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Results for “Probability Distribution Function (PDF)”

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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All sourcesNASA NTRSDOE OSTI
NASA NTRS1988

Bayes classification of terrain cover using normalized polarimetric data

The normalized polarimetric classifier (NPC) which uses only the relative magnitudes and phases of the polarimetric data is proposed for discrimination of terrain elements. The probability density functions (PDFs) of polarimetric data are assumed to have a complex Gaussian distribution, and the marginal PDF of the normalized polarimetric data is derived by adopting the Euclidean norm as the normalization function. The general form of the distance measure for the NPC is also obtained. It is demonstrated that for polarimetric data with an arbitrary PDF, the distance measure of NPC will be independent of the normalization function selected even when the classifier is mistrained. A complex Gaussian distribution is assumed for the polarimetric data consisting of grass and tree regions. The probability of error for the NPC is compared with those of several other single-feature classifiers. The classification error of NPCs is shown to be independent of the normalization function.

Yueh, H. A.↗
NASA NTRS2019

TURBOMAT: A Probabilistic Turbomachinery Aeroelastic Analysis Tool

An integration of aeroelastic analysis procedures with probabilistic analysis methods enables us to design safe reliable engines with quantified reliability. Towards this goal, a graphical user interface (GUI) based tool that integrates the codes Aeroelastic analysis of propfans (ASTROP2) and Numerical Evaluation of Stochastic Structures Under Stress (NESSUS) is developed. The tool entitled TURBOMachinery Aeroelastic Analysis Tool (TURBOMAT), is developed utilizing the MATrix Laboratory (Matlab) Guide (Graphical User Interface Development) tool box. TURBOMAT provides a user friendly computational environment for rapid assessment of Turbomachinery blades flutter characteristics, subjected to uncertain loading conditions with variability in material and aerodynamic properties. The tool is seen as an education tool for new students and young engineers starting their careers in structural Aeroelasticity who want to learn and understand aeroelastic aspects of turbomachinery components, fans, compressors and turbines, including uncertainties in loading and material properties.A typical fan blade configuration geometry was chosen to demonstrate the tool. The results are presented in the form of probabilistic density function (PDF), the cumulative distribution function (CDF) and sensitivity factors. Both first order fast probability integration (FPI) and the Monto Carlo (MC) techniques are used in the analysis and compared. The tool enabled us to quantify blade flutter reliability as well as the ranking of uncertain variables and their importance to blade flutter response.

Aeroelastic Analysis↗
NASA NTRS1985

An APL program for the distribution of energy deposition by charged particles passing through thin absorbers

An APL program which numerically evaluates the probability density function (PDF) for the energy deposited in a thin absorber by a charged particle is proposed, with application to the construction, pointing, and control of spacecraft. With this program, the PDF of the restricted energy loss distribution of Watts (1973) is derived, and Vavilov's (1957) distribution is obtained by proper parameter selection. The method is demonstrated with the example of the effect of charged particle induced radiation on the Hubble Space Telescope (HST) pointing accuracy. A Monte Carlo study simulates the photon noise caused by charged particles passing through the photomultiplier tube window, and the stochastic variation of energy loss is introduced into the simulation by generating random energy losses from a power law distribution. The program eliminates annoying loop procedures, and model parameter sensitivity can be studied using the graphical output.

Howell, L. W.↗
NASA NTRS1989

K-distribution and polarimetric terrain radar clutter

A multivariate K-distribution is proposed to model the statistics of fully polarimetric radar data from earth terrain with polarizations HH, HV, VH, and VV. In this approach, correlated polarizations of radar signals, as characterized by a covariance matrix, are treated as the sum of N n-dimensional random vectors; N obeys the negative binomial distribution with a parameter alpha and mean N-bar. Subsequently, an n-dimensional K-distribution, with either zero or nonzero mean, is developed in the limit of infinite N-bar or illuminated area. The probability density function (PDF) of the K-distributed vector normalized by its Euclidean norm is independent of the parameter alpha and is the same as that derived from a zero-mean Gaussian-distributed random vector.

Yueh, S. H.↗
NASA NTRS1988

Remote sensing of earth terrain

Two monographs and 85 journal and conference papers on remote sensing of earth terrain have been published, sponsored by NASA Contract NAG5-270. A multivariate K-distribution is proposed to model the statistics of fully polarimetric data from earth terrain with polarizations HH, HV, VH, and VV. In this approach, correlated polarizations of radar signals, as characterized by a covariance matrix, are treated as the sum of N n-dimensional random vectors; N obeys the negative binomial distribution with a parameter alpha and mean bar N. Subsequently, and n-dimensional K-distribution, with either zero or non-zero mean, is developed in the limit of infinite bar N or illuminated area. The probability density function (PDF) of the K-distributed vector normalized by its Euclidean norm is independent of the parameter alpha and is the same as that derived from a zero-mean Gaussian-distributed random vector. The above model is well supported by experimental data provided by MIT Lincoln Laboratory and the Jet Propulsion Laboratory in the form of polarimetric measurements.

Kong, J. A.↗
NASA NTRS1973

Statistical properties of two sine waves in Gaussian noise.

A detailed study is presented of some statistical properties of a stochastic process that consists of the sum of two sine waves of unknown relative phase and a normal process. Since none of the statistics investigated seem to yield a closed-form expression, all the derivations are cast in a form that is particularly suitable for machine computation. Specifically, results are presented for the probability density function (pdf) of the envelope and the instantaneous value, the moments of these distributions, and the relative cumulative density function (cdf).

Esposito, R.↗
NASA NTRS1992

Long-time behavior of material-surface curvature in isotropic turbulence

The behavior at large times of the curvature of material elements in turbulence is investigated using Lagrangian velocity-gradient time series obtained from direct numerical simulations of isotropic turbulence. The main objectives are: to study the asymptotic behavior of the pdf curvature as a function of initial curvature and shape; and to establish whether the curvature of an initially plane material element goes to a stationary probability distribution. The evidence available in the literature about the asymptotic curvature-pdf of initially flat surfaces is ambiguous, and the conjecture is that it is quasi-stationary. In this work several material-element ensembles of different initial curvatures and shapes are studied. It is found that, at long times the moments of the logarithm of curvature are independent of the initial pdf of curvature. This, it is argued, supports the view that the curvature attains a stationary distribution at long times. It is also shown that, irrespective of initial shape or curvature, the shape of any material element at long times is cylindrical with a high probability.

Girimaji, S. S.↗
NASA NTRS2022

Markovian Statistical Model of Cloud Optical Thickness. Part I: Theory and Examples

We present a generalization of the binary-value Markovian model previously used for statistical characterization of cloud masks to a continuous-value model describing 1D fields of cloud optical thickness (COT). This model has simple functional expressions and is specified by four parameters: the cloud fraction, the autocorrelation (scale) length, and the two parameters of the normalized probability density function of (non-zero) COT values (this PDF is assumed to have gamma-distribution form). Cloud masks derived from this model by separation between the values above and below some threshold in COT appear to have the same statistical properties as in binary-value model described in our previous publications. We demonstrate the ability of our model to generate examples of various cloud-field types by using it to statistically imitate actual cloud observations made by the Research Scanning Polarimeter (RSP) during two field experiments.

binary-value Markovian model↗
NASA NTRS2004

CDF and PDF Comparison Between Humacao, Puerto Rico and Florida

The knowledge of the atmospherics phenomenon is an important part in the communication system. The principal factor that contributes to the attenuation in a Ka band communication system is the rain attenuation. We have four years of tropical region observations. The data in the tropical region was taken in Humacao, Puerto Rico. Previous data had been collected at various climate regions such as desserts, template area and sub-tropical regions. Figure 1 shows the ITU-R rain zone map for North America. Rain rates are important to the rain attenuation prediction models. The models that predict attenuation generally are of two different kinds. The first one is the regression models. By using a data set these models provide an idea of the observed attenuation and rain rates distribution in the present, past and future. The second kinds of models are physical models which use the probability density functions (PDF).

Gonzalez-Rodriguez, Rosana↗
NASA NTRS1974

System and method for character recognition

A character recognition system is disclosed in which each character in a retina, defining a scanning raster, is scanned with random lines uniformly distributed over the retina. For each type of character to be recognized the system stores a probability density function (PDF) of the random line intersection lengths and/or a PDF of the random line number of intersections. As an unknown character is scanned, the random line intersection lengths and/or the random line number of intersections are accumulated and based on a comparison with the prestored PDFs a classification of the unknown character is performed.

Hong, J. P.↗
NASA NTRS2022

Non-Gaussian Distributions of TOA SW Flux as Observed by MISR and CERES

The Top of Atmosphere (TOA) shortwave (SW) flux, converted from Terra Multi angle Imaging‐SpectroRadiometer (MISR) narrow band albedos, is compared with that measured from Clouds and the Earth’s Radiant Energy System (CERES). We describe the probability density function (PDF) of the monthly TOA SW flux and how the statistical third moment, skewness, can impact the quantification of the flux. The PDF of the SW flux is not normally distributed but positively skewed. In both sets of observations, the near-global (80S-80N) median value of the SW flux is ≈3 W/m 2 less than the mean value, due to the positive skewness of the distribution. The near-global mean TOA SW flux converted from MISR is about 7 W/m 2 (≈7%) less than CERES measured flux during the last two decades. Surprisingly, hemispheric asymmetry exists with TOA SW observations from Terra platform. SH reflects 3.92 W/m 2 and 1.15 W/m 2 more mean SW flux than NH, from MISR and CERES Single Scanner Footprint products, respectively. We can infer that the offsetting by morning clouds in the SH is greater than the effect of hemispheric imbalance of SW flux caused by different land masses in two hemispheres. While the characteristics of the two SW fluxes broadly agree with each other, differences in the regional PDF from two different SW fluxes are substantially different over high cloud regions and high altitude regions. Our analysis shows that some parts of the different skewness from two measurements may be attributed to the different calibration of the radiance anisotropy over high cloud scenes.

satellite measurements↗
NASA NTRS2012

Statistical Orbit Determination using the Particle Filter for Incorporating Non-Gaussian Uncertainties

The tracking of space objects requires frequent and accurate monitoring for collision avoidance. As even collision events with very low probability are important, accurate prediction of collisions require the representation of the full probability density function (PDF) of the random orbit state. Through representing the full PDF of the orbit state for orbit maintenance and collision avoidance, we can take advantage of the statistical information present in the heavy tailed distributions, more accurately representing the orbit states with low probability. The classical methods of orbit determination (i.e. Kalman Filter and its derivatives) provide state estimates based on only the second moments of the state and measurement errors that are captured by assuming a Gaussian distribution. Although the measurement errors can be accurately assumed to have a Gaussian distribution, errors with a non-Gaussian distribution could arise during propagation between observations. Moreover, unmodeled dynamics in the orbit model could introduce non-Gaussian errors into the process noise. A Particle Filter (PF) is proposed as a nonlinear filtering technique that is capable of propagating and estimating a more complete representation of the state distribution as an accurate approximation of a full PDF. The PF uses Monte Carlo runs to generate particles that approximate the full PDF representation. The PF is applied in the estimation and propagation of a highly eccentric orbit and the results are compared to the Extended Kalman Filter and Splitting Gaussian Mixture algorithms to demonstrate its proficiency.

Mashiku, Alinda↗
NASA NTRS1987

Statistics of Sxy estimates

The statistics of Sxy estimates derived from orthogonal-component measurements are examined. Based on results of Goodman (1957), the probability density function (pdf) for Sxy(f) estimates is derived, and a closed-form solution for arbitrary moments of the distribution is obtained. Characteristic functions are used to derive the exact pdf of Sxy(tot). In practice, a simple Gaussian approximation is found to be highly accurate even for relatively few degrees of freedom. Implications for experiment design are discussed, and a maximum-likelihood estimator for a posterior estimation is outlined.

Freilich, M. H.↗
NASA NTRSDate not supplied

Adding GPU Support to the Markov Chain Monte Carlo Code Catmip

In geophysics, we are confronted with many under-determined inverse problems. For example, all of our observations of earthquakes are made at the Earth’s surface. So, when we try to infer how slip during an earthquake evolves in space and time, we find that there are many potential slip histories that are consistent with our limited observations and our understanding of earthquake physics. One way to approach these problems is with Bayesian analysis which allows us to infer the ensemble of all potential slip models that satisfy the observations and our prior knowledge of earthquake physics. In Bayesian analysis, our prior knowledge is known as the prior probability density function or prior PDF, the fit to the data is known as the data likelihood, and the target PDF that satisfies both the prior PDF and data likelihood is known as the posterior PDF. However, simulating the posterior PDF typically requires using Markov Chain Monte Carlo (MCMC) to draw tens of billions of random realizations of earthquake slip models, which may not be computationally feasible. To make this and similar geophysical inversions computationally tractable, we developed the Cascading Adaptive Transitional Metropolis In Parallel (CATMIP) algorithm. CATMIP is an efficient parallel Markov Chain Monte Carlo (MCMC) sampler that is used for model fitting and uncertainty quantification in geophysics. Example use cases are earthquake rupture modeling, determining mineral composition on Mars, reconstructing the history of ocean salinity, and historical earthquake relocation. CATMIP employs many parallel instances of the Metropolis algorithm for sampling in a transitioning framework. Transitioning is a process in which a set of random samples at equilibrium with a known probability density function (PDF) are used as seeds for the Markov chains to sample successive target PDFs that incrementally move the distribution from the starting seeds to the final desired PDF that describes the relative plausibility of potential values for the model parameters. The algorithm is implemented as a Master-Worker model employing MPI for communication. The worker processes are loosely coupled with global parameters periodically optimized by the master process. This provides a very high amount of parallelism with little communication between updates. During the presentation we will discuss the history of the algorithm and elaborate the earthquake rupture modeling use case for the CATMIP package. Our first step toward GPU optimization was to optimize the code for the CPU. CPU profiling revealed that most of the compute time is spent in calls to level 2 BLAS routines and calls to GSL random number generators. We revised the algorithm to employ level 3 BLAS routines instead. In our presentation we will describe how this was accomplished. Adding GPU support to CATMIP consisted mostly of replacing the calls to GSL with calls to GPU vendor-provided library routines. A small number of loops were directly implemented in CUDA. In the presentation will provide implementation details. Finally, we will discuss methods for profiling and opportunities for further optimizing GPU execution. By creating a code with the flexibility to run on either a CPU or GPU architecture, CATMIP can be used on systems ranging from large CPU-based HPC environments to single servers with GPU acceleration and everything in between.

HECC↗
NASA NTRS1979

The reflection of impulses from a nonlinear random sea

A model of the reflection of radar impulses from the sea at near-vertical incidence is used to account for non-Gaussian ocean waves statistics. The joint probability density function (pdf), of wave height and slope, is calculated according to the theory of Longuet-Higgins (1963) on the distribution of variables in a 'weakly nonlinear' random era. The long-crested approximation is made, a Phillips wave spectrum is assumed, and the Gram-Charlier series is truncated after skewness terms. It is found that the height and height-slope skewness coefficients bear the ratio 1:2 and that the derived impulse response and conditional cross section versus wave height are in excellent agreement with previous observations. Finally, it is suggested that the empirically determined and theoretically predicted sea state bias be corrected for in the routine processing of satellite radar altimeter data.

Jackson, F. C.↗
NASA NTRS1995

Coupled Monte Carlo Probability Density Function/ SPRAY/CFD Code Developed for Modeling Gas-Turbine Combustor Flows

The success of any solution methodology for studying gas-turbine combustor flows depends a great deal on how well it can model various complex, rate-controlling processes associated with turbulent transport, mixing, chemical kinetics, evaporation and spreading rates of the spray, convective and radiative heat transfer, and other phenomena. These phenomena often strongly interact with each other at disparate time and length scales. In particular, turbulence plays an important role in determining the rates of mass and heat transfer, chemical reactions, and evaporation in many practical combustion devices. Turbulence manifests its influence in a diffusion flame in several forms depending on how turbulence interacts with various flame scales. These forms range from the so-called wrinkled, or stretched, flamelets regime, to the distributed combustion regime. Conventional turbulence closure models have difficulty in treating highly nonlinear reaction rates. A solution procedure based on the joint composition probability density function (PDF) approach holds the promise of modeling various important combustion phenomena relevant to practical combustion devices such as extinction, blowoff limits, and emissions predictions because it can handle the nonlinear chemical reaction rates without any approximation. In this approach, mean and turbulence gas-phase velocity fields are determined from a standard turbulence model; the joint composition field of species and enthalpy are determined from the solution of a modeled PDF transport equation; and a Lagrangian-based dilute spray model is used for the liquid-phase representation with appropriate consideration of the exchanges of mass, momentum, and energy between the two phases. The PDF transport equation is solved by a Monte Carlo method, and existing state-of-the-art numerical representations are used to solve the mean gasphase velocity and turbulence fields together with the liquid-phase equations. The joint composition PDF approach was extended in our previous work to the study of compressible reacting flows. The application of this method to several supersonic diffusion flames associated with scramjet combustor flow fields provided favorable comparisons with the available experimental data. A further extension of this approach to spray flames, three-dimensional computations, and parallel computing was reported in a recent paper. The recently developed PDF/SPRAY/computational fluid dynamics (CFD) module combines the novelty of the joint composition PDF approach with the ability to run on parallel architectures. This algorithm was implemented on the NASA Lewis Research Center's Cray T3D, a massively parallel computer with an aggregate of 64 processor elements. The calculation procedure was applied to predict the flow properties of both open and confined swirl-stabilized spray flames.

Source record↗
NASA NTRS2004

Picturing Data With Uncertainty

NASA is in the business of creating maps for scientific purposes to represent important biophysical or geophysical quantities over space and time. For example, maps of surface temperature over the globe tell scientists where and when the Earth is heating up; regional maps of the greenness of vegetation tell scientists where and when plants are photosynthesizing. There is always uncertainty associated with each value in any such map due to various factors. When uncertainty is fully modeled, instead of a single value at each map location, there is a distribution expressing a set of possible outcomes at each location. We consider such distribution data as multi-valued data since it consists of a collection of values about a single variable. Thus, a multi-valued data represents both the map and its uncertainty. We have been working on ways to visualize spatial multi-valued data sets effectively for fields with regularly spaced units or grid cells such as those in NASA's Earth science applications. A new way to display distributions at multiple grid locations is to project the distributions from an individual row, column or other user-selectable straight transect from the 2D domain. First at each grid cell in a given slice (row, column or transect), we compute a smooth density estimate from the underlying data. Such a density estimate for the probability density function (PDF) is generally more useful than a histogram, which is a classic density estimate. Then, the collection of PDFs along a given slice are presented vertically above the slice and form a wall. To minimize occlusion of intersecting slices, the corresponding walls are positioned at the far edges of the boundary. The PDF wall depicts the shapes of the distributions very dearly since peaks represent the modes (or bumps) in the PDFs. We've defined roughness as the number of peaks in the distribution. Roughness is another useful summary information for multimodal distributions. The uncertainty of the multi-valued data can also be interpreted by the number of peaks and the widths of the peaks as shown by the PDF walls.

Kao, David↗
NASA NTRS2003

Non-Gaussian PDF Modeling of Turbulent Boundary Layer Fluctuating Pressure Excitation

The purpose of the study is to investigate properties of the probability density function (PDF) of turbulent boundary layer fluctuating pressures measured on the exterior of a supersonic transport aircraft. It is shown that fluctuating pressure PDFs differ from the Gaussian distribution even for surface conditions having no significant discontinuities. The PDF tails are wider and longer than those of the Gaussian model. For pressure fluctuations upstream of forward-facing step discontinuities and downstream of aft-facing step discontinuities, deviations from the Gaussian model are more significant and the PDFs become asymmetrical. Various analytical PDF distributions are used and further developed to model this behavior.

Steinwolf, Alexander↗