Empirical Fitting Procedure Research Articles

Seascape origin of Richards growth.

First proposed as an empirical rule over half a century ago, the Richards growth equation has been frequently invoked in population modeling and pandemic forecasting. Central to this model is the advent of a fractional exponent γ, typically fitted to the data. While various motivations for this nonanalytical form have been proposed, it is still considered foremost an empirical fitting procedure. Here, we find that Richards-like growth laws emerge naturally from generic analytical growth rules in a distributed population, upon inclusion of (i) migration (spatial diffusion) among different locales, and (ii) stochasticity in the growth rate, also known as "seascape noise." The latter leads to a wide (power law) distribution in local population number that, while smoothened through the former, can still result in a fractional growth law for the overall population. This justification of the Richards growth law thus provides a testable connection to the distribution of constituents of the population.

Applying an in-situ calibration method of spectral line shape to determine flame temperature of methane and ethylene spherically expanding flames

Spectral line shape caused by the collision between probing molecules and surrounding species is an important parameter that affects the accuracy of combustion diagnostics. Here, we report an in-situ calibration method of the pressure broadening coefficients using spherically expanding flames over continuously increasing flame pressure. A desirable isothermal condition is produced in the spherical expanding flame with unity Lewis number, in which the thermal and molecular diffusions are balanced during flame propagation. Laminar burning velocities and Markstein length of methane-air and ethylene-air flames are measured to understand their thermodynamic and transport properties. Mid-infrared absorption spectra of carbon dioxide at 4.2 µm are recorded from 0.3 atm to 5 atm in a single run of the experiment. Pressure broadening parameters are extracted using an empirical fitting procedure. After extending to wide ranges of flame conditions, the flame temperatures of the methane-air and ethylene-air flames are correctly determined using the in-situ-determined line shape parameters. The flame temperature deviation from adiabatic flame temperature is imperceptible in the weakly stretched quasi-steady spherically expanding flames. Such unity Lewis number spherical expanding flames are proven to be near-ideal reactors for spectroscopic calibration in combustion diagnostics.

Effects of stochastic parametrization on extreme value statistics.

Extreme geophysical events are of crucial relevance to our daily life: they threaten human lives and cause property damage. To assess the risk and reduce losses, we need to model and probabilistically predict these events. Parametrizations are computational tools used in the Earth system models, which are aimed at reproducing the impact of unresolved scales on resolved scales. The performance of parametrizations has usually been examined on typical events rather than on extreme events. In this paper, we consider a modified version of the two-level Lorenz'96 model and investigate how two parametrizations of the fast degrees of freedom perform in terms of the representation of extreme events. One parametrization is constructed following Wilks [Q. J. R. Meteorol. Soc. 131, 389-407 (2005)] and is constructed through an empirical fitting procedure; the other parametrization is constructed through the statistical mechanical approach proposed by Wouters and Lucarini [J. Stat. Mech. Theory Exp. 2012, P03003 (2012); J. Stat. Phys. 151, 850-860 (2013)]. The two strategies show different advantages and disadvantages. We discover that the agreement between parametrized models and true model is in general worse when looking at extremes rather than at the bulk of the statistics. The results suggest that stochastic parametrizations should be accurately and specifically tested against their performance on extreme events, as usual optimization procedures might neglect them.

Ultrafast Spectral Photoresponse of Bilayer Graphene: Optical Pump-Terahertz Probe Spectroscopy.

Photoinduced terahertz conductivity Δσ(ω) of Bernal stacked bilayer graphene (BLG) with different dopings is measured by time-resolved optical pump terahertz probe spectroscopy. The real part of photoconductivity Δσ(ω) (ΔσRe(ω)) is positive throughout the spectral range 0.5-2.5 THz in low-doped BLG. This is in sharp contrast to Δσ(ω) for high-doped bilayer graphene where ΔσRe(ω) is negative at low frequency and positive on the high frequency side. We use Boltzmann transport theory to understand quantitatively the frequency dependence of Δσ(ω), demanding the energy dependence of different scattering rates such as short-range impurity scattering, Coulomb scattering, carrier-acoustic phonon scattering, and substrate surface optical phonon scattering. We find that the short-range disorder scattering dominates over other processes. The calculated photoconductivity captures very well the experimental conductivity spectra as a function of lattice temperature varying from 300 to 4 K, without any empirical fitting procedures adopted so far in the literature. This helps us to understand the intraband conductivity of photoexcited hot carriers in 2D materials.

Merging Empirical Valence Bond Theory with Quantum Chemistry to Model Proton Transfer Processes in Water

Proton transfer processes in water are of fundamental importance for, among others, electrochemical proton discharge. Empirical valence bond (EVB) approaches were shown in the past to be a versatile tool for modeling complex phenomena such as proton discharge at metal electrodes. By replacing empirical fitting procedures with on-the-fly quantum chemistry (QC) calculations, we arrive at a transferable and systematically tunable description of proton transfer in water with EVB.

Atomistic simulation of trivalent ions doped in the hexagonal LuMnO3 ferroelectric phase

The interatomic potentials for multiferroic hexagonal LuMnO3 (h-LMO) in its ferroelectric (FE) phase was obtained for empirical fitting procedure and applied to study the effects of a series of trivalent metal and rare-earth dopants on its structural and ferroelectric distortions. The potential parameters found accurately reproduce lattice parameters (error below 0.3%), interatomic distances, pressure effect on the lattice parameters, and tilting angles of the MnO5 polyhedron. We investigated doping at Mn, Lu1 and Lu2 atom sites. It was found that the doped Mn atom site is the most energetically advantageous site for both the metal (Al, Ga, Fe and Cr) and rare-earth (La, Nb, Sm, Eu, Gd, Ho, Er, Yb, Y and Sc) atoms studied. The M-O3/O4 distances decrease when the Mn3+ ion site is doped by trivalent metal ions. The effects on the FE distortion of the material under doping and changes in the tilting angle of the M/MnO5 polyhedron were analyzed.

Screened hybrid functional applied to 3d0→3d8transition-metal perovskites LaMO3(M = Sc–Cu): Influence of the exchange mixing parameter on the structural, electronic, and magnetic properties

We assess the performance of the Heyd-Scuseria-Ernzerhof (HSE) screened hybrid density functional scheme applied to the perovskite family LaMO3 (M=Sc-Cu) and discuss the role of the mixing parameter alpha (which determines the fraction of exact Hartree-Fock exchange included in the density functional theory (DFT) exchange-correlation functional) on the structural, electronic, and magnetic properties. The physical complexity of this class of compounds, manifested by the largely varying electronic characters (band/Mott-Hubbard/charge-transfer insulators and metals), magnetic orderings, structural distortions (cooperative Jahn-Teller like instabilities), as well as by the strong competition between localization/delocalization effects associated with the gradual filling of the t_2g and e_g orbitals, symbolize a critical and challenging case for theory. Our results indicates that HSE is able to provide a consistent picture of the complex physical scenario encountered across the LaMO3 series and significantly improve the standard DFT description. The only exceptions are the correlated paramagnetic metals LaNiO3 and LaCuO3, which are found to be treated better within DFT. By fitting the ground state properties with respect to alpha we have constructed a set of 'optimum' values of alpha from LaScO3 to LaCuO3: it is found that the 'optimum' mixing parameter decreases with increasing filling of the d manifold (LaScO3: 0.25; LaTiO3 & LaVO3: 0.10-0.15; LaCrO3, LaMnO3, and LaFeO3: 0.15; LaCoO3: 0.05; LaNiO3 & LaCuO3: 0). This trend can be nicely correlated with the modulation of the screening and dielectric properties across the LaMO3 series, thus providing a physical justification to the empirical fitting procedure.

Recent progress in theoretical analysis of vibrational sum frequency generation spectroscopy

This article summarizes the computational analysis of the vibrational sum frequency generation (SFG) spectroscopy with molecular dynamics simulation. The analysis allows direct comparison of experimental SFG spectra and microscopic interface structure obtained by molecular simulation, and thereby obviates empirical fitting procedures of the observed spectra. In the theoretical formulation, the frequency-dependent nonlinear susceptibility of an interface is calculated in two ways, based on the energy representation and time-dependent representation. The application to aqueous interfaces revealed a number of new insights into the local structure of electrolyte interfaces and the interpretation of SFG spectroscopy.

Interatomic potential parameters for potassium tetrachlorozincate and their application to modelling its phase transformations.

An empirical fitting procedure is applied to derive interatomic potential parameters for a model phase transition system, namely potassium tetrachlorozincate (K(2)ZnCl(4)). The derived potential is found to reliably model the known crystallographic structure for the ferroelectric and paraelectric phases of this compound. Potential transferability is demonstrated by applying the parameters derived to the optimization of the known molecular structure for a similar inorganic system (rubidium tetrachlorozincate).

Model-Free Analysis of Stretched Relaxation Dispersions

Nuclear magnetic relaxation dispersion (NMRD) measurements can provide valuable information about the dynamics and structure of macromolecular solutions and other complex fluids. A large number of1H NMRD studies of water in concentrated protein solutions and in semisolid biological samples have been reported. The observed dispersion usually extends over a wide frequency range and then cannot be described by a Lorentzian spectral density function. We propose here a model-free approach for analyzing such stretched dispersion profiles. Unlike the traditional empirical fitting procedures, the model-free approach is based on rigorous theory and produces parameters with well-defined physical significance. The model-free approach is validated with the aid of synthetic relaxation data, showing that it is robust and accurate, and is then applied to new water1H NMRD data from solutions of the protein bovine pancreatic trypsin inhibitor (BPTI). By separating the static and dynamic information content of the relaxation dispersion, the model-free analysis shows that the dramatic salt effect observed in BPTI solutions is due almost entirely to a slowing down of protein rotation with little change of protein structure. An analysis of the same data in terms of the empirical dispersion function used in most1H NMRD studies leads to a qualitatively different picture. We demonstrate that this widely used dispersion function is unphysical and that its parameters do not have the physical meaning usually ascribed to them.

Deriving empirical potentials for molecular ionic materials

AbstractThe procedure for deriving interatomic potentials for molecular ionic materials, using empirical fitting procedures, is described. Potentials are obtained for carbonates, phosphates and perchlorates, and used to calculate crystal and lattice properties which are compared with available experimental data.

Computer simulation of defects in KTaO3.

Using atomistic computer simulation methods we propose a general overview of the point defect structure of ${\mathrm{KTaO}}_{3}$. A set of short-range potential parameters was derived using an empirical fitting procedure. These parameters were then applied within the framework of the shell model. We employed a two-region strategy in combination with the Mott-Littleton technique for defect calculations. Intrinsic defects including Frenkel and Schottky-type disorder, polarons, and reduction/oxidation mechanisms are treated in the study as well as a great number of extrinsic defects. The way these ions are introduced into the crystal, their agglomeration into defect complexes and possible off-center displacements are investigated. It is found that in ${\mathrm{KTaO}}_{3}$, intrinsic disorder is of minor significance and small electron polarons possibly are stable. The calculations support the idea that reduction is significantly influenced by preexisting oxygen vacancies, which result from charge compensation of accidental acceptor impurities in the material. Extrinsic monovalent ions substitute on the potassium site for which no charge compensation is necessary. Ions of other valencies are incorporated on both cation lattice sites which then gives rise to a charge self-compensation, in some cases combined with the formation of oxygen vacancies.

The correction for TDS in time-of-flight Laue diffraction. II. An empirical fitting procedure

A simple method of correcting the diffraction pattern for the effect of thermal diffuse scattering in the case of time-of-flight neutron Laue diffraction is outlined. The method, based on empirical peak fitting around the Bragg peak positions, is shown to be straightforward and particularly appropriate for this data collection method. The corrected data are valuable in terms of both Bragg intensity extraction, achieved through extrapolation to the Bragg peak positions, and the appearence of the corrected diffraction pattern throughout reciprocal space. The possible use of the corrected diffraction pattern in the quantitative analysis of, for example, diffuse scattering arising from disordered or non-stoichiometric materials is discussed, along with the potential use of a library of empirical TDS correction models.

Reanalysis of the Am I level spectrum and the nuclear quadrupole moments of Am-isotopes

The fine structure (fs) and hyperfine structure (hfs) level scheme of Am I is reanalysed using a semi empirical fitting procedure which incorporates experimental data. Especially new laserspectroscopic measurements of the hfs of some electronic transitions in the Am atom enables us to make a more detailed analysis of 5f 7 7s 7p fine structure in Am I. In particular a relation is given between theB-factor values and the value of the nuclear electric quadrupole moment of241Am independent of a calibration by results of nuclear spectroscopy.

Optimisation of efficiency extrapolation functions in radioactivity standardisation

When radioactivity is measured by 4πβ-γ efficiency extrapolation the data is commonly fitted to a linear function, but also to a polynomial function of second or third order if this proves necessary. In order to justify the empirical fitting procedure, we compared the experimental results with some functions which could be expected theoretical grounds. The most suitable function was a linear function at high 4π counting efficiencies joined on to a second-order function at low 4π counting efficiencies, while a third-order polynomial function without second-order term produced accurate results regardless of the fitting region.

Interatomic potentials:: Derivation of parameters for binary oxides and their use in ternary oxides

In this paper interatomic potentials for ionic oxides are derived in a consistent way using an empirical fitting procedure. Shell-model parameters are obtained which are transferable providing anion polarisability is allowed to vary with structure and host cation. Changes in cation coordination are reflected in a modification of the cation-anion short-range interaction parameters. Parameters are derived for a range of binary oxides and the application of these parameters to the modelling of ternary oxides is discussed.

Conjoint Analysis: A Comparative Analysis of Specification Tests for the Utility Function

The focus of this paper is on determining appropriate combination rules for idiosyncratic ordinal utility functions in conjoint measurement. An axiomatic diagnosis is used which is based on explanatory criteria rather than goodness-of-fit or predictive criteria. Experimental results are presented comparing the selection of combination rules based on axiomatic tests and empirical fitting procedures for additive (x + y + z) and distributive (x(y + z)) combinations of three factors. A number of conflicting diagnoses result from the simultaneous applications of axiomatic tests and goodness-of-fit criteria, suggesting the importance of testing certain necessary conditions for simple polynomial combination rules, such as additive and distributive rules. Because the axiomatic approach is effective in rejecting inappropriate functional forms, we argue that it should complement rather than replace the empirical fitting diagnostic approaches.

Ab initio treatment of the Renner-Teller effect for the X 2 B 1 and A 2 A 1 electronic states of NH2

A large AO basis including atom-centred f- and s- and p-type bond functions is employed to obtain potential curves for the X 2 B 1 and A 2 A 1 states of NH2 at close to the full CI level of treatment. Two different matrix methods are then applied to describe the Renner-Teller coupling between the bending vibrational states of the two electronic species; excellent agreement is found between the two sets of theoretical results. Use of a simple kinetic energy operator with a constant reduced mass leads to a term value spectrum which agrees to within 100 cm-1 for the measured locations of all low lying vibronic levels in this system and to within 300 cm-1 of the highest lying such species observed to date. Similarly good agreement is found for the ND2 and NHD isotopic species, whereby in these cases the amount of vibronic coupling is, not surprisingly, found to be markedly less than in NH2 itself. Equilibrium bond lengths and angles of this system are calculated to agree to within 0·01 Å and 1° of their observed values and computed vibrational frequencies are found to agree to within 3 per cent of their measured counterparts. The ab initio T e value for the A 2 A 1-X 2 B 1 transition is 11 350 cm-1, in very good accord with the value recently reported by Jungen, Hallin and Merer of 11 294 cm-1 based on an empirical fitting procedure to the observed results, but the calculated barrier height of 980 cm-1 in the A 2 A 1 state is 250 cm-1 greater than the corresponding value obtained from the JHM empirical fit. Furthermore such discrepancies between ab initio CI and empirically fitted potential data become much greater for small bond angles beyond ϑ = 90°, whereby it is argued that uncertainties in the form of the bending kinetic energy operator for large amplitude motion as well as failure to consider a low-energy 2 B 2 state in NH2 which undergoes a Jahn-Teller intersection with 2 A 1 in the neighbourhood of ϑ = 70° make it difficult for empirical fitting procedures to obtain high accuracy for the potential curves in the small angle bending region. Finally the computed intensity distributions for the A 2 A 1-X 2 B 1 transition in this system are found to be in almost quantitative agreement both with what has been measured and also with what is obtained in the JHM fitting procedure.