scholarly journals Quantile mechanics II: changes of variables in Monte Carlo methods and GPU-optimised normal quantiles

2014 ◽  
Vol 25 (2) ◽  
pp. 177-212 ◽  
Author(s):  
WILLIAM T. SHAW ◽  
THOMAS LUU ◽  
NICK BRICKMAN
Keyword(s):  
Monte Carlo ◽  
Monte Carlo Methods ◽  
Distribution Functions ◽  
Cumulative Distribution ◽  
Processing Unit ◽  
Model Risk ◽  
Change Of Variables ◽  
Special Cases ◽  
Wide Range ◽  
Branch Divergence

With financial modelling requiring a better understanding of model risk, it is helpful to be able to vary assumptions about underlying probability distributions in an efficient manner, preferably without the noise induced by resampling distributions managed by Monte Carlo methods. This paper presents differential equations and solution methods for the functions of the form Q(x) = F−1(G(x)), where F and G are cumulative distribution functions. Such functions allow the direct recycling of Monte Carlo samples from one distribution into samples from another. The method may be developed analytically for certain special cases, and illuminate the idea that it is a more precise form of the traditional Cornish–Fisher expansion. In this manner the model risk of distributional risk may be assessed free of the Monte Carlo noise associated with resampling. The method may also be regarded as providing both analytical and numerical bases for doing more precise Cornish–Fisher transformations. Examples are given of equations for converting normal samples to Student t, and converting exponential to normal. In the case of the normal distribution, the change of variables employed allows the sampling to take place to good accuracy based on a single rational approximation over a very wide range of sample space. The avoidance of branching statements is of use in optimal graphics processing unit (GPU) computations as it avoids the effect of branch divergence. We give a branch-free normal quantile that offers performance improvements in a GPU environment while retaining the best precision characteristics of well-known methods. We also offer models with low probability branch divergence. Comparisons of new and existing forms are made on Nvidia GeForce GTX Titan and Tesla C2050 GPUs. We argue that in both single- and double-precisions, the change-of-variables approach offers the most GPU-optimal Gaussian quantile yet, working faster than the Cuda 5.5 built-in function.

scholarly journals Monte Carlo Calculations of the Radial Distribution Functions for a Proton?Electron Plasma

1965 ◽  
Vol 18 (2) ◽  
pp. 119 ◽  
Author(s):  
AA Barker
Keyword(s):  
Monte Carlo ◽  
Crystal Lattice ◽  
Radial Distribution ◽  
Lattice Theory ◽  
Electron Plasma ◽  
Distribution Functions ◽  
Radial Distribution Functions ◽  
Monte Carlo Calculations ◽  
Wide Range ◽  
General Method

A general method is presented for computation of radial distribution functions for plasmas over a wide range of temperatures and densities. The method uses the Monte Carlo technique applied by Wood and Parker, and extends this to long-range forces using results borrowed from crystal lattice theory. The approach is then used to calculate the radial distribution functions for a proton-electron plasma of density 1018 electrons/cm3 at a temperature of 104 OK. The results show the usefulness of the method if sufficient computing facilities are available.

scholarly journals Hygroscopicity distribution concept for measurement data analysis and modeling of aerosol particle mixing state with regard to hygroscopic growth and CCN activation

2010 ◽  
Vol 10 (15) ◽  
pp. 7489-7503 ◽  
Author(s):  
H. Su ◽  
D. Rose ◽  
Y. F. Cheng ◽  
S. S. Gunthe ◽  
A. Massling ◽  
...  
Keyword(s):  
Chemical Composition ◽  
Measurement Data ◽  
Particle Diameter ◽  
Distribution Functions ◽  
Cumulative Distribution ◽  
Cloud Condensation Nucleus ◽  
Mathematical Framework ◽  
Hygroscopic Growth ◽  
Mixing State ◽  
Wide Range

Abstract. This paper presents a general concept and mathematical framework of particle hygroscopicity distribution for the analysis and modeling of aerosol hygroscopic growth and cloud condensation nucleus (CCN) activity. The cumulative distribution function of particle hygroscopicity, H(κ, Dd) is defined as the number fraction of particles with a given dry diameter, Dd, and with an effective hygroscopicity parameter smaller than the parameter κ. From hygroscopicity tandem differential mobility analyzer (HTDMA) and size-resolved CCN measurement data, H(κ, Dd) can be derived by solving the κ-Köhler model equation. Alternatively, H(κ, Dd) can be predicted from measurement or model data resolving the chemical composition of single particles. A range of model scenarios are used to explain and illustrate the concept, and exemplary practical applications are shown with HTDMA and CCN measurement data from polluted megacity and pristine rainforest air. Lognormal distribution functions are found to be suitable for approximately describing the hygroscopicity distributions of the investigated atmospheric aerosol samples. For detailed characterization of aerosol hygroscopicity distributions, including externally mixed particles of low hygroscopicity such as freshly emitted soot, we suggest that size-resolved CCN measurements with a wide range and high resolution of water vapor supersaturation and dry particle diameter should be combined with comprehensive HTDMA measurements and size-resolved or single-particle measurements of aerosol chemical composition, including refractory components. In field and laboratory experiments, hygroscopicity distribution data from HTDMA and CCN measurements can complement mixing state information from optical, chemical and volatility-based techniques. Moreover, we propose and intend to use hygroscopicity distribution functions in model studies investigating the influence of aerosol mixing state on the formation of cloud droplets.

scholarly journals Spatial interpolation of rain-field dynamic time-space evolution based on radar rainfall data

2020 ◽  
Vol 51 (3) ◽  
pp. 521-540
Author(s):  
Peng Liu ◽  
Yeou-Koung Tung
Keyword(s):  
Temporal Distribution ◽  
Radar Data ◽  
Indicator Kriging ◽  
Distribution Functions ◽  
Cumulative Distribution ◽  
Radar Rainfall ◽  
Time Space ◽  
Geostatistical Approach ◽  
Wide Range ◽  
Dynamic Time

Abstract Accurate and reliable measurement and prediction of the spatial and temporal distribution of rain field over a wide range of scales are important topics in hydrologic investigations. In this study, a geostatistical approach was adopted. To estimate the rainfall intensity over a study domain with the sample values and the spatial structure from the radar data, the cumulative distribution functions (CDFs) at all unsampled locations were estimated. Indicator kriging (IK) was used to estimate the exceedance probabilities for different preselected threshold levels, and a procedure was implemented for interpolating CDF values between the thresholds that were derived from the IK. Different probability distribution functions of the CDF were tested and their influences on the performance were also investigated. The performance measures and visual comparison between the observed rain field and the IK-based estimation suggested that the proposed method can provide good results of the estimation of indicator variables and is capable of producing a realistic image.

On intersection volumes of confidence hyper-ellipsoids and two geometric Monte Carlo methods

2021 ◽  
Vol 0 (0) ◽  
Author(s):  
Nima Rabiei ◽  
Elias G. Saleeby
Keyword(s):  
Monte Carlo ◽  
Monte Carlo Methods ◽  
Numerical Experiments ◽  
High Probability ◽  
Statistical Decision ◽  
Time Series Clustering ◽  
Sample Mean ◽  
Wide Range ◽  
Geometric Type ◽  
Statistical Decision Making

Abstract The intersection or the overlap region of two n-dimensional ellipsoids plays an important role in statistical decision making in a number of applications. For instance, the intersection volume of two n-dimensional ellipsoids has been employed to define dissimilarity measures in time series clustering (see [M. Bakoben, T. Bellotti and N. M. Adams, Improving clustering performance by incorporating uncertainty, Pattern Recognit. Lett. 77 2016, 28–34]). Formulas for the intersection volumes of two n-dimensional ellipsoids are not known. In this article, we first derive exact formulas to determine the intersection volume of two hyper-ellipsoids satisfying a certain condition. Then we adapt and extend two geometric type Monte Carlo methods that in principle allow us to compute the intersection volume of any two generalized convex hyper-ellipsoids. Using the exact formulas, we evaluate the performance of the two Monte Carlo methods. Our numerical experiments show that sufficiently accurate estimates can be obtained for a reasonably wide range of n, and that the sample-mean method is more efficient. Finally, we develop an elementary fast Monte Carlo method to determine, with high probability, if two n-ellipsoids are separated or overlap.

scholarly journals THE BIVARIATE EXTENSION OF AMOROSO DISTRIBUTION

2020 ◽  
Vol 4 (2) ◽  
pp. 261-283
Author(s):  
David Sam Jayakumar ◽  
A Sulthan ◽  
W Samuel
Keyword(s):  
Maximum Likelihood ◽  
Distribution Functions ◽  
Cumulative Distribution ◽  
Characteristic Functions ◽  
Estimation Of Parameters ◽  
Numerical Application ◽  
Hazard Functions ◽  
Maximum Likelihood Approach ◽  
Special Cases ◽  
Likelihood Approach

This paper introduces the bivariate extension of the amoroso distribution and its density function is expressed in terms of hyper-geometric function. The standard amoroso distribution, cumulative distribution functions, conditional distributions, and its moments are also derived. The Product moments, Co-variance, correlations, and Shannon’s differential entropy are also shown. Moreover, the generating functions such as moment, Cumulant, Characteristic functions are expressed in Fox-wright function, and the Survival, hazard, and Cumulative hazard functions are also computed. The special cases of the bivariate amoroso distribution are also discussed and nearly 780 bivariate mixtures of distributions can be derived. Finally, the two-dimensional probability surfaces are visualized for the selected special cases and we also showed the estimation of parameters by the method of maximum likelihood approach, and the constrained maximum likelihood approach is also computed by using Non-linear Programming with a numerical application

Monte Carlo methods

2018 ◽  
Author(s):  
Joseph F. Boudreau ◽  
Eric S. Swanson
Keyword(s):  
Monte Carlo ◽  
Monte Carlo Methods ◽  
Random Pattern ◽  
Multivariate Distributions ◽  
Wide Range ◽  
Probability And Statistics ◽  
Continuous And Discrete Variables ◽  
Multidimensional Integrals ◽  
Selection Of ◽  
Standard Library

Monte Carlo methods are those designed to obtain numerical answers with the use of random numbers . This chapter discusses random engines, which provide a pseudo-random pattern of bits, and their use in for sampling a variety of nonuniform distributions, for both continuous and discrete variables. A wide selection of uniform and nonuniform variate generators from the C++ standard library are reviewed, and common techniques for generating custom nonuniform variates are discussed. The chapter presents the uses of Monte Carlo to evaluate integrals, particularly multidimensional integrals, and then introduces the important method of Markov chain Monte Carlo, suitable for solving a wide range of scientific problems that require the sampling of complicated multivariate distributions. Relevant topics in probability and statistics are also introduced in this chapter. Finally, the topics of thermalization, autocorrelation, multimodality, and Gibbs sampling are presented.

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