Least-squares Procedure Research Articles

Reduced-order subscales for POD models

In this work the Reduced-Order Subscales for Proper Orthogonal Decomposition models are presented. The basic idea consists in splitting the full-order solution into the part which can be captured by the reduced-order model and the part which cannot, the subscales, for which a model is required. The proposed model for the subscales is defined as a linear function of the solution of the reduced-order model. The coefficients of this linear function are obtained by comparing the solution of the full-order model with the solution of the reduced-order model for the same initial conditions, which, for convenience, are evaluated in the snapshots used to train the original reduced-order-model. The difference between both solutions are the subscales, for which a model can be built using a least-squares procedure. The subscales are then introduced as a correction in the reduced-order model, resulting in an important improvement in accuracy. The enhanced reduced-order model is tested in several numerical examples. These practical cases show that the use of the subscales leads to more accurate solutions, successfully corrects errors introduced by hyper-reduction, and allows to solve complex flow problems using a reduced number of degrees of freedom.

Using Least Squares for Error Propagation

The method of least-squares (LS) has a built-in procedure for estimating the standard errors (SEs) of the adjustable parameters in the fit model: They are the square roots of the diagonal elements of the covariance matrix. This means that one can use least-squares to obtain numerical values of propagated errors by defining the target quantities as adjustable parameters in an appropriate LS fit model. Often this will be an exact, weighted, nonlinear fit, requiring special precautions to circumvent program idiosyncrasies and extract the desired a priori SEs. These procedures are reviewed for several commercial programs and illustrated specifically for the KaleidaGraph program. Examples include the estimation of ΔH°, ΔS°, ΔG°, and K°(T) and their SEs from K° (equilibrium constant) values at two temperatures, with and without uncertainty in T, which is included using the effective variance method, a general-purpose LS procedure for including uncertainty in independent variables. In some cases, the target quantities can be obtained from the original data analysis, by redefining the fit model to include the quantity of interest as an adjustable parameter, automatically handling correlation problems. Examples include the uncertainty in the fit function itself, line areas from spectral line profile data, and the analysis of spectrophotometric data for complex formation.

Uniaxial Compressive Response and Constitutive Modeling of Selected Polymers Over a Wide Range of Strain Rates

The present work deals with constitutive modeling of the compressive stress–strain response of selected polymers at strain rates from 10−3 to nearly 103 s−1. Six different commercially available extruded polymers—ABS, HDPE, PC, POM, PP and PVC—are tested at room temperature. Cylindrical specimens with a slenderness ratio (=length/diameter) of 0.5 are used in high strain-rate tests, and those with the slenderness ratios of 1.0 and 2.0 are used in low and intermediate strain-rate tests. High strain-rate compressive stress–strain loops up to a strain of nearly 0.08 are obtained on a standard split Hopkinson pressure bar. Low and intermediate strain-rate compressive ones are measured on an Instron testing machine. By fitting experimental loading stress–strain data to a modified Ramberg–Osgood equation, material parameters are uniquely determined using a linear least-squares procedure. Experimental results indicate that all polymers tested exhibit intrinsic dynamic viscoelastic–plastic characteristics and a higher elastic after-effect following complete unloading. It is shown that the modified Ramberg–Osgood constitutive model is appropriate for describing the monotonic loading compressive stress–strain relations of the three semi-crystalline polymers over a wide range of strain rates. The advantages and limitations of the constitutive model are also discussed.

Application of long-chain alcohols as faecal markers to estimate diet composition of horses and cattle fed with herbaceous and woody species

Utilization of long-chain alcohols (LCOH) as diet composition markers in horses and cattle was assessed in a study conducted with 12 mature crossbreed mares (385±47 kg BW) and six adult non-lactating cows (499±36 kg BW) of Asturiana de los Valles breed. The LCOH data were combined with alkane and long-chain fatty acid (LCFA) data to test the applicability of combining these markers to estimate diet composition. Animals were randomly divided into groups of three animals and received a daily total amount of 1.0 kg dry matter/100 kg BW of diets composed of different proportions of ryegrass (Lolium perenne) and woody species (Ulex gallii and heather). Diet composition was estimated from even-chain LCOH (C20-OH to C30-OH) combined or not with alkane (C25-C31 and C33) and/or LCFA (C22-FA to C28-FA, C30-FA, C32-FA and C34-FA) concentrations in diet components and faeces by least-squares procedures, using marker faecal concentrations uncorrected for incomplete faecal recovery (FR0) or corrected using mean recoveries across diets within animal species (FR1). Results showed large differences between plant species in their LCOH profiles, and that these markers offered additional discriminatory information to that provided by alkanes and LCFA. The LCOH markers were incompletely recovered in the faeces of both animal species. In cattle, LCOH FR tended to increase with carbon-chain length in a linear manner in both diets (P<0.001), whereas in horses overall data showed a curvilinear relationship between these variables. Combination of LCOH, LCFA and alkanes resulted in more accurate diet estimates. Correction of faecal LCOH concentrations to incomplete FR led to more accurate diet composition estimates in both animal species. Results obtained in this study suggest the usefulness of LCOH markers combined with alkanes and LCFA to estimate diet composition of horses and cattle grazing mixed grassy–woody plant communities.

From direct-space discrepancy functions to crystallographic least squares.

Crystallographic least squares are a fundamental tool for crystal structure analysis. In this paper their properties are derived from functions estimating the degree of similarity between two electron-density maps. The new approach leads also to modifications of the standard least-squares procedures, potentially able to improve their efficiency. The role of the scaling factor between observed and model amplitudes is analysed: the concept of unlocated model is discussed and its scattering contribution is combined with that arising from the located model. Also, the possible use of an ancillary parameter, to be associated with the classical weight related to the variance of the observed amplitudes, is studied. The crystallographic discrepancy factors, basic tools often combined with least-squares procedures in phasing approaches, are analysed. The mathematical approach here described includes, as a special case, the so-called vector refinement, used when accurate estimates of the target phases are available.

Synthesis, Characterization and Crystal Structure of Ethyl 2-amino-4-(4-fluorophenyl)-6-phenylcyclohexa-1,3-diene-1-carboxylate

Ethyl 2-amino-4-(4-fluorophenyl)-6-phenylcyclohexa-1,3-diene-1-carboxylate was synthesized from ethyl 6-(4-bromophenyl)-4-(4-fluorophenyl)-2-oxocyclohex-3-ene-1-carboxylate and ammonium acetate in glacial acetic acid. The synthesized compound was characterized by elemental analysis, FT-IR, thermogravimetric analysis (TGA), differential thermal analysis (DTA), UV-Visible, and single-crystal X-ray diffraction. The title compound, ethyl 2-amino-4-(4-fluorophenyl)-6-phenylcyclohexa-1,3-diene-1-carboxylate, C21H2OFNO2, crystallizes in the orthorhombic space group Pbca with the following unit-cell parameters: a = 17.417(2), b = 9.7287(9), c = 21.014(2) Å, and Z = 8. The crystal structure was solved by direct methods using single-crystal X-ray diffraction data collected at room temperature and refined by full-matrix least-squares procedures to a final R-value of 0.0644 for 1616 observed reflections. An intramolecular N–H…O hydrogen bond generates an S(6) graph-set motif. In the crystal, molecules are linked by N–H…O hydrogen bonds, forming a two-dimensional network.

A Comparative Analysis of Speed Profile Models for Ankle Pointing Movements: Evidence that Lower and Upper Extremity Discrete Movements are Controlled by a Single Invariant Strategy.

Little is known about whether our knowledge of how the central nervous system controls the upper extremities (UE), can generalize, and to what extent to the lower limbs. Our continuous efforts to design the ideal adaptive robotic therapy for the lower limbs of stroke patients and children with cerebral palsy highlighted the importance of analyzing and modeling the kinematics of the lower limbs, in general, and those of the ankle joints, in particular. We recruited 15 young healthy adults that performed in total 1,386 visually evoked, visually guided, and target-directed discrete pointing movements with their ankle in dorsal–plantar and inversion–eversion directions. Using a non-linear, least-squares error-minimization procedure, we estimated the parameters for 19 models, which were initially designed to capture the dynamics of upper limb movements of various complexity. We validated our models based on their ability to reconstruct the experimental data. Our results suggest a remarkable similarity between the top-performing models that described the speed profiles of ankle pointing movements and the ones previously found for the UE both during arm reaching and wrist pointing movements. Among the top performers were the support-bounded lognormal and the beta models that have a neurophysiological basis and have been successfully used in upper extremity studies with normal subjects and patients. Our findings suggest that the same model can be applied to different “human” hardware, perhaps revealing a key invariant in human motor control. These findings have a great potential to enhance our rehabilitation efforts in any population with lower extremity deficits by, for example, assessing the level of motor impairment and improvement as well as informing the design of control algorithms for therapeutic ankle robots.

Variable Selection for Semiparametric Partially Linear Covariate-Adjusted Regression Models

In this article, the partially linear covariate-adjusted regression models are considered, and the penalized least-squares procedure is proposed to simultaneously select variables and estimate the parametric components. The rate of convergence and the asymptotic normality of the resulting estimators are established under some regularization conditions. With the proper choices of the penalty functions and tuning parameters, it is shown that the proposed procedure can be as efficient as the oracle estimators. Some Monte Carlo simulation studies and a real data application are carried out to assess the finite sample performances for the proposed method.

Validation of a two-dimensional modeling of an externally irradiated slurry photoreactor

A two-dimensional model describing the behavior of a batch cylindrical photoreactor has been validated through a series of photocatalytic reactivity runs. The light intensity radiation field produced by 1–6 external UV fluorescent lamps was modeled together with Langmuir–Hinshelwood kinetics and 4-nitrophenol (NP) and Degussa P25 TiO2 were used as the probe molecule and the catalyst, respectively. The experiments were carried out under different conditions of light intensity, photocatalyst amount and substrate concentration. The model parameters were the kinetic constant of substrate adsorption, desorption and degradation and the exponent of the power law expressing the kinetics dependence on the light intensity. A least-squares best fitting procedure was used to fit the experimental data to the model and find the four required parameters. A satisfactory agreement was obtained between experimental and calculated values of the observed pseudo-first order kinetic constant, with a coefficient of determination R2=0.83. Notably, the present model is of general interest for cylindrical slurry photoreactors, where different molecules can be oxidized in the presence of several photocatalysts and it can be used, for instance, to accurately compare the performance of a set of catalysts.

An algorithm to decompose ground reaction forces and moments from a single force platform in walking gait

In walking experimental conditions, subjects are sometimes unable to perform two steps on two different forceplates. This leads the authors to develop methods for discerning right and left ground reaction data while they are summed during the double support in walking. The aim of this study is to propose an adaptive transition function that considers the walking speed and ground reaction forces (GRF). A transition function is used to estimate left and right side GRF signals in double support. It includes a shape coefficient adjusted using single support GRF parameters. This shape coefficient is optimized by a non-linear least-square curve-fitting procedure to match the estimated signals with real GRF. A multiple regression is then performed to identify GRF parameters of major importance selected to compute the right and left GRF of the double support. Relative RMSE (RMSER), maximum GRF differences normalized to body mass and differences of center of pressure (CoP) are computed between real and decomposed signals. During double support, RMSER are 6%, 18%, 3.8%, 4.3%, 3%, and 12.3% for anterior force, lateral force, vertical force, frontal moment, sagittal moment and transverse moment, respectively. Maximum GRF differences normalized to body mass are lower than 1N/kg and mean CoP difference is 0.0135m, when comparing real to decomposed signals during double support. This work shows the accuracy of an adaptive transition function to decompose GRF and moment of right and left sides. This method is especially useful to accurately discern right and left GRF data in single force platform configurations.

A NEW DOMAIN DECOMPOSITION PARALLEL ALGORITHM FOR CONVECTION–DIFFUSION PROBLEM

Basing on overlapping domain decomposition, we construct a new parallel algorithm combined the method of subspace correction with least-squares procedure for solving time-dependent convection–diffusion problem. This algorithm is fully parallel. We analyze the convergence of approximate solution, and study the dependence of the convergent rate on the spacial mesh size, time increment, iteration number and sub-domains overlapping degree. Both theoretical analysis and numerical results suggest that only one or two iterations are needed to reach to given accuracy at each time step.

Variations in Abundance Enhancements in Impulsive Solar Energetic-Particle Events and Related CMEs and Flares

We study event-to-event variations in the abundance enhancements of the elements He through Pb for Fe-rich impulsive solar energetic-particle (SEP) events, and their relationship with properties of associated coronal mass ejections (CMEs) and solar flares. Using a least-squares procedure we fit the power-law enhancement of element abundances as a function of their mass-to-charge ratio A/Q to determine both the power and the coronal temperature (which determines Q) in each of 111 impulsive SEP events identified previously. Individual SEP events with the steepest element enhancements, e.g. ∼ (A/Q)6, tend to be smaller, lower-fluence events with steeper energy spectra that are associated with B- and C-class X-ray flares, with cooler (∼ 2.5 MK) coronal plasma, and with narrow (< 100∘), slower (< 700 km s−1) CMEs. On the other hand, higher-fluence SEP events have flatter energy spectra, less-dramatic heavy-element enhancements, e.g. ∼ (A/Q)3, and come from somewhat hotter coronal plasma (∼ 3.2 MK) associated with C-, M-, and even X-class X-ray flares and with wider CMEs. Enhancements in 3He/4He are uncorrelated with those in heavy elements. However, events with 3He/4He≥0.1 are even more strongly associated with narrow, slow CMEs, with cooler coronal plasma, and with B- and C-class X-ray flares than are other Fe-rich impulsive SEP events with smaller enhancements of 3He.

Site-selective high-resolution X-ray absorption spectroscopy and high-resolution X-ray emission spectroscopy of cobalt nanoparticles.

The special (macroscopic) properties of nanoparticles are mainly due to their large surface-to-volume ratio. Thus, the separate characterization of geometric and electronic properties of surface and bulk would be favorable for a better understanding of the properties of nanoparticles. Because of the chemical sensitivity of X-ray fluorescence lines, in particular those involving higher lying electronic states, high-resolution fluorescence-detected X-ray absorption spectra (HRFD-XAS) offer these opportunities. In this study, three types of wet-chemically synthesized Co nanoparticles, ∼6 nm in diameter with varying thicknesses of a protective shell, were investigated at the ID26 beamline of the European Synchrotron Radiation Facility. HRFD-XAS spectra at the Co K-edge, that is, X-ray absorption near-edge structure (HRFD-XANES) and extended X-ray absorption fine structure (HRFD-EXAFS) spectra, were recorded via detection of the Kβ1,3 fluorescence at specific energies. As these spectra are only partly site-selective due to a strong overlap of the emission lines, a numerical procedure was applied based on a least-squares fitting procedure, realized by singular value decomposition. The detailed analysis of the obtained site-selective spectra, regarding chemical composition and crystallographic phase, using measured and simulated FEFF9-based reference spectra, showed that the metallic core had mainly hexagonal close-packed structure with lattice constants matching bulk Co; the spectra for the shell could be satisfactorily fitted by a mixture of CoO and CoCO3; however, with an obvious need for at least a third compound. To obtain additional information about ligands attached to Co, valence-to-core X-ray emission spectra (VTC-XES) using the Kβ2,5 and the satellite structure Kβ″ and VTC-XANES spectra thereof were also recorded, by which the former results are confirmed. Further on, FEFF simulations indicate that a Co-N compound is a very likely candidate for the third component. The presented results clearly show that VTC-XES and HRFD-XAS are suitable tools for the detailed specification of the core and the surface of nanoparticles, in particular upon realizing "real" site-selectivity for XANES and EXAFS with a general strategy applicable to a wide range of systems.

Modelling functional additive quantile regression using support vector machines approach

This work deals with conditional quantiles estimation when several functional covariates are involved, via a support vector machines nonparametric methodology. We establish weak consistency of this estimator. To fit the additive components, we use an ordinary backfitting procedure combined with an iterative reweighted least-squares procedure to solve the penalised minimisation problem. This procedure makes it possible to derive a split sample method for choosing the hyper-parameters of the model. The performances of the proposed technique, in terms of forecast accuracy, are evaluated through simulation and a real dataset study.

Accurate equilibrium structures of fluoro- and chloroderivatives of methane

This work is a systematic study of molecular structure of fluoro-, chloro-, and fluorochloromethanes. For the first time, the accurate ab initio structure is computed for 10 molecules (CF4, CClF3, CCl2F2, CCl3F, CHClF2, CHCl2F, CH2F2, CH2ClF, CH2Cl2, and CCl4) at the coupled cluster level of electronic structure theory including single and double excitations augmented by a perturbational estimate of the effects of connected triple excitations [CCSD(T)] with all electrons being correlated and Gaussian basis sets of at least quadruple-ζ quality. Furthermore, when possible, namely for the molecules CH2F2, CH2Cl2, CH2ClF, CHClF2, and CCl2F2, accurate semi-experimental equilibrium (rSEe) structure has also been determined. This is achieved through a least-squares structural refinement procedure based on the equilibrium rotational constants of all available isotopomers, determined by correcting the experimental ground-state rotational constants with computed ab initio vibration–rotation interaction constants and electronic g-factors. The computed and semi-experimental equilibrium structures are in excellent agreement with each other, but the rSEe structure is generally more accurate, in particular for the CF and CCl bond lengths. The carbon–halogen bond length is discussed within the framework of the ligand close-packing model as a function of the atomic charges. For this purpose, the accurate equilibrium structures of some other molecules with alternative ligands, such as CH3Li, CF3CCH, and CF3CN, are also computed.

Synthesis, Characterization, and Crystal Structure of Negundoside (2′-p-Hydroxybenzoyl Mussaenosidic Acid)

The structure of title compound Negundoside (2′-p-hydroxybenzoyl mussaenosidic acid) was established by spectral and X-ray diffraction studies. The compound crystallizes in the monoclinic crystal system with space group P21 having unit cell parameters: a=11.6201 (5) Å, b=9.2500 (4) Å, c=12.2516 (5) Å, β=97.793 (4)°, and Z=2. The crystal structure was solved by direct method using single crystal X-ray diffraction data collected at room temperature and refined by full-matrix least-squares procedures to a final R value of 0.0520 for 3389 observed reflections.

STRUCTURE OF 2-AMINO-5-OXO-4-p-TOLYL-4,5-DIHYDROPYRANO[3,2-c]CHROMENE-3-CARBONITRILE

Normal 0 The compound 2-Amino-5-oxo-4- p -tolyl-4,5-dihydropyrano[3,2- c ]chromene-3-carbonitrile , crystallizes in the monoclinic space group P121/c1 with the unit-cell parameters: a= 9.1330(7), b= 13.1343(9), c= 13.1945(8) A, β = 91.746(4) o and Z = 4. The crystal structure was solved by direct methods using single-crystal X-ray diffraction data collected at room temperature and refined by full-matrix least-squares procedures to a final R-value of 0.0537 for 1830 observed reflections. The molecules within the unit cell are stabilized by C-H..O, N-H..O N-H..N and C-H.. π type of hydrogen bonding.

DMF SOLVATED CRYSTAL STRUCTURE OF 3,3,6,6-TETRAMETHYL-9-(3-CHLORO PHENYL)-3,4,6,7,9,10-HEXAHYDROACRIDINE-1,8-DIONE

Normal 0 The dimethylformamide solvated crystal structure of 3,3,6,6-tetramethyl-9-(3-chlorophenyl)-3,4,6,7,9,10-hexahydroacridine-1,8-dione (C 26 H 33 Cl N 2 O 3 ) crystallizes in the triclinic space group P -1with unit cell parameters: a=9.0814(4), b=11.5650(5), c=12.1354(5) Ǻ, a =91.806(3)o, β=96.450(3)o, g = 104.565(4)o and Z = 2. The crystal structure was solved by direct methods and refined by full-matrix least-squares procedures to a final R -value of 0.0 56 for 2691 observed reflections. The central ring of the acridinedione system adopts a boat conformation and the two outer rings adopt sofa conformations. The solvent moiety (Dimethylformamide) is involved in the existence of intra and intermolecular C-H...O interactions.

Elastic behavior and pressure-induced structure evolution of topaz up to 45 GPa

The behavior of a natural topaz, Al2.00Si1.05O4.00(OH0.26F1.75), has been investigated by means of in situ single-crystal synchrotron X-ray diffraction up to 45 GPa. No phase transition or change in the compressional regime has been observed within the pressure-range investigated. The compressional behavior was described with a third-order Birch–Murnaghan equation of state (III-BM-EoS). The III-BM-EoS parameters, simultaneously refined using the data weighted by the uncertainties in P and V, are as follows: KV = 158(4) GPa and KV′ = 3.3(3). The confidence ellipse at 68.3 % (Δχ2 = 2.30, 1σ) was calculated starting from the variance–covariance matrix of KV and K′ obtained from the III-BM-EoS least-square procedure. The ellipse is elongated with a negative slope, indicating a negative correlation of the parameters KV and KV′, with KV = 158 ± 6 GPa and KV′ = 3.3 ± 4. A linearized III-BM-EoS was used to obtain the axial-EoS parameters (at room-P), yielding: K(a) = 146(5) GPa [βa = 1/(3K(a)) = 0.00228(6) GPa−1] and K′(a) = 4.6(3) for the a-axis; K(b) = 220(4) GPa [βb = 0.00152(4) GPa−1] and K′(b) = 2.6(3) for the b-axis; K(c) = 132(4) GPa [βc = 0.00252(7) GPa−1] and K′(c) = 3.3(3) for the c-axis. The elastic anisotropy of topaz at room-P can be expressed as: K(a):K(b):K(c) = 1.10:1.67:1.00 (βa:βb:βc = 1.50:1.00:1.66). A series of structure refinements have been performed based on the intensity data collected at high pressure, showing that the P-induced structure evolution at the atomic scale is mainly represented by polyhedral compression along with inter-polyhedral tilting. A comparative analysis of the elastic behavior and P/T-stability of topaz polymorphs and “phase egg” (i.e., AlSiO3OH) is carried out.

X-ray Crystallography of bis(O-ethylxanthato)bis(3,5-dimethyl pyridine)nickel(II)

Normal 0 A six-coordinated Ni(II) complex, bis( O -ethylxanthato)bis(3, 5-dimethylpyridine)nickel(II), [ C 20 H 28 N 2 NiO 2 S 4 ], has been synthesized by the reaction of bis( O -ethylxanthato)nickel(II) with 3,5-dime thyl pyridine in acetone. The Ni atom in the title complex is octahedrally coordinated within a trans-N 2 S 4 donor set, with the Ni atom located on a centre of inversion. The structure of the title compound was elucidated by a single-crystal X-ray diffraction method. The compound crystallizes in the triclinic space group P-1 with unit-cell parameters: a = 7.1750(3) A , b = 9.3864(5) A , c = 9.6914(4) A and α = 84.962(4)°, β = 73.017(4)°, γ = 75.172(4)° . The crystal structure was solved by direct methods and refined by full-matrix least-squares procedures to a final R -value of 0.0238 for 2162 observed reflections. The asymmetric unit comprises half a molecule. The pyridine ring is coplanar, and is held almost perpendicular to the dithiocarbonato group. Molecules in the crystal are packed together to form layers.