arXiv subjects

Wolfgang Trutschnig

Publications and source records attributed to Wolfgang Trutschnig.

24 records · Page 2Linked to original sources

arXiv2020

On weak conditional convergence of bivariate Archimedean and Extreme Value copulas, and consequences to nonparametric estimation

Looking at bivariate copulas from the perspective of conditional distributions and considering weak convergence of almost all conditional distributions yields the notion of weak conditional convergence. At first glance, this notion of convergence for copulas might seem far too restrictive to be of any practical importance - in fact, given samples of a copula $C$ the corresponding empirical copulas do not converge weakly conditional to $C$ with probability one in general. Within the class of Archimedean copulas and the class of Extreme Value copulas, however, standard pointwise convergence and weak conditional convergence can even be proved to be equivalent. Moreover, it can be shown that every copula $C$ is the weak conditional limit of a sequence of checkerboard copulas. After proving these three main results and pointing out some consequences we sketch some implications for two recently introduced dependence measures and for the nonparametric estimation of Archimedean and Extreme Value copulas.

math.ST↗

arXiv2019

A copula-based measure for quantifying asymmetry in dependence and associations

Asymmetry is an inherent property of bivariate associations and therefore must not be ignored. The currently applicable dependence measures mask the potential asymmetry of the underlying dependence structure by implicitly assuming that quantity Y is equally dependent on quantity X, and vice versa, which is generally not true. We introduce the copula-based dependence measure qad that quantifies asymmetry. Specifically, qad is applicable in general situations, is sensitive to noise in data, detects asymmetry in dependence and reliably quantifies the information gain/predictability of quantity Y given knowledge of quantity X, and vice versa. Using real-world data sets, we demonstrate the relevance of asymmetry in associations. Asymmetry in dependence is a novel category of information that provides substantial information gain in analyses of bivariate associations.

stat.AP↗

arXiv2018

A sharp inequality for Kendall's $τ$ and Spearman's $ρ$ of Extreme-Value Copulas

We derive a new (lower) inequality between Kendall's tau? and Spearman's rho? for two-dimensional Extreme-Value Copulas, show that this inequality is sharp in each point and conclude that the comonotonic and the product copula are the only Extreme-Value Copulas for which the well-known lower Hutchinson-Lai inequality is sharp.

math.ST↗

arXiv2018

On all Pickands Dependence Functions whose corresponding Extreme-Value-Copulas have Spearman $ρ$ (Kendall $τ$) identical to some value $v \in [0,1]$

We answer an open question posed by the second author at the Salzburg workshop on Dependence Models and Copulas in 2016 concerning the size of the family $\mathcal{A}^ρ_v$ ($\mathcal{A}^τ_v$) of all Pickands dependence functions $A$ whose corresponding Extreme-Value-Copulas have Spearman $ρ$ (Kendall $τ$) equal to some arbitrary, fixed value $v \in [0,1]$. After determining compact sets $Ω^ρ_v, Ω^τ_v \subseteq [0,1] \times [\frac{1}{2},1]$ containing the graphs of all Pickands dependence functions from the classes $\mathcal{A}^ρ_v$ and $\mathcal{A}^τ_v$ respectively, we then show that both sets are best possible.

math.ST↗

arXiv2017

On the exact region determined by Kendall's tau and Spearman's rho

Using properties of shuffles of copulas and tools from combinatorics we solve the open question about the exact region $Ω$ determined by all possible values of Kendall's $τ$ and Spearman's $ρ$. In particular, we prove that the well-known inequality established by Durbin and Stuart in 1951 is only sharp on a countable set with sole accumulation point $(-1,-1)$, give a simple analytic characterization of $Ω$ in terms of a continuous, strictly increasing piecewise concave function, and show that $Ω$ is compact and simply connected but not convex. The results also show that for each $(x,y)\in Ω$ there are mutually completely dependent random variables whose $τ$ and $ρ$ values coincide with $x$ and $y$ respectively.

math.ST↗

arXiv2015

Why GPS makes distances bigger than they are

Global Navigation Satellite Systems (GNSS), such as the Global Positioning System (GPS), are among the most important sensors for movement analysis. GPS is widely used to record the trajectories of vehicles, animals and human beings. However, all GPS movement data are affected by both measurement and interpolation error. In this article we show that measurement error causes a systematic bias in distances recorded with a GPS: the distance between two points recorded with a GPS is -- on average -- bigger than the true distance between these points. This systematic `overestimation of distance' becomes relevant if the influence of interpolation error can be neglected, which is the case for movement sampled at high frequencies. We provide a mathematical explanation of this phenomenon and we illustrate that it functionally depends on the autocorrelation of GPS measurement error ($C$). We argue that $C$ can be interpreted as a quality measure for movement data recorded with a GPS. If there is strong autocorrelation any two consecutive position estimates have very similar error. This error cancels out when average speed, distance or direction are calculated along the trajectory. Based on our theoretical findings we introduce a novel approach to determine $C$ in real-world GPS movement data sampled at high frequencies. We apply our approach to a set of pedestrian and a set of car trajectories. We find that the measurement error in the data is strongly spatially and temporally autocorrelated and give a quality estimate of the data. Finally, we want to emphasize that all our findings are not limited to GPS alone. The systematic bias and all its implications are bound to occur in any movement data collected with absolute positioning if interpolation error can be neglected.

physics.ins-det↗