Self-consistent Estimates Research Articles

Multi-scale homogenization-based modeling of semi-crystalline polymers

This paper presents a finite-strain, multi-scale constitutive model for semi-crystalline polymers, accounting explicitly for the current state and evolution of the underlying crystallographic, lamellar and morphological texture. Specifically, a semi-crystalline polymer is modeled as a two-scale composite, assumed to be, at the larger length scale, an aggregate of randomly distributed grains that, at the smaller length scale, are made up of alternating layers of an amorphous and a crystalline phase. The model incorporates finite elasticity for the amorphous phase and crystallographic hardening for the crystalline phase. The instantaneous effective response of this composite is determined by means of multi-scale homogenization methods, consisting in the use of a “linear comparison composite” (LCC) with the same internal structure as the actual nonlinear composite, with local properties that are optimally chosen via suitably designed variational principles. The effective properties of the resulting two-scale LCC are obtained through a “sequential” homogenization procedure, involving the exact solution for the effective behavior of the lamellar grains and a self-consistent estimate for the aggregate. The latter results are also used to establish evolution laws for the appropriate internal variables in the material. The predictions of the model for the macroscopic response and texture evolution in high-density polyethylene are confronted with available experimental results and compared with those of earlier models.

Self-consistent estimation of censored quantile regression

The principle of self-consistency has been employed to estimate regression quantile with randomly censored response. The asymptotic studies for this type of approach was established only recently, partly due to the complex forms of the current self-consistent estimators of censored regression quantiles. Of interest, how the self-consistent estimation of censored regression quantiles is connected to the alternative martingale-based approach still remains uncovered. In this paper, we propose a new formulation of self-consistent censored regression quantiles based on stochastic integral equations. The proposed representation of censored regression quantiles entails a clearly defined estimation procedure. More importantly, it greatly simplifies the theoretical investigations. We establish the large sample equivalence between the proposed self-consistent estimators and the existing estimator derived from martingale-based estimating equations. The connection between the new self-consistent estimation approach and the available self-consistent algorithms is also elaborated.

A Ground State Monte Carlo Approach for Studies of Dipolar Systems with Rotational Degrees of Freedom

We have developed a path integral ground state Monte Carlo (PIGSMC) algorithm for quantum simulations of rotating dipolar molecules, using a highly accurate sixth-order algorithm. The method allows us to calculate unbiased estimates of ground state properties of dipolar molecules in a variety of geometries, with or without an external electric field. To demonstrate the capability of the approach, we calculate the orientational phase diagram of a one dimensional lattice system of rotating point dipoles in the absence of any external electric fields. We find that for finite lattice size, this system exhibits an order–disorder transition at finite dipolar interaction strength in contrast to the well-known orientational disorder of the corresponding one dimensional O(3) quantum rotor models. Comparison of the quantum Monte Carlo results with a self-consistent field estimate of the phase transition shows the emergence of an ordered phase at non-zero dipolar strength, confirming the symmetry breaking role of the anisotropic dipole–dipole interaction.

Self-consistent estimation of mean response functions and their derivatives

Many methods have been developed for the nonparametric estimation of a mean response function, but most of these methods do not lend themselves to simultaneous estimation of the mean response function and its derivatives. Recovering derivatives is important for analyzing human growth data, studying physical systems described by differential equations, and characterizing nanoparticles from scattering data. In this article the authors propose a new compound estimator that synthesizes information from numerous pointwise estimators indexed by a discrete set. Unlike spline and kernel smooths, the compound estimator is infinitely differentiable; unlike local regression smooths, the compound estimator is self-consistent in that its derivatives estimate the derivatives of the mean response function. The authors show that the compound estimator and its derivatives can attain essentially optimal convergence rates in consistency. The authors also provide a filtration and extrapolation enhancement for finite samples, and the authors assess the empirical performance of the compound estimator and its derivatives via a simulation study and an application to real data. The Canadian Journal of Statistics 39: 280–299; 2011 © 2011 Statistical Society of Canada Plusieurs methodes ont ete developpees pour l'estimation non parametrique d'une fonction de reponse moyenne. Cependant, la majorite de ces methodes ne permettent pas l'estimation simultanee de la fonction de reponse moyenne et de ses derivees. L'obtention des derivees est importante lorsque nous analysons les donnees sur la croissance humaine, nous etudions des systemes physiques decrits par des equations differentielles ou encore lorsque nous caracterisons les nanoparticules a partir de donnees de diffusion. Dans cet article, les auteurs proposent une nouvel estimateur compose qui combine l'information provenant de plusieurs estimateurs ponctuels indices de faon discrete. Contrairement aux lisseurs par splines ou par noyau, l'estimateur compose est infiniment differentiable. Contrairement aussi aux lisseurs par regression locale, l'estimateur compose est autocoherent dans le sens que ses derivees estiment les derivees de la fonction de reponse moyenne. Les auteurs montrent que l'estimateur compose, ainsi que ses derivees, atteignent essentiellement le taux de convergence optimale. Les auteurs fournissent aussi des ajustements par filtration ou extrapolation pour les petits echantillons. a l'aide d'une etude de simulation, ils mesurent aussi la performance empirique de l'estimateur compose et de ses derivees. Ils l'appliquent aussi a un vrai jeu de donnees. La revue canadienne de statistique 39: 280–299; 2011 © 2011 Societe statistique du Canada

Bounds and self-consistent estimates for elastic constants of polycrystals composed of orthorhombics or crystals with higher symmetries

Methods for computing Hashin-Shtrikman bounds and related self-consistent estimates of elastic constants for polycrystals composed of crystals having orthorhombic symmetry have been known for about three decades. However, these methods are underutilized, perhaps because of some perceived difficulties with implementing the necessary computational procedures. Several simplifications of these techniques are introduced, thereby reducing the overall computational burden, as well as the complications inherent in mapping out the Hashin-Shtrikman bounding curves. The self-consistent estimates of the effective elastic constants are very robust, involving a quickly converging iteration procedure. Once these self-consistent values are known, they may then be used to speed up the computations of the Hashin-Shtrikman bounds themselves. It is shown furthermore that the resulting orthorhombic polycrystal code can be used as well to compute both bounds and self-consistent estimates for polycrystals of higher-symmetry tetragonal, hexagonal, and cubic (but not trigonal) materials. The self-consistent results found this way are shown to be the same as those obtained using the earlier methods, specifically those methods designed specially for each individual symmetry type. But the Hashin-Shtrikman bounds found using the orthorhombic code are either the same or (more typically) tighter than those found previously for these special cases (i.e., tetragonal, hexagonal, and cubic). The improvement in the Hashin-Shtrikman bounds is presumably due to the additional degrees of freedom introduced into the available search space.

Self-Consistent Method for Density Estimation

Summary The estimation of a density profile from experimental data points is a challenging problem, which is usually tackled by plotting a histogram. Prior assumptions on the nature of the density, from its smoothness to the specification of its form, allow the design of more accurate estimation procedures, such as maximum likelihood. Our aim is to construct a procedure that makes no explicit assumptions, but still providing an accurate estimate of the density. We introduce the self-consistent estimate: the power spectrum of a candidate density is given, and an estimation procedure is constructed on the assumption, to be released a posteriori, that the candidate is correct. The self-consistent estimate is defined as a prior candidate density that precisely reproduces itself. Our main result is to derive the exact expression of the self-consistent estimate for any given data set, and to study its properties. Applications of the method require neither priors on the form of the density nor the subjective choice of parameters. A cut-off frequency, akin to a bin size or a kernel bandwidth, emerges naturally from the derivation. We apply the self-consistent estimate to artificial data generated from various distributions and show that it reaches the theoretical limit for the scaling of the square error with the size of the data set.

Nonuniform cratering of the Moon and a revised crater chronology of the inner Solar System

We model the cratering of the Moon and terrestrial planets from the present knowledge of the orbital and size distribution of asteroids and comets in the inner Solar System, in order to refine the crater chronology method. Impact occurrences, locations, velocities and incidence angles are calculated semi-analytically, and scaling laws are used to convert impactor sizes into crater sizes. Our approach is generalizable to other moons or planets. The lunar cratering rate varies with both latitude and longitude: with respect to the global average, it is about 25% lower at (±65°N, 90°E) and larger by the same amount at the apex of motion (0°N, 90°W) for the present Earth–Moon separation. The measured size-frequency distributions of lunar craters are reconciled with the observed population of near-Earth objects under the assumption that craters smaller than a few kilometers in diameter form in a porous megaregolith. Varying depths of this megaregolith between the mare and highlands is a plausible partial explanation for differences in previously reported measured size-frequency distributions. We give a revised analytical relationship between the number of craters and the age of a lunar surface. For the inner planets, expected size-frequency crater distributions are calculated that account for differences in impact conditions, and the age of a few key geologic units is given. We estimate the Orientale and Caloris basins to be 3.73 Ga old, and the surface of Venus to be 240 Ma old. The terrestrial cratering record is consistent with the revised chronology and a constant impact rate over the last 400 Ma. Better knowledge of the orbital dynamics, crater scaling laws and megaregolith properties are needed to confidently assess the net uncertainty of the model ages that result from the combination of numerous steps, from the observation of asteroids to the formation of craters. Our model may be inaccurate for periods prior to 3.5 Ga because of a different impactor population, or for craters smaller than a few kilometers on Mars and Mercury, due to the presence of subsurface ice and to the abundance of large secondaries, respectively. Standard parameter values allow for the first time to naturally reproduce both the size distribution and absolute number of lunar craters up to 3.5 Ga ago, and give self-consistent estimates of the planetary cratering rates relative to the Moon.

The mechanical properties of a transverse isotropic voided material

The mechanical properties of a transverse isotropic honeycomb porous material were investigated by the experiment and the finite element method. The relative density dependence of the macroscopic elastoplastic constants along the in-plane and out of plane directions (e.g., Young’s modulus, Poisson’s ratio, yielding stress, etc.) was systematically studied. Good agreement between the experimental and numerical results was observed. In addition, the three dimensional numerical results were compared with simplified foam models and self-consistent estimates, showing the necessity to improve available models for foams in order to achieve a better predictive capability. At the same time, the elastic–plastic failure mechanism was discussed in detail.

GEOMETRIC AND DYNAMICAL MODELS OF REVERBERATION MAPPING DATA

We present a general method to analyze reverberation mapping data that provides both estimates for the black hole mass and for the geometry and dynamics of the broad line region (BLR) in active galactic nuclei (AGN). Our method directly infers the spatial and velocity distribution of the BLR from the data, allowing us to easily derive a velocity-resolved transfer function and allowing for a self-consistent estimate of the black hole mass without a virial coefficient. We obtain estimates and reasonable uncertainties of the BLR model parameters by implementing a Markov Chain Monte Carlo algorithm using the formalism of Bayesian probability theory. We use Gaussian Processes to interpolate the the continuum light curve data and create mock light curves that can be fitted to the data. We test our method by creating simulated reverberation mapping data-sets with known true parameter values and by trying to recover these parameter values using our models. We are able to recover the parameters with realistic uncertainties that depend upon the variability of the AGN and the quality of the reverberation mapping campaign. With a geometry model we can recover the mean radius of the BLR to within ~0.1dex random uncertainty for simulated data with an integrated line flux uncertainty of 1.5%, while with a dynamical model we can recover the black hole mass and the mean radius to within ~0.05dex random uncertainty, for simulated data with a line profile average signal to noise ratio of 4 per spectral pixel. These uncertainties do not include modeling errors, which are likely to be present in the analysis of real data, and should therefore be considered as lower limits to the accuracy of the method.

Inhomogeneity of the first and second statistical moments of stresses inside the heterogeneities of random structure matrix composites

In this paper linearly thermoelastic composite media are treated, which consist of a homogeneous matrix containing a statistically homogeneous random set of heterogeneities. Effective properties (such as compliance, thermal expansion, stored energy) as well as the first statistical moments of stresses in the phases are estimated for the general case of nonhomogeneity of the thermoelastic inclusion properties. The micromechanical approach is based on the generalization of the “multiparticle effective field” method (MEFM, see for references Buryachenko, Appl. Mech. Rev. (2001), 54, 1–47), previously proposed for the estimation of stress field averages in the phases. The method exploits as a background the new general integral equation proposed by the author before and makes it possible to abandon the use of the central concept of classical micromechanics such as effective field hypothesis as well as their satellite hypothesis of “ellipsoidal symmetry”. The implicit recursion representations of the effective thermoelastic properties and stress concentration factor are expressed through some building blocks described by numerical solutions for both the one and two inclusions inside the infinite medium subjected to the inhomogeneous effective fields evaluated from subsequent self-consistent estimations. One also estimates the inhomogeneous statistical moments of local stress fields which are extremely useful for understanding the evolution of nonlinear phenomena such as plasticity, creep, and damage. Just at some additional assumptions (such as an effective field hypothesis) the involved tensors can be expressed through the Green function, Eshelby tensor and external Eshelby tensor. These estimated inhomogeneities of effective fields lead to the detection of fundamentally new effects for the local stresses inside the heterogeneities.

The nonparametric maximum likelihood estimator for middle-censored data

In this note, we consider data subjected to middle censoring where the variable of interest becomes unobservable when it falls within an interval of censorship. We demonstrate that the nonparametric maximum likelihood estimator (NPMLE) of distribution function can be obtained by using Turnbull's (1976) EM algorithm or self-consistent estimating equation ( Jammalamadaka and Mangalam, 2003) with an initial estimator which puts mass only on the innermost intervals. The consistency of the NPMLE can be established based on the asymptotic properties of self-consistent estimators (SCE) with mixed interval-censored data ( Yu et al., 2000, 2001).

Slip rates and seismic moment deficits on major active faults in mainland China

[1] The Chinese mainland features widespread active faults and intensive seismic activity; both can be described in terms of slip rates on these faults. Previous studies of fault slip rates in mainland China have focused on individual faults or fault segments, and large discrepancies exist among results derived from different methods. Here we derive a self-consistent estimate of the slip rates on all major faults in mainland China using the recently updated GPS data and an elastic block model. The predicted slip rates are high in western and low in eastern China, ranging from 25.1 ± 2.0 mm/yr in the Himalayan Thrust System to less than 1.0 mm/yr in north China. Using these slip rates, we estimated the rates of moment accumulation on the major fault zones and compared them with the seismic moment released on each fault zone using the Chinese historical earthquake catalog that extends for more than 2000 years in many regions. The results show nine seismic zones with large moment deficits (unreleased moment). Future refinement of GPS measurements and earthquake history will allow better estimates of slip rates on individual faults and better assessment of earthquake hazard on these faults in mainland China.

Simultaneous marginal survival estimators when doubly censored data is present

A doubly censoring scheme occurs when the lifetimes T being measured,from a well-known time origin, are exactly observed within a window [L, R] of observational time and are otherwise censored either from above (right-censored observations)or below (left-censored observations). Sample data consists on the pairs (U, δ)where U = min{R, max{T, L}} and δ indicates whether T is exactly observed (δ = 0),right-censored (δ = 1) or left-censored (δ = −1). We are interested in the estimation of the marginal behaviour of the three random variables T, L and R based on the observed pairs (U, δ).We propose new nonparametric simultaneous marginal estimators Ŝ(T) , Ŝ(L) and Ŝ(R) for the survival functions of T, L and R, respectively, by means of an inverse-probability-of-censoring approach. The proposed estimators Ŝ(T) , Ŝ(L) and Ŝ(R) are not computationally intensive, generalize the empirical survival estimator and reduce to the Kaplan-Meier estimator in the absence of left-censored data. Furthermore,Ŝ(T) is equivalent to a self-consistent estimator, is uniformly strongly consistent and asymptotically normal. The method is illustrated with data from a cohort of drug users recruited in a detoxification program in Badalona (Spain). For these data we estimate the survival function for the elapsed time from starting IV-drugs to AIDS diagnosis, as well as the potential follow-up time. A simulation study is discussed to assess the performance of the three survival estimators for moderate sample sizes and different censoring levels.

Ram pressure stripping in a galaxy formation model - I. A novel numerical approach

We develop a new numerical approach to describe the action of ram pressure stripping (RPS) within a semi-analytic model of galaxy formation and evolution which works in combination with non-radiative hydrodynamical simulations of galaxy clusters. The new feature in our method is the use of the gas particles to obtain the kinematical and thermodynamical properties of the intragroup and intracluster medium (ICM). This allows a self-consistent estimation of the RPS experienced by satellite galaxies. We find that the ram pressure (RP) in the central regions of clusters increases approximately one order of magnitude between z = 1 and 0, consistent with the increase in the density of the ICM. The mean RP experienced by galaxies within the virial radius increases with decreasing redshift. In clusters with virial masses M_vir ~10^15 h^-1 M_Sun, over 50 per cent of satellite galaxies have experienced RP ~10^(-11) h^2 dyn cm^-2 or higher for z <= 0.5. In smaller clusters (M_vir ~10^14 h^-1 M_Sun) the mean RP are approximately one order of magnitude lower at all redshifts. RPS has a strong effect on the cold gas content of galaxies for all cluster masses. At z = 0, over 70 per cent of satellite galaxies within the virial radius are completely depleted of cold gas. For the more massive clusters the fraction of depleted galaxies is already established at z ~ 1, whereas for the smaller clusters this fraction increases appreciably between z = 1 and 0. This indicates that the rate at which the cold gas is stripped depends on the virial mass of the host cluster. Compared to our new approach, the use of an analytic profile to describe the ICM results in an overestimation of the RP larger than 50 per cent for z >= 0.5.

Theoretical investigation of an energetic fullerene derivative

A self-consistent estimation method for the thermochemical properties of N-methyl-3-(2',4',6'-trinitrobenzene)-fulleropyrrolidine (MTNBFP) is presented. This method is based on enthalpy of formation (Delta(f)H(m)(minus sign in circle)) and enthalpy of combustion obtained from BLYP/DNP calculations of the total energies and frequencies for MTNBFP. The enthalpy of formation was calculated by an optimized set of isodesmic reactions given the available experimental Delta(f)H(m)(minus sign in circle) of relative compounds. MTNBFP has a high enthalpy of formation, 2782.2 kJ/mol. Detonation velocity and detonation pressure were also presented in terms of Kamlet and Jacobs equations. Drop hammer impact sensitivity tests and blasting point per 5 s tests indicate MTNBFP may be a potential candidate primary explosive. To understand the test results well, we proposed a series of chemical reaction mechanisms and interpreted the relationship between impact sensitivity and electronic structures from the viewpoint of nitro group charge, electrostatic potential, and vibrational modes.

Measuring inter-pulse intervals in sperm whale clicks: Consistency of automatic estimation methods

Sperm whale clicks are characterized by a multi-pulsed structure. The time lag between consecutive pulses, i.e., the inter-pulse interval (IPI), is related to the size of the sound production organ such that its measurement provides a means to acoustically estimate the size of individual whales. Due to off-axis effects the identification of pulses is, however, not always straightforward, and automatic measurement methods provide not only more objective estimation, but may also facilitate IPI estimation in cases where single click measurements are ambiguous. In particular, averaging measurements over a time series of clicks from the same whale could enhance the discrimination of time invariant pulses. The authors developed two automatic methods of automatic IPI measurement based on waveform and autocorrelation averaging and compared their accuracy and consistency with other previously used methods. Manual measurement by an experienced operator provided the most self-consistent estimates. The autocorrelation averaging technique had the best overall performance of the automated methods achieving a very similar performance to manual measurement. On some recordings cepstrum averaging methods converged when autocorrelation did not. Therefore, applying both of these automated methods and choosing the best of the two are recommended.

Microstructural effects on the overall poroelastic properties of saturated porous media

At the macroscopic scale, the quasi-static deformation of an elastic porous medium saturated by an incompressible Newtonian fluid is described by the well-known Biot's model, which involves four effective parameters. In this work, the three effective poroelastic properties and the permeability of two periodic microstructures of saturated cohesive granular media, i.e. simple cubic (SC) and body-centered cubic (BCC) arrays of overlapping spheres, are computed by solving, over the representative elementary volume, boundary-value problems arising from the homogenization process. The influence of microstructure properties, i.e. solid volume fraction, arrangement of spheres, number of contacts as well as the intrinsic properties of the solid phase on the overall properties, is highlighted. Numerical results are then compared with rigorous bounds, self-consistent estimations, exact expansions and experimental results on ceramics and metals available in the literature. Finally, the capability of the obtained results on such periodic microstructures to describe the poroelastic properties of real porous media is discussed.

An inverse-probability-weighted approach to the estimation of distribution function with middle-censored data

In this article, we propose an inverse-probability-weighted (IPW) estimator of distribution function for middle-censored data. By Jammalamadaka and Mangalam (2003), the IPW estimator is the nonparametric maximum likelihood estimator (NPMLE) when all censored intervals contain at least one uncensored observation. The asymptotic properties of the IPW estimator are derived. A simulation study is conducted to compare the performance between the IPW estimator and the self-consistent estimator (SCE). Simulation results indicate that the performance of the IPW estimator is close to that of the SCE.

Data Sensitivity of the ECCO State Estimate in a Regional Setting

Abstract The Estimating the Circulation and Climate of the Ocean (ECCO) consortium provides a framework in which the adjoint method of data assimilation is applied to a general circulation model to provide a dynamically self-consistent estimate of the time-varying ocean state, which is constrained by observations. In this study, the sensitivity of the solution to the constraints provided by various datasets is investigated in a regional setting in the North Pacific. Four assimilation experiments are performed, which vary by the data used as constraints and the relative weights associated with these data. The resulting estimates are compared to two of the assimilated datasets as well as to data from two time series stations not used as constraints. These comparisons demonstrate that increasing the weights of the subsurface data provides overall improvement in the model–data consistency of the estimate of the state of the North Pacific Ocean. However, some elements of the solution are degraded. This could result from incompatibility between datasets, possibly because of hidden biases, or from errors in the model physics made more evident by the increased weight on subsurface data. The adjustments to the control parameters of surface forcing and initial conditions necessary to obtain the more accurate fit to the data are found to be within prior error bars.

Estimation of Multiple Response Rates in Phase II Clinical Trials with Missing Observations

Responses are often correlated in clinical trials. A patient who is not evaluable for a response may still provide some information to the response through his or her status on other responses. Under the assumption of missing at random, we propose the utilization of a self-consistent estimator. We show that the proposed estimators are more efficient than the conventional estimators by asymptotic relative efficiency and simulation study. An example from a Phase II clinical trial on children with chronic myelogenous leukemia is provided.