Laplace and Mellin transform for reconstructing the probability distribution by a limited amount of information基于有限信息量重构概率分布的 Laplace 与 Mellin 变换
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Niu LZ, Di Paola M, Pirrotta A, Xu W, 2024. Laplace and Mellin transform for reconstructing the probability distribution by a limited amount of information. Probabilistic Engineering Mechanics, 78: 103700.DOI: 10.1016/j.probengmech.2024.103700
首先对于单边概率密度函数 (probability density function, PDF) ，介绍了一类采用 Laplace 变换重构随机变量概率密度函数的方法。这类方法定义了新的复数量，称为移位特征函数，可采用经典 Fourier 级数展开来计算概率密度函数。然后通过重新定义双边 Laplace 变换，将该方法扩展到处理双面边概率密度函数。即使逆 Laplace 变换中积分沿虚轴离散化，这一新定义仍适用。为进行对比，还介绍了一类基于 Mellin 变换的双边复分数阶矩新定义，解决了概率密度函数重构过程中出现的零点奇异性。关键词: Laplace 变换, Fourier 变换, 平移特征函数, 复分数阶矩, Fokker-Planck-Kolmogorov 方程A method for reconstructing the Probability Density Function (PDF) of a random variable using the Laplace transform is first introduced for one-sided PDFs. This approach defines new complex quantities, referred as Shifted Characteristic Functions, which allow the PDF to be computed using a classical Fourier series expansion. The method is then extended to handle double-sided PDFs by redefining the double-sided Laplace transform. This new definition remains applicable even when the integral in the inverse Laplace transform is discretized along the imaginary axis. For comparison, a new definition of double-sided Complex Fractional Moments based on Mellin transform is also introduced, addressing the singularity at zero that arises during PDF reconstruction.Keywords: Laplace transform; Fourier transform; Shift characteristic function; Complex fractional moment; FPK equation.图 1: b = π/Δθ = 15.7, μ = -2, ν = 0.5, Δθ = 0.2, m = 78, mΔθ ≈ bFig. 1. b = π/Δθ = 15.7, μ = -2, ν = 0.5, Δθ = 0.2, m = 78, and mΔθ ≈ b
图 2: β = 0, Δθ = 0.2, m = 78, ν = 2, b = π/Δθ = 15.7 下的混叠问题Fig. 2. Aliasing problem with β = 0, Δθ = 0.2, m = 78, ν = 2, b = π/Δθ = 15.7
图 3: (a) 响应的精确解与重构解; (b) 响应 Laplace 变换的实部与虚部Fig. 3. (a) Black dashed line is the exact X (t), and the red comtinuous line is the reconstruct X (t); (b) Continuous line is the A_x^L (s_k), and the dashed line is the imaginary part B_x^L (s_k), ν = 1
图 4: (a) 响应的精确解与重构解; (b) 响应 Laplace 变换的实部与虚部Fig. 4. (a) Black dashed line is the exact X (t), and the red comtinuous line is the reconstruct X (t); (b) Continuous line is the A_x^L (s_k), and the dashed line is the imaginary part B_x^L (s_k), α = 0.2, ν = 1
图 5: (a) 响应的精确解与重构解; (b) 响应 Laplace 变换的实部与虚部Fig. 5. (a) Black dashed line is the exact X (t), and the red comtinuous line is the reconstruct X (t); (b) Continuous line is the A_x^L (s_k), and the dashed line is the imaginary part B_x^L (s_k), α = 0.2, c = 0.5, ν = 1
图 6: (a) 移位特征函数的实部; (b) 移位特征函数的虚部Fig. 6. (a) Real part of SCF; (b) Imaginary part of SCF when t ϵ [0, 15]
Fig. 7. (a) Reconstructed PDF when t = 0, 5, 10, 15 and q = 0.01; (b) Reconstructed PDF when q = 0, 0.01, 0.05, 0.1 and t = 10
Fig. 8. β = 1
Fig. 9. ρ = 2
图 10: 平移双边移位特征函数: (a) 实部; (b) 虚部Fig. 10. Translated double-sided SCF with the initial β = 0.1, and Δβ = 0.3, the translated β = 0.4: (a) Real part; (b) Imaginary part
图 11: 平移双边移位特征函数: (a) 实部; (b) 虚部Fig. 11. Translated double-sided SCF with the initial ρ = 0.5, and Δρ = -0.2, the translated ρ = 0.7: (a) Real part; (b) Imaginary part
作者信息 | Authors
牛立志 Li-Zhi Niu, 共同通讯作者 (Corresp.)西北工业大学 (Northwestern Polytechnical University) 数学与统计学院Email: niulizhi902@mail.nwpu.edu.cn
Mario Di Paola, 共同通讯作者 (Corresp.)意大利巴勒莫大学 (Universita Degli Studi di Palermo) 工程系Email: mario.dipaola@unipa.it
Antonina Pirrotta, 共同通讯作者 (Corresp.)意大利巴勒莫大学 (Universita Degli Studi di Palermo) 工程系Email: antonina.pirrotta@unipa.it
徐伟 Wei Xu, 共同通讯作者 (Corresp.)西北工业大学 (Northwestern Polytechnical University) 数学与统计学院Email: weixunpu@nwpu.edu.cn
律梦泽 M.Z. Lyu | 编辑 (Ed)
P.D. Spanos | 审校 (Rev)
陈建兵 J.B. Chen | 审校 (Rev)
彭勇波 Y.B. Peng | 审校 (Rev)
