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This monograph was prepared to give the practicing engineer a clear understanding of probability and statistics with special consideration to problems frequently encountered in aerospace engineering. It is conceived to be both a desktop reference and a refresher for aerospace engineers in government and industry. It could also be used as a supplement to standard texts for in-house training courses on the subject.
Based on statistical mechanics, a macroscopically homogeneous system, i.e., a single phase in the present context, is composed of many independent configurations that the system embraces. The macroscopical properties of the system are determined by the properties and statistical probabilities of those configurations with respect to external conditions. The volume of a single phase is thus the weighted sum of the volumes of all configurations. Consequently, the derivative of the volume to temperature of a single phase depends on both the derivatives of the volumes of every configuration to temperature and the derivatives of their statistical probabilities to temperature, with the latter introducing nonlinear emergent behaviors. It is shown that the derivative of the volume to the temperature of the single phase can be negative, i.e., negative thermal expansion, due to the symmetry-breaking non-ground-state configurations with smaller volumes than that of the ground-state configuration and the rapid increase of the statistical probabilities of the former, and negative thermal expansion can be predicted without fitting parameters from the zentropy theory that combines quantum mechanics and statistical mechanics with the free energy of each configuration predicted from quantum mechanics and the partition function of each configuration calculated from its free energy.
Computer simulation study to determine normality or nonnormality on any given population of sampled data - statistical probability distribution
The values of the Normalized Radar Backscattering Cross Section (NRCS), sigma (o), obtained by a scatterometer are random variables whose variance is a known function of the expected value. The probability density function can be obtained from the normal distribution. Models for the expected value obtain it as a function of the properties of the waves on the ocean and the winds that generated the waves. Point estimates of the expected value were found from various statistics given the parameters that define the probability density function for each value. Random intervals were derived with a preassigned probability of containing that value. A statistical test to determine whether or not successive values of sigma (o) are truly independent was derived. The maximum likelihood estimates for wind speed and direction were found, given a model for backscatter as a function of the properties of the waves on the ocean. These estimates are biased as a result of the terms in the equation that involve natural logarithms, and calculations of the point estimates of the maximum likelihood values are used to show that the contributions of the logarithmic terms are negligible and that the terms can be omitted.
Methods for the recovery of winds near the surface of the ocean from measurements of the normalized radar backscattering cross section must recognize and make use of the statistics (i.e., the sampling variability) of the backscatter measurements. Radar backscatter values from a scatterometer are random variables with expected values given by a model. A model relates backscatter to properties of the waves on the ocean, which are in turn generated by the winds in the atmospheric marine boundary layer. The effective wind speed and direction at a known height for a neutrally stratified atmosphere are the values to be recovered from the model. The probability density function for the backscatter values is a normal probability distribution with the notable feature that the variance is a known function of the expected value. The sources of signal variability, the effects of this variability on the wind speed estimation, and criteria for the acceptance or rejection of models are discussed. A modified maximum likelihood method for estimating wind vectors is described. Ways to make corrections for the kinds of errors found for the Seasat SASS model function are described, and applications to a new scatterometer are given.
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Launching pad operations for missiles - statistical probability and prelaunch problems
Reliability estimation of redundant systems for determining statistical probability of mission success
Statistical-mechanical estimate of orbital collision probabilities
Statistical probability distribution of first occurrence and first recurrence times of given level crossing in continuous random process
Statistical probability and sampled data method for compressing data from deep space probes
Strong coupled rotational excitation problem in atom-diatomic molecule scattering system, discussing transition probability matrices statistical analysis
We investigate limits on the extinction values of Type Ia supernovae (SNe Ia) to statistically determine the most probable color excess, E(B – V), with galactocentric distance, and use these statistics to determine the absorption-to-reddening ratio, R{sub V}, for dust in the host galaxies. We determined pixel-based dust mass surface density maps for 59 galaxies from the Key Insight on Nearby Galaxies: a Far-infrared Survey with Herschel (KINGFISH). We use SN Ia spectral templates to develop a Monte Carlo simulation of color excess E(B – V) with R{sub V} = 3.1 and investigate the color excess probabilities E(B – V) with projected radial galaxy center distance. Additionally, we tested our model using observed spectra of SN 1989B, SN 2002bo, and SN 2006X, which occurred in three KINGFISH galaxies. Finally, we determined the most probable reddening for Sa–Sap, Sab–Sbp, Sbc–Scp, Scd–Sdm, S0, and irregular galaxy classes as a function of R/R{sub 25}. We find that the largest expected reddening probabilities are in Sab–Sb and Sbc–Sc galaxies, while S0 and irregular galaxies are very dust poor. We present a new approach for determining the absorption-to-reddening ratio R{sub V} using color excess probability functions and find values of R{sub V} = 2.71 ± 1.58 for 21 SNe Ia observed in Sab–Sbp galaxies, and R{sub V} = 1.70 ± 0.38, for 34 SNe Ia observed in Sbc–Scp galaxies.
The identification of the parameters of the optimal control model from input-output data of the human operator is considered. Accepting the basic structure of the model as a cascade of a full-order observer and a feedback law, and suppressing the inherent optimality of the human controller, the parameters to be identified are the feedback matrix, the observer gain matrix, and the intensity matrices of the observation noise and the motor noise. The identification of the parameters is a statistical problem, because the system and output are corrupted by noise, and therefore the solution must be based on the statistics (probability density function) of the input and output data of the human operator. However, based on the statistics of the input-output data of the human operator, no distinction can be made between the observation and the motor noise, which shows that the model suffers from overparameterization.
With more and more application of digital I and C systems in nuclear power plants (NPP), more and more sensors are also required. Large amount of sensors really can make contribution to the condition monitoring of systems and equipment in NPP, whereas the heavy use also increases the malfunction risk of sensor self. Therefore the reliability and effectiveness of sensors must be guaranteed first. Thus the condition of sensors should be monitored first in a NPP. Since the precise mathematical models of sensors in complex systems are relative difficult to establish, however quantity of operating data can be easily obtained due to the wide application of digital I and C systems, thus this article adopts principal component analysis (PCA) to carry out condition monitoring for sensors. The PCA model developed for sensors is proved to be effective with operating data acquired from a real NPP. It can detect and isolate the fault when an artificial fault is imposed to the normal data. Then the monitoring results are further processed by another control limit (second confidence limit) besides T 2 and Q statistics (first confidence limit) in PCA. The value of second confidence limit is calculated based on the probability statistics of T 2 and Q statistics. And simulation tests show that the application of the second confidence limit can greatly reduce false alarms in the PCA model and the model performance can be greatly improved with the application of false alarms reducing method. (authors)
Invasive perennial pepperweed (Lepidium latifolium) has spread rapidly throughout the western United States in the past fifteen years. Pepperweed outcompetes many native species for water and nutrients, disturbing sensitive ecosystems. The purpose of this study is to map the contemporary distribution of pepperweed throughout the wetland ecosystem of the restored South San Francisco Bay Salt Ponds and create a habitat suitability model to predict future spread. Pepperweed reflectance data were collected in-situ with the GER 1500 spectroradiometer along with presence and absence pepperweed GPS data points. A Spectral Angle Mapper (SAM) classification algorithm (Method A) was used to distinguish pepperweed spectra and map its distribution on an EO-1 Hyperion image of the study area. Similarly, a supervised classification was run on an ASTER image of the study area. A GIS multivariate habitat suitability model was created to predict areas most susceptible to pepperweed colonization. Variables incorporated into the model were tidal extent, propensity for disturbance, pond salinity, proximity to levees and channels, and terrain curvature. A Generalized Additive Model (GAM) was also used to generate a suitability map (Method B) and investigate the statistical probability that each variable contributed to predict pepperweed spread. Results from the GAM revealed distance to channels, distance to disturbance factors, extreme pond salinity and curvature as statistically significant in determining the locations of suitable pepperweed habitats
Initial state selected dynamics of the Ne + NeH{sup +}(v{sub 0} = 0, j{sub 0} = 0) → NeH{sup +} + Ne reaction is investigated by quantum and statistical quantum mechanical (SQM) methods on the ground electronic state. The three-body ab initio energies on a set of suitably chosen grid points have been computed at CCSD(T)/aug-cc-PVQZ level and analytically fitted. The fitting of the diatomic potentials, computed at the same level of theory, is performed by spline interpolation. A collinear [NeHNe]{sup +} structure lying 0.72 eV below the Ne + NeH{sup +} asymptote is found to be the most stable geometry for this system. Energies of low lying vibrational states have been computed for this stable complex. Reaction probabilities obtained from quantum calculations exhibit dense oscillatory structures, particularly in the low energy region and these get partially washed out in the integral cross section results. SQM predictions are devoid of oscillatory structures and remain close to 0.5 after the rise at the threshold thus giving a crude average description of the quantum probabilities. Statistical cross sections and rate constants are nevertheless in sufficiently good agreement with the quantum results to suggest an important role of a complex-forming dynamics for the title reaction.
Accounting for the statistical geometric and material variability of structures in analysis has been a topic of considerable research for the last 30 years. The determination of quantifiable measures of statistical probability of a desired response variable, such as natural frequency, maximum displacement, or stress, to replace experience-based "safety factors" has been a primary goal of these studies. There are, however, several problems associated with their satisfactory application to realistic structures, such as bladed disks in turbomachinery. These include the accurate definition of the input random variables (rv's), the large size of the finite element models frequently used to simulate these structures, which makes even a single deterministic analysis expensive, and accurate generation of the cumulative distribution function (CDF) necessary to obtain the probability of the desired response variables. The research presented here applies a methodology called probabilistic dynamic synthesis (PDS) to solve these problems. The PDS method uses dynamic characteristics of substructures measured from modal test as the input rv's, rather than "primitive" rv's such as material or geometric uncertainties. These dynamic characteristics, which are the free-free eigenvalues, eigenvectors, and residual flexibility (RF), are readily measured and for many substructures, a reasonable sample set of these measurements can be obtained. The statistics for these rv's accurately account for the entire random character of the substructure. Using the RF method of component mode synthesis, these dynamic characteristics are used to generate reduced-size sample models of the substructures, which are then coupled to form system models. These sample models are used to obtain the CDF of the response variable by either applying Monte Carlo simulation or by generating data points for use in the response surface reliability method, which can perform the probabilistic analysis with an order of magnitude less computational effort. Both free- and forced-response analyses have been performed, and the results indicate that, while there is considerable room for improvement, the method produces usable and more representative solutions for the design of realistic structures with a substantial savings in computer time.