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Wind Energy Resources
Published in Radian Belu, Fundamentals and Source Characteristics of Renewable Energy Systems, 2019
The distribution law mvM(θ), given by Equation (5.81), can be numerically integrated between two given values of θ to obtain the probability that the wind direction is found within a particular angle sector. Various methods are employed to compute the 3N parameters on which the mixture of von Mises distribution depends, the least squares (LS) method being the most common. Figure 5.5 is showing the fitted von Mise distributions to the wind direction time series collected at two locations in western Nevada, for the period from 2003 to 2008.
Directional Statistics of Preferential Orientations of Two Shapes in Their Aggregate and Its Application to Nanoparticle Aggregation
Published in Technometrics, 2018
Ali Esmaieeli Sikaroudi, David A. Welch, Taylor J. Woehl, Roland Faller, James E. Evans, Nigel D. Browning, Chiwoo Park
For a symmetric shape category, the equivalence (9) holds in θ, and the probability density function of θ should have the following symmetries: Therefore, if f has a mode at γ ∈ [0, π/2], it also has the modes at − γ, − π + γ and π − γ. A von-Mises distribution is popularly used to describe a unimodal probability density of angular data (Mardia et al. 2012). We take a mixture of four von Mises distributions with equal weights to represent the four modes caused by the four-way symmetry, where cosh ( · ) is a hyperbolic cosine function, and γ ∈ [0, π/2]. One can easily check that the density function satisfies the symmetry (11) as desired. Note that the normalization (10) applies for mirroring θ onto the first quadrant [0, π/2], and f has the same density for all quadrants. Therefore, the density function of the normalized angle is simply four times of f, where . One can show , so it is a valid probability density function. The two parameters γ and κ can be estimated by the maximum likelihood estimation described in Section 4.1, and the goodness of fit test for the estimated parameters can be performed by the method described in Section 4.2. Sections 4.3 and 4.4 describe the statistical hypotheses testing problems to test the two hypotheses that we mentioned in the beginning of this section.