Gaussian Random Variates Research Articles

Fast Simulation of Additive Generalized Gaussian Noise Environments

Due to the complexity and associated inordinate computing time required to generate generalized Gaussian random variates, it is necessary to find faster ways to simulate events caused by additive generalized Gaussian noise. Following a concise review of the two known exact methods of generating generalized Gaussian random variates, an exact weighted counting approach is investigated for rapid estimation of the probability of events caused by the presence of additive generalized Gaussian noise. The weighted counting method reduces simulation times typically by 104 times, and will allow simulation of complex systems not previously feasible.

A New Method for Generating Gaussian Random Variates With Clarke's Autocorrelation

We present a new method for efficient and accurate generation of a discrete-time series of N complex Gaussian random variates with Clarke's autocorrelation. When the product f <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">D</sub> N is small (f <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">D</sub> is the Doppler frequency normalized with respect to the sampling frequency), the proposed method provides variates with accurate autocorrelation over the full lag range at computational complexity equal to that of the well-known inverse discrete Fourier transform method. In addition, the new method is a better practical alternative to the optimum Karhunen-Loève expansion, particularly for large N values.

A Bayesian non-parametric Potts model with application to pre-surgical FMRI data

The Potts model has enjoyed much success as a prior model for image segmentation. Given the individual classes in the model, the data are typically modeled as Gaussian random variates or as random variates from some other parametric distribution. In this article, we present a non-parametric Potts model and apply it to a functional magnetic resonance imaging study for the pre-surgical assessment of peritumoral brain activation. In our model, we assume that the Z-score image from a patient can be segmented into activated, deactivated, and null classes, or states. Conditional on the class, or state, the Z-scores are assumed to come from some generic distribution which we model non-parametrically using a mixture of Dirichlet process priors within the Bayesian framework. The posterior distribution of the model parameters is estimated with a Markov chain Monte Carlo algorithm, and Bayesian decision theory is used to make the final classifications. Our Potts prior model includes two parameters, the standard spatial regularization parameter and a parameter that can be interpreted as the a priori probability that each voxel belongs to the null, or background state, conditional on the lack of spatial regularization. We assume that both of these parameters are unknown, and jointly estimate them along with other model parameters. We show through simulation studies that our model performs on par, in terms of posterior expected loss, with parametric Potts models when the parametric model is correctly specified and outperforms parametric models when the parametric model in misspecified.

On the Generation of Correlated Gaussian Random Variates by Inverse DTF

In this paper the problem of generating a stationary band-limited Gaussian random vector with arbitrary complex autocorrelation by the inverse discrete Fourier transform (IDTF) algorithm is considered. Instead of using the classical frequency mask (FM), determined from samples of the (band-limited) target power spectral density (PSD) of the process, a new FM is obtained by matching the autocorrelation obtained with the IDFT algorithm to a desired arbitrary autocorrelation. Example results presented show that the new FM is able to significantly increase the autocorrelation accuracy of the generated process at no additional online computational cost.

Generating generalized inverse Gaussian random variates by fast inversion

The inversion method for generating non-uniformly distributed random variates is a crucial part in many applications of Monte Carlo techniques, e.g., when low discrepancy sequences or copula based models are used. Unfortunately, closed form expressions of quantile functions of important distributions are often not available. The (generalized) inverse Gaussian distribution is a prominent example. It is shown that algorithms that are based on polynomial approximation are well suited for this distribution. Their precision is close to machine precision and they are much faster than root finding methods like the bisection method that has been recently proposed.

Generating inverse Gaussian random variates by approximation

The inverse Gaussian distribution is a useful distribution with important applications. But there is less discussion in the literature on sampling of this distribution. The method given in [Atkinson, A.C., 1982. The simulation of generalized inverse Gaussian and hyperbolic random variables. SIAM Journal on Scientific and Statistical Computing 3(4), 502–515] is based on rejection method where some (uniform) random numbers from the sample are discarded. This feature makes it difficult to take advantage of the low discrepancy sequences which have important applications. In [Michael, J., Schucany, W., Haas, R., 1976. Generating random variates using transformations with multiple roots. The American Statistician 30(2), 88–90], Michael et al. give a method to generate random variables with inverse Gaussian distribution. In their method, two pseudorandom numbers uniformly distributed on (0, 1) are needed in order to generate one inverse Gaussian random variable. In this short paper, we present a new method, based on direct approximate inversion, to generate the inverse Gaussian random variables. In this method, only one pseudorandom number is needed in generating one inverse Gaussian random variate. This method enables us to make use of the better convergence of low discrepancy sequence than the pseudorandom sequence. Numerical results show the superiority of low discrepancy sequence than the pseudorandom sequence in simulating the mean of the inverse Gaussian distribution by using our sampling method. Further application of this method in exotic option pricing under the normal inverse Gaussian model is under investigation.

Adaptive Newton-based multivariate smoothed functional algorithms for simulation optimization

In this article, we present three smoothed functional (SF) algorithms for simulation optimization. While one of these estimates only the gradient by using a finite difference approximation with two parallel simulations, the other two are adaptive Newton-based stochastic approximation algorithms that estimate both the gradient and Hessian. One of the Newton-based algorithms uses only one simulation and has a one-sided estimate in both the gradient and Hessian, while the other uses two-sided estimates in both quantities and requires two simulations. For obtaining gradient and Hessian estimates, we perturb each parameter component randomly using independent and identically distributed (i.i.d) Gaussian random variates. The earlier SF algorithms in the literature only estimate the gradient of the objective function. Using similar techniques, we derive two unbiased SF-based estimators for the Hessian and develop suitable three-timescale stochastic approximation procedures for simulation optimization. We present a detailed convergence analysis of our algorithms and show numerical experiments with parameters of dimension 50 on a setting involving a network of M / G /1 queues with feedback. We compare the performance of our algorithms with related algorithms in the literature. While our two-simulation Newton-based algorithm shows the best results overall, our one-simulation algorithm shows better performance compared to other one-simulation algorithms.

Excitation energy transfer in the LHC-II trimer: a model based on the new 2.72 Å structure

Energy transfer of the light harvesting complex LHC-II trimer, extracted from spinach, was studied in the Q(y) region at room temperature by femtosecond transient absorption spectroscopy. Configuration interaction exciton method [Linnanto et al. (1999) J Phys Chem B 103: 8739-8750] and 2.72 A structural information reported by Liu et al. was used to calculate spectroscopic properties and excitation energy transfer rates of the complex. Site energies of the pigments and coupling constants of pigment pairs in close contact were calculated by using a quantum chemical configuration interaction method. Gaussian random variation of the diagonal and off-diagonal exciton matrix elements was used to account for inhomogeneous broadening. Rate calculations included only the excitonic states initially excited and probed in the experiments. A kinetic model was used to simulate time and wavelength dependent absorption changes after excitation on the blue side of the Q(y) transition and compared to experimentally recorded rates. Analysis of excitonic wavefunctions allowed identification of pigments initially excited and probed into later. It was shown that excitation of the blue side of the Q(y) band of a single LHC-II complex results in energy transfer from chlorophyll b's of the lumenal side to chlorophyll a's located primarly on one of the monomers of the stromal side.

An empirically based path loss model for wireless channels in suburban environments

We present a statistical path loss model derived from 1.9 GHz experimental data collected across the United States in 95 existing macrocells. The model is for suburban areas, and it distinguishes between different terrain categories. Moreover, it applies to distances and base antenna heights not well-covered by existing models. The characterization used is a linear curve fitting the decibel path loss to the decibel-distance, with a Gaussian random variation about that curve due to shadow fading. The slope of the linear curve (corresponding to the path loss exponent, /spl gamma/) is shown to be a random variate from one macrocell to another, as is the standard deviation /spl sigma/ of the shadow fading. These two parameters are statistically modeled, with the dependencies on base antenna height and terrain category made explicit. The resulting path loss model applies to base antenna heights from 10 to 80 m, base-to-terminal distances from 0.1 to 8 km, and three distinct terrain categories.

On the simulation of biological diffusion processes

Many phenomena of interest in biology can be modeled using diffusion processes satisfying a stochastic differential euqation. We consider first the stochastic differential equation d X = μ Xd t + σ X d W, where W is a standard Wiener process and representing a population growth process. This is simulated using both a strong Euler scheme involving normal pseudorandom numbers and a weak Euler scheme using Bernoulli pseudorandom numbers. Results are given for the mean of X(1) and its 95% confidence intervals for various numbers of simulations. It is found that there are no significant differences between the results obtained by these two schemes at a particular value of the time step, but that the weak scheme takes less computer time than the strong scheme. We also consider the process satisfying d X=(− γ 1 X + γ 2(1 − X))d t + √ X(1 − X)d W, representing a gene frequency under the influence of random mating and mutation. It is similarly found that the results of simulation by the two schemes are not significantly different. It is concluded that in simulations of many biological diffusion processes it is often advantageous to employ a scheme involving Bernoulli rather than Gaussian random variates not only because it involves fewer machine arithmetic operations but also because problems with large jumps that sometimes occur with extreme values of normal variates are less likely, thus enabling one to employ a larger time step with a concomitant saving in machine time.

On real parameter stability margins and their computation

A stability margin for linear control systems subject to real parameter variations can be computed from the constraints arising from the conventional Routh-Hurwitz criteria. The proposed stability margin is similar to the real-μ criterion, but with the bound on the real parameter variations given by a differentiable function. If the uncertain parameters are gaussian random variates, then an upper bound on the probability of instability is constructed. A computational procedure is proposed based on polynomial continuation.

Spatial variability of soil penetrometer measurements at the mesoscopic scale

Laboratory penetrometer measurements are reported for 10 undisturbed cores from two soil types: a Typic Eutrocrept and an Aeric Ochraqualf. Resistance to penetration was measured at numerous points within each core, which permitted an analysis of spatial variation at the mesoscopic scale (10 −3–10 0 m). The objectives were to examine the extent of autocorrelation in the data, determine the optimum number of penetrations per sample, and identify physical properties influencing variability. Two different penetrometers were used: a constant-load needle penetrometer and a constant-rate micro-penetrometer. Data were expressed as a percentage of the maximum depth attainable with the needle penetrometer (PMP), and as the slope ( m) and intercept ( n) of linearized percent penetrability vs. tip pressure relationships for the micro-penetrometer. No correlation was found between the m and n parameters, suggesting they represent independent physical properties contributing to soil strength. The PMP and m measurements were positively skewed, whereas the n measurements were normally distributed. None of the parameters exhibited any spatial autocorrelation. The average fractal dimension for all three parameters was 2.89 ± 0.09, indicating Gaussian random variation. For a given bulk density, the mean PMP decreased exponentially with decreasing moisture content. Conversely, at constant water potential, mean values of m and n decreased linearly with increasing bulk density. Coefficients of variation (CV) were relatively uniform for the n parameter (11.5–20.1%), but depended upon soil moisture and bulk density in the case of PMP and m, respectively. The CV for PMP increased exponentially with decreasing soil moisture, while a curvilinear relationship was found between the variation in m and bulk density. This was modelled with a second order polynomial, which predicted a maximum CV of 36.1% at 1.14 Mg m −3 bulk density. The amount of variability encountered was greater than previously reported for laboratory penetrometer measurements. The fact that this variation was not independent for two of the three parameters investigated, has important implications for sampling at the mesoscopic scale. Before a study is undertaken, preliminary measurements should be made to determine the optimum number of samples for the parameter under consideration. Sampling requirements for soil penetrability appear to increase with decreasing moisture content and bulk density.

Scattering of electromagnetic waves from a randomly perturbed quasiperiodic surface

The scattering of electromagnetic waves from a randomly perturbed quasiperiodic surface is studied for active remote sensing of plowed fields. The Kirchoff approximation is used. The narrow-band Gaussian random variation around the spatial frequency of the sinusoidal variation is used to introduce the quasiperiodicity. The physical optics integral is evaluated to obtain closed form solutions for coherent and incoherent bistatic scattering coefficients. In the geometrical optics limit, it is shown that the bistatic scattering coefficients are proportional to the probability of the occurrence of the slopes which will specularly reflect the incident wave to the observation direction. The theoretical results are illustrated for the various cases by plotting backscattering cross sections as a function of the angle of incidence. It is shown that there is a large difference between the cases where the incident wave vector is parallel or perpendicular to the row direction. When the incident wave vector is perpendicular to the row direction, the maximum value of the backscattering cross section does not necessary occur at normal incidence. The scattering coefficients can be interpreted as a convolution of the scattering patterns for the sinusoidal and the random rough surfaces. For the backscattering cross sections we observe the occurrence of peaks whose relative magnitudes and the locations are explained in terms of the scattering patterns for sinusoidal surfaces.