Statistical Moment Method Research Articles

Statistical methods of parameter estimation for deterministically chaotic time series

We discuss the possibility of applying some standard statistical methods (the least-square method, the maximum likelihood method, and the method of statistical moments for estimation of parameters) to deterministically chaotic low-dimensional dynamic system (the logistic map) containing an observational noise. A "segmentation fitting" maximum likelihood (ML) method is suggested to estimate the structural parameter of the logistic map along with the initial value x(1) considered as an additional unknown parameter. The segmentation fitting method, called "piece-wise" ML, is similar in spirit but simpler and has smaller bias than the "multiple shooting" previously proposed. Comparisons with different previously proposed techniques on simulated numerical examples give favorable results (at least, for the investigated combinations of sample size N and noise level). Besides, unlike some suggested techniques, our method does not require the a priori knowledge of the noise variance. We also clarify the nature of the inherent difficulties in the statistical analysis of deterministically chaotic time series and the status of previously proposed Bayesian approaches. We note the trade off between the need of using a large number of data points in the ML analysis to decrease the bias (to guarantee consistency of the estimation) and the unstable nature of dynamical trajectories with exponentially fast loss of memory of the initial condition. The method of statistical moments for the estimation of the parameter of the logistic map is discussed. This method seems to be the unique method whose consistency for deterministically chaotic time series is proved so far theoretically (not only numerically).

Pharmacokinetic study of carbamazepine and its carbamazepine 10,11-epoxide metabolite in a group of female epileptic patients under chronic treatment

Background Although epileptic crises are equally frequent in women and men, several factors cause female epileptics to present a series of gender-specific problems. To date, few studies have been published on the kinetics of carbamazepine (CBZ) and carbamazepine 10,11-epoxide (CBZ-E) active metabolite in a Mexican population, and no information for epileptic women of reproductive age is available. The aim of the present work was to study the pharmacokinetic behavior of this group of women during steady state. Methods Fourteen epileptic women under chronic treatment receiving only the anticonvulsant CBZ to control their crises were studied. A blood sample was taken before breakfast, before the morning dose of 200 mg, and after the dose at 1, 2, 3, 4, 5, and 8 h. Serum was separated by centrifugation at 1,350× g. Serum concentrations of carbamazepine (CBZ) and of the metabolite carbamazepine 10,11-epoxide (CBZ-E) were measured by HPLC. Pharmacokinetic parameters were calculated by statistical moment method after obtaining serum concentrations. Results Maximum time (T max) for CBZ was reached at 2.72±0.71 h and for CBZ-E, it was 3.60±0.79 h. C max for CBZ was 7.30±2.30 μg/mL, while C min for CBZ was 6.30±2.49. Maximum serum values for CBZ-E were 1.01±0.57, equivalent to 13.80% of CBZ; t 1 2 value for CBZ and CBZ-E was 18.20 and 16.10 h, respectively. AUC values for CBZ and metabolite were 70.33±17.10 μg/L/h and 9.20±2.50 μg/L/h, respectively. CBZ and CBZ-E clearance did not show differences and were 0.37 mL/kg/min and 0.40 mL/kg/min, respectively. Extraction index for serum concentrations of CBZ and CBZ-E AUC CBZ/AUC CBZ-E was 0.13; positive correlation was observed between serum concentrations of CBZ and E-CBZ, with r = 0.94. Conclusions The schedule we suggest for therapeutic monitoring of serum concentrations of CBZ in chronic treatments is 3 h for maximum peak concentration of C max after dose administration and for minimum peak concentration, C min prior to subsequent administration of the dose.

Thermodynamic quantities of metals investigated by an analytic statistical moment method

The thermodynamic properties of metals are studied by including explicitly the anharmonic effects of the lattice vibrations going beyond the quasiharmonic approximations. The free energy, thermal lattice expansion coefficients, mean-square atomic displacements, and specific heats at the constant volume and those at the constant pressure, ${C}_{v}$ and ${C}_{p},$ are derived in closed analytic forms in terms of the power moments of the atomic displacements. The analytical formulas give highly accurate values of the thermodynamic quantities, which are comparable to those of the molecular dynamics or Monte Carlo simulations for a wide temperature range. The present formalism is well suited to calculate the thermodynamic quantities of metals and alloys by including the many body electronic effects and by combining it with the first-principles approaches.

Calculation of thermodynamic quantities of metals and alloys by statistical moment method

The thermodynamic quantities of metals and alloys are studied using the moment method in the quantum statistical mechanics, going beyond the quasi-harmonic approximations. Including the power moments of the atomic displacements up to the fourth order, the free energy, specific heats Cv and Cp, mean square atomic displacements and thermal lattice expansion coefficients are derived explicitly in terms of the second and fourth order vibrational coupling constants. The thermodynamic quantities are calculated both for cubic (fcc and bcc) and closed packed hexagonal (hcp) metals. The Lennard–Jones type of potentials as well as the effective interatomic potentials derived from the tight-binding (TB) total energy calculation scheme are used. The calculated thermodynamic quantities are favorably compared with the experimental results. For the calculations of alloys, we investigate the effects of thermal lattice vibration on the long range order (LRO) parameter and order–disorder transitions of the ordered binary alloys.

RELIABILITY SENSITIVITY FOR ROTOR–STATOR SYSTEMS WITH RUBBING

On the basis of the dynamic equations of the Jeffcott rotor–stator model with imbalance, the reliability sensitivity of the rotor–stator systems with rubbing is examined. A statistical fourth moment method is developed to determine the first four moments of system response and state function. The distribution function of the system state function is approximately determined by the standard normal distribution functions using the Edgeworth series technique. The reliability and reliability sensitivity are obtained and the effect on reliability and reliability sensitivity of shaft stiffness and damping, stator stiffness and damping, radial clearance and stator radial stiffness is studied. Numerical results are also presented and discussed.

Additivity of statistical moments in the exponentially modified Gaussian model of chromatography

A homologous series of saturated fatty acids ranging from C 10 to C 22 was separated by reversed-phase capillary liquid chromatography. The resultant zone profiles were found to be fit best by an exponentially modified Gaussian (EMG) function. To compare the EMG function and statistical moments for the analysis of the experimental zone profiles, a series of simulated profiles was generated by using fixed values for retention time and different values for the symmetrical ( σ) and asymmetrical ( τ) contributions to the variance. The simulated profiles were modified with respect to the integration limits, the number of points, and the signal-to-noise ratio. After modification, each profile was analyzed by using statistical moments and an iteratively fit EMG equation. These data indicate that the statistical moment method is much more susceptible to error when the degree of asymmetry is large, when the integration limits are inappropriately chosen, when the number of points is small, and when the signal-to-noise ratio is small. The experimental zone profiles were then analyzed by using the statistical moment and EMG methods. Although care was taken to minimize the sources of error discussed above, significant differences were found between the two methods. The differences in the second moment suggest that the symmetrical and asymmetrical contributions to broadening in the experimental zone profiles are not independent. As a consequence, the second moment is not equal to the sum of σ 2 and τ 2, as is commonly assumed. This observation has important implications for the elucidation of thermodynamic and kinetic information from chromatographic zone profiles.

Application of statistical moment method to thermodynamic quantities of metals and alloys

The thermodynamic quantities of metals and alloys are studied using the moment method in the quantum statistical mechanics, going beyond the harmonic approximation, and fully taking into account the anharmonicity of lattice vibrations. Including the power moments of the atomic displacements up to the fourth order, the free energy, specific heats C v and C p, mean square relative displacements and thermal lattice expansion coefficients are given explicitly in terms of the second and fourth order vibrational coupling constants. The effects of anharmonicity of lattice vibration on the long range order (LRO) parameter and order-disorder transitions of the binary alloys are investigated in detail. The changes in the root mean square atomic displacements upon the order-disorder phase transition are evaluated and compared with the available experimental results. We also discuss the melting transitions of metals and alloys within the framework of the present statistical moment method, and estimate the melting temperatures through the limiting temperature of the crystalline stability.

Quantifying Uncertainty in CFD

Quantifying Uncertainty in CFD

Approach for Input Uncertainty Propagation and Robust Design in CFD Using Sensitivity Derivatives1

An implementation of the approximate statistical moment method for uncertainty propagation and robust optimization for quasi 1-D Euler CFD code is presented. Given uncertainties in statistically independent, random, normally distributed input variables, first-and second-order statistical moment procedures are performed to approximate the uncertainty in the CFD output. Efficient calculation of both first- and second-order sensitivity derivatives is required. In order to assess the validity of the approximations, these moments are compared with statistical moments generated through Monte Carlo simulations. The uncertainties in the CFD input variables are also incorporated into a robust optimization procedure. For this optimization, statistical moments involving first-order sensitivity derivatives appear in the objective function and system constraints. Second-order sensitivity derivatives are used in a gradient-based search to successfully execute a robust optimization. The approximate methods used throughout the analyses are found to be valid when considering robustness about input parameter mean values.

Calculation of thermodynamic quantities of metals and alloys by the statistical moment method

The thermodynamic properties of metals and alloys are studied using the moment method in the statistical dynamics, which allows us to take into account the anharmonicity of thermal lattice vibrations and size mismatch of constituent atoms, going beyond the quasi-harmonic approximation. Within the fourth-order moment approximation, the free energy and equilibrium lattice spacing of the binary alloys are given explicitly in terms of the effective pair potentials and the second- and fourth-order vibrational constants. The long-range order (LRO) parameter η and order-disorder transition temperatures (ODTs) Tc of binary alloys are obtained by solving the explicit transcendental equations. The numerical calculations of thermodynamic quantities for Cu3Au and β-CuZn alloys show that the inclusion of the anharmonicity of lattice vibration plays an essentially important role in determining the phase stabilities of metals and alloys.

Reliability Analysis for Rotor Rubbing

The dynamic equations of the Jeffcott rotor with imbalance are concerned. The reliability of rotor systems with rubbing is examined. A statistical fourth moment method is developed to determine the first four moments of the system response and the state function. The distribution function of the system state function is approximately determined by using the Edgeworth series technique. Its reliability is obtained. The effect on the reliability of the shaft stiffness, the external damping coefficient, the degree of imbalance and the stator stiffness is studied. Numerical results are also presented and discussed.

Kinetic modeling of methyl methacrylate free-radical polymerization initiated by tetraphenyl biphosphine

This study presents a semi-empirical kinetic model for the prediction of monomer conversion and average molecular weight during pseudo-living radical polymerization of methyl methacrylate with tetraphenyl biphosphine as the initiator in bulk, using UV light as the energy source. The model incorporates the chain-transfer to the initiator, which has been shown to be an important reaction for controlling the molecular weight of formed polymers. To solve the set of proposed kinetic equations the method of statistical moments was employed. The kinetic parameters estimated by fitting the initiator conversion data and monomer conversion data were used for modeling the molecular weight profiles, which were in a fair agreement with the experimentally measured average molecular weights. The pseudo-living radical polymerization scheme is characterized by having reversible termination steps. The proposed circular reaction mechanism was lumped in the empirical power-law reaction rate expression, which seemed to be successfully employed in the model development. The modeling results may be used for the characterization of the polymerization process studied as well as for the prediction of the polymerization behavior at monomer conversions up to 30%.

Application of Statistical Moment Method to Thermodynamic Properties of Metals at High Pressures

The moment method in statistical dynamics is used to study the thermodynamic properties of metals taking into account the anharmonicity effects of the lattice vibrations and hydrostatic pressures. The explicit expressions of the lattice constant, thermal expansion coefficient, and the specific heats C v and C p of cubic (fcc) metals are derived within the fourth order moment approximation. The thermodynamic quantities of Al, Au, Ag, Cu, and Pt metals are calculated as a function of the pressure, and they are in good agreement with the corresponding experimental results. The effective pair potentials work well for the calculations of transition and noble metals, compared to those of the s p -valence metals. For obtaining better agreement of the thermodynamic quantities of metals like Al, it is required at least to use the more sophisticated electronic many body potentials. In general, it has been shown that the anharmonicity effects of lattice vibration play a dominant role in determining the thermodynamic...

Determination of the diffusion coefficient of 1-(2′-pyridylazo)-2-naphthol in ethanol–water solutions using flow injection and nuclear magnetic resonance techniques

The values of the diffusion coefficient of PAN (1-(2′-pyridylazo)-2-naphthol) in various ethanol–water solutions at 298 K were determined using both flow injection (FI) and nuclear magnetic resonance (NMR) techniques. The transient FI concentration curves monitored by the detector were described by the axially dispersed plug flow model with an axial dispersion coefficient obeying Taylor's theory. The diffusion coefficient of PAN determined in 40–100% (v/v) ethanol solutions in water by both curve-fitting and the statistical moments method varied in the range from 2.43×10 −10 to 8.31×10 −10 m 2 s −1. As expected, the diffusion coefficient values obtained by curve-fitting were found to be more reliable. A pulsed field gradient (PFG) spin echo NMR experiment was also used to measure the diffusion coefficient of PAN in the same solutions. The NMR results were found to follow the same trend as the FI results though they were from 5% to 16% lower in value. This deviation was attributed to the association effects facilitated by the experimental conditions under which the NMR measurements were taking place, i.e. in quiet solutions and at considerably higher concentrations than those used by the FI technique. The diffusion coefficient of PAN in pure water was calculated as 2.21×10 −10 m 2 s −1 by extrapolating the FI results to zero ethanol concentration. The results reported in the present investigation can be used for studying the electrochemical properties of PAN in ethanol–water solutions as well as for elucidating the sensing mechanism of an optode based on immobilisation of PAN into Nafion membranes.

Influence of Heat Transfer on the Kinetic Process Involved during the Adsorption of Soluble Organic Substances on to Porous Polymeric Sorbents

During adsorption from the gaseous phase on to solid sorbents, it is known that heat transfer generally induces a mass transfer process. The present investigation was concerned with the solution of some questions relating to the heat transfer phenomenon during adsorption from the aqueous phase. In so doing, the thermodynamic and kinetic characteristics of the adsorption of benzene derivatives from aqueous solution by polymeric sorbents, i.e. Polysorb 60/100 and sample G-5, have been studied using the method of statistical moments for calculating the heat diffusion contribution to the general mass diffusion process. This method consists of the mathematical analysis of the temperature dependence of the kinetic diffusion curves on the basis of a linear adsorption isotherm. The contribution of the kinetics of heat transfer to the first sum of the moment of the kinetic curve may be defined as: [Formula: see text] where Mlq is that part of the first moment which represents heat diffusion and MID is the diffusion part of the first moment. The results obtained for the different systems investigated show the existence of a heat transfer effect on the kinetic adsorption process. This effect which retards the heat transfer process may be expressed in terms of the final rate of heat dispersion. The external contribution of heat transfer to the general mass transfer has been evaluated as 3–11% for various research systems. It has been shown that the evaluated contribution of the heat transfer is larger during adsorption by hydrophilic polymeric sorbents than when hydrophobic ones are employed. This fact may be connected with the different chemical composition and density of the sorbents.

First passage of uncertain single degree-of-freedom nonlinear oscillators

The first passage of uncertain single degree-of-freedom nonlinear oscillators subjected to random excitation is examined. Statistical fourth moment method is developed to determine first four moments of system response and state function. Distribution function of the system state function is approximately determined by the standard normal distribution functions using Edgeworth series technique and the reliability is obtained.

Stochastic analysis of time-variant nonlinear dynamic systems. Part 1: The Fokker-Planck-Kolmogorov equation approach in stochastic mechanics

The probabilistic description and analysis of the response of time-invariant nonlinear dynamic systems driven by stochastic processes is usually treated by means of evaluation of statistical moments and cumulants of the response. The background of these methods is the Fokker-Planck-Kolmogorov (FPK) equation for a probability density function or the Pugachev equation for a characteristic function, respectively. The exact solutions of these equations are obtained only for isolated cases. For engineering probabilistic analysis of a complex nonlinear systems, different mixed (hybrid) methods in these cases are used. In this study a ‘benchmark’ solution is obtained on the basis of the FPK equation in conjunction with the method of statistical moments for nonlinear mechanical system with colored parametric excitations. In Part 1 (this part), an exact solution of FPK equation on the basis of asymptotic analysis of nonlinear dynamic behavior of parametric excitation system is discussed. In Parts 2 and 3, applications of this method to stochasticity and stability analysis of nonlinear time-variant systems are considered. A comparison with the accuracy of different statistical methods is discussed. In Parts 4 and 5, a method of stochastic analysis of relativistic and quantum dynamic systems is described on the basis of a generalized stochastic Hamilton-Jacobi equations on a differential manifold as Riemanian geometry. This involves the task of relativistic navigation and dissipative quantum models of a nonlinear parametric oscillator in the presence of stochastic excitations on a differential manifold with different metric tensors of the space-time continuum.

Magnetic ground-state properties and spectral distributions. I. X-ray-absorption spectra.

We develop the method of statistical moments for polarized spectra. In addition to the total intensity measured by sum rules, the average transition energy can be studied. A general procedure is given to remove core-level excitations from the spectrum operator and to obtain purely ground-state properties. The potential of the methods is shown by a rederivation of the x-ray-absorption sum rules and branching-ratio rules for the isotropic spectrum and linear and circular dichroism. Insight is obtained into the origins of changes in the first moment of the isotropic spectrum. They measure short-range correlation functions of crystal and exchange fields.

Hydroquinone oxidation kinetics in adsorptive liquid chromatographic beds

Catalytic properties of chromatographic silicas modified with ferric ions using a batch preparation process are compared with those modified in situ in a column on the basis of the hydroquinone oxidation to benzoquinone in a liquid chromatographic reactor. This is done through application of a statistical moment method for kinetic parameter estimation from reactor chromatograms. The first absolute moment of the overall elution profile is utilized to evaluate the pseudo-first order rate constants for oxidation both with and without consideration of hydroquinone sorption processes. In general, hydroquinone adsorption processes are found to have a significant role in the overall process and affect the intrinsic reaction rate measurements. On comparing the two modified silicas, it is found that the influence of adsorption in models is less pronounced with the batch-treated material. The decrease in reaction rate effects with the material prepared by a batch process can be attributed to increased quenching of strongly adsorptive sites on the silica surface as a result of prolonged exposure to metal ions in solution. Retention behavior is consistent with a decrease in the influence of hydrogen bonding hydroxyl groups on the batch-treated material.

Rate process analysis in the liquid chromatographic reactor: an application of the first statistical moment

A statistical moment method based on the overall elution profile is developed for a (pseudo-) first-order chemical reaction occurring in a chromatographic reactor. Application of this method is illustrated for a hydroquinone oxidation catalyzed by an iron-modified chromatographic silica in an organic environment. The effects of some mass-transfer processes on intrinsic kinetics for reactions occurring in a liquid chromatographic column are also examined. A simple reaction model and a model incorporating finite adsorption and desorption rates together with reaction are both developed