Order Joint Moments Research Articles

Design of Gaussian inputs for Wiener model identification

We develop a tractable algorithms for finding the optimal power spectral density of the Gaussian input excitation for identifying a Wiener model. This problem is known as a difficult problem for two reasons. Firstly, the estimation accuracy depends on the higher order joint moments of the potentially infinitely many past samples of the input signal. In addition, the covariancematrix of the parameter estimates is thought to be a highly non-convex function of the power spectral density function. In this contribution we show that under Gaussian assumption it is possible to completely parameterize the set of all admissible information matrices with only a finite number of parameters. We present a convex algorithm to solve the D-optimal design problem. This idea can be extended further to design Gaussian mixture designs.

Generalized Estimating Equations to Binary Probit Model

Inference in generalized linear mixed models with multivariate random effects is often made cumbersome by the high-dimensional intractable integrals involved in the marginal likelihood. This article presents an inferential methodology based on the generalized estimating equations for the probit latent traits models. This method belonging to the broad class of semi parametric approaches involves marginal joint moments of order 1 and 2, which has analytical expression. The different results are illustrated with a simulation study.

Performance of translation approach for modeling correlated non-normal variables

It is common to construct a consistent multivariate distribution from non-normal marginals and Pearson product–moment correlations using the well known translation approach. A practical variant of this approach is to match the Spearman rank correlations of the measured data, rather than the Pearson correlations. In this paper, the performance of these translation methods is evaluated based on their abilities to match the following exact solutions from one benchmark bivariate example where the joint distribution is known: (1) high order joint moments, (2) joint probability density functions (PDFs), and (3) probabilities of failure. It is not surprising to find significant errors in the joint moments and PDFs. However, it is interesting to observe that the Pearson and Spearman methods produce very similar results and neither method is consistently more accurate or more conservative than the other in terms of probabilities of failure. In addition, the maximum error in the probability of failure may not be associated with a large correlation. It can happen at an intermediate correlation.

Yule–Walker type estimators in periodic bilinear models: strong consistency and asymptotic normality

This paper studies the covariance structure and the asymptotic properties of Yule–Walker (YW) type estimators for a bilinear time series model with periodically time-varying coefficients. We give necessary and sufficient conditions ensuring the existence of moments up to eighth order. Expressions of second and third order joint moments, as well as the limiting covariance matrix of the sample moments are given. Strong consistency and asymptotic normality of the YW estimator as well as hypotheses testing via Wald’s procedure are derived. We use a residual bootstrap version to construct bootstrap estimators of the YW estimates. Some simulation results will demonstrate the large sample behavior of the bootstrap procedure.

Joint statistics of natural frequencies of stochastic dynamic systems

The description of real-life engineering structural systems is usually associated with some amount of uncertainty in specifying material properties, geometric parameters and boundary conditions. In the context of structural dynamics it is necessary to consider joint probability distribution of the natural frequencies in order to account for these uncertainties. Current methods to deal with such problems are dominated by approximate perturbation methods. In this paper a new approach based on an asymptotic approximation of multidimensional integrals is proposed. A closed-form expression for general order joint moments of arbitrary number of natural frequencies of linear stochastic systems is derived. The proposed method does not employ the small-randomness and Gaussian random variable assumption usually used in the perturbation based methods. Joint distributions of the natural frequencies are investigated using numerical examples and the results are compared with Monte Carlo simulation.

Random eigenvalue problems revisited

The description of real-life engineering structural systems is associated with some amount of uncertainty in specifying material properties, geometric parameters, boundary conditions and applied loads. In the context of structural dynamics it is necessary to consider random eigenvalue problems in order to account for these uncertainties. Within the engineering literature, current methods to deal with such problems are dominated by approximate perturbation methods. Some exact methods to obtain joint distribution of the natural frequencies are reviewed and their applicability in the context of real-life engineering problems is discussed. A new approach based on an asymptotic approximation of multi-dimensional integrals is proposed. A closed-form expression for general order joint moments of arbitrary numbers of natural frequencies of linear stochastic systems is derived. The proposed method does not employ the ‘small randomness’ assumption usually used in perturbation based methods. Joint distributions of the natural frequencies are investigated using numerical examples and the results are compared with Monte Carlo simulation.

On Marginal Quasi-Likelihood Inference in Generalized Linear Mixed Models

In view of the cumbersome and often intractable numerical integrations required for a full likelihood analysis, several suggestions have been made recently for approximate inference in generalized linear mixed models (GLMMs). Two closely related approximate methods are the penalized quasi-likelihood (PQL) method and the marginal quasi-likelihood (MQL) method. The PQL approach generally produces biased estimates for the regression effects and the variance component of the random effects. Recently, some corrections have been proposed to remove these biases. But the corrections appear to be satisfactory only when the variance component of the random effects is small. The MQL approach has also been used for inference in the GLMMs. This approach requires the computations of the joint moments of the clustered observations, up to order four. But the derivation of these moments are not easy. Consequently, different “working” formulas have been used, especially for the mean and covariance matrix of the observations, which may not lead to desirable estimates. In this paper, we use a small variance component (of the random effects) approach and develop the MQL estimating equations for the parameters based on the joint moments of order up to four. The proposed approach thus avoids the use of the so-called “working” covariance and higher order moment matrices, leading to better estimates for the regression and the overdispersion parameters, in the sense of efficiency in particular.

Determining the joint moments of a motor drive's harmonic current phasors

It has been previously demonstrated that if a sufficiently large number of harmonic current sources are operating, a complete probabilistic characterization of the resulting power system-wide harmonic voltages can be found with knowledge of only the first and second order joint moments of each harmonic current's rectangular components. However, little consideration has been given to the problem of determining these joint moments for specific types of harmonic sources. This paper presents a method for analytically determining the joint moments of the resolved harmonic current components produced by two motor drives which utilize a phase-controlled rectifier. It is shown that when given the load torque as a function of the randomly varying motor speed, the means, variances and covariance of each harmonic current phasor's rectangular components can be expressed in terms of the probability density function of the motor speed.

Structural modal interaction with combination internal resonance under wide-band random excitation

In this paper the non-linear interaction of a three degree of freedom structural model subjected to a wide-band random excitation is examined. The non-linearity of the system results in different critical regions of internal resonance and this has a significant effect on the response statistics. With reference to the combination internal resonance of the summed type the system response is analyzed by using the Fokker-Planck equation approach together with a non-Gaussian closure scheme. The non-Gaussian closure is based on the cumulant properties of order greater than three. As a first order approximation the scheme yields 209 first order differential equations in first through fourth order joint moments of the response co-ordinates. The analysis is carried out with the aid of the computer algebra software MACSYMA. The response statistics are determined, numerically in the time and frequency (internal detuning) domains. Contrary to the Gaussian closure scheme, the non-Gaussian closure solution yields a strictly stationary response in addition to a number of complex response characteristics not previously reported in the literature of the area of non-linear random vibration. These include multiple solutions and jump phenomena (jump and collapse in the response mean squares) at internal detuning slightly shifted from the exact internal resonance condition. At exact internal resonance the system response possesses a unique limit cycle in a stochastic sense. The regions of multiple solutions are defined in terms of system parameters (damping ratios and non-linear coupling parameter) and excitation spectral density level.

Approximate joint photocount statistics for Gaussian light with arbitrary spectral shape

An approximate formula is found for the joint moment generating function of integrated intensity fluctuations of Gaussian light in two time intervals. Some lower−order joint moments generated by this approximate formula are compared to their exact values known in case of Lorentzian light and good accuracy is obtained.