Archimedean copulas derived from Morgenstern utility functions

The (additive) generator of an Archimedean copula - as well as the inverse of the generator - is a strictly decreasing and convex function, while Morgenstern utility functions (applying to risk averse decision makers) are nondecreasing and concave. This provides a basis for deriving either a generator of Archimedean copulas, or its inverse, from a Morgenstern utility function. If we derive the generator in this way, dependence properties of an Archimedean copula that are often taken to be desirable, match with generally sought after properties of the corresponding utility function. It is shown how well known copula families are derived from established utility functions. Also, some new copula families are derived, and their properties are discussed. If, on the other hand, we instead derive the inverse of the generator from the utility function, there is a link between the magnitude of measures of risk attitude (like the very common Arrow-Pratt coefficient of absolute risk aversion) and the strength of dependence featured by the corresponding Archimedean copula

Geometric realizations and duality for Dahmen-Micchelli modules and De Concini-Procesi-Vergne modules

We give an algebraic description of several modules and algebras related to the vector partition function, and we prove that they can be realized as the equivariant K-theory of some manifolds that have a nice combinatorial description. We also propose a more natural and general notion of duality between these modules, which corresponds to a Poincar\'e duality-type correspondence for equivariant K-theory.Comment: Final version, to appear on Discrete and Computational Geometr

Analytical approximation to the multidimensional Fokker--Planck equation with steady state

The Fokker--Planck equation is a key ingredient of many models in physics, and related subjects, and arises in a diverse array of settings. Analytical solutions are limited to special cases, and resorting to numerical simulation is often the only route available; in high dimensions, or for parametric studies, this can become unwieldy. Using asymptotic techniques, that draw upon the known Ornstein--Uhlenbeck (OU) case, we consider a mean-reverting system and obtain its representation as a product of terms, representing short-term, long-term, and medium-term behaviour. A further reduction yields a simple explicit formula, both intuitive in terms of its physical origin and fast to evaluate. We illustrate a breadth of cases, some of which are `far' from the OU model, such as double-well potentials, and even then, perhaps surprisingly, the approximation still gives very good results when compared with numerical simulations. Both one- and two-dimensional examples are considered.Comment: Updated version as publishe

From Quantum B\"acklund Transforms to Topological Quantum Field Theory

We derive the quantum analogue of a B\"acklund transformation for the quantised Ablowitz-Ladik chain, a space discretisation of the nonlinear Schr\"odinger equation. The quantisation of the Ablowitz-Ladik chain leads to the $q$-boson model. Using a previous construction of Baxter's Q-operator for this model by the author, a set of functional relations is obtained which matches the relations of a one-variable classical B\"acklund transform to all orders in $\hbar$. We construct also a second Q-operator and show that it is closely related to the inverse of the first. The multi-B\"acklund transforms generated from the Q-operator define the fusion matrices of a 2D TQFT and we derive a linear system for the solution to the quantum B\"acklund relations in terms of the TQFT fusion coefficients.Comment: 29 pages,4 figures (v3: published version

Data-adaptive harmonic spectra and multilayer Stuart-Landau models

Harmonic decompositions of multivariate time series are considered for which we adopt an integral operator approach with periodic semigroup kernels. Spectral decomposition theorems are derived that cover the important cases of two-time statistics drawn from a mixing invariant measure. The corresponding eigenvalues can be grouped per Fourier frequency, and are actually given, at each frequency, as the singular values of a cross-spectral matrix depending on the data. These eigenvalues obey furthermore a variational principle that allows us to define naturally a multidimensional power spectrum. The eigenmodes, as far as they are concerned, exhibit a data-adaptive character manifested in their phase which allows us in turn to define a multidimensional phase spectrum. The resulting data-adaptive harmonic (DAH) modes allow for reducing the data-driven modeling effort to elemental models stacked per frequency, only coupled at different frequencies by the same noise realization. In particular, the DAH decomposition extracts time-dependent coefficients stacked by Fourier frequency which can be efficiently modeled---provided the decay of temporal correlations is sufficiently well-resolved---within a class of multilayer stochastic models (MSMs) tailored here on stochastic Stuart-Landau oscillators. Applications to the Lorenz 96 model and to a stochastic heat equation driven by a space-time white noise, are considered. In both cases, the DAH decomposition allows for an extraction of spatio-temporal modes revealing key features of the dynamics in the embedded phase space. The multilayer Stuart-Landau models (MSLMs) are shown to successfully model the typical patterns of the corresponding time-evolving fields, as well as their statistics of occurrence.Comment: 26 pages, double columns; 15 figure

Improved method for SNR prediction in machine-learning-based test

This paper applies an improved method for testing the signal-to-noise ratio (SNR) of Analogue-to-Digital Converters (ADC). In previous work, a noisy and nonlinear pulse signal is exploited as the input stimulus to obtain the signature results of ADC. By applying a machine-learning-based approach, the dynamic parameters can be predicted by using the signature results. However, it can only estimate the SNR accurately within a certain range. In order to overcome this limitation, an improved method based on work is applied in this work. It is validated on the Labview model of a 12-bit 80 Ms/s pipelined ADC with a pulse- wave input signal of 3 LSB noise and 7-bit nonlinear rising and falling edges

Functional Poisson approximation in Kantorovich-Rubinstein distance with applications to U-statistics and stochastic geometry

A Poisson or a binomial process on an abstract state space and a symmetric function $f$ acting on $k$-tuples of its points are considered. They induce a point process on the target space of $f$. The main result is a functional limit theorem which provides an upper bound for an optimal transportation distance between the image process and a Poisson process on the target space. The technical background are a version of Stein's method for Poisson process approximation, a Glauber dynamics representation for the Poisson process and the Malliavin formalism. As applications of the main result, error bounds for approximations of U-statistics by Poisson, compound Poisson and stable random variables are derived, and examples from stochastic geometry are investigated.Comment: Published at http://dx.doi.org/10.1214/15-AOP1020 in the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org

Simplified Pair Copula Constructions --- Limits and Extensions

So called pair copula constructions (PCCs), specifying multivariate distributions only in terms of bivariate building blocks (pair copulas), constitute a flexible class of dependence models. To keep them tractable for inference and model selection, the simplifying assumption that copulas of conditional distributions do not depend on the values of the variables which they are conditioned on is popular. In this paper, we show for which classes of distributions such a simplification is applicable, significantly extending the discussion of Hob{\ae}k Haff et al. (2010). In particular, we show that the only Archimedean copula in dimension d \geq 4 which is of the simplified type is that based on the gamma Laplace transform or its extension, while the Student-t copula is the only one arising from a scale mixture of Normals. Further, we illustrate how PCCs can be adapted for situations where conditional copulas depend on values which are conditioned on

redicting dynamic specifications of ADCs with a low-quality digital input signal

A new method is presented to test dynamic parameters of Analogue-to-Digital Converters (ADC). A noisy and nonlinear pulse is applied as the test stimulus, which is suitable for a multi-site test environment. The dynamic parameters are predicted using a machine-learning-based approach. A training step is required in order to build the mapping function using alternate signatures and the conventional test parameters, all measured on a set of converters. As a result, for industrial testing, only a simple signature-based test is performed on the Devices-Under-Test (DUTs). The signature measurements are provided to the mapping function that is used to predict the conventional dynamic parameters. The method is validated by simulation on a 12-bit 80 Ms/s pipelined ADC with a pulse wave input signal of 3 LSB noise and 7-bit nonlinear rising and falling edges. The final results show that the estimated mean error is less than 4% of the full range of the dynamic specifications