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Introduction
Published in Christophe Ley, Thomas Verdebout, Modern Directional Statistics, 2017
Christophe Ley, Thomas Verdebout
Directional statistics is a branch of statistics dealing with observations that are directions. In most cases, these observations lie on the circumference of the unit circle of R $ {\mathbb{R}} $ 2 (one then speaks of circular statistics) or on the surface of the unit hypersphere of R $ {\mathbb{R}} $ p for p ≥ 3 (implying the terminology of spherical statistics)1. Data of this type typically arise in meteorology (wind directions), astronomy (directions of cosmic rays or stars), earth sciences (location of an earthquake’s epicentre on the surface of the earth) and biology (circadian rhythms, studies of animal navigation), to cite but these. The key difficulty when dealing with such data is the curvature of the sample space since the unit hypersphere or circle is a non-linear manifold. This can readily be seen on a very simple example. Imagine two points on the sphere of R $ {\mathbb{R}} $ 3 and consider their average. This point will in general not lie on the sphere. This reasoning of course extends to several points on the sphere, entailing that the basic concept of sample mean needs to be adapted in order to yield a true mean direction on the sphere. Thus, the wheel has to be reinvented for virtually every classical concept from multivariate statistics.
The importance of riparian plant orientation in river flow: implications for flow structures and drag
Published in Journal of Ecohydraulics, 2018
Richard J. Boothroyd, Richard J. Hardy, Jeff Warburton, Timothy I. Marjoribanks
Directional statistics quantify directional differences between the form drag force and drag coefficient for the defoliated and foliated plants over the entire range of plant orientations assessed. A key difference is shown, with the mean direction when defoliated (form drag force 71°; drag coefficient 50°) different than when foliated (form drag force 318°, drag coefficient 327°). Furthermore, Figure 11 shows clear axes of maximum and minimum symmetry for the foliated drag response, with the defoliated drag response characterized by a small number of discrete spikes. These differences are quantified through the circular variance, angular deviation, circular skewness, and circular kurtosis (Table 2). If drag response values were spread evenly around the circle, the circular variance would equal unity. For both the form drag force and drag coefficient, circular variance values are close to unity, however the value is consistently lower for the defoliated plant. The angular deviation is analogous to the linear standard deviation but bounded between 0 and (Berens 2009). Angular deviation values for the defoliated plant are slightly lower than for the foliated plant, although both fall towards the upper bound. Circular skewness values for the foliated plant are slightly further away from 0, indicating less symmetry in the drag response around the mean direction for the foliated plant than for the defoliated plant. Finally, the similar values of circular kurtosis indicate that neither dataset are strongly peaked. Descriptive and directional statistics show differences in the drag response with changes in plant orientation, most notably in the different directionality in the drag response between the defoliated and foliated plants.
Tomographic scanning of rock joint roughness
Published in Applied Earth Science, 2022
Next we determine overall joint plane orientation via the orientational tensor method of directional statistics (Woodcock 1977). All the triangle normals from the joint portion of the mesh are gathered into a second-rank tensor whose principal eigenvector is identified as the normal of the effective overall joint plane. (NB the indices i,j=x,y,z denote Cartesian coordinate space.)
Understanding flood regime changes of the Mahanadi River
Published in ISH Journal of Hydraulic Engineering, 2023
Poulomi Ganguli, Yamini Rama Nandamuri, Chandranath Chatterjee
We determine the persistence in peak discharge timing using directional statistics. The mean flood date, temporal variability and the persistence in flood timing are evaluated for both methods of flood samplings at respective stream gauges (Figure 6). For the majority of sites, the mean flood dates are temporally concentrated around the mid of August for both MMF and PTF series (Figure 6a,b) except for Andhiarkore and Naraj, which showed the mean flood dates are close to September, i.e. 29th August and 27th August for the MMF events (Figure 6c). While Andhiarkore is located at Region I, Naraj is situated in the delta region. Further, we observe that considering both methods of flood samples, the PTF series shows a stronger tendency in temporal clustering of mean flood dates around mid of August (Figure 6b,c) and higher circular variance than that of the MMF series (Figure 6d). A larger circular variance implies wide dispersion of the streamflow record. A large value of circular variance among PTF samples for most of the sites indicates a wide dispersion than the MMF sample, which is not captured well solely by employing a single measure of dispersion, i.e. persistence in flood timing due to a large number of overlapping values of the neighbouring gauges (Figures 6a,b). Overall, we infer that the timing of floods is highly persistent across MRB. Our results agree with an earlier assessment (Burn and Whitfield 2018), which has shown that the pluvial discharge regime typically shows a larger persistence in peak discharge timing than other flow regimes. The southwest monsoon season spanning from June to September is the flood-rich month for the rain-fed river basins of India. The average monsoon onset days over the basin are slightly earlier than the 15th of June (Raymond et al. 2020b), while the peak months of rainfall are July and August. In particular, the temporal clustering of flood timing in August suggests the unimodal nature of flood seasonality. This implies, even if the onset of monsoon over the basin is slightly earlier than the middle of June, considering ranges of catchment characteristics (Berghuijs et al. 2016; Gaál et al. 2012; Haga et al. 2005), the peak runoff responses are apparent after a few days’ of time lags. Typically, the spatial and temporal distribution of runoff, and the storm duration of flood-producing rainfall are dictated by catchment response time (Gericke and Smithers 2014). The prevalence of fluvial floods during monsoon season in rain-dominated basins of peninsular India (Jena et al. 2014), compounded by heavy rainfall, indicates extreme rainfall is a dominant driver for the flood.