On Buffon Machines and Numbers

The well-know needle experiment of Buffon can be regarded as an analog (i.e., continuous) device that stochastically "computes" the number 2/pi ~ 0.63661, which is the experiment's probability of success. Generalizing the experiment and simplifying the computational framework, we consider probability distributions, which can be produced perfectly, from a discrete source of unbiased coin flips. We describe and analyse a few simple Buffon machines that generate geometric, Poisson, and logarithmic-series distributions. We provide human-accessible Buffon machines, which require a dozen coin flips or less, on average, and produce experiments whose probabilities of success are expressible in terms of numbers such as, exp(-1), log 2, sqrt(3), cos(1/4), aeta(5). Generally, we develop a collection of constructions based on simple probabilistic mechanisms that enable one to design Buffon experiments involving compositions of exponentials and logarithms, polylogarithms, direct and inverse trigonometric functions, algebraic and hypergeometric functions, as well as functions defined by integrals, such as the Gaussian error function.Comment: Largely revised version with references and figures added. 12 pages. In ACM-SIAM Symposium on Discrete Algorithms (SODA'2011

Modeling electricity spot prices - Combining mean-reversion, spikes and stochastic volatility

Starting with the liberalization of electricity trading, this market grew rapidly over the last decade. However, while spot and future markets are rather liquid nowadays, option trading is still limited. One of the potential reasons for this is that the spot price process of electricity is still puzzling researchers and practitioners. In this paper, we propose an approach to model spot prices that combines mean-reversion, spikes and stochastic volatility. Thereby we use different mean-reversion rates for 'normal' and 'extreme' (spike) periods. Another feature of the model is its ability to capture correlation structures of electricity price spikes. Furthermore, all model parameters can easily be estimated with help of historical data. Consequently, we argue that this model does not only extend academic literature on electricity spot price modeling, but is also suitable for practical purposes, e.g. as underlying price model for option pricing. --Electricity,Energy markets,Lévy processes,Mean-reversion,Spikes,Stochastic volatility,GARCH

Spatial and temporal dynamics of malaria transmission in rural western Kenya

ABSTRACT: BACKGROUND: Understanding the impact of reducing Plasmodium falciparum malaria transmission requires estimates of the relationship between health outcomes and exposure to infectious mosquitoes. However, measures of exposure such as mosquito density and entomological inoculation rate (EIR) are generally aggregated over large areas and time periods, biasing the outcome-exposure relationship. There are few studies examining the extent and drivers of local variation in malaria exposure in endemic areas. METHODS: We describe the spatio-temporal dynamics of malaria transmission intensity measured by mosquito density and EIR in the KEMRI/CDC health and demographic surveillance system using entomological data collected during 2002-2004. Geostatistical zero inflated binomial and negative binomial models were applied to obtain location specific (house) estimates of sporozoite rates and mosquito densities respectively. Model-based predictions were multiplied to estimate the spatial pattern of annual entomological inoculation rate, a measure of the number of infective bites a person receive per unit of time. The models included environmental and climatic predictors extracted from satellite data, harmonic seasonal trends and parameters describing space-time correlation. RESULTS: Anopheles gambiae s.l was the main vector species accounting for 86% (n=2309) of the total collected mosquitoes with the remainder being Anopheles funestus. Sixty eight percent (757/1110) of the surveyed houses had no mosquitoes. Distance to water bodies, vegetation and day temperature were significantly associated with mosquito density. Overall annual point estimates of EIR were 6.7, 9.3 and 9.6 infectious bites per annum for 2002, 2003 and 2004 respectively. Monthly mosquito density and EIR varied over the study period peaking in May during the wet season. The predicted and observed densities and EIR showed a strong seasonal and spatial pattern over the study area. CONCLUSIONS: Spatio-temporal maps of malaria transmission intensity obtained in this study are not only useful in understanding variability in malaria epidemiology over small areas but also provides a high resolution exposure surface that can be used to analyse the impact of malaria exposure on mortalit

Power Spectra of X-ray Binaries

The interpretation of Fourier spectra in the time domain is critically examined. Power density spectra defined and calculated in the time domain are compared with Fourier spectra in the frequency domain for three different types of variability: periodic signals, Markov processes and random shots. The power density spectra for a sample of neutron stars and black hole binaries are analyzed in both the time and the frequency domains. For broadband noise, the two kinds of power spectrum in accreting neutron stars are usually consistent with each other, but the time domain power spectra for black hole candidates are significantly higher than corresponding Fourier spectra in the high frequency range (10--1000 Hz). Comparing the two kinds of power density spectra may help to probe the intrinsic nature of timing phenomena in compact objects.Comment: 21 pages, 10 figures, to appear in Astrophysical Journa

Penalized nonparametric mean square estimation of the coefficients of diffusion processes

We consider a one-dimensional diffusion process $(X_t)$ which is observed at $n+1$ discrete times with regular sampling interval $\Delta$. Assuming that $(X_t)$ is strictly stationary, we propose nonparametric estimators of the drift and diffusion coefficients obtained by a penalized least squares approach. Our estimators belong to a finite-dimensional function space whose dimension is selected by a data-driven method. We provide non-asymptotic risk bounds for the estimators. When the sampling interval tends to zero while the number of observations and the length of the observation time interval tend to infinity, we show that our estimators reach the minimax optimal rates of convergence. Numerical results based on exact simulations of diffusion processes are given for several examples of models and illustrate the qualities of our estimation algorithms.Comment: Published at http://dx.doi.org/10.3150/07-BEJ5173 in the Bernoulli (http://isi.cbs.nl/bernoulli/) by the International Statistical Institute/Bernoulli Society (http://isi.cbs.nl/BS/bshome.htm