based on non-destructive assessment of the fibre content and orientation,” Cement and Concrete Composites, vol. 83, pp. 222-238, 2017

M.R. Jolly, A. Prabhakar, B. Sturzu, K. Hollstein, R. Singh, S. Thomas, P. Foote, A. Shaw, “Review of Non-destructive Testing (NDT) Techniques and their Applicability to Thick Walled Composites,” Procedia CIRP, vol. 38, pp. 129-136, 2015

Y. Ameur, H. Hedenmalm, N. Makarov, “Random normal matrices and Ward identities,” The Annals of Probability, vol. 43, issue 3, pp. 1157-1201, 2015.

W. He, G. Li, P. Hao, Y. Zeng, “Maximum entropy method-based reliability analysis with correlated input variables via hybrid dimension-reduction method,” Journal of Mechanical Design, vol. 141, issue 10, 101405, 2019.

A.A. Hassen, H. Taheri, U.K. Vaidya, “Non-destructive investigation of thermoplastic reinforced composites,” Composites Part B: Engineering, vol. 97, pp. 244-254, 2016.

K. Marhadi, S. Venkataraman, S.S. Pai, “Quantifying uncertainty in statistical distribution of small sample data using Bayesian inference of unbounded Johnson distribution,” International Journal of Reliability and Safety, vol. 6, issue 4, pp. 311-337, 2012.

T. Nishizu, A. Takezawa, M. Kitamura “Eigenfrequecy-based damage identification method for non-destructive testing based on topology optimization,” Engineering Optimization, vol. 49, issue 3, pp. 417-433, 2015.

V. Eremenko, A. Zaporozhets, V. Isaenko, K. Babikova, “Application of Wavelet Transform for Determining Diagnostic Signs,” CEUR Workshop Proceedings, vol. 2387, pp. 202-214, 2019.

G. Barbato, G. Genta, A. Germak, R. Levi, G. Vicario, “Treatment of experimental data with discordant observations: issues in empirical identification of distribution,” Measurement Science Review, vol. 12, issue 4, pp. 133-140, 2012.

N.I.E. Farhana, M.S. Abdul Majid, M.P. Paulraj, E. Ahmadhilmi, M.N. Fakhzan, A.G. Gibson, “A novel vibration based non-destructive testing for predicting glass fibre/matrix volume fraction in composites using a neural network model,” Composite Structures, vol. 144, pp. 96-107, 2016.

A. Katunin, K. Dragan, M. Dziendzikowski, “Damage identification in aircraft composite structures: A case study using various non-destructive testing techniques,” Composite Structures, vol. 127, pp. 1-19, 2015.

M. Pavlovic, M. Doycinovic, S. Martinovic, M. Vlahovic, Z. Stevic, T. Volkov Husovic, “Non destructive monitoring of cavitation erosion of cordierite based coatings,” Composites Part B: Engineering, vol. 97, pp. 84-91, 2016.

M. Alatefi, S. Ahmad, M. Alkahtani, “Performance evaluation using multivariate non-normal process capability,” Processes, vol. 7, 833, 2019.

C. Mineo, S.G. Pierce, P.I. Nicholson, I. Cooper, “Robotic path planning for non-destructive testing – A custom MATLAB toolbox approach,” Robotics and Computer-Integrated Manufacturing, vol. 37, pp. 1-12, 2016.

X. Zhang, S. Du, “A linear Dirichlet mixture model for decomposing scenes: Application to analyzing urban functional zonings,” Remote Sensing of Environment, vol. 169, pp. 37-49, 2015.

C.-O. Amedee-Manesme, F. Barthelemy, D. Maillard, “Computation of the corrected Cornish–Fisher expansion using the response surface methodology: application to VaR and CVaR,” Annals of Operations Research, vol. 281, issue 1-2, pp. 423-453, 2019.

I. Holmes, M.T. Lacey, B.D. Wick, “Bloom’s inequality: commutators in a two-weight setting,” Archiv der Mathematik, vol. 106, issue 1, pp. 53-63, 2016.

M. Lints, S. Dos Santos, A. Salupere, “Solitary waves for non-destructive testing applications: Delayed nonlinear time reversal signal processing optimization,” Wave Motion, vol. 71, pp. 101-112, 2017.

P.J.A. Cayton and D.S. Mapa, Time-varying conditional Johnson SU density in value-at-risk (VaR) methodology, 2012, [Online]. Available:

W.C. Torrez, J.T. Durham and R.D. Trueblood, “Performance measures for neural nets using Johnson distributions,” Proceedings of the IEEE International Conference on Neural Networks, San Francisco, CA, USA, 28 March-1 April, 1993, pp. 506-510.

P. Duchene, S. Chaki, A. Ayadi. P. Kraqczak, “A review of non-destructive techniques used for mechanical damage assessment in polymer composites,” Journal of Material Science, vol. 53, pp. 7915-7938, 2018.

M.C. Porcu, L. Pieczonka, A. Frau, W.J. Staszewski, F. Aymerich, “Assessing the scaling subtraction method for impact damage detection in composite plates,” Journal of Nondestructive Evaluation, vol. 36, 33, 2017.

H. Hayashi, Y. Kurita and T. Tsuji, “A non-Gaussian approach for biosignal classification based on the Johnson SU translation,” Proceedings of the 2015 IEEE 8th International Workshop on Computational Intelligence and Applications (IWCIA), Hiroshima, Japan, 6-7 November, 2015, pp. 115-120.

L. Lin, Y. Saad, C. Yang, “Approximating Spectral Densities of Large Matrices,” SIAM Review, vol. 58, issue 1, pp. 34-65, 2016.

A.P. Bartok, G. Csanyi, “Gaussian approximation potentials: A brief tutorial introduction,” Quantum Chemistry, vol. 115, issue 16, pp. 1051-1057, 2015.

S. Anitha and B.M. Ramesh, “Network reconfiguration for loss minimization by using Johnson’s algorithm,” Proceedings of the 2018 4th International Conference on Electrical Energy Systems (ICEES), Chennai, India, 7-9 February, 2018, pp. 680-684.

R.H. Storer, J.J. Swain, S. Venkatraman and J.R. Wilson, “Comparison of methods for fitting data using Johnson translation distributions,” Proceedings of the 1998 Winter Simulation Conference, San Diego, CA, USA, 12-14 December, 1998, pp. 476-481.

S. Nakagawa, P.C.D. Johnson, H. Schielzeth, “The coefficient of determination R2 and intra-class correlation coefficient from generalized linear mixed-effects models revisited and expanded,” J. R. Soc. Interface, vol. 14, issue 134, 2017.

G. Letac, L. Mattner, M. Piccioni, “The median of an exponential family and the normal law,” Statistics & Probability Letters, vol. 133, pp. 38-41, 2018.

M. Baak, S. Gadatsch, R. Harrington, W. Verkerke, “Interpolation between multi-dimensional histograms using a new non-linear moment morphing method,” Nuclear Instruments and Methods in Physics Research Section A: Accelerators, Spectrometers, Detectors and Associated Equipment, vol. 771, pp. 39-48, 2015.

O.S. Sonbul, A.N. Kalashnikov, “Determining the operating distance of air ultrasound range finders: calculations and experiments,” International Journal of Computing, vol. 13, issue 2, pp. 125-131, 2014.

A.O. Zaporozhets, O.O. Redko, V.P. Babak, V.S. Eremenko, V.M. Mokiychuk, “Method of indirect measurement of oxygen concentration in the air,” Naukovyi Visnyk Natsionalnoho Hirnychoho Universytetu, issue 5, pp. 105-114, 2018.

Y. Zhang, H. Liu, Y. Huang, Z. Deng, “A nonlinear ARMA-GARCH model with Johnson Su innovations and its application to sea clutter modeling,” IEEE Access, vol. 6, pp. 11888-118969, 2018.

K.K. Phoon and J. Ching, “Multivariate model for soil parameters based on Johnson distributions,” Proceedings of the 2018 Foundation Engineering in the Face of Uncertainty: Honoring Fred H. Kulhawy, San Diego, USA, 3-7 March, 2013, pp. 337-353.

V.P. Babak, V.M. Mokiychuk, A.A. Zaporozhets, A.A. Redko, “Improving the efficiency of fuel combustion with regard to the uncertainty of measuring oxygen concentration,” Eastern-European Journal of Enterprise Technologies, vol. 6, no. 8(84), pp. 54-59, 2016.

L. Zhang, H. Cheng, H. He, Q. Zhou and P. Zeng, “Johnson system based reliability evaluation of composite power system with wind farms,” Proceedings of the International Conference on Renewable Power Generation (RPG 2015), Beijing, China, 17-18 October 2015, p. 1-6.


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