Kurtosis ordering of the generalized secant hyperbolic distribution: a technical note

Two major generalizations of the hyperbolic secant distribution have been proposed in the statistical literature which both introduce an additional parameter that governs the kurtosis of the generalized distribution. The generalized hyperbolic secant (GHS) distribution was introduced by Harkness and Harkness (1968) who considered the p-th convolution of a hyperbolic secant distribution. Another generalization, the so-called generalized secant hyperbolic (GSH) distribution was recently suggested by Vaughan 2002). In contrast to the GHS distribution, the cumulative and inverse cumulative distribution function of the GSH distribution are available in closedform expressions. We use this property to proof that the additional shape parameter of the GSH distribution is actually a kurtosis parameter in the sense of van Zwet (1964). --kurtosis ordering,hyperbolic secant distribution

Some results on weak and strong tail dependence coefficients for means of copulas

Copulas represent the dependence structure of multivariate distributions in a natural way. In order to generate new copulas from given ones, several proposals found its way into statistical literature. One simple approach is to consider convex-combinations (i.e. weighted arithmetic means) of two or more copulas. Similarly, one might consider weighted geometric means. Consider, for instance, the Spearman copula, defined as the geometric mean of the maximum and the independence copula. In general, it is not known whether weighted geometric means of copulas produce copulas, again. However, applying a recent result of Liebscher (2006), we show that every weighted geometric mean of extreme-value copulas produces again an extreme-value copula. The second contribution of this paper is to calculate extremal dependence measures (e.g. weak and strong tail dependence coe±cients) for (weighted) geometric and arithmetic means of two copulas. --Tail Dependence,Extreme-value copulas,arithmetic and geometric mean

Kurtosis transformation and kurtosis ordering

Leptokurtic distributions can be generated by applying certain non-linear transformations to a standard normal random variable. Within this work we derive general conditions for these transformations which guarantee that the generated distributions are ordered with respect to the partial ordering of van Zwet for symmetric distributions and the partial ordering of MacGillivray for arbitrary distributions. In addition, we propose a general power transformation which nests the H-, J- and K-transformations which have already been proposed in the literature. Within this class of power transformations the above mentioned condition can be easily verified and the power can be interpreted as parameter of leptokurtosis. --Power kurtosis transformation,leptokurtosis,kurtosis ordering

Constructing symmetric generalized FGM copulas by means of certain univariate distributions

In this paper we focus on symmetric generalized Fairlie-Gumbel-Morgenstern (or symmetric Sarmanov) copulas which are characterized by means of so-called generator functions. In particular, we introduce a class of generator functions which is based on univariate distributions with certain properties. Some of the generator functions from the literature are recovered. Moreover two new generators are suggested, implying two new copulas. Finally, the opposite way around, it is exemplarily shown how to calculate the univariate distribution which belongs to a given copula generator function. --

Skewness by splitting the scale parameter

There are several possibilities to introduce skewness into a symmetric distribution. One of these procedures applies two dfferent parameters of scale - with possibly different weights - to the positive and the negative part of a symmetric density. Within this work we show that this technique incorporates a well-defined parameter of skewness, i.e. that the generated distributions are skewed to the right (left) if the parameter of skewness takes values less (greater) than one. Secondly, we prove that the skewness parameter is compatible with the skewness ordering of van Zwet (1964) which is the strongest ordering in the hierarchy of orderings discussed by Oja (1981). Hence, the generated (skewed) distributions can be ordered by the skewness parameter. --Skewness,skewness to the right,skewness ordering,score function.

Kurtosis modelling by means of the J-transformation

The H-family of distributions or H-distributions, introduced by Tukey (1960, 1977), are generated by a single transformation of the standard normal distribution and allow for leptokurtosis represented by the parameter h. Alternatively, Haynes, MacGillivray and Mengersen (1997) generated leptokurtic distributions by applying the K-transformation to the normal distribution. In this study we propose a third transformation - the so-called J-transformation - and derive some properties of this transformation. Moreover, so-called elongation generating functions (EGF’s) are introduced. By means of EGF's we are able to visualize the strength of tail elongation and to construct new transformations. Finally, we compare the three transformations towards their goodness-of-fit in the context of financial return data. --kurtosis,variable transformation,normal transformation,tail elongation.

Photochemistry of Hydrocarbon Radicals

Hydrocarbon radicals are ideal model systems to understand the dynamics of chemical reactions. In addition they are of considerable importance in high-energy environments, such as combustion engines or hydrocarbon crackers. In order to understand their reactions, we deposit a known amount of energy in the radicals by laser excitation and observe the subsequent dynamics by two experimental methods, a) picosecond time-resolved photoelectron spectroscopy and b) nanosecond photoionization of hydrogen atoms, the major reaction product in many radical reactions.Recent results for three radicals are summarized and discussed: allyl (C3H5), propargyl (C3H3), and ethyl (C2H5). We report the major reaction products, the pathways leading to these products, their rates of formation, and translational energy releases

Zero-lag long-range synchronization of Hodgkin-Huxley neurons is enhanced by dynamical relaying : poster presentation

Background The synchrony hypothesis postulates that precise temporal synchronization of different pools of neurons conveys information that is not contained in their firing rates. The synchrony hypothesis had been supported by experimental findings demonstrating that millisecond precise synchrony of neuronal oscillations across well separated brain regions plays an essential role in visual perception and other higher cognitive tasks [1]. Albeit, more evidence is being accumulated in favour of its role as a binding mechanism of distributed neural responses, the physical and anatomical substrate for such a dynamic and precise synchrony, especially zero-lag even in the presence of non-negligible delays, remains unclear. Here we propose a simple network motif that naturally accounts for zero-lag synchronization for a wide range of temporal delays [3]. We demonstrate that zero-lag synchronization between two distant neurons or neural populations can be achieved by relaying the dynamics via a third mediating single neuron or population. Methods We simulated the dynamics of two Hodgkin-Huxley neurons that interact with each other via an intermediate third neuron. The synaptic coupling was mediated through alpha-functions. Individual temporal delays of the arrival of pre-synaptic potentials were modelled by a gamma distribution. The strength of the synchronization and the phase-difference between each individual pairs were derived by cross-correlation of the membrane potentials. Results In the regular spiking regime the two outer neurons consistently synchronize with zero phase lag irrespective of the initial conditions. This robust zero-lag synchronization naturally arises as a consequence of the relay and redistribution of the dynamics performed by the central neuron. This result is independent on whether the coupling is excitatory or inhibitory and can be maintained for arbitrarily long time delays (see Fig. 1). Conclusion We have presented a simple and extremely robust network motif able to account for the isochronous synchronization of distant neural elements in a natural way. As opposed to other possible mechanisms of neural synchronization, neither inhibitory coupling, gap junctions nor precise tuning of morphological parameters are required to obtain zero-lag synchronized neuronal oscillation

Photonic Delay Systems as Machine Learning Implementations

Nonlinear photonic delay systems present interesting implementation platforms for machine learning models. They can be extremely fast, offer great degrees of parallelism and potentially consume far less power than digital processors. So far they have been successfully employed for signal processing using the Reservoir Computing paradigm. In this paper we show that their range of applicability can be greatly extended if we use gradient descent with backpropagation through time on a model of the system to optimize the input encoding of such systems. We perform physical experiments that demonstrate that the obtained input encodings work well in reality, and we show that optimized systems perform significantly better than the common Reservoir Computing approach. The results presented here demonstrate that common gradient descent techniques from machine learning may well be applicable on physical neuro-inspired analog computers

Was ist eigentlich aus der neuen Lernkultur geworden? Ein Blick auf Instruktionsansätze mit Potenzial zur Veränderung kulturell geteilter Lehr- und Lernskripts

In den letzten Jahren wurde in der Bildungsdiskussion erneut der Ruf nach einer neuen Lernkultur laut. In der neuen Lernkultur soll das Lernen aktiv, konstruktiv, selbstbestimmt, motiviert, sozial und flexibel von Medien unterstützt sein. In diesem Artikel werden fünf Instruktionsansätze, die im Laufe der letzten Jahre neu oder weiter entwickelt wurden, dahingehend analysiert, inwieweit bzw. wie sie zentrale Aspekte der neuen Lernkultur theoretisch elaborieren und damit für empirische Untersuchungen fruchtbar machen. Ein wichtiges Fazit ist, dass die Ansätze durch die vorgenommenen theoretischen Ausdifferenzierungen zumindest in einzelnen Bereichen Potenzial zur Stimulation empirischer Forschung haben und Beiträge zur Etablierung einer neuen Lernkultur im Sinne einer Veränderung kulturell geteilter Lehr- und Lernskripts leisten können. (DIPF/Orig.)Over the last years demands for the development of a new culture of learning were raised in the debate on education. According to that approach, learning was to be active, constructive, self-determined, motivated, social and flexibly supported by media. The authors focus on five instructional approaches which were conceived and developed in recent years, and analyze in how far and in what way crucial aspects of the new culture of learning are elaborated theoretically, thus providing a fruitful basis for empirical studies. The major results show that, due to their theoretical differentiations, these approaches have, at least in parts, the potential to stimulate empirical research and that they may contribute to the establishment of a new culture of learning aiming at culturally shared teaching and learning scripts. (DIPF/Orig.