Empirical probability

Summary

In probability theory and statistics, the empirical probability, relative frequency, or experimental probability of an event is the ratio of the number of outcomes in which a specified event occurs to the total number of trials, i.e., by means not of a theoretical sample space but of an actual experiment. More generally, empirical probability estimates probabilities from experience and observation. Given an event A in a sample space, the relative frequency of A is the ratio \tfrac m n, m being the number of outcomes in which the event A occurs, and n being the total number of outcomes of the experiment. In statistical terms, the empirical probability is an estimator or estimate of a probability. In simple cases, where the result of a trial only determines whether or not the specified event has occurred, modelling using a binomial distribution might be appropriate and then the empirical estimate is the maximum likeliho

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This thesis studies the phenomenon of jamming in granular media, such as when a salt shaker gets clogged. We use modern instrumentation, like X-ray synchrotron tomography, to look inside real jamming experiments. High performance computers allow simulating mathematical models of jamming, but we are also able to treat some of them just using paper and pencil. One main part of this thesis consists of an experimental validation of the distinct-element-method (DEM). In this model, grains are modeled separately, their trajectories obey Newton's laws of motion and a model of the contacts between grains is given. Real experiments of jamming of glass beads flowing out of a container were carried out. 3D snapshots of the interior of the media were taken using X-ray synchrotron tomography. These snapshots were computer processed using state of the art image analysis. It was found that 3D DEM is capable of predicting quite well the final positions of the grains of the real experiments. Indeed, in cases of instant jamming (jamming without a substantial previous flow of beads) the simulations agree well with the real experiments. However, in cases of non instant jamming, because of chaotic behavior of the model and the system, the results do not agree. Furthermore, a sensitivity analysis to grain location and size perturbations was carried out. In a second part, we describe results on 2D DEM simulations of jamming in a hopper. We focus on the jamming probability J, the average time T before jamming and the average number ψ of beads falling through the hole when jamming occurs. These quantities were related to global parameters such as the number of grains, the hole size, the friction coefficient, grain length or the angle of the hopper (in opposition to fine-scale parameters that are the positions and radii of the grains). In agreement with intuition, a monotonic behavior of J and ψ as a function of the number of grains, the hole size, the friction coefficient was found. However, surprising results were also found such as the non-monotonicity of the average number of beads falling through the hole when jamming occurs as a function of the grain length and the hopper angle. In the third part, we study simple probabilistic 2D models called SPM, in which non-interacting particles move with constant speed towards the center of a circular sector. Formulas giving the jamming probability or the average time before jamming when jamming occurs as a function of global parameters were found. SPM and 2D DEM were compared and a locally good correspondence between the global parameters of the two was established. SPM led us to study some combinatorial problems, in particular two bi-indexed recurrence sequences. One gives the number of ways of placing identical balls in fixed-size numbered urns and the other the number of subsets of a given ordered set without a certain number of consecutive elements. Several different ways of computing the sequences, each advantageous in certain cases, were found.

EPFL2009

Particle shape versus friction in granular jamming

Thomas Liebling, Lionel Pournin, Michel Tsukahara

What is the influence of sand grain shapes on an hourglass accuracy, and how does it affect the jamming probability? We study such a question using numerical simulation and our recently introduced spheropolyhedral particles. Spheropolyhedra constitute a very rich class of shapes that naturally generalize the traditional spheres used in the Distinct Element Model. Our experiment consists in pouring particles into a container pierced with a circular opening, and to note whether they flow or jam. By repeating this experiment, one can evaluate the jamming probability of a given granular assembly. We show here that intuition is misleading: while the jamming probability is clearly dependent on friction, it does not show a dependence on particle sharpness.

AIP2009

Correcting Jackknife Confidence Intervals

This paper presents a method for constructing confidence intervals using the jackknife. Asymptotic expansions are used to assess and reduce the skewness. It is proved that, although the error in the coverage rate remains of order O(1/n) for two-sided intervals, the main term due to the skewness in the distribution is eliminated, which constitute an improvement to the normal approximation that is important in the context of small samples. Several simulations are presented, with different distributions and functionals, and compared to the empirical accuracy obtained with a normal approximation. It is observed that our method produces in many cases a better empirical accuracy than the intervals due to a normal approximation, without requiring any knowledge on the underlying distribution.

1998

EPFL2009