47 research outputs found
A link between Kendall's tau, the length measure and the surface of bivariate copulas, and a consequence to copulas with self-similar support
Working with shuffles we establish a close link between Kendall's tau, the
so-called length measure, and the surface area of bivariate copulas and derive
some consequences. While it is well-known that Spearman's rho of a bivariate
copula A is a rescaled version of the volume of the area under the graph of A,
in this contribution we show that the other famous concordance measure,
Kendall's tau, allows for a simple geometric interpretation as well - it is
inextricably linked to the surface area of A.Comment: 12 pages, 3 figure
Baire category results for quasi–copulas
AbstractThe aim of this manuscript is to determine the relative size of several functions (copulas, quasi–
copulas) that are commonly used in stochastic modeling. It is shown that the class of all quasi–copulas that
are (locally) associated to a doubly stochastic signed measure is a set of first category in the class of all quasi–
copulas. Moreover, it is proved that copulas are nowhere dense in the class of quasi-copulas. The results are
obtained via a checkerboard approximation of quasi–copulas
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 (). We argue that 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
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.Comment: 17 pages, 8 figures, submitted to IJGI