Potential functions in mathematical pattern recognition
Mathematical pattern recognition, using potential functions superposition methods for reconstructing probability density from samples
Meisel, W. S.↗
SEARCH · Engineering Papers
Results for “probability density function”
Search indexed NASA NTRS and DOE OSTI research on propulsion, heat transfer, battery materials and energy systems. Follow report and document links to the original sources.
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Multivariate Density Estimation and Remote Sensing
Current efforts to develop methods and computer algorithms to effectively represent multivariate data commonly encountered in remote sensing applications are described. While this may involve scatter diagrams, multivariate representations of nonparametric probability density estimates are emphasized. The density function provides a useful graphical tool for looking at data and a useful theoretical tool for classification. This approach is called a thunderstorm data analysis.
Scott, D. W.↗
Probabilistic simulation of stress concentration in composite laminates
A computational methodology is described to probabilistically simulate the stress concentration factors in composite laminates. This new approach consists of coupling probabilistic composite mechanics with probabilistic finite element structural analysis. The probabilistic composite mechanics is used to probabilistically describe all the uncertainties inherent in composite material properties while probabilistic finite element is used to probabilistically describe the uncertainties associated with methods to experimentally evaluate stress concentration factors such as loads, geometry, and supports. The effectiveness of the methodology is demonstrated by using it to simulate the stress concentration factors in composite laminates made from three different composite systems. Simulated results match experimental data for probability density and for cumulative distribution functions. The sensitivity factors indicate that the stress concentration factors are influenced by local stiffness variables, by load eccentricities and by initial stress fields.
Chamis, C. C.↗
Probabilistic Simulation of Stress Concentration in Composite Laminates
A computational methodology is described to probabilistically simulate the stress concentration factors (SCF's) in composite laminates. This new approach consists of coupling probabilistic composite mechanics with probabilistic finite element structural analysis. The composite mechanics is used to probabilistically describe all the uncertainties inherent in composite material properties, whereas the finite element is used to probabilistically describe the uncertainties associated with methods to experimentally evaluate SCF's, such as loads, geometry, and supports. The effectiveness of the methodology is demonstrated by using is to simulate the SCF's in three different composite laminates. Simulated results match experimental data for probability density and for cumulative distribution functions. The sensitivity factors indicate that the SCF's are influenced by local stiffness variables, by load eccentricities, and by initial stress fields.
Chamis, C. C.↗
Cosmological density fluctuations and large-scale structure From N-point correlation functions to the probability distribution
Knowledge of N-point correlation functions for all N allows one to invert and obtain the probability distribution of mass fluctuations in a fixed volume. The hierarchical sequence of higher order is applied to correlations with dimensionless amplitudes suggested by the BBGKY equations. The resulting distribution is significantly non-Gaussian, even for quite small mean square fluctuations. The qualitative and to some degree quantitative results are to a large degree independent of the exact sequence of amplitudes. An ensemble of such models compared with N-body simulations fails in detail to account for the low-density frequency distribution.
Fry, J. N.↗
Using harmonic oscillators to determine the spot size of Hermite-Gaussian laser beams
The similarity of the functional forms of quantum mechanical harmonic oscillators and the modes of Hermite-Gaussian laser beams is illustrated. This functional similarity provides a direct correlation to investigate the spot size of large-order mode Hermite-Gaussian laser beams. The classical limits of a corresponding two-dimensional harmonic oscillator provide a definition of the spot size of Hermite-Gaussian laser beams. The classical limits of the harmonic oscillator provide integration limits for the photon probability densities of the laser beam modes to determine the fraction of photons detected therein. Mathematica is used to integrate the probability densities for large-order beam modes and to illustrate the functional similarities. The probabilities of detecting photons within the classical limits of Hermite-Gaussian laser beams asymptotically approach unity in the limit of large-order modes, in agreement with the Correspondence Principle. The classical limits for large-order modes include all of the nodes for Hermite Gaussian laser beams; Sturm's theorem provides a direct proof.
Steely, Sidney L.↗
Probability density distribution of velocity differences at high Reynolds numbers
Recent understanding of fine-scale turbulence structure in high Reynolds number flows is mostly based on Kolmogorov's original and revised models. The main finding of these models is that intrinsic characteristics of fine-scale fluctuations are universal ones at high Reynolds numbers, i.e., the functional behavior of any small-scale parameter is the same in all flows if the Reynolds number is high enough. The only large-scale quantity that directly affects small-scale fluctuations is the energy flux through a cascade. In dynamical equilibrium between large- and small-scale motions, this flux is equal to the mean rate of energy dissipation epsilon. The pdd of velocity difference is a very important characteristic for both the basic understanding of fully developed turbulence and engineering problems. Hence, it is important to test the findings: (1) the functional behavior of the tails of the probability density distribution (pdd) represented by P(delta(u)) is proportional to exp(-b(r) absolute value of delta(u)/sigma(sub delta(u))) and (2) the logarithmic decrement b(r) scales as b(r) is proportional to r(sup 0.15) when separation r lies in the inertial subrange in high Reynolds number laboratory shear flows.
Praskovsky, Alexander A.↗
Helicity fluctuations in incompressible turbulent flows
Results from direct numerical simulations of several homogeneous flows and fully developed turbulent channel flow indicate that the probability distribution function (pdf) of relative helicity density exhibits at most a 20 percent deviation from a flat distribution. Isotropic flows exhibit a slight helical nature but the presence of mean strain in homogeneous turbulence suppresses helical behavior. All the homogeneous turbulent flows studied show no correlation between relative helicity density and the dissipation of turbulent kinetic energy. The channel flow simulations indicate that, except for low-dissipation regions near the outer edge of the buffer layer, there is no tendency for the flow to be helical. The strong peaks in the relative helicity density pdf and the association of these peaks with regions of low dissipation found in previous simulations by Pelz et al.(1985) are not observed.
Rogers, Michael M.↗
On minimizing the probability of misclassification for linear feature selection
The use of techniques for feature selection permits treatment of classification problems in spaces of reduced dimensions. A method is considered of linear feature selection for n-dimensional observation vectors which belong to one of two populations, where each population is described by a known multivariate normal density function. More specifically, the problem of finding a 1xn transformation matrix B for which the probability of misclassification with respect to the one-dimensional transformed density functions was minimized was considered. Theoretical results are presented which give rise to a numerically tractable expression for the variation in the probability of misclassification with respect to B. Using this expression a computational procedure is discussed for obtaining a B which minimizes the probability of misclassification. Preliminary numerical results are discussed.
Guseman, L. F., Jr.↗
On Orbital Elements of Extrasolar Planetary Candidates and Spectroscopic Binaries
We estimate probability densities of orbital elements, periods, and eccentricities, for the population of extrasolar planetary candidates (EPC) and, separately, for the population of spectroscopic binaries (SB) with solar-type primaries. We construct empirical cumulative distribution functions (CDFs) in order to infer probability distribution functions (PDFs) for orbital periods and eccentricities. We also derive a joint probability density for period-eccentricity pairs in each population. Comparison of respective distributions reveals that in all cases EPC and SB populations are, in the context of orbital elements, indistinguishable from each other to a high degree of statistical significance. Probability densities of orbital periods in both populations have P(exp -1) functional form, whereas the PDFs of eccentricities can he best characterized as a Gaussian with a mean of about 0.35 and standard deviation of about 0.2 turning into a flat distribution at small values of eccentricity. These remarkable similarities between EPC and SB must be taken into account by theories aimed at explaining the origin of extrasolar planetary candidates, and constitute an important clue us to their ultimate nature.
Stepinski, T. F.↗
On differentiating the probability of error in multipopular feature selection
A method of linear feature selection for n dimensional observation vectors which belong to one of m populations is presented. Each population has a known apriori probability and is described by a known multivariate normal density function. Specifically we consider the problem of finding a k x n matrix B of rank k (k n) for which the transformed probability of misclassification is minimized. Providing that the transformed a posterior probabilities are distinct theoretical results are obtained which, for the case k = l, give rise to a numerically tractable formula for the derivative of the probability of misclassification. It is shown that for the two population problem this condition is also necessary. The dependence of the minimum probability of error on the a priori probabilities is investigated. The minimum probability of error satisfies a uniform Lipschitz condition with respect to the a priori probabilities.
Peters, B. C.↗
LFSPMC: Linear feature selection program using the probability of misclassification
The computational procedure and associated computer program for a linear feature selection technique are presented. The technique assumes that: a finite number, m, of classes exists; each class is described by an n-dimensional multivariate normal density function of its measurement vectors; the mean vector and covariance matrix for each density function are known (or can be estimated); and the a priori probability for each class is known. The technique produces a single linear combination of the original measurements which minimizes the one-dimensional probability of misclassification defined by the transformed densities.
Guseman, L. F., Jr.↗
Gravitational microlensing - Powerful combination of ray-shooting and parametric representation of caustics
We present a combination of two very different methods for numerically calculating the effects of gravitational microlensing: the backward-ray-tracing that results in two-dimensional magnification patterns, and the parametric representation of caustic lines; they are in a way complementary to each other. The combination of these methods is much more powerful than the sum of its parts. It allows to determine the total magnification and the number of microimages as a function of source position. The mean number of microimages is calculated analytically and compared to the numerical results. The peaks in the lightcurves, as obtained from one-dimensional tracks through the magnification pattern, can now be divided into two groups: those which correspond to a source crossing a caustic, and those which are due to sources passing outside cusps. We determine the frequencies of those two types of events as a function of the surface mass density, and the probability distributions of their magnitudes. We find that for low surface mass density as many as 40 percent of all events in a lightcurve are not due to caustic crossings, but rather due to passings outside cusps.
Wambsganss, J.↗
Generalized Maximum Entropy
A long standing mystery in using Maximum Entropy (MaxEnt) is how to deal with constraints whose values are uncertain. This situation arises when constraint values are estimated from data, because of finite sample sizes. One approach to this problem, advocated by E.T. Jaynes [1], is to ignore this uncertainty, and treat the empirically observed values as exact. We refer to this as the classic MaxEnt approach. Classic MaxEnt gives point probabilities (subject to the given constraints), rather than probability densities. We develop an alternative approach that assumes that the uncertain constraint values are represented by a probability density {e.g: a Gaussian), and this uncertainty yields a MaxEnt posterior probability density. That is, the classic MaxEnt point probabilities are regarded as a multidimensional function of the given constraint values, and uncertainty on these values is transmitted through the MaxEnt function to give uncertainty over the MaXEnt probabilities. We illustrate this approach by explicitly calculating the generalized MaxEnt density for a simple but common case, then show how this can be extended numerically to the general case. This paper expands the generalized MaxEnt concept introduced in a previous paper [3].
Cheeseman, Peter↗
On the normality and accuracy of simulated random processes.
Efficient methods of simulating stationary and nonstationary random processes and envelopes, by using a series of sine or cosine functions or by using the fast Fourier transform, have been proposed previously. Without applying the central limit theorem, it is shown in this paper that the simulated random processes are asymptotically Gaussian processes as the number of terms, N, of sine or cosine functions approaches infinity. The accuracy of the first-order probability densities of the simulated random processes is investigated by using the fast Fourier transform. Numerical results are computed with respect to the variation of the number of terms, N, of sine or cosine functions used for simulation. It is shown that within the practical range of N, such as 500, the accuracy is remarkably satisfactory even outside the region of 3 standard deviations.
Yang, J.-N.↗
Optimal Control via Self-Generated Stochasticity
The problem of global maxima of functionals has been examined. Mathematical roots of local maxima are the same as those for a much simpler problem of finding global maximum of a multi-dimensional function. The second problem is instability even if an optimal trajectory is found, there is no guarantee that it is stable. As a result, a fundamentally new approach is introduced to optimal control based upon two new ideas. The first idea is to represent the functional to be maximized as a limit of a probability density governed by the appropriately selected Liouville equation. Then, the corresponding ordinary differential equations (ODEs) become stochastic, and that sample of the solution that has the largest value will have the highest probability to appear in ODE simulation. The main advantages of the stochastic approach are that it is not sensitive to local maxima, the function to be maximized must be only integrable but not necessarily differentiable, and global equality and inequality constraints do not cause any significant obstacles. The second idea is to remove possible instability of the optimal solution by equipping the control system with a self-stabilizing device. The applications of the proposed methodology will optimize the performance of NASA spacecraft, as well as robot performance.
Zak, Michail↗
A quasi-Newton approach to optimization problems with probability density constraints
A quasi-Newton method is presented for minimizing a nonlinear function while constraining the variables to be nonnegative and sum to one. The nonnegativity constraints were eliminated by working with the squares of the variables and the resulting problem was solved using Tapia's general theory of quasi-Newton methods for constrained optimization. A user's guide for a computer program implementing this algorithm is provided.
Tapia, R. A.↗
Optimum quantization.
Optimum uniform quantizer structures for digital data with known continuous probability density noting telemetering of stationary Gaussian Markov process functions
Kurtenbach, A. J.↗