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

The genetic architecture of the human cerebral cortex

The cerebral cortex underlies our complex cognitive capabilities, yet little is known about the specific genetic loci that influence human cortical structure. To identify genetic variants that affect cortical structure, we conducted a genome-wide association meta-analysis of brain magnetic resonance imaging data from 51,665 individuals. We analyzed the surface area and average thickness of the whole cortex and 34 regions with known functional specializations. We identified 199 significant loci and found significant enrichment for loci influencing total surface area within regulatory elements that are active during prenatal cortical development, supporting the radial unit hypothesis. Loci that affect regional surface area cluster near genes in Wnt signaling pathways, which influence progenitor expansion and areal identity. Variation in cortical structure is genetically correlated with cognitive function, Parkinson's disease, insomnia, depression, neuroticism, and attention deficit hyperactivity disorder

Study of Gaussian processes, Lévy processes and infinitely divisible distributions

Thesis (Ph.D.)--Boston UniversityPLEASE NOTE: Boston University Libraries did not receive an Authorization To Manage form for this thesis or dissertation. It is therefore not openly accessible, though it may be available by request. If you are the author or principal advisor of this work and would like to request open access for it, please contact us at [email protected]. Thank you.In this thesis, we study distribution functions and distributional-related quantities for various stochastic processes and probability distributions, including Gaussian processes, inverse Levy subordinators, Poisson stochastic integrals, non-negative infinitely divisible distributions and the Rosenblatt distribution. We obtain analytical results for each case, and in instances where no closed form exists for the distribution, we provide numerical solutions. We mainly use two methods to analyze such distributions. In some cases, we characterize distribution functions by viewing them as solutions to differential equations. These are used to obtain moments and distributions functions of the underlying random variables. In other cases, we obtain results using inversion of Laplace or Fourier transforms. These methods include the Post-Widder inversion formula for Laplace transforms, and Edgeworth approximations. In Chapter 1, we consider differential equations related to Gaussian processes. It is well known that the heat equation together with appropriate initial conditions characterize the marginal distribution of Brownian motion. We generalize this connection to finite dimensional distributions of arbitrary Gaussian processes. In Chapter 2, we study the inverses of Levy subordinators. These processes are non-Markovian and their finite-dimensional distributions are not known in closed form. We derive a differential equation related to these processes and use it to find an expression for joint moments. We compute numerically these joint moments in Chapter 3 and include several examples. Chapter 4 considers Poisson stochastic integrals. We show that the distribution function of these random variables satisfies a Kolmogorov-Feller equation, and we describe the regularity of solutions and numerically solve this equation. Chapter 5 presents a technique for computing the density function or distribution function of any non-negative infinitely divisible distribution based on the Post-Widder method. In Chapter 6, we consider a distribution given by an infinite sum of weighted gamma distributions. We derive the Levy-Khintchine representation and show when the tail of this sum is asymptotically normal. We derive a Berry-Essen bound and Edgeworth expansions for its distribution function. Finally, in Chapter 7 we look at the Rosenblatt distribution, which can be expressed as a infinite sum of weighted chi-squared distributions. We apply the expansions in Chapter 6 to compute its distribution function.2031-01-0

Distribution functions of Poisson random integrals: analysis and computation

We want to compute the cumulative distribution function of a one-dimensional Poisson stochastic integral ()=∫0()() , where N is a Poisson random measure with control measure n and g 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.This research was partially supported by the NSF grants DMS-0706786, and DGE-0221680. (DMS-0706786 - NSF; DGE-0221680 - NSF)First author draf