The high-precision estimation of a multimodal probability density function is a difficult problem in many engineering fields. We propose a new method to improve the estimation accuracy based on the fractional moment-based maximum entropy method with a nonlinear transformation and a multi-peak recognition method. For the translation parameters in the nonlinear transformation, three approaches, such as sample-based least square polynomial fitting technique, sample-based kernel density estimation and classical maximum entropy method, are presented to determine the parameters. By adjusting the translation parameter, the valley of the probability density function curve can be translated to the position with a larger slope, and the distance between adjacent peaks is enlarged to avoid the wrong fitting form of the probability density function curve with multiple peaks. After the parameters of the transformation are obtained, the fractional moment-based maximum entropy method is applied to predict the probability density function of the transformed performance function. Two numerical examples are used to verify the accuracy and stability of the proposed method. Two engineering examples are introduced to illustrate the applicability and efficiency of the proposed method in the real-life engineering setting. It is concluded that the proposed method uses fewer moments with less additional calculation costs, and has good computational efficiency and applicability for the modeling of multimodal probability density functions, compared with the classical fractional moment-based maximum entropy method.

多模态概率密度函数的高精度估计是许多工程领域的难题。我们提出了一种基于分数阶矩的最大熵方法和非线性变换和多峰识别方法来提高估计精度的新方法。对于非线性变换中的平移参数，提出了基于样本的最小二乘多项式拟合技术、基于样本的核密度估计和经典的最大熵方法三种方法来确定参数。通过调整平移参数，可以将概率密度函数曲线的谷底平移到斜率较大的位置，并扩大相邻峰之间的距离，避免多峰概率密度函数曲线拟合形式错误。得到变换的参数后，应用基于分数阶矩的最大熵方法对变换后的性能函数的概率密度函数进行预测。两个数值算例验证了所提方法的准确性和稳定性。引入了两个工程实例来说明所提出的方法在现实工程环境中的适用性和效率。得出的结论是，所提出的方法使用的矩较少，额外的计算成本较低，对于多模态概率密度函数的建模具有良好的计算效率和适用性，