Advertisement

Structural and Multidisciplinary Optimization

, Volume 57, Issue 1, pp 95–113 | Cite as

Kernel density estimation with bounded data

  • Young-Jin Kang
  • Yoojeong NohEmail author
  • O-Kaung Lim
RESEARCH PAPER

Abstract

The uncertainties of input variables are quantified as probabilistic distribution functions using parametric or nonparametric statistical modeling methods for reliability analysis or reliability-based design optimization. However, parametric statistical modeling methods such as the goodness-of-fit test and the model selection method are inaccurate when the number of data is very small or the input variables do not have parametric distributions. To deal with this problem, kernel density estimation with bounded data (KDE-bd) and KDE with estimated bounded data (KDE-ebd), which randomly generates bounded data within given input variable intervals for given data and applies them to generate density functions, are proposed in this study. Since the KDE-bd and KDE-ebd use input variable intervals, they attain better convergence to the population distribution than the original KDE does, especially for a small number of given data. The KDE-bd can even deal with a problem that has one data with input variable bounds. To verify the proposed method, statistical simulation tests were carried out for various numbers of data using multiple distribution types and then the KDE-bd and KDE-ebd were compared with the KDE. The results showed the KDE-bd and KDE-ebd to be more accurate than the original KDE, especially when the number of data is less than 10. It is also more robust than the original KDE regardless of the quality of given data, and is therefore more useful even if there is insufficient data for input variables.

Keywords

Kernel density estimation Nonparametric statistical modeling Interval approach Nonparametric distribution Bounded data Intersection area 

Notes

Acknowledgments

This work was supported by the National Research Foundation of Korea (NRF) grant, funded by the Korean Government (NRF-2015R1A1A3A04001351) and by the Technology Innovation Program (10048305, Launching Plug-in Digital Analysis Framework for Modular System Design) and the Human Resources Development program (No. 20164030201230) of the Korea Institute of Energy Technology Evaluation and Planning (KETEP) grant funded by the Ministry of Trade, Industry and Energy. This support is greatly appreciated.

References

  1. Agarwal H, Renaud JE, Preston EL, Padmanabhan D (2004) Uncertainty quantification using evidence theory in multidisciplinary design optimization. Reliab Eng Syst Saf 85(1):281–294CrossRefGoogle Scholar
  2. Akaike H (1974) A new look at the statistical model identification. IEEE Trans Autom Control 19(6):716–723MathSciNetCrossRefzbMATHGoogle Scholar
  3. Analytical Methods Committee (1989) Robust statistics-how not to reject outliers. Part 1. Basic concepts. Analyst 114(12):1693–1697CrossRefGoogle Scholar
  4. Anderson TW, Darling DA (1952) Asymptotic theory of certain goodness of fit criteria based on stochastic processes. Ann Math Stat 23(2):193–212MathSciNetCrossRefzbMATHGoogle Scholar
  5. Ayyub BM, McCuen RH (2012) Probability, statistics, and reliability for engineers and scientists. CRC Press, FloridazbMATHGoogle Scholar
  6. Betrie GD, Sadiq R, Morin KA, Tesfamariam S (2014) Uncertainty quantification and integration of machine learning techniques for predicting acid rock drainage chemistry: a probability bounds approach. Sci Total Environ 490:182–190CrossRefGoogle Scholar
  7. Betrie GD, Sadiq R, Nichol C, Morin KA, Tesfamariam S (2016) Environmental risk assessment of acid rock drainage under uncertainty: the probability bounds and PHREEQC approach. J Hazard Mater 301:187–196CrossRefGoogle Scholar
  8. Burnham KP, Anderson DR (2004) Multimodel inference: understanding AIC and BIC in model selection. Sociol Methods Res 33(2):261–304MathSciNetCrossRefGoogle Scholar
  9. Chen S (2015) Optimal bandwidth selection for kernel density functionals estimation. J Probab Stat 2015:21MathSciNetCrossRefGoogle Scholar
  10. Cho SG, Jang J, Kim S, Park S, Lee TH, Lee M, Choi JS, Kim HW, Hong S (2016) Nonparametric approach for uncertainty-based multidisciplinary design optimization considering limited data. Struct Multidiscip Optim 54(6):1671–1688CrossRefGoogle Scholar
  11. Cowling A, Hall P (1996) On pseudodata methods for removing boundary effect in kernel density estimation. J R Stat Soc Ser B Methodol 58(3):551–563MathSciNetzbMATHGoogle Scholar
  12. Cox M, Harris P (2003) Up a GUM tree? Try the full monte! National Physical Laboratory, TeddingtonGoogle Scholar
  13. Eldred MS, Agarwal H, Perez VM, Wojtkiewicz SF Jr, Renaud JE (2007) Investigation of reliability method formulations in DAKOTA/UQ. Struct Infrastruct Eng 3(3):199–213CrossRefGoogle Scholar
  14. Frigge M, Hoaglin DC, Lglewicz B (1989) Some implementations of the boxplot. Am Stat 43(1):50–54Google Scholar
  15. Gabauer W (2000) Manual of codes of practice for the determination of uncertainties in mechanical tests on metallic materials, the determination of uncertainties in tensile testing. UNCERT COP7 report, Project SMT4-CT97-2165Google Scholar
  16. Gasser T, Müller HG (1979) Kernel estimation of regression functions. Smoothing Techniques for Curve Estimation 757:23–68MathSciNetCrossRefzbMATHGoogle Scholar
  17. Guidoum AC (2015) Kernel estimator and bandwidth selection for density and its derivatives. Department of Probabilities & Statistics, Faculty of Mathematics, University of Science and Technology Houari Boumediene, Algeria, https://cran.r-project.org/web/packages/kedd/vignettes/kedd.pdf
  18. Hansen BE (2009) Lecture notes on nonparametrics. University of Wisconsin-Madison, WI, USA, http://www.ssc.wisc.edu/~bhansen/718/NonParametrics1.pdf
  19. Hardle W, Marron JS, Wand MP (1990) Bandwidth choice for density derivatives. J R Stat Soc Ser B Methodol 52(1):223–232MathSciNetzbMATHGoogle Scholar
  20. Jang J, Cho SG, Lee SJ, Kim KS, Hong JP, Lee TH (2015) Reliability-based robust design optimization with kernel density estimation for electric power steering motor considering manufacturing uncertainties. IEEE Trans Magn 51(3):1–4CrossRefGoogle Scholar
  21. Jones MC, Kappenman RF (1992) On a class of kernel density estimate bandwidth selectors. Scand J Stat 19(4):337–349MathSciNetzbMATHGoogle Scholar
  22. Jung JH, Kang YJ, Lim OK, Noh Y (2017) A new method to determine the number of experimental data using statistical modeling methods. J Mech Sci Technol 31(6):2901–2910CrossRefGoogle Scholar
  23. Kang YJ, Lim OK, Noh Y (2016) Sequential statistical modeling for distribution type identification. Struct Multidiscip Optim 54(6):1587–1607CrossRefGoogle Scholar
  24. Kang YJ, Hong JM, Lim OK, Noh Y (2017) Reliability analysis using parametric and nonparametric input modeling methods. J Comput Struct Eng Inst Korea 30(1):87–94CrossRefGoogle Scholar
  25. Karanki DR, Kushwaha HS, Verma AK, Ajit S (2009) Uncertainty analysis based on probability bounds (P-box) approach in probabilistic safety assessment. Risk Anal 29(5):662–675CrossRefGoogle Scholar
  26. Karunamuni RJ, Alberts T (2005a) On boundary correction in kernel density estimation. Stat Methodol 2(3):191–212MathSciNetCrossRefzbMATHGoogle Scholar
  27. Karunamuni RJ, Alberts T (2005b) A generalized reflection method of boundary correction in kernel density estimation. Can J Stat 33(4):497–509MathSciNetCrossRefzbMATHGoogle Scholar
  28. Karunamuni RJ, Zhang S (2008) Some improvements on a boundary corrected kernel density estimator. Stat Probab Lett 78(5):499–507MathSciNetCrossRefzbMATHGoogle Scholar
  29. Marron JS, Ruppert D (1994) Transformations to reduce boundary bias in kernel density estimation. J R Stat Soc Ser B Methodol 56(4):653–671MathSciNetzbMATHGoogle Scholar
  30. Montgomery DC, Runger GC (2003) Applied statistics and probability for engineers (3rd edition). Wiley, New YorkGoogle Scholar
  31. Noh Y, Choi KK, Lee I (2010) Identification of marginal and joint CDFs using Bayesian method for RBDO. Struct Multidiscip Optim 40(1):35–51MathSciNetCrossRefzbMATHGoogle Scholar
  32. Schindler A (2011) Bandwidth selection in nonparametric kernel estimation. PhD Thesis. Göttingen, Georg-August Universität, DissGoogle Scholar
  33. Schuster EF (1985) Incorporating support constraints into nonparametric estimators of densities. Commun StatTheory Methods 14(5):1123–1136MathSciNetCrossRefzbMATHGoogle Scholar
  34. Schwarz (1978) Estimating the dimension of a model. Ann Stat 6(2):461–464MathSciNetCrossRefzbMATHGoogle Scholar
  35. Scott DW, Terrell GR (1987) Biased and unbiased cross-validation in density estimation. J Am Stat Assoc 82(400):1131–1146MathSciNetCrossRefzbMATHGoogle Scholar
  36. Shah H, Hosder S, Winter T (2015) Quantification of margins and mixed uncertainties using evidence theory and stochastic expansions. Reliab Eng Syst Saf 138:59–72CrossRefGoogle Scholar
  37. Sheather SJ (2004) Density estimation. Stat Sci 19(4):588–597CrossRefzbMATHGoogle Scholar
  38. Sheather SJ, Jones MC (1991) A reliable data-based bandwidth selection method for kernel density estimation. J R Stat Soc Ser B Methodol 53(3):683–690MathSciNetzbMATHGoogle Scholar
  39. Silverman BW (1986) Density estimation for statistics and data analysis, vol 26. CRC press, LondonCrossRefzbMATHGoogle Scholar
  40. Tucker WT, Ferson S (2003) Probability bounds analysis in environmental risk assessment. Applied Biomathematics, Setauket, New York, http://www.ramas.com/pbawhite.pdf
  41. Tukey JW (1977) Exploratory data analysis. Pearson, New YorkzbMATHGoogle Scholar
  42. Verma AK, Srividya A, Karanki DR (2010) Reliability and safety engineering. Springer, LondonCrossRefGoogle Scholar
  43. Wand MP, Jones MC (1994) Kernel smoothing. CRC press, LondonzbMATHGoogle Scholar
  44. Yao W, Chen X, Quyang Q, Van Tooren M (2013) A reliability-based multidisciplinary design optimization procedure based on combined probability and evidence theory. Struct Multidiscip Optim 48(2):339–354MathSciNetCrossRefGoogle Scholar
  45. Youn BD, Jung BC, Xi Z, Kim SB, Lee WR (2011) A hierarchical framework for statistical model calibration in engineering product development. Comput Methods Appl Mech Eng 200(13):1421–1431Google Scholar
  46. Zhang Z, Jiang C, Han X, Hu D, Yu S (2014) A response surface approach for structural reliability analysis using evidence theory. Adv Eng Softw 69:37–45CrossRefGoogle Scholar

Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2017

Authors and Affiliations

  1. 1.School of Mechanical EngineeringPusan National UniversityPusanSouth Korea

Personalised recommendations