Three-dimensional Probability Density Functions Research Articles

Effects of noise on transfer entropy estimation for damage detection

Recent research has shown that transfer entropy can be effectively employed as a feature for nonlinearity detection and linear damage identification. However, computation of transfer entropy requires the estimation of non-parametric one-, two-, and three-dimensional probability density functions. Therefore, small random perturbations caused by noise could lead to large variances in transfer entropy estimates. In this paper, we evaluate the effect of input and output noise on estimation of transfer entropy, and how noise, in turn, affects the capability of transfer entropy as a damage detector in a structural health monitoring (SHM) application. A damage index from the transfer entropy is computed from the response of a simulated multi-degree-of-freedom oscillator subject to linear and nonlinear stiffness changes in the presence of various noise influences. Damage indices are also evaluated for an experimental frame structure. Based on the computational study, we find that input noise lessens the sensitivity of the damage feature by diminishing the ability of the non-parametric density estimators to produce low variance transfer entropy estimations. Despite this reduced capability, an input that has no deterministic component can still detect a 25% stiffness loss in the computational example employed. Output noise has a greater impact on the feature's ability to accurately estimate the transfer entropy, such that a signal-to-noise threshold of approximately 30 dB leads to a greatly diminished ability to detect damage. Despite these noise effects, all damage cases tested on an experimental frame structure were detectable using the transfer entropy damage index.

Collision Probability for Larger Bodies Having Nonlinear Relative Motion

d = cross-sectional radius of torus dz = subvolume height h = distance above x–y plane of torus P = collision probability PR = collision probability rate R = radius of torus r = polar coordinate in symmetrized encounter plane t = time Vz = in-track velocity in symmetrized encounter frame x, y, z = coordinates used in probability density function = contour integration parameter = three-dimensional probability density function = standard deviation in symmetrized encounter frame x;y;z = standard deviations of relative position uncertainty for each axis in diagonal frame

Three-Dimensional Probability Density Functions via Tomographic Inversion

In many experimental observation systems where the goal is to record a three-dimensional observation of an object, or a set of objects, a lower-dimensional projection of the intended subject is obtained. In some situations only the statistical properties of such objects are desired: the three-dimensional probability density function. This article demonstrates that under special symmetries this function can be obtained from either a one- or two-dimensional probability density function which has been obtained from the observed, projected data. Standard tomographic theorems can be used to guarantee the uniqueness of this function, and a natural basis set can be used in computing the three-dimensional function from the one- or two-dimensional projection. The theory of this inversion is explored using theoretical and computational methods with examples of data taken from scientific experiments.

On the Calculation of Stationary Solutions of Multi-Dimensional Fokker–Planck Equations by Orthogonal Functions

In this paper, nonlinear stochastic systems are investigatedvia associated Fokker–Planck equations. Their stationary solutions arecalculated by expansions into orthogonal functions, e.g. especiallyadjusted polynomials and Fourier series. The weighting functions of thenew polynomials are obtained by the application of the stochasticaveraging method. The proposed analysis is demonstrated with severalexamples. The first one is a two-dimensional problem of nonlinearoscillators driven by white noise. The second one describes two-massoscillators with independent coloured noise excitations leading tosix-dimensional probability density functions. The next example ispresenting a system driven by both harmonic and stochastic excitationleading to three-dimensional probability density functions. Finally,oscillators with dry friction characteristics are examined.

Information theory and electron density

The intriguing concept of inherent uncertainty of probability schemes in information theory and statistical inference is applied to the molecular electron density. The electron density function is treated as a multimodal, three-dimensional probability density function describing the distribution of the electrons of a molecule in real space. A simple theory is proposed to introduce the amount of information associated with perturbations of the nuclear geometry such as molecular vibrations and reaction paths, in particular. It is shown by computations that the amount of information associated with the normal modes of vibration is related to the reduced mass. The proposed theory also suggests a novel Riemannian nuclear configuration space which is completely defined by the observable electron density of a molecular system.

Maximum entropy estimate of the solution to the inverse problem of acoustic scattering

The propagation of forced time-harmonic vibrations with small amplitude through a special class of random viscoelastic bodies is considered. These media are assumed to be composed ofM (M=1, 2, 3,...) distinct and linearly viscoelastic materials (phases). All these phases are randomly dispersed in the infinite volume of the body in the form of grains, perfectly bonded together. The vibrations are excited by the oscillations of an infinitely extended plane the oscillations being caused by some applied periodic force. Assuming that the equivalent homogenization method is applicable, the mean displacement field is found to be in the form of an attenuated plane wave. The inverse problem was formulated as the task of finding a probabilistic description of such local structure of the body and displacement field simultaneously that guarantees the average displacement field to be in the form of the plane wave. The method of solving this problem by the use of the Jaynes' principle of minimum prejudice was proposed and outlined. Equations were deduced for determination of a three-dimensional probability density function concerning the local structure of the body and displacement field simultaneously.

Computer-aided evaluation of gastric proteoglycans by high-resolution gas chromatography

A convenient procedure is described for the simultaneous separation and identification of monosaccharides and aminomonosaccharides constituting the proteoglycans of gastric mucosa. Computer assistance was used to obtain significant information from gas chromatographic data on proteoglycan modifications induced by anti-inflammatory drugs. Glass capillary chromatography was used for the separation and identification of saccharides and aminomonosaccharides of hydrolyzed mucus samples scraped from the stomachs of control and treated rats. The data, transmitted to a 48 K personal computer, were processed to obtain three-dimensional probability density functions or subjected to factor analysis to evaluate the suitability of the method to evidence modifications at the proteoglycan level induced by drugs, orally administered to rats. Indomethacin, zolimidine and acetylsalicylic acid were used as models for the applications.