Abstract
In general, although some random variables such as wind speed, temperature, and load are known to have multimodal distributions, input or output random variables are considered to follow unimodal distributions without assessing the unimodality or multimodality of distributions from samples. In uncertainty analysis, estimating unimodal distribution as multimodal distribution or vice versa can lead to erroneous analysis results. Thus, whether a distribution is unimodal or multimodal must be assessed before the estimation of distributions. In this paper, the bimodality coefficient (BC) and Hartigan’s dip statistic (HDS), which are representative methods for assessing multimodality, are introduced and compared. Then, a combined HDS with BC method is proposed. The proposed method has the advantages of both BC and HDS by using the skewness and kurtosis of samples as well as the dip statistic through a link function between the BC values in BC and significance level in HDS. To verify the performance of the proposed method, statistical simulation tests were conducted to evaluate the multimodality for various unimodal, bimodal, and trimodal models. The implementation of the proposed method to real engineering data is shown through case studies. The results demonstrate that the proposed method is more accurate, robust, and reliable than the BC and original HDS alone.
Highlights
In the field of engineering, most random variables are treated as having unimodal distributions with one mode so that the estimated probability density function (PDF) or cumulative distribution function (CDF) with unimodality from samples is applied to probabilistic statistical analysis or design methods such as reliability analysis, reliability-based design optimization, and robust design [1, 2]
If sufficient data are available, the performances of HDSw/bimodality coefficient (BC)(Quad) and HDSw/BC(Irr) are similar for unimodal distributions and the performance of HDSw/BC(Irr) is better than that of HDSw/BC(Quad) for multimodal distributions. us, users are recommended to employ either HDSw/BC(Quad) or HDSw/BC(Irr) for insufficient data according to the type of errors that need to be reduced and employ HDSw/BC(Irr) for sufficient data
To overcome the contrast limitations of the BC and Hartigan’s dip statistic (HDS), HDSw/BC was proposed. e proposed HDSw/BC uses the BC values calculated from sample skewness and kurtosis to determine the optimum significance level through quadratic or irrational functions and identifies the modality of the data using a dip statistic of the original HDS determined by the distribution shapes. us, HDSw/BC methods are more reliable and stable than the BC and more useful and accurate than the original HDS. e quadratic function affects the significance level to produce the smallest significance level; HDSw/BC(Quad) has the highest probability to judge the data as unimodality
Summary
In the field of engineering, most random variables are treated as having unimodal distributions with one mode so that the estimated probability density function (PDF) or cumulative distribution function (CDF) with unimodality from samples is applied to probabilistic statistical analysis or design methods such as reliability analysis, reliability-based design optimization, and robust design [1, 2]. If users know in advance whether a random variable has a unimodal or multimodal distribution, they can select the appropriate parametric or nonparametric modeling method [1, 2, 10]. HDS is a more accurate, reliable, and appropriate method for assessing a unimodality or multimodality, it slowly converges to the modality of the true model as sample size increases, and its results are sensitive to the significance level. Is study proposed a new modality assessment method (HDS with BC), which was first applied in the field of engineering, by combining the existing modality estimation methods HDS and BC. It was verified that the proposed method is more accurate, reliable, and quickly converges to the true unimodality or multimodality through the assessment of multimodality of unimodal, bimodal, and trimodal distributions in simulations and case studies of real measurements and engineering data. It can be highly likely to be used in the engineering field
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