Design of Hamiltonian Monte Carlo for perfect simulation of general continuous distributions

Hamiltonian Monte Carlo (HMC) is an efficient method of simulating smooth distributions and has motivated the widely used No-U-turn Sampler (NUTS) and software Stan. We build on NUTS and the technique of “unbiased sampling” to design HMC algorithms that produce perfect simulation of general continuous distributions that are amenable to HMC. Our methods enable separation of Markov chain Monte Carlo convergence error from experimental error, and thereby provide much more powerful MCMC convergence diagnostics than current state-of-the-art summary statistics which confound these two errors. Objective comparison of different MCMC algorithms is provided by the number of derivative evaluations per perfect sample point. We demonstrate the methodology with applications to normal, t and normal mixture distributions up to 100 dimensions, and a 12-dimensional Bayesian Lasso regression. HMC runs effectively with a goal of 20 to 30 points per trajectory. Numbers of derivative evaluations per perfect sample point range from 390 for a univariate normal distribution to 12,000 for a 100-dimensional mixture of two normal distributions with modes separated by six standard deviations, and 22,000 for a 100-dimensional t-distribution with four degrees of freedom

A study of waves in the earth's bow shock

The perturbation vectors of waves up and downstream from the region of maximum compression in the bow shock were examined on OGO-5 under particularly steady solar wind conditions. The polarization of the upstream waves was RH, circular and of the downstream waves LH, elliptical in the spacecraft frame. By observing that the polarization of the waves remained unchanged as the shock motion swept the wave structure back and forth across the satellite three times in eight minutes, it was found that the waves were not stationary in the shock frame. A study of the methods of determining the shock normal indicates that the normal estimated from a shock model should be superior to one based upon magnetic coplanarity. The propagation vectors of the waves examined did not coincide with the shock model normal, the average magnetic field, or the plasma flow velocity. However, the major axis of the polarization ellipse of the downstream wave was nearly parallel to the upstream propagation vector

Adiabatic Motion of a Quantum Particle in a Two-Dimensional Magnetic Field

The adiabatic motion of a charged, spinning, quantum particle in a two - dimensional magnetic field is studied. A suitable set of operators generalizing the cinematical momenta and the guiding center operators of a particle moving in a homogeneous magnetic field is constructed. This allows us to separate the two degrees of freedom of the system into a {\sl fast} and a {\sl slow} one, in the classical limit, the rapid rotation of the particle around the guiding center and the slow guiding center drift. In terms of these operators the Hamiltonian of the system rewrites as a power series in the magnetic length \lb=\sqrt{\hbar c\over eB} and the fast and slow dynamics separates. The effective guiding center Hamiltonian is obtained to the second order in the adiabatic parameter \lb and reproduces correctly the classical limit.Comment: 17 pages, LaTe

Bulletin No. 97 - Report on the Southern Utah Experiment Station

When, in the early part of the year 1905 , the management of the Southern Utah Experiment Farm was turned over to the officials of the Utah Agricultural Experiment Station, in compliance with an act of the legislature of that year a number of experiments were under way, principally variety tests with orchard and small fruit. The State Board of Horticulture had been in charge of this farm since its establishment in 1899, and had succeeded in making out of it not only a place that will indicate the varieties of fruits adapted to that climate and soil , but also a farm that in neatness and careful arrangement can be an instructive model to anyone contemplating engaging in that industry. The new management decided not to make any radical change in the tests that were under way , but to continue them to such a conclusion a will yield the valuable results that were to come out of them. New tests or investigations are to be started with the same crop without interfering with them as variety tests, and upon the ground still available or which has not yet been set out into orchard, vineyard, or other permanent crops. Such it is planned to do as the problems suggest themselves, and as means are available

Comparison of Howland and General Impedance Converter (GIC) circuit based current sources for bio-impedance measurements

The current source is a key component in bio-impedance measurement systems. The accuracy of the current source can be measured in terms of its output impedance together with other parameters, with certain applications demanding extremely high output impedance. This paper presents an investigation and comparison of different current source designs based on the Enhanced Howland circuit combined with a General Impedance Converter (GIC) circuit using both ideal and non-ideal operational amplifiers. Under differing load conditions two different settings of the GIC are evaluated and the results are compared to show its performance settings. Whilst the study has shown that over a wide bandwidth (i.e. 100Hz-100MHz) the output impedance is limited, operation over a more limited range offers output impedance in the Giga-ohm range, which can be considered as being infinite

Poor intestinal permeability in mildly stunted Nepali children: associations with weaning practices and Giardia lamblia infection

Studies in the Gambia, using the lactulose-mannitol dual-sugar intestinal permeability test (lactulose:mannitol ratio) as a non-invasive way of investigating mucosal damage, have shown that food malabsorption is significantly associated with early growth retardation. In this cross-sectional study, 210 poor urban Nepali children, 0-60 months old, were recruited and measured for height or length and weight, 167 were examined for intestinal permeability and 173 for parasite infection. Weaning and morbidity data were collected from 172 caretakers. Children were mildly stunted (mean height-for-age z-score -1·45) and underweight (mean weight-for-age z-score -1·62). The lactulose:mannitol ratio (0·26) was poorer than that of UK children (0·12), but similar to that found in Bengali children of the same age (0·24). Two stages of weaning, the onset supplementary feeding (6 months) and the cessation of breast-feeding (23 months), were shown to have differential impact. In children currently breast-feeding, the duration of supplementation was negatively related to lactose (P<0·001) and lactose:lactulose values (P<0·0001), indicating lactose maldigestion. In children who had ceased breast-feeding, a longer period of lactation was associated with poorer intestinal permeability (P=0·031), and poorer height-for-age (P=0·024), which was an unexpected result. No significant relationships were found between intestinal permeability and growth, or with morbidity and helminth infection, except in children with Giardia lamblia who had worse lactulose:mannitol ratios than those without (0·43 v. 0·25 respectively, P=0·014). It is likely that insults to the gut (e.g. Giardia) and challenges to the immune system (weaning) have a different impact in early and late infancy

Correlated quantum percolation in the lowest Landau level

Our understanding of localization in the integer quantum Hall effect is informed by a combination of semi-classical models and percolation theory. Motivated by the effect of correlations on classical percolation we study numerically electron localization in the lowest Landau level in the presence of a power-law correlated disorder potential. Careful comparisons between classical and quantum dynamics suggest that the extended Harris criterion is applicable in the quantum case. This leads to a prediction of new localization quantum critical points in integer quantum Hall systems with power-law correlated disorder potentials. We demonstrate the stability of these critical points to addition of competing short-range disorder potentials, and discuss possible experimental realizations.Comment: 15 pages, 12 figure

Perfect simulation from unbiased simulation

We show that any application of the technique of unbiased simulation becomes perfect simulation when coalescence of the two coupled Markov chains can be practically assured in advance. This happens when a fixed number of iterations is high enough that the probability of needing any more to achieve coalescence is negligible; we suggest a value of $10^{-20}$. This finding enormously increases the range of problems for which perfect simulation, which exactly follows the target distribution, can be implemented. We design a new algorithm to make practical use of the high number of iterations by producing extra perfect sample points with little extra computational effort, at a cost of a small, controllable amount of serial correlation within sample sets of about 20 points. Different sample sets remain completely independent. The algorithm includes maximal coupling for continuous processes, to bring together chains that are already close. We illustrate the methodology on a simple, two-state Markov chain and on standard normal distributions up to 20 dimensions. Our technical formulation involves a nonzero probability, which can be made arbitrarily small, that a single perfect sample point may have its place taken by a "string" of many points which are assigned weights, each equal to $\pm 1$, that sum to~$1$. A point with a weight of $-1$ is a "hole", which is an object that can be cancelled by an equivalent point that has the same value but opposite weight $+1$.Comment: 17 pages, 4 figures; for associated R scripts, see https://github.com/George-Leigh/PerfectSimulatio