Multivariate Skew-Normal Generalized Hyperbolic distribution and its properties

Investigating surrogate safety measures at midblock pedestrian crossings using multivariate models with roadside camera data

This study aims to evaluate and compare Surrogate Safety Measures (SSMs) at five midblock Rectangular Rapid Flashing Beacons (RRFB) and two midblock Pedestrian Hybrid Beacons (PHB) sites in Florida using extensive video data collected over the study period of July to November 2021. Computer vision and data processing resulted in four pedestrian SSMs, namely spatial gap, temporal gap, relative time to collision (RTTC) and Post Encroachment Time (PET). An initial investigation of the SSMs using Mann-Whitney-Wilcoxon tests revealed significant differences in the SSM values across different treatment types and hours of the day. Additionally, univariate regression of spatial gap, and multivariate regression of temporal gap, RTTC and PET revealed significant differences of SSMs across RRFB and PHB sites. The study considered both linear and non-linear (gamma, inverse Gaussian and lognormal) regression models. After considering various traffic and operational parameters, the data were aggregated for each pedestrian-vehicle interaction on each lane to create a total of 395 observations. The SSMs included average spatial gap, temporal gap, RTTC and PET for each interaction of pedestrian and vehicle on each lane. The results indicated that non-linear models performed better than the linear models. Moreover, the presence of the PHB, weekday, signal activation, lane count, pedestrian speed, vehicle speed, land use mix, morning period and pedestrian starting position from the sidewalk have been found to be significant determinants of the SSMs. Results also suggest temporal SSMs increase at the PHB sites compared to the RRFB sites, indicating an improvement of traffic safety at PHB sites. However, the spatial gap decreased for PHB sites compared to the RRFB sites, which suggests that pedestrians tend to start to cross the RRFB sites when they perceive vehicles to be further away than at the PHB sites.

Skewness-Based Projection Pursuit as an Eigenvector Problem in Scale Mixtures of Skew-Normal Distributions

This paper addresses the projection pursuit problem assuming that the distribution of the input vector belongs to the flexible and wide family of multivariate scale mixtures of skew normal distributions. Under this assumption, skewness-based projection pursuit is set out as an eigenvector problem, described in terms of the third order cumulant matrix, as well as an eigenvector problem that involves the simultaneous diagonalization of the scatter matrices of the model. Both approaches lead to dominant eigenvectors proportional to the shape parametric vector, which accounts for the multivariate asymmetry of the model; they also shed light on the parametric interpretability of the invariant coordinate selection method and point out some alternatives for estimating the projection pursuit direction. The theoretical findings are further investigated through a simulation study whose results provide insights about the usefulness of skewness model-based projection pursuit in the statistical practice.

On the scale mixtures of multivariate skew slash distributions

In this paper, the scale mixtures of multivariate skew slash distributions is introduced. The probability density function with some additional properties are discussed. The first four order moments, skewness and kurtosis of this distribution are calculated. Furthermore, the first two moments of its quadratic forms are obtained. In particular, the linear transformation, stochastic representation and hierarchical representation are studied. In the end, the EM algorithm is proposed.