Hermite Polynomial Expansion Research Articles

Estimation of Ordered Response Models with Sample Selection

We introduce two new Stata commands for the estimation of an ordered response model with sample selection. The opsel command uses a standard maximum-likelihood approach to fit a parametric specification of the model where errors are assumed to follow a bivariate Gaussian distribution. The snpopsel command uses the semi-nonparametric approach of Gallant and Nychka (1987, Econometrica 55: 363–390) to fit a semiparametric specification of the model where the bivariate density function of the errors is approximated by a Hermite polynomial expansion. The snpopsel command extends the set of Stata routines for semi-nonparametric estimation of discrete response models. Compared to the other semi-nonparametric estimators, our routine is relatively faster because it is programmed in Mata. In addition, we provide new postestimation routines to compute linear predictions, predicted probabilities, and marginal effects. These improvements are also extended to the set of semi-nonparametric Stata commands originally written by Stewart (2004, Stata Journal 4: 27–39) and De Luca (2008, Stata Journal 8: 190–220). An illustration of the new opsel and snpopsel commands is provided through an empirical application on self-reported health with selectivity due to sample attrition.

Analytical Pairs Trading Under Different Assumptions on the Spread and Ratio Dynamics

We demonstrate how arbitrarily sized long/short baskets whose portfolio value is modelled with spread or ratio of any asset weighting can be treated as a sequential stopping problem. In particular, when the underlying data generating process follows an Ornstein-Uhlenbeck, Cox-Ingersoll-Ross or GARCH diffusion, we derive closed form expressions for mean and variance of trade time and trade return (with transaction costs). From these expressions the risk and return characteristics are examined when the objective functions aim to maximisise expected return and Sharpe ratio. The SDE where possible are estimated using exact maximum likelihood, Euler and Hermite Polynomial Expansion. The estimation parameters and their sensitivity can be used to rank potential trades and understand the risk/return profi le involved with this style of investment.

Multimodal Physical Activity Recognition by Fusing Temporal and Cepstral Information

A physical activity (PA) recognition algorithm for a wearable wireless sensor network using both ambulatory electrocardiogram (ECG) and accelerometer signals is proposed. First, in the time domain, the cardiac activity mean and the motion artifact noise of the ECG signal are modeled by a Hermite polynomial expansion and principal component analysis, respectively. A set of time domain accelerometer features is also extracted. A support vector machine (SVM) is employed for supervised classification using these time domain features. Second, motivated by their potential for handling convolutional noise, cepstral features extracted from ECG and accelerometer signals based on a frame level analysis are modeled using Gaussian mixture models (GMMs). Third, to reduce the dimension of the tri-axial accelerometer cepstral features which are concatenated and fused at the feature level, heteroscedastic linear discriminant analysis is performed. Finally, to improve the overall recognition performance, fusion of the multimodal (ECG and accelerometer) and multidomain (time domain SVM and cepstral domain GMM) subsystems at the score level is performed. The classification accuracy ranges from 79.3% to 97.3% for various testing scenarios and outperforms the state-of-the-art single accelerometer based PA recognition system by over 24% relative error reduction on our nine-category PA database.

Estimation of Ordered Response Models with Sample Selection

We introduce two new Stata commands for the estimation of an ordered response model with sample selection. The opsel command uses a standard maximum likelihood (ML) approach to fit a parametric specification of the model where errors are assumed to follow a bivariate Gaussian distribution. The snpopsel command uses the semi-nonparametric (SNP) approach of Gallant and Nychka (1987) to fit a semiparametric specification of the model where the bivariate density function of the errors is approximated by a Hermite polynomial expansion. The snpopsel command extends the set of Stata routines for SNP estimation of discrete response models. Compared to the other SNP estimators, our routine is relatively faster because it is programmed in MATA. In addition, we provide new post-estimation routines to compute linear predictions, predicted probabilities and marginal effects. These improvements are also extended to the set of SNP Stata commands originally written by Stewart (2004) and De Luca (2008). An illustration of the new opsel and snpopsel commands is provided through an empirical application on self-reported health with selectivity due to sample attrition.

Ion energization by wave–particle interactions: Comparison of spectral and particle simulation solutions of the Vlasov equation

The evolution of an ensemble of charged particles is given by the Vlasov equation with a prescribed electromagnetic field. We choose a steady uniform magnetic field and a perpendicular electric field that oscillates in space and time. We solve the Vlasov equation with both a Fourier–Hermite spectral method and a particle simulation. The aim is to compare the effectiveness of the two approaches in the description of the ion energization in a geophysical context. We validate both solutions with an analytic result for a spatially homogeneous oscillating electric field. We show that the convergence of the Hermite polynomial expansion is greatly improved with the appropriate velocity scaling. The relationship between particle dynamics and the features of the velocity distribution function is discussed. We show that the energization of the ion distribution is related to stochastic heating arising from the chaotic dynamics associated with the equations of motion for the particles in the given fields.

Gyrokinetic Study of the Local Entropy Dynamics in Turbulent Plasmas with Zonal Flow

An entropy balance equation that maintains the dynamics in the radial direction is derived to study the role of zonal flows in turbulent transport. The equation describes the dynamics not only of the entropy density production related to zonal flow and local heat flux, but also of entropy density convection. We investigated these spatio-temporal dynamics of entropy in ion temperature gradient driven turbulence based on a global gyrokinetic Vlasov simulations in a slab geometry. The contribution of zonal flow to entropy production is observed to be small in the strong instability case, whereas the zonal flow energy is sufficiently large. We also found that the entropy density with n = 2 in the Hermite polynomial expansion in velocity space plays an important role in convecting the entropy density, causing turbulent spreading.

Some results on Gaussian Besov–Lipschitz spaces and Gaussian Triebel–Lizorkin spaces

In this paper we define Besov–Lipschitz and Triebel–Lizorkin spaces in the context of Gaussian harmonic analysis, the harmonic analysis of Hermite polynomial expansions. We study inclusion relations among them, some interpolation results and continuity results of some important operators (the Ornstein–Uhlenbeck and the Poisson–Hermite semigroups and the Bessel potentials) on them. We also prove that the Gaussian Sobolev spaces L α p ( γ d ) are contained in them. The proofs are general enough to allow extensions of these results to the case of Laguerre or Jacobi expansions and even further in the general framework of diffusion semigroups.

Beyond the dynamic density functional theory for steady currents: Application to driven colloidal particles in a channel

Motivated by recent studies on the dynamics of colloidal solutions in narrow channels, we consider the steady state properties of an assembly of noninteracting particles subject to the action of a traveling potential moving at a constant speed, while the solvent is modeled by a heat bath at rest in the laboratory frame. Here, since the description we propose takes into account the inertia of the colloidal particles, it is necessary to consider the evolution of both positions and momenta and study the governing equation for the one-particle phase-space distribution. First, we derive the asymptotic form of its solutions as an expansion in Hermite polynomials and their generic properties, such as the force and energy balance, and then we particularize our study to the case of an inverted parabolic potential barrier. We numerically obtain the steady state density and temperature profile and show that the expansion is rapidly convergent for large values of the friction constant and small drifting velocities. On the one hand, the present results confirm the previous studies based on the dynamic density functional theory (DDFT): On the other hand, when the friction constant is large, it display effects such as the presence of a wake behind the barrier and a strong inhomogeneity in the temperature field which are beyond the DDFT description.

Hermite Polynomials Expansions for Discrete-Time Nonlinear Filtering

A finite-dimensional approximation to general discrete-time non linear filtering problems is provided. It consists in a direct approximation to the recursive Bayes formula, based on a Hermite polynomials expansion of the transition density of the signal process. Tha approximation is in the sense of convergence, in a suitable weighted norm, to the conditional density of a signal process given the observations. The choise of the norm is in turn made so as to guarantee also the convergence of the conditional moments as well as to allow the evaluation of an upper bound for the approximation error.

CLT for [formula omitted] moduli of continuity of Gaussian processes

Let G = { G ( x ) , x ∈ R 1 } be a mean zero Gaussian process with stationary increments and set σ 2 ( | x − y | ) = E ( G ( x ) − G ( y ) ) 2 . Let f be a symmetric function with E f 2 ( η ) < ∞ , where η = N ( 0 , 1 ) . When σ 2 ( s ) is concave or when σ 2 ( s ) = s r , 1 < r ≤ 3 / 2 , lim h ↓ 0 ∫ a b f ( G ( x + h ) − G ( x ) σ ( h ) ) d x − ( b − a ) E f ( η ) Φ ( h , σ ( h ) , f , a , b ) = law N ( 0 , 1 ) where Φ ( h , σ ( h ) , f , a , b ) is the variance of the numerator. This result continues to hold when σ 2 ( s ) = s r , 3 / 2 < r < 2 , for certain functions f , depending on the nature of the coefficients in their Hermite polynomial expansion. The asymptotic behavior of Φ ( h , σ ( h ) , f , a , b ) at zero is described in a very large number of cases.

Order series method for forecasting non‐Gaussian time series

AbstractA new forecasting non‐Gaussian time series method based on order series transformation properties has been proposed. The proposed method improves Yu's method without using Hermite polynomial expansion to process nonlinear instantaneous transformations and provides acceptable forecasting accuracy. Copyright © 2007 John Wiley &amp; Sons, Ltd.

Interpolation and approximation in [formula omitted

Assume a standard Brownian motion W = ( W t ) t ∈ [ 0 , 1 ] , a Borel function f : R → R such that f ( W 1 ) ∈ L 2 , and the standard Gaussian measure γ on the real line. We characterize that f belongs to the Besov space B 2 , q θ ( γ ) ≔ ( L 2 ( γ ) , D 1 , 2 ( γ ) ) θ , q , obtained via the real interpolation method, by the behavior of a X ( f ( X 1 ) ; τ ) ≔ ∥ f ( W 1 ) - P X τ f ( W 1 ) ∥ L 2 , where τ = ( t i ) i = 0 n is a deterministic time net and P X τ : L 2 → L 2 the orthogonal projection onto a subspace of ‘discrete’ stochastic integrals x 0 + ∑ i = 1 n v i - 1 ( X t i - X t i - 1 ) with X being the Brownian motion or the geometric Brownian motion. By using Hermite polynomial expansions the problem is reduced to a deterministic one. The approximation numbers a X ( f ( X 1 ) ; τ ) can be used to describe the L 2 -error in discrete time simulations of the martingale generated by f ( W 1 ) and (in stochastic finance) to describe the minimal quadratic hedging error of certain discretely adjusted portfolios.

Seasonal Modeling of Multivariate Distributions of Metocean Parameters With Application to Marine Operations

Statistical information about the joint occurrence of metocean parameters is of importance for many offshore activities. For instance, in marine operations, environmental limitations may be brought about by both wind and wave conditions. Thus, knowledge of their joint occurrence is important as the persistence duration (i.e., the duration of the sea state persistence above or beneath a given level) and the seasonal dependence of wind and waves appear to be of large interest. However, such a modeling becomes difficult as the number of considered variables increases, especially when utilizing a common parameterization of some conditional distributions. This paper proposes a general methodology that aims at modeling seasonal joint distributions of n such parameters from their correlation structure and the n marginal distributions fitted by generalized gamma ones. Two methods are proposed in order to derive an approximate joint distribution from the modeled margins. The first one matches the correlation matrix only, whereas the second one, which is based on a multivariate Hermite polynomials expansion of the multinormal distribution, is able to match joint moments of order higher than two. However, more restrictive conditions are shown by the latter. An application to the simple example of the joint occurrence of significant wave height and the mean wind velocity at the 10m elevation is used to illustrate the methods. Eventually, examples of applications like simultaneous persistence of wind and wave conditions as well as seastate forecasting from statistics are given.

Non-Gaussian simulation using Hermite polynomials expansion and maximum entropy principle

The non-Gaussian simulation method using Hermite polynomials expansion presented in a previous article is improved. It is aimed to simulate the paths of a strictly stationary non-Gaussian process given the N-first moments of its one-dimensional marginal distribution and its autocorrelation function. The present new model consists in using the maximum entropy principle in order to determine its marginal distribution. This allows to obtain new results of convergence. The convergences of this new model are then examined and the method is illustrated by some examples.

Quantum hydrodynamic equations and quantum-hierarchy decoupling scheme.

There is a need to extract the relevant device physics from exact quantum transport formulations, and hence to reduce the complexity of quantum transport simulations for practical applications. We use a Hermite polynomial expansion to drastically reduce the number of degrees of freedom associated with the momentum variables. The result is a quantum hierarchy in real space. We also give a general procedure for the quantum-hierarchy decoupling scheme to derive the quantum hydrodynamic (QHD) and quantum drift-diffusion transport equations. We present some numerical results for the quantum hierarchy. A rigorous foundation of a decoupling procedure is given whereby the lower-order equations are renormalized in terms of a self-consistent effective potential, quantum diffusion coefficient, and moments, endowed with all the quantum corrections to order variant Planck's over h(2). Our decoupling scheme is based on the general expression of Tr H(n) to order variant Planck's over h(2), valid at all temperatures without the need for expansion in terms of the small parameter and high temperature assumption. This is very important conceptually since existing QHD formulations, using expansion to order h(2), are based on a Boltzmann distribution with the restrictive assumption of a small parameter, which is not valid in abrupt heterojunction semiconductor devices. They also fail to account for important quantum nonlinearity in the form of nonequilibrium quantum corrections. These nonequilibrium quantum corrections are expected to play a major role in approximating the coherence manifested by the highly nonlinear current-voltage characteristics of resonant tunneling structures.

Method of moments for the dilute granular flow of inelastic spheres.

Some peculiar features of granular materials (smooth, identical spheres) in rapid flow are the normal pressure differences and the related anisotropy of the velocity distribution function f((1)). Kinetic theories have been proposed that account for the anisotropy, mostly based on a generalization of the Chapman-Enskog expansion [N. Sela and I. Goldhirsch, J. Fluid Mech. 361, 41 (1998)]. In the present paper, we approach the problem differently by means of the method of moments; previously, similar theories have been constructed for the nearly elastic behavior of granular matter but were not able to predict the normal pressures differences. To overcome these restrictions, we use as an approximation of the f((1)) a truncated series expansion in Hermite polynomials around the Maxwellian distribution function. We used the approximated f((1)) to evaluate the collisional source term and calculated all the resulting integrals; also, the difference in the mean velocity of the two colliding particles has been taken into account. To simulate the granular flows, all the second-order moment balances are considered together with the mass and momentum balances. In balance equations of the Nth-order moments, the (N+1)th-order moments (and their derivatives) appear: we therefore introduced closure equations to express them as functions of lower-order moments by a generalization of the "elementary kinetic theory," instead of the classical procedure of neglecting the (N+1)th-order moments and their derivatives. We applied the model to the translational flow on an inclined chute obtaining the profiles of the solid volumetric fraction, the mean velocity, and all the second-order moments. The theoretical results have been compared with experimental data [E. Azanza, F. Chevoir, and P. Moucheront, J. Fluid Mech. 400, 199 (1999); T. G. Drake, J. Fluid Mech. 225, 121 (1991)] and all the features of the flow are reflected by the model: the decreasing exponential profile of the solid volumetric fraction, the parabolic shape of the mean velocity, the constancy of the granular temperature and of its components. Besides, the model predicts the normal pressures differences, typical of the granular materials.

Non-Gaussian simulation using Hermite polynomial expansion: convergences and algorithms

Mathematical justifications are given for a Monte Carlo simulation technique based on memoryless transformations of Gaussian processes. Different types of convergences are given for the approaching sequence. Moreover an original numerical method is proposed in order to solve the functional equation yielding the underlying Gaussian process autocorrelation function.

Hydrodynamic modes of a sheared granular flow from the Boltzmann and Navier–Stokes equations

The initial growth rates for the hydrodynamic modes of the shear flow of a three-dimensional collection of inelastic spheres is analyzed using two models. The first is the generalized Navier–Stokes equations, derived for the shear flow of inelastic spheres using the Chapman–Enskog procedure, where the energy equation has an additional dissipation term due to inelastic collisions. The second is the solution of the linearized Boltzmann equation, where the distribution function in the base state is determined using a Hermite polynomial expansion in the velocity moments. For perturbations with variations in the velocity and gradient directions, it is found that the solutions obtained by two procedures are qualitatively similar, though there are quantitative differences. For perturbations with variations in the vorticity direction, it is found that there are qualitative differences in the predictions for the initial growth rate of the perturbations.

Analysis of stochastic gradient identification of Wiener-Hammerstein systems for nonlinearities with Hermite polynomial expansions

This paper investigates the statistical behavior of a sequential adaptive gradient search algorithm for identifying an unknown Wiener-Hammerstein (1958) system (WHS) with Gaussian inputs. The WHS nonlinearity is assumed to be expandable in a series of orthogonal Hermite polynomials. The sequential procedure uses (1) a gradient search for the unknown coefficients of the Hermite polynomials, (2) an LMS adaptive filter to partially identify the input and output linear filters of the WHS, and (3) the higher order terms in the Hermite expansion to identify each of the linear filters. The third step requires the iterative solution of a set of coupled nonlinear equations in the linear filter coefficients. An alternative scheme is presented if the two filters are known a priori to be exponentially shaped. The mean behavior of the various gradient recursions are analyzed using small step-size approximations (slow learning) and yield very good agreement with Monte Carlo simulations. Several examples demonstrate that the scheme provides good estimates of the WHS parameters for the cases studied.

Hydrodynamic modes of a three-dimensional sheared granular material

The hydrodynamic modes of a three-dimensional sheared granular flow are determined by solving the linearised Boltzmann equation. The steady state is determined using an expansion in the parameter ε=(1− e) 1/2, and terms correct to O( ε 4) are retained in the expansion. The distribution function is expressed as the product of a Gaussian distribution and an expansion in Hermite polynomials, and the coefficients in the expansion are determined by solving the Boltzmann equation for the steady flow. A basis set consisting of 14 functions, containing products of Hermite polynomials upto fourth order, were used for calculating the steady distribution function. In order to determine the decay rate of the hydrodynamic modes, small perturbations in the form of Fourier modes in the spatial directions and a Hermite polynomial expansion in the particle velocities, were placed on the base state, and the initial growth rates of these perturbations were determined. The number of solutions for the initial growth rates depend on the number of basis functions used for defining the perturbations. However, it was found that the initial growth rates of the hydrodynamic modes showed small variations when the number of basis functions was increased from 10 to 20. The initial growth rates for the hydrodynamic modes showed unusual behaviour in the flow and the vorticity directions. In the flow directions, the scaling laws previously obtained for a two-dimensional system were recovered in this case as well. In the vorticity direction, it was found that all five growth rates were real, and proportional to m in the limit m→0, where m is the wave number in the vorticity direction. In addition, two of the growth rates are positive, indicating that there are two unstable modes in this direction.