An Adaptive Gaussian Mixture Model for structural reliability analysis using convolution search technique

Non-parametric probability density estimation has gained popularity due to its flexibility and ease of use without requiring prior assumptions about distribution types. Notable examples include Kernel Density Estimation, Gaussian Mixture Model (GMM), the Mellin transform, and the Generalized Distribution Reconstruction (GDR) method, etc. However, these methods can encounter issues such as tail oscillation and sensitivity to initial guesses, particularly in the context of structural reliability analysis. To improve accuracy, this paper proposes an Adaptive Gaussian Mixture Model method. This method uses the inverse Fourier relationship between the Characteristic Function (CF) and the Probability Density Function (PDF), combined with a convolution search technique for parameter estimation. First, a more accurate expression for the CF is introduced, where the undetermined parameters are specified based on the numerically estimated CF curve. Then, a convolution search domain is developed to determine these parameters, including weight coefficients, mean domain, and standard deviation domain. Compared to the conventional methods for parameter estimation, the proposed convolution search technique can effectively avoid the problems of overfitting and initial parameter sensitivity. Using these parameters, the PDF is reconstructed and evolves into an Adaptive Gaussian Mixture Model. Numerical investigations are conducted to validate the efficacy of the proposed method, with comparisons made to the Mellin transform, GDR, Classic GMM, and other parametric methods.