ABSTRACT
Distributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 168 8.4 Cylindrical Distributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 168
8.4.1 The Johnson-Wehrly Distribution . . . . . . . . . . . . . . . . . . . . . . 168 8.4.2 The Weibull-von Mises Distribution . . . . . . . . . . . . . . . . . . . . 169 8.4.3 Gamma-von Mises Distribution . . . . . . . . . . . . . . . . . . . . . . . . . 170 8.4.4 Generalized Gamma-von Mises Distribution . . . . . . . . . . . 171 8.4.5 Sine-Skewed Weibull-von-Mises Distribution . . . . . . . . . . . 172 8.4.6 Parameter Estimation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173
8.5 Application 1: Quantification of the Speed/Turning Angle Patterns of a Flying Bird . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173
8.6 Application of Cylindrical Distributions 2: How Trees Are Expanding Crowns . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 174 8.6.1 Crown Asymmetry in Boreal Forests . . . . . . . . . . . . . . . . . . . . 176 8.6.2 Crown Asymmetry Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 177 8.6.3 Results of the Cylindrical Models . . . . . . . . . . . . . . . . . . . . . . . 180
8.7 Concluding Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 184 Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 184 Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 185
Methods and
Correlation/covariance is a fundamental concept, and multivariate analyses often begin by investigating their presence in given multivariate data. In particular, if an objective variable is not a scalar but a vector with correlations, to construct a statistical model, we need multivariate probability distributions that can flexibly capture the correlations between the objective variables. For example, the multivariate Gaussian distribution is used in the Gaussian process and the geostatistical process (Cressie & Wikle, 2011) to express stochastic uncertainty in temporally or spatially correlated objective variables.
