Self-consistent Estimates Research Articles

Estimating the Term Structure with Linear Regressions: Getting to the Roots of the Problem

Linear estimators of the affine term structure model are inconsistent since they cannot reproduce the factors used in estimation. This is a serious handicap empirically, giving a worse fit than the conventional MLE estimator that ensures consistency. We show that a simple self-consistent estimator can be constructed using the eigenvalue decomposition of a regression estimator. The remaining parameters of the model follow analytically. The fit of this model is virtually indistinguishable from that of the MLE. We apply the method to estimate the model of the U.S. Treasury yields and a joint model of the U.S. and German yield curves.

Hill–Mandel condition and bounds on lower symmetry elastic crystals

Despite advances in contemporary micromechanics, there is a void in the literature on a versatile method for estimating the effective properties of polycrystals comprising of highly anisotropic single crystals belonging to lower symmetry class. Basing on variational principles in elasticity and the Hill–Mandel homogenization condition, we propose a versatile methodology to fill this void. It is demonstrated that the bounds obtained using the Hill–Mandel condition are tighter than the Voigt and Reuss [1,2] bounds, the Hashin–Shtrikman [3] bounds as well as a recently proposed self-consistent estimate by Kube and Arguelles [4] even for polycrystals with highly anisotropic single crystals.

Scaling and excitation of combined convection in a rapidly rotating plane layer

The optimum (to my mind) scaling of the combined thermal and compositional convection in a rapidly rotating plane layer is proposed.This scaling follows from self-consistent estimates of typical physical quantities. Similarity coefficients are introduced for the ratio convection dissipation/convection generation (s) and the ratio thermal convection/compositional convection (r). The third new and most important coefficient δ is the ratio of the characteristic size normal to the axis of rotation to the layer thickness. The faster the rotation, the lower δ. In the case of the liquid Earth core, δ ~ 10–3 substitutes for the generally accepted Ekman number (E ~ 10–15) and s ~ 10–6 substitutes for the inverse Rayleigh number 1/Ra ~ 10–30. It is found that, at turbulent transport coefficients, number s and the Prandtl number are on the order of unity for any objects and δ is independent of transport coefficients. As a result of expansion in powers of δ, an initially 3D system of six variables is simplified to an almost 2D system of four variables without δ. The problem of convection excitation in the main volume is algebraically solved and this problem for critical values is analytically solved. Dispersion relations and general expressions for critical wavenumbers, numbers s (which determine Rayleigh numbers), other critical parameters, and asymptotic solutions are derived. Numerical estimates are made for the liquid cores in the planets that resemble the Earth. Further possible applications of the results obtained are proposed for the interior of planets, moons, their oceans, stars, and experimental objects.

Bounds and self-consistent estimates of the elastic constants of polycrystals

The Hashin–Shtrikman bounds on the elastic constants have been previously calculated for polycrystalline materials with crystallites having general elastic symmetry (triclinic crystallite symmetry). However, the calculation of tighter bounds and the self-consistent estimates of these elastic constants has remained unsolved. In this paper, a general theoretical expression for the self-consistent elastic constants is formulated. An iterative method is used to solve the expression for the self-consistent estimates. Each iteration of the solution gives the next tighter set of bounds including the well-known Voigt–Reuss and Hashin–Shtrikman bounds. Thus, all of the bounds on the elastic constants and the self-consistent estimates for any crystallite symmetry are obtained in a single, computationally efficient procedure. The bounds and self-consistent elastic constants are reported for several geophysical materials having crystallites of monoclinic and triclinic symmetries.

Estimation of Enthalpy of Formation of Liquid Transition Metal Alloys: A Modified Prescription Based on Macroscopic Atom Model of Cohesion

The enthalpy of formation Δo H f is an important thermodynamic quantity, which sheds significant light on fundamental cohesive and structural characteristics of an alloy. However, being a difficult one to determine accurately through experiments, simple estimation procedures are often desirable. In the present study, a modified prescription for estimating Δo H f L of liquid transition metal alloys is outlined, based on the Macroscopic Atom Model of cohesion. This prescription relies on self-consistent estimation of liquid-specific model parameters, namely electronegativity (ϕ L) and bonding electron density (n b L ). Such unique identification is made through the use of well-established relationships connecting surface tension, compressibility, and molar volume of a metallic liquid with bonding charge density. The electronegativity is obtained through a consistent linear scaling procedure. The preliminary set of values for ϕ L and n b L , together with other auxiliary model parameters, is subsequently optimized to obtain a good numerical agreement between calculated and experimental values of Δo H f L for sixty liquid transition metal alloys. It is found that, with few exceptions, the use of liquid-specific model parameters in Macroscopic Atom Model yields a physically consistent methodology for reliable estimation of mixing enthalpies of liquid alloys.

A Combined Experimental and Modelling Approach to Elastic–Plastic Crack Driving Force Calculation in the Presence of Residual Stresses

Since all residual stress measurement methods have inherent limitations, it is normally impractical to completely characterise a three-dimensional residual stress field by experimental means. This lack of complete information makes it difficult to incorporate measured residual stress data into the analysis of elastic–plastic fracture without resorting to simplified methods such as the Failure Assessment Diagram (FAD) approach. We propose a technique in which the complete residual stress field is reconstructed from measurements and used in finite element analysis of the fracture process. Residual elastic strains and stresses in three-point bend fracture specimens were measured using neutron diffraction and an iterative method was used to generate a self-consistent estimate of the complete residual stress field. This enabled calculation of the J contour integral for a specimen acted on by both residual stress and an externally-applied load, allowing the interaction between residual and applied stress to be observed in detail.

Generalized self-consistent electroelastic estimation of piezoelectric nanocomposites accounting for fiber section shape under antiplane shear

Amicromechanicsmodel composed of a three-phase confocal elliptical cylinder of the nanocompositeswith interface effect is proposed.Ageneralized self-consistent method for the piezoelectric nanocomposites accounting for fiber section shape under far-field mechanical–electrical loads is presented based on the model. By using the theory of Gurtin–Murdoch surface/interface and the conformal mapping technique, a closedform solution of the effective electroelastic constants is obtained. The present solution can be degenerated into the existing solution. The results show that the effective electroelastic constants are dramatically size dependent when the size of the fiber is on the order of nanometers. The effective electroelastic constants decrease monotonically with the fiber section aspect ratio \({\gamma}\) increasing from 0 to 1. The elastic constant and dielectric constant obtained by the present solution are very similar to the results obtained by the classical electroelastic theory, whereas the piezoelectric coupling constant obtained by the present solution is very different from the results obtained by the classical electroelastic theory.

Global-mean marine δ13C and its uncertainty in a glacial state estimate

A paleo-data compilation with 492 δ13C and δ18O observations provides the opportunity to better sample the Last Glacial Maximum (LGM) and infer its global properties, such as the mean δ13C of dissolved inorganic carbon. Here, the paleo-compilation is used to reconstruct a steady-state water-mass distribution for the LGM, that in turn is used to map the data onto a 3D global grid. A global-mean marine δ13C value and a self-consistent uncertainty estimate are derived using the framework of state estimation (i.e., combining a numerical model and observations). The LGM global-mean δ13C is estimated to be 0.14‰ ± 0.20‰ at the two standard error level, giving a glacial-to-modern change of 0.32‰ ± 0.20‰. The magnitude of the error bar is attributed to the uncertain glacial ocean circulation and the lack of observational constraints in the Pacific, Indian, and Southern Oceans. To halve the error bar, roughly four times more observations are needed, although strategic sampling may reduce this number. If dynamical constraints can be used to better characterize the LGM circulation, the error bar can also be reduced to 0.05 to 0.1‰, emphasizing that knowledge of the circulation is vital to accurately map δ13C in three dimensions.

Towards the optimization of the thermal–hydraulic performance of gyrotron collectors

Different configurations of water-cooled Cu collector for gyrotrons are investigated using the StarCCM+CFD code, aimed at optimizing its thermal–hydraulic (TH) performance. Although the current collectors show a good performance, the collector can be subjected to transient heat loads, due to the spent electron beam, of up to several tens of MW/m2, and there is an interest to increase the gyrotron output power in the future. Furthermore, an optimized cooling will lead to improved reliability and lifetime of the collector. Starting from a hypervapotron (HV)-like collector, characterized by 100+ deep rectangular cavities with aspect ratio (AR)=3, we present in the first part of the paper a single-cavity steady-state parametric analysis of the effect of AR on the heat exhaust capabilities. The investigation is then extended to other collector designs, including circumferential ribs and dimples, in order to assess the options for further improvements of the TH performance. The peak Cu temperature is computed by the code and its minimization is the target of the present optimization exercise. A self-consistent estimate of the heat transfer coefficient between collector and coolant is also obtained, which could be useful for fatigue and lifetime assessments. In the second part of the paper the most promising collector geometries identified in the first part are analyzed in the case of a transient heat load (vertical sweeping), first at the level of a single spatial period of the collector structure, then at the full-collector level. The results of the TH transient analysis are compared with both the results of the first part and with the transient purely thermal analysis of the full collector, showing for all geometries considered in this study a room for cooling efficiency improvement with respect to the HV-like design with AR=3, at least in the operating conditions considered for this study (V∼4m/s, almost 100°C sub-cooling).

An Automatic Mosaicking Algorithm for the Generation of a Large-Scale Forest Height Map Using Spaceborne Repeat-Pass InSAR Correlation Magnitude

This paper describes an automatic mosaicking algorithm for creating large-scale mosaic maps of forest height. In contrast to existing mosaicking approaches through using SAR backscatter power and/or InSAR phase, this paper utilizes the forest height estimates that are inverted from spaceborne repeat-pass cross-pol InSAR correlation magnitude. By using repeat-pass InSAR correlation measurements that are dominated by temporal decorrelation, it has been shown that a simplified inversion approach can be utilized to create a height-sensitive measure over the whole interferometric scene, where two scene-wide fitting parameters are able to characterize the mean behavior of the random motion and dielectric changes of the volume scatterers within the scene. In order to combine these single-scene results into a mosaic, a matrix formulation is used with nonlinear least squares and observations in adjacent-scene overlap areas to create a self-consistent estimate of forest height over the larger region. This automated mosaicking method has the benefit of suppressing the global fitting error and, thus, mitigating the “wallpapering” problem in the manual mosaicking process. The algorithm is validated over the U.S. state of Maine by using InSAR correlation magnitude data from ALOS/PALSAR and comparing the inverted forest height with Laser Vegetation Imaging Sensor (LVIS) height and National Biomass and Carbon Dataset (NBCD) basal area weighted (BAW) height. This paper serves as a companion work to previously demonstrated results, the combination of which is meant to be an observational prototype for NASA’s DESDynI-R (now called NISAR) and JAXA’s ALOS-2 satellite missions.

Application of nonlinear methods to the study of ionospheric plasma

Most of the processes taking place in the auroral region of Earth's ionosphere are reflected in a variety of dynamic forms of the aurora borealis. In order to study these processes it is necessary to consider temporary and spatial variations of the characteristics of ionospheric plasma. Most traditional methods of classical physics are applicable mainly for stationary or quasi-stationary phenomena, but dynamic regimes, transients, fluctuations, selfsimilar scaling could be considered using the methods of nonlinear dynamics. Special interest is the development of the methods for describing the spatial structure and the temporal dynamics of auroral ionosphere based on the ideas of percolation theory and fractal geometry. The fractal characteristics (the Hausdorff fractal dimension and the index of connectivity) of Hall and Pedersen conductivities are used to the description of fractal patterns in the ionosphere. To obtain the self-consistent estimates of the parameters the Hausdorff fractal dimension and the index of connectivity in the auroral zone, an additional relation describing universal behavior of the fractal geometry of percolation at the critical threshold is applied. Also, it is shown that Tsallis statistics can be used to study auroral ionosphere

Study of a temperature gradient metamorphism of snow from 3-D images: time evolution of microstructures, physical properties and their associated anisotropy

Abstract. We carried out a study to monitor the time evolution of microstructural and physical properties of snow during temperature gradient metamorphism: a snow slab was subjected to a constant temperature gradient in the vertical direction for 3 weeks in a cold room, and regularly sampled in order to obtain a series of three-dimensional (3-D) images using X-ray microtomography. A large set of properties was then computed from this series of 3-D images: density, specific surface area, correlation lengths, mean and Gaussian curvature distributions, air and ice tortuosities, effective thermal conductivity, and intrinsic permeability. Whenever possible, specific attention was paid to assess these properties along the vertical and horizontal directions, and an anisotropy coefficient defined as the ratio of the vertical over the horizontal values was deduced. The time evolution of these properties, as well as their anisotropy coefficients, was investigated, showing the development of a strong anisotropic behavior during the experiment. Most of the computed physical properties of snow were then compared with two analytical estimates (self-consistent estimates and dilute beds of spheroids) based on the snow density, and the size and anisotropy of the microstructure through the correlation lengths. These models, which require only basic microstructural information, offer rather good estimates of the properties and anisotropy coefficients for our experiment without any fitting parameters. Our results highlight the interplay between the microstructure and physical properties, showing that the physical properties of snow subjected to a temperature gradient cannot be described accurately using only isotropic parameters such as the density and require more refined information. Furthermore, this study constitutes a detailed database on the evolution of snow properties under a temperature gradient, which can be used as a guideline and a validation tool for snow metamorphism models at the micro- or macroscale.

ExELS: an exoplanet legacy science proposal for the ESA Euclid mission - II. Hot exoplanets and sub-stellar systems

The Exoplanet Euclid Legacy Survey (ExELS) proposes to determine the frequency of cold exoplanets down to Earth mass from host separations of ~1 AU out to the free-floating regime by detecting microlensing events in Galactic Bulge. We show that ExELS can also detect large numbers of hot, transiting exoplanets in the same population. The combined microlensing+transit survey would allow the first self-consistent estimate of the relative frequencies of hot and cold sub-stellar companions, reducing biases in comparing "near-field" radial velocity and transiting exoplanets with "far-field" microlensing exoplanets. The age of the Bulge and its spread in metallicity further allows ExELS to better constrain both the variation of companion frequency with metallicity and statistically explore the strength of star-planet tides. We conservatively estimate that ExELS will detect ~4100 sub-stellar objects, with sensitivity typically reaching down to Neptune-mass planets. Of these, ~600 will be detectable in both Euclid's VIS (optical) channel and NISP H-band imager, with ~90% of detections being hot Jupiters. Likely scenarios predict a range of 2900-7000 for VIS and 400-1600 for H-band. Twice as many can be expected in VIS if the cadence can be increased to match the 20-minute H-band cadence. The separation of planets from brown dwarfs via Doppler boosting or ellipsoidal variability will be possible in a handful of cases. Radial velocity confirmation should be possible in some cases, using 30-metre-class telescopes. We expect secondary eclipses, and reflection and emission from planets to be detectable in up to ~100 systems in both VIS and NISP-H. Transits of ~500 planetary-radius companions will be characterised with two-colour photometry and ~40 with four-colour photometry (VIS,YJH), and the albedo of (and emission from) a large sample of hot Jupiters in the H-band can be explored statistically.

Thermal Postbuckling of Imperfect Circular Functionally Graded Material Plates: Examination of Voigt, Mori–Tanaka, and Self-Consistent Schemes

Thermal postbuckling of solid circular plates made of a through-the-thickness functionally graded material (FGM) is analyzed in this paper. Initial imperfection of the plate is also taken into account. Each thermomechanical property of the plate is assumed to be a function of the temperature and thickness coordinate. Equivalent properties of the FGM media are obtained based on three different homogenization schemes, namely, Voigt rule, Mori–Tanaka scheme, and self-consistent estimate. Temperature profile is assumed to be through-the-thickness direction only. The solution of the heat conduction equation is obtained using an iterative central finite difference scheme. Various types of thermal loadings, such as uniform temperature rise, temperature specified at surfaces, and heat flux, are considered. Nonlinear equilibrium equations of the plate are obtained by means of the conventional Ritz method. Solution of the resulting nonlinear equilibrium equations and temperature distribution are obtained simultaneously at each step of heating. It is shown that response of a perfect clamped FGM plate is of the bifurcation type of buckling with stable postbuckling equilibrium branch, whereas imperfect clamped and perfect/imperfect simply supported FGM plates do not reveal the bifurcation type of instability through the nonuniform heating process. Furthermore, amplitude of initial imperfection is an important factor on the equilibrium path of FGM circular plates, especially for simply supported ones.

Micromechanical effective medium modeling of metamaterials of the Willis form

The unique behavior of acoustic metamaterials (AMM) results from deeply sub-wavelength structures with hidden degrees of freedom rather than the inherent material properties of their constituents. This distinguishes AMM from classical composite or cellular materials and also complicates attempts to model their overall response. This is especially true when sub-wavelength structures yield anisotropic effective material response, a key feature of AMM devices designed using transformation acoustics. Further, previous work has shown that the dynamic response of heterogeneous materials must include coupling between the overall strain and momentum fields [Milton and Willis, Proc. R. Soc. A 463, 855–880, (2007)]. A micromechanical homogenization model of the overall Willis constitutive equations is presented to address these difficulties. The model yields a low-volume-fraction estimate of anisotropic and frequency-dependent effective properties in the long-wavelength limit. The model employs volume averages of the dyadic Green’s function calculating the particle displacement resulting from a unit force source. This Green’s function is shown to be analogous to one that determines the particle velocity in a fluid resulting from a unit dipole moment. The predicted effective properties for isotropic materials with spherical inclusions fall within the Hashin-Shtrikman bounds and agree with self-consistent estimates. [Work supported by ONR.]

Holographic estimates of the deconfinement temperature

The problem of self-consistent estimates of the deconfinement temperature \(T_\mathrm{c}\) in the framework of the bottom–up holographic approach to QCD is scrutinized. It is shown that the standard soft wall model gives \(T_\mathrm{c}\) for the planar gluodynamics around 260 MeV in good agreement with the lattice data. The extensions of the soft wall model adjusted for descriptions of realistic meson spectra result in a broad range of the predictions. This uncertainty is related with a poor experimental information on the radially excited mesons.

A self-consistent estimator of survival function with interval-censored and left-truncated data

Interval censoring refers to a situation in which, Ti∗, the time to occurrence of an event of interest is only known to lie in an interval [Li∗,Ri∗]. In some cases, the variable Ti∗ also suffers left-truncation. The nonparametric maximum likelihood estimator (NPMLE) of the survival function of Ti∗ can be obtained by using an EM algorithm of Turnbull (1976). One disadvantage of the NPMLE is that it is not uniquely defined in the innermost intervals. In this article, we propose a self-consistent estimator (SCE), which does not require interpolation. Furthermore, we show that the NPMLE is also an SCE. We establish the consistency of the SCE under certain conditions, which implies that the NPMLE is also a consistent estimator. A simulation study is conducted to compare the performance between the SCE and the NPMLE.

Microstructure-based calculations and experimental results for sound absorbing porous layers of randomly packed rigid spherical beads

Acoustics of stiff porous media with open porosity can be very effectively modelled using the so-called Johnson-Champoux-Allard-Pride-Lafarge model for sound absorbing porous media with rigid frame. It is an advanced semi-phenomenological model with eight parameters, namely, the total porosity, the viscous permeability and its thermal analogue, the tortuosity, two characteristic lengths (one specific for viscous forces, the other for thermal effects), and finally, viscous and thermal tortuosities at the frequency limit of 0 Hz. Most of these parameters can be measured directly, however, to this end specific equipment is required different for various parameters. Moreover, some parameters are difficult to determine. This is one of several reasons for the so-called multiscale approach, where the parameters are computed from specific finite-element analyses based on some realistic geometric representations of the actual microstructure of porous material. Such approach is presented and validated for layers made up of loosely packed small identical rigid spheres. The sound absorption of such layers was measured experimentally in the impedance tube using the so-called two-microphone transfer function method. The layers are characterised by open porosity and semi-regular microstructure: the identical spheres are loosely packed by random pouring and mixing under the gravity force inside the impedance tubes of various size. Therefore, the regular sphere packings were used to generate Representative Volume Elements suitable for calculations at the micro-scale level. These packings involve only one, two, or four spheres so that the three-dimensional finite-element calculations specific for viscous, thermal, and tortuous effects are feasible. In the proposed geometric packings, the spheres were slightly shifted in order to achieve the correct value of total porosity which was precisely estimated for the layers tested experimentally. Finally, in this paper some results based on the self-consistent estimates are also provided.

Self-Consistent Density Estimation

Estimating a continuous density function from a finite set of data points is an important tool in many scientific disciplines. Popular nonparametric density estimators include histograms and kernel density methods. These methods require the researcher to control the degree of smoothing inherent in an estimated function. In a recent approach, a new method for nonparametric density estimation was proposed that finds the estimate self-consistently, that is without requiring the researcher to choose a smoothing parameter a priori. In this article, we outline the basic ideas of the self-consistent density estimator, and we present a Stata implementation of the method. In addition, we present results of Monte Carlo simulations that show that the self-consistent estimator performs better than other methods, especially for larger data samples.

Nonparametric Estimation for Random Censored Data Based on Ranking Set Sampling

In the case where the population distribution is unknown, the Kaplan–Meier estimator of the reliability function based on a ranked set sample with random right-censored data is first proposed. It is shown to be a unique self-consistent estimator. Then, the censored RSS estimator of the population mean is constructed. A simulation study is conducted to compare the performance of the proposed estimators with the corresponding estimators based on a simple random sample. It is shown that the ranked set sampling has higher efficiency. Finally, the proposed method is applied to a renal carcinoma study.