Springer Nature is making SARS-CoV-2 and COVID-19 research free. View research | View latest news | Sign up for updates

Sequential statistical modeling method for distribution type identification

  • 359 Accesses

  • 16 Citations


To accurately analyze behavior of mechanical system, accurate statistical modeling of input variables is necessary by identifying probabilistic distributions of input variables. These distributions are generally determined by applying goodness-of-fit (GOF) tests or model selection methods to the given data on the input variables. However, GOF tests only accept or reject the hypothesis that a candidate distribution is appropriate to represent the given data. The model selection methods determine the best-fit distribution for the given data among various candidate distributions but do not provide any information about the adequacy of using the identified distribution to represent the given data. Therefore, in this paper, a sequential statistical modeling (SSM) method is proposed. The SSM method uses a GOF test to select appropriate candidate distributions from among all possible distributions and then identifies the best-fit distribution from among the selected candidate distributions using a model selection method. The adequacy of the identified best-fit distribution is verified by using an area metric that measures the intersection area between the probability density function (PDF) of the best-fit distribution and the data distribution. This metric can be used to analyze the similarities between the PDFs of the candidate distributions. In statistical simulation tests, it was observed that the SSM method identified correct distributions more accurately and conservatively than the GOF tests or model selection methods alone.

This is a preview of subscription content, log in to check access.

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9
Fig. 10
Fig. 11
Fig. 12
Fig. 13


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

  2. Anderson TW, Darling DA (1952) Asymptotic theory of certain goodness of fit criteria based on stochastic processes. Ann Math Stat 23(2): 193–212

  3. Ayyub BM, McCuen RH (2012) Probability, statistics, and reliability for engineers and scientists. CRC Press, Florida

  4. Budynas RG, Nisbett JK (2010) Shingley’s mechanical engineering design (9th edition), chap. 6. McGraw-Hill, New York

  5. Burnham KP, Anderson DR (2004) Multimodel inference: understanding AIC and BIC in model selection. Sociol Methods Res 33(2):261–304

  6. D’Agostino RB, Stephens MA (1986) Goodness-of-fit techniques. Marcel-Dekker, New York

  7. Findley DF (1991) Counterexamples to parsimony and BIC. Ann Inst Stat Math 43(3):505–514

  8. Hogg RV, Tanis E (2009) Probability and Statistical inference (8th edition). Prentice Hall, New Jersey

  9. Kahraman C, Kerre EE, Bozbura FT (2012) Uncertainty modeling in knowledge engineering and decision making, world scientific

  10. Kang YJ, Noh Y (2015) Comparison study of statistical modeling methods for identifying distribution types, 11th World Congress on Structural and Multidisciplinary Optimisation, Sydney, Australia, 2015

  11. MatWeb (2016), www.matweb.com/. Accessed 2016 Feb 2

  12. Noh Y, Choi KK, Lee I (2010) Identification of marginal and Joint CDFs using Bayesian method for RBDO. Struct Multidiscip Optim 40(1):35–51

  13. Scott DW (1979) On optimal and data-based histograms. Bimoetrika 66(3):605–610

  14. Silverman BW (1986) Density estimation for statistics and data analysis, Vol. 26. CRC press, London, p 45

  15. Socie D (2014) Probabilistic Statistical Simulations Technical Background, eFatigue LLC, 2008, https://www.efatigue.com/probabilistic/background/statsim.html#Cor, April, 2014

  16. Soong TT (2004) Fundamentals of probability and statistics for engineers. John & Wiley & Sons Ltd., England

  17. Sugiura N (1978) Further analysis of the data by akaike’s information criterion and the finite corrections. Commun Stat Theory Methods 7(1):13–26

  18. Wand MP, Jones MC (1994) Kernel smoothing. CRC press, London

  19. Yang JS, Kang TJ (1998) The properties of cotton yarn spun from the fiber mixture of the multimodal distribution (I). J Korean Fiber Soc 35(5):272–28

  20. Youn BD, Jung BC, Xi Z, Kim SB, Lee WR (2011) A hierarchical framework for statistical model calibration in engineering product development. Comput Methods in Appl Mech Eng 200:1421–1431

Download references


This work was supported by the National Research Foundation of Korea (NRF-2015R1A1A3A04001351) funded by the government of Korea and by the Technology Innovation Program (10048305, Launching Plug-In Digital Analysis Framework for Modular System Design). This support is greatly appreciated.

Author information

Correspondence to Yoojeong Noh.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Kang, Y., Lim, O. & Noh, Y. Sequential statistical modeling method for distribution type identification. Struct Multidisc Optim 54, 1587–1607 (2016). https://doi.org/10.1007/s00158-016-1567-2

Download citation


  • Sequential statistical modeling method
  • Goodness-of-fit test
  • Model selection method
  • Area metric