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Kernel density estimation with bounded data

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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.

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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.

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Authors and Affiliations

  1. School of Mechanical Engineering, Pusan National University, Pusan, 609-735, South Korea

    Young-Jin Kang, Yoojeong Noh & O-Kaung Lim

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  1. Young-Jin Kang
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  2. Yoojeong Noh
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  3. O-Kaung Lim
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Correspondence to Yoojeong Noh.

Appendix 1: Silverman’s rule of thumb

Appendix 1: Silverman’s rule of thumb

The Silverman’s rule of thumb is a method which minimizes an objective function, mean integrated squared error (MISE), and it is probably the most popular one among the bandwidth selection methods (Schindler 2011). It assumes that true density is normally distributed therefore Silverman’s rule will compute a bandwidth close to optimal if a random variable X is reasonably close to the normal distribution (Silverman 1986; Hansen 2009). It defines according to various kernel functions as follows (Hansen 2009).

$$ h=\widehat{\sigma}{C}_{\nu }(k){n}^{-1/\left(2\nu +1\right)} $$
(18)

where C ν (k) is the constant from Table 12, and ν is the order of the kernel.

Table 12 Constants of Silverman’s rule (Hansen 2009)
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Kang, YJ., Noh, Y. & Lim, OK. Kernel density estimation with bounded data. Struct Multidisc Optim 57, 95–113 (2018). https://doi.org/10.1007/s00158-017-1873-3

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