Properties and numerical evaluation of the Rosenblatt distribution

This paper studies various distributional properties of the Rosenblatt distribution. We begin by describing a technique for computing the cumulants. We then study the expansion of the Rosenblatt distribution in terms of shifted chi-squared distributions. We derive the coefficients of this expansion and use these to obtain the L\'{e}vy-Khintchine formula and derive asymptotic properties of the L\'{e}vy measure. This allows us to compute the cumulants, moments, coefficients in the chi-square expansion and the density and cumulative distribution functions of the Rosenblatt distribution with a high degree of precision. Tables are provided and software written to implement the methods described here is freely available by request from the authors.Comment: Published in at http://dx.doi.org/10.3150/12-BEJ421 the Bernoulli (http://isi.cbs.nl/bernoulli/) by the International Statistical Institute/Bernoulli Society (http://isi.cbs.nl/BS/bshome.htm

Distribution functions of Poisson random integrals: Analysis and computation

We want to compute the cumulative distribution function of a one-dimensional Poisson stochastic integral I(\krnl) = \displaystyle \int_0^T \krnl(s) N(ds), where $N$ is a Poisson random measure with control measure $n$ and \krnl is a suitable kernel function. We do so by combining a Kolmogorov-Feller equation with a finite-difference scheme. We provide the rate of convergence of our numerical scheme and illustrate our method on a number of examples. The software used to implement the procedure is available on demand and we demonstrate its use in the paper.Comment: 28 pages, 8 figure

Numerical Computation of First-Passage Times of Increasing Levy Processes

Let $\{D(s), s \geq 0\}$ be a non-decreasing L\'evy process. The first-hitting time process $\{E(t) t \geq 0\}$ (which is sometimes referred to as an inverse subordinator) defined by $E(t) = \inf \{s: D(s) > t \}$ is a process which has arisen in many applications. Of particular interest is the mean first-hitting time $U(t)=\mathbb{E}E(t)$. This function characterizes all finite-dimensional distributions of the process $E$. The function $U$ can be calculated by inverting the Laplace transform of the function $\widetilde{U}(\lambda) = (\lambda \phi(\lambda))^{-1}$, where $\phi$ is the L\'evy exponent of the subordinator $D$. In this paper, we give two methods for computing numerically the inverse of this Laplace transform. The first is based on the Bromwich integral and the second is based on the Post-Widder inversion formula. The software written to support this work is available from the authors and we illustrate its use at the end of the paper.Comment: 31 Pages, 7 sections, 11 figures, 2 table

The Error is the Feature: how to Forecast Lightning using a Model Prediction Error

Despite the progress within the last decades, weather forecasting is still a challenging and computationally expensive task. Current satellite-based approaches to predict thunderstorms are usually based on the analysis of the observed brightness temperatures in different spectral channels and emit a warning if a critical threshold is reached. Recent progress in data science however demonstrates that machine learning can be successfully applied to many research fields in science, especially in areas dealing with large datasets. We therefore present a new approach to the problem of predicting thunderstorms based on machine learning. The core idea of our work is to use the error of two-dimensional optical flow algorithms applied to images of meteorological satellites as a feature for machine learning models. We interpret that optical flow error as an indication of convection potentially leading to thunderstorms and lightning. To factor in spatial proximity we use various manual convolution steps. We also consider effects such as the time of day or the geographic location. We train different tree classifier models as well as a neural network to predict lightning within the next few hours (called nowcasting in meteorology) based on these features. In our evaluation section we compare the predictive power of the different models and the impact of different features on the classification result. Our results show a high accuracy of 96% for predictions over the next 15 minutes which slightly decreases with increasing forecast period but still remains above 83% for forecasts of up to five hours. The high false positive rate of nearly 6% however needs further investigation to allow for an operational use of our approach.Comment: 10 pages, 7 figure

U.S. Foreign Assistance to Latin America and the Caribbean

This report concerns the trends in U.S. assistance to the Latin America and Caribbean region generally reflect the trends and rationales for U.S. foreign aid programs globally. Aid to the region increased during the 1960's with the Alliance for Progress and during the 1980s with aid to Central America. Since 2000, aid levels have increased, especially in the Andean region as the focus has shifted from Cold War issues to counternarcotics and security assistance. Current aid levels to Latin America and the Caribbean comprise about 11.8% of the worldwide FY2006 bilateral aid budget. Amounts requested for FY2007 would reduce this ratio to 10.6%