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Four research projects are summarized which include: (1) the generation of random numbers on the IBM 360/44, (2) statistical tests used to check out random number generators, (3) Specht density estimators, and (4) use of estimators of probability density functions in analyzing large amounts of data.
A database of over 1100 silicate glass compositions and densities was compiled and used to evaluate the efficacy of an algorithm for estimating the density of silicate glass compositions. We sought to develop a parsimonious algorithm based on the additivity of partial molar volumes of individual oxide components weighted by their mole fraction in a glass composition. Bound molar volumes were used for oxides in which the density of the oxide bound in a glass matrix was previously determined. The bound molar volumes were known for oxides covering 97.5 mole percent of the database compositional space. The measured glass densities were plotted against the estimated glass densities and a linear regression yielded an R 2 adj. = 0.95 and a slope and intercept of approximately one and zero, respectively. This regression suggests that glass densities estimated by the algorithm, within analysis uncertainty, are equal to the measured densities of the glasses. In addition to the development of the density estimation, we corroborated many of the referenced bound molar volume data used in the parameterization of the estimation algorithm via linear regression of the individual partial molar volumes versus the inverse measured densities (specific volumes) of the glasses in the database.
Estimates of snow and firn density are required for satellite-altimetry-based retrievals of ice sheet mass balance that rely on volume-to-mass conversions. Therefore, biases and errors in presently used density models confound assessments of ice sheet mass balance and by extension ice sheet contribution to sea level rise. Despite this importance, most contemporary firn densification models rely on simplified semi-empirical methods, which are partially reflected by significant modeled density errors when compared to observations. In this study, we present a new drifting-snow compaction scheme that we have implemented into SNOWPACK, a physics-based land surface snow model. We show that our new scheme improves existing versions of SNOWPACK by increasing simulated near-surface (defined as the top 10 m) density to be more in line with observations (near-surface bias reduction from −44.9 to −5.4 kg m−3). Furthermore, we demonstrate high-quality simulation of near-surface Antarctic snow and firn density at 122 observed density profiles across the Antarctic ice sheet, as indicated by reduced model biases throughout most of the near-surface firn column when compared to two semi-empirical firn densification models (SNOWPACK mean bias=−9.7 kg m−3, IMAU-FDM mean bias=−32.5 kg m−3, GSFC-FDM mean bias=15.5 kg m−3). Notably, our analysis is restricted to the near surface where firn density is most variable due to accumulation and compaction variability driven by synoptic weather and seasonal climate variability. Additionally, the GSFC-FDM exhibits lower mean density bias from 7–10 m (SNOWPACK bias=−22.5 kg m−3, GSFC-FDM bias=10.6 kg m−3) and throughout the entire near surface at high-accumulation sites (SNOWPACK bias=−31.4 kg m−3, GSFC-FDM bias=−4.7 kg m−3). However, we found that the performance of SNOWPACK did not degrade when applied to sites that were not included in the calibration of semi-empirical models. This suggests that SNOWPACK may possibly better represent firn properties in locations without extensive observations and under future climate scenarios, when firn properties are expected to diverge from their present state.
The Analytic Predictor Corrector (APC) and Energy Controller (EC) atmospheric guidance concepts were adapted to control an interplanetary vehicle aerobraking in the Martian atmosphere. Changes are made to the APC to improve its robustness to density variations. These changes include adaptation of a new exit phase algorithm, an adaptive transition velocity to initiate the exit phase, refinement of the reference dynamic pressure calculation and two improved density estimation techniques. The modified controller with the hybrid density estimation technique is called the Mars Hybrid Predictor Corrector (MHPC), while the modified controller with a polynomial density estimator is called the Mars Predictor Corrector (MPC). A Lyapunov Steepest Descent Controller (LSDC) is adapted to control the vehicle. The LSDC lacked robustness, so a Lyapunov tracking exit phase algorithm is developed to guide the vehicle along a reference trajectory. This algorithm, when using the hybrid density estimation technique to define the reference path, is called the Lyapunov Hybrid Tracking Controller (LHTC). With the polynomial density estimator used to define the reference trajectory, the algorithm is called the Lyapunov Tracking Controller (LTC). These four new controllers are tested using a six degree of freedom computer simulation to evaluate their robustness. The MHPC, MPC, LHTC, and LTC show dramatic improvements in robustness over the APC and EC.
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.
This paper introduces a novel anomaly detection approach tailored for time series data with exclusive reliance on normal events during training. Our key innovation lies in the application of kernel-density estimation (KDE) to scrutinize reconstruction errors, providing an empirically derived probability distribution for normal events post-reconstruction. This non-parametric density estimation technique offers a nuanced understanding of anomaly detection, differentiating it from prevalent threshold-based mechanisms in existing methodologies. In post-training, events are encoded, decoded, and evaluated against the estimated density, providing a comprehensive notion of normality. In addition, we propose a data augmentation strategy involving variational autoencoder-generated events and a smoothing step for enhanced model robustness. The significance of our autoencoder-based approach is evident in its capacity to learn normal representation without prior anomaly knowledge. Through the KDE step on reconstruction errors, our method addresses the versatility of anomalies, departing from assumptions tied to larger reconstruction errors for anomalous events. Our proposed likelihood measure then distinguishes normal from anomalous events, providing a concise yet comprehensive anomaly detection solution. The extensive experimental results support the feasibility of our proposed method, yielding significantly improved classification performance by nearly 10% on the UCR benchmark data.
A report is given to supplement the progress report of June 17, 1977. In that progress report gamma, lognormal, and Rayleigh probability density functions were fitted to the times between lightning flashes in the storms of 9/12/75, 8/26/75, and 7/13/76 by the maximum likelihood method. The goodness of fit is checked by the Kolmogoroff-Smirnoff test. Plots of the estimated densities along with normalized histograms are included to provide a visual check on the goodness of fit. The lognormal densities are the most peaked and have the highest tails. This results in the best fit to the normalized histogram in most cases. The Rayleigh densities have too broad and rounded peaks to give good fits. In addition, they have the lowest tails. The gamma densities fall inbetween and give the best fit in a few cases.
PHD general-purpose classifier computer program. Uses Bayesian methods to classify vectors of real numbers, based on combination of statistical techniques that include multivariate density estimation, Parzen density kernels, and EM (Expectation Maximization) algorithm. By means of simple graphical interface, user trains classifier to recognize two or more classes of data and then use it to identify new data. Written in ANSI C for Unix systems and optimized for online classification applications. Embedded in another program, or runs by itself using simple graphical-user-interface. Online help files makes program easy to use.
Gaussian type densities estimated by representing unknown density in terms of convolution expansion of Gaussian probability density and arbitrary distribution, discussing convergence rates
Methods for anomaly detection of new physics processes are often limited to low-dimensional spaces due to the difficulty of learning high-dimensional probability densities. Particularly at the constituent level, incorporating desirable properties such as permutation invariance and variable-length inputs becomes difficult within popular density estimation methods. In this work, we introduce a permutation-invariant density estimator for particle physics data based on diffusion models, specifically designed to handle variable-length inputs. We demonstrate the efficacy of our methodology by utilizing the learned density as a permutation-invariant anomaly detection score, effectively identifying jets with low likelihood under the background-only hypothesis. To validate our density estimation method, we investigate the ratio of learned densities and compare to those obtained by a supervised classification algorithm.
The Cassini spacecraft was launched on October 15, 1997 by a Titan 4B launch vehicle. After an interplanetary cruise of almost seven years, it arrived at Saturn on June 30, 2004. In 2005, Cassini completed three flybys of Enceladus, a small, icy satellite of Saturn. Observations made during these flybys confirmed the existence of water vapor plumes in the south polar region of Enceladus. Five additional low-altitude flybys of Enceladus were successfully executed in 2008-9 to better characterize these watery plumes. During some of these Enceladus flybys, the spacecraft attitude was controlled by a set of three reaction wheels. When the disturbance torque imparted on the spacecraft was predicted to exceed the control authority of the reaction wheels, thrusters were used to control the spacecraft attitude. Using telemetry data of reaction wheel rates or thruster on-times collected from four low-altitude Enceladus flybys (in 2008-10), one can reconstruct the time histories of the Enceladus plume jet density. The 1 sigma uncertainty of the estimated density is 5.9-6.7% (depending on the density estimation methodology employed). These plume density estimates could be used to confirm measurements made by other onboard science instruments and to support the modeling of Enceladus plume jets.
The development of next-generation molten salt reactors relies on accurate knowledge of the thermophysical properties of the candidate coolant and fueled molten salts. These thermophysical properties include density, viscosity, thermal conductivity, and heat capacity. Because of difficulties in measuring thermophysical properties of molten salts, there are many gaps in the current state of thermophysical property knowledge of these salts, particularly those that contain actinides or beryllium. Therefore, leveraging modeling techniques to estimate unknown molten salt thermophysical properties and guide future experimental measurements has high value for the nuclear industry. Here, the densities of molten fluoride pseudo-ternary salt systems, which are of interest to the nuclear industry, were estimated using Redlich-Kister expansion and Muggianu interpolation techniques. The pseudo-ternary systems considered for estimation in this study were NaF-LiF-ZrF 4 , LiF-BeF 2 -ZrF 4 , LiF-BeF 2 -ThF 4 , NaF-LiF-BeF 2 , NaF-KF-BeF 2 , NaF-ZrF 4 -UF 4 , and NaF-BeF 2 -UF 4 . This Redlich-Kister estimation approach accounts for nonideal mixing behavior based on pseudo-binary subsystem interaction parameters determined from experimentally measured pseudo-binary system density data sets. The Redlich-Kister estimation was compared with the method of additive molar volumes, which assumes ideal mixing. Additionally, the Redlich-Kister approach was used to determine previously unknown binary and ternary interaction parameters based on experimentally measured density data sets for select pseudo-ternary salt systems. The results of this study show improvement in density estimation using the Redlich-Kister approach for all systems considered compared with estimation by additive molar volumes. Furthermore, this analysis allowed for the estimation of nonideal density behavior in experimentally unstudied ZrF 4 -UF 4 and BeF 2 -UF 4 , as well as the quantification of ternary interaction in NaF-LiF-ZrF 4 , NaF-BeF 2 -UF 4 , and NaF-ZrF 4 -UF 4 .
Abstract Mixture density networks (MDNs) can be used to generate posterior density functions of model parameters $$\varvec{\theta }$$ θ given a set of observables $${\mathbf {x}}$$ x . In some applications, training data are available only for discrete values of a continuous parameter $$\varvec{\theta }$$ θ . In such situations, a number of performance-limiting issues arise which can result in biased estimates. We demonstrate the usage of MDNs for parameter estimation, discuss the origins of the biases, and propose a corrective method for each issue.
Direct and cross power spectral density estimation of discrete data using fast Fourier transform with digital filter solution
A new co-added IUE echelle spectrum of the bright DA white dwarf CD -38 deg 10980, together with a newly determined radial velocity for this star, indicate that the sharp lined Si and C absorption features seen in the UV are clearly circumstellar in origin. Absorption in both excited and ground state transitions occurs at a velocity displaced by -12.1 +/- 2.0 km.s with respect to the photospheric velocity. Weak features due to the Si IV doublet are seen at a velocity intermediate between that of the circumstellar features and the photosphere. First time estimates of column densities for excited and ground states of C II, Si II, and Si III are derived. These quantities are used with electron density estimates derived from these species to determine the location and physical conditions of the circumstellar gas in the vicinity of CD -38 deg 10980. If collisional excitation alone is responsible for the excited levels of Si III observed in CD -38 deg 10980, then electron densities in the circumstellar gas must exceed 10(exp 9)/cu cm. Substantially lower electron densities are possible if the circumstellar gas is located near enough to the star so that photoexcitation is the dominant process responsible for the excited lines seen in the UV. Strong limits are placed on the photospheric abundance of Si and C in the star itself. These limits are in sharp contrast to the theoretical predictions of radiative levitation in which Si, but not C, is expected in the photosphere of a white dwarf such as CD -38 deg 10980. The interstellar line of sight to CD -38 deg 10980 is also investigated.
The nominal launch date of the Mars Observer spacecraft is 09 - 16 - 02, with Mars encounter on 08 - 19 - 93 and the mapping orbit established on 12 - 06 - 93. Because this orbit is nearly circular at low altitude (361 km), both gravity and atmospheric drag will dominate the orbital evolution and are also the major error sources. This paper provides: (1) a summary of atmospheric density and errors for the Mars Observer orbit and time frame, (2) an assessment of spacecraft-position errors using current density estimates, and (3) an estimate of atmospheric density modeling improvement directly from the Doppler data acquired during flight operations. With an improved gravity field model, reconstructed spacecraft-position errors are at the level of 0.5 km (1sigma). However, when predicting the spacecraft's position over 14 days, the density error (at the 90-percent confidence level) dominates, yielding a down-track error of almost 70 km. By modeling the mean atmospheric density, this error can be reduced by at least a factor of two.
Aerobraking has been used four times to decrease the apoapsis of a spacecraft in a captured orbit around a planetary body with a significant atmosphere utilizing atmospheric drag to decelerate the spacecraft. While aerobraking requires minimum fuel, the long time required for aerobraking requires both a large operations staff, and large Deep Space Network resources. A study to automate aerobraking has been sponsored by the NASA Engineering and Safety Center to determine initial feasibility of equipping a spacecraft with the onboard capability for autonomous aerobraking, thus saving millions of dollars incurred by a large aerobraking operations workforce and continuous DSN coverage. This paper describes the need for autonomous aerobraking, the development of the Autonomous Aerobraking Development Software that includes an ephemeris estimator, an atmospheric density estimator, and maneuver calculation, and the plan forward for continuation of this study.
This paper describes the orbit reconstruction of Mars Express (MEX) with the specific goal of estimating the atmospheric density near periapsis and evaluating its variability and predictability. The desired outcome is to validate the covariance analysis assumption of atmospheric variability for the 2005 NASA Mars Reconnaissance Orbiter (MRO), as well as evaluate the accuracy of the density estimates output from the atmosphere model used by MRO. Topics covered include the MRO atmosphere model, MEX orbit determination and post-fit Doppler residuals, and atmosphere trending statistics gleaned from the orbit reconstructions.