Estimating wind speed probability distribution by diffusion-based kernel density method

Clustering by fast search and find of density peaks via heat diffusion

“…It provides better performance and is consistent with the true density, whereas Eq. 3 is inconsistent [31,33]. The superior performance and fast evaluation of KDEs via diffusion is discussed in [32].…”

“…Furthermore, Eq. 13 can be solved using fast Fourier transform and takes O(n log 2 n) operations [31,33]. For small bandwidth, Eq.…”

“…2 or 3 for estimating the density of a given dataset, we propose to use the KDE via the heat diffusion method [31]. Heat diffusion method views the kernel density estimate as a unique solution to the diffusion partial differential equation, which evolves for a time t proportional to the kernel bandwidth h [31][32][33].The interpretation of KDE via heat diffusion derives from the concept of the Weiner process, W , a continuous time stochastic process where the next stage is directly calculated by the previous state, such that 1. The preparatory probability is equally distributed through the d-dimensional data points {d 1 , d 2 , d 3 , ...,d n }.…”

“…Optimal bandwidth is obtained as a fixed-point solution to a recursion and can be estimated using the fast cosine transform without considering the normality assumption on the distribution [31][32][33]. Botev, et al [31] proposed a unique solution of the nonlinear equation to adaptively find the optimal bandwidth t for KDE,…”

Probabilistic load flow considering correlations of input variables following arbitrary distributions

“…In [19], polynomial normal transformation was applied to establish probability distribution models of input variables, but the usage of moments makes this approach quite similar with series expansion methods. With the limitations of parametric estimation, nonparametric estimation such as kernel density estimation was employed to analyze the probability features of wind power [20] and PV generation [21], and achieved considerable results. But the kernel density estimation methods shown in viewed literatures were based on the fixed bandwidth theory, which yields significant error in marginal intervals with extremely high probability.…”