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NASA NTRS1983

Conic sectors for sampled-data feedback systems

The conic-sector analysis of the closed-loop stability and robustness of a multivariable-analog-system controller based on sampled-data feedback compensation is investigated. Conic sectors and sampled-data feedback systems are defined, and the existence of a conic sector containing a sampled-data operator is established mathematically. An example is presented to prove that the conic sector is computable and gives sufficient conditions of closed-loop stability. A procedure for determining sampled-data-operator gain is also derived.

Thompson, P. M.↗

NASA NTRS1975

Multivariate Analysis, Retrieval, and Storage System (MARS). Volume 6: MARS System - A Sample Problem (Gross Weight of Subsonic Transports)

The Mars system is a tool for rapid prediction of aircraft or engine characteristics based on correlation-regression analysis of past designs stored in the data bases. An example of output obtained from the MARS system, which involves derivation of an expression for gross weight of subsonic transport aircraft in terms of nine independent variables is given. The need is illustrated for careful selection of correlation variables and for continual review of the resulting estimation equations. For Vol. 1, see N76-10089.

Hague, D. S.↗

DOE OSTI2024

Sampling Size Optimization for Bioburden Density Estimation in Planetary Protection

Planetary protection (PP) is a discipline that focuses on minimizing the biological contamination of spacecraft to ensure compliance with international policy. Precise estimation of bioburden - the total number of microbes in or on spacecraft hardware – and the bioburden density are of utmost importance for PP. Such estimation is the way concordance with requirements is demonstrated, and it is critical for quantifying the potential risk of inadvertently contaminating other planetary bodies. Although a suite of molecular techniques have been used to thoroughly characterize and profile the microbiome of various cleanroom environments and spacecraft, the gold standard remains the physical enumeration of microbes via culturing of samples directly taken from spacecraft and associated surfaces. However, due to technical, budgetary, and programmatic constraints, only a manageable portion (around 10%) of the entire spacecraft surface is directly sampled with cotton swabs or wipes. To generate the bioburden current best estimate (CBE) for components not directly verifiable, the accepted approach is to apply a NASA-defined bioburden estimate based on the components’ manufacturing or assembly environment. This approach utilizes a prespecified bioburden density estimation that applies a maximum value across the total surface area of the specified component. For hardware components that underwent similar assembly processes, an implied bioburden is adopted for all components, based on a direct verification of a representative component within the same lot. Once all components have a CBE, the bioburden estimates are generated. In previous publication [ 1], we have shown that statistical risks quantifying the accuracy of the estimates for sampled, prespecified, and implied components can be derived and ranked. For mean squared error (MSE) function, the risks are available analytically and hence a cost function can be obtained to optimize the risks with respect to the sampling area and sampling cost. Since the sampling area and sampling cost are two complimentary variables, their sum will have a well-defined minimum. This paper presents the multivariate optimization of the integrated risk of an empirical Bayes estimator to determine the optimal sampling schedule for a given number of components. It is assumed that given a number of components, N, the bioburden density for each component can either be sampled, implied, or prespecified. The multivariate optimization searches through different options to sample, imply or prespecify the bioburden density for a component, and account for the component’s surface area and cost of sampling. The idea of the optimization is based on the observation that the statistical risk of using an estimator is a monotonically decreasing function of the sampled area. The larger the sampled area, the lower the risk of using the estimator as the estimator becomes more and more accurate as the sampling area increases. On the other hand, the cost of sampling is monotonically increasing as the sampled surface grows. This makes the risk and total cost of sampling complimentary variables which can be counterbalanced to achieve an optimal overall value with respect to the sampled surface. In this paper, the integrated risk has been used to quantify the accuracy of the estimator. This risk has been selected because it depends on neither the true value of the parameter nor on the collected data. The cost of each sample was also available to obtain the total cost of sampling of N components. The paper will present the results based on computer-simulated data as well as the data collected during the InSight mission. The computer-simulated data have N components with randomly generated total areas and each component assigned to one of the three categories according to the method of estimating of bioburden density: sampled, implied, or prespecified. The cost of sampling is also available. The cost of sampling is estimated based on a cost model provided by the planetary protection group at JPL. For this paper, the overall cost was assumed to be a linear function of exposure. The optimization process finds the allocation of the components to the three categories that minimizes the tradeoff between integrated risk and total cost. For the InSight data, a set of components is selected representing all three categories, and optimization is performed to determine if the performed allocation was optimal or if a better allocation could have been obtained. To the best of our knowledge, this work is the first attempt not only perform an accurate estimation of bioburden density but also do it in an optimal way.

97 - MATHEMATICS AND COMPUTING↗

NASA NTRS1981

Computer program documentation for the patch subsampling processor

The programs presented are intended to provide a way to extract a sample from a full-frame scene and summarize it in a useful way. The sample in each case was chosen to fill a 512-by-512 pixel (sample-by-line) image since this is the largest image that can be displayed on the Integrated Multivariant Data Analysis and Classification System. This sample size provides one megabyte of data for manipulation and storage and contains about 3% of the full-frame data. A patch image processor computes means for 256 32-by-32 pixel squares which constitute the 512-by-512 pixel image. Thus, 256 measurements are available for 8 vegetation indexes over a 100-mile square.

Nieves, M. J.↗

DOE OSTI2024

Search for ${\text {Z}{}{}} {\text {Z}{}{}} $ and ${\text {Z}{}{}} {\text {H}{}{}} $ production in the ${\text {b}{}{}} {\bar{{\text {b}{}{}}}{}{}} {\text {b}{}{}} {\bar{{\text {b}{}{}}}{}{}} $ final state using proton-proton collisions at $\sqrt{s}=13\,\text {Te}\hspace{-.08em}\text {V} $

A search for ${\text {Z}{}{}} {\text {Z}{}{}} $ and ${\text {Z}{}{}} {\text {H}{}{}} $ production in the ${\text {b}{}{}} {\bar{{\text {b}{}{}}}{}{}} {\text {b}{}{}} {\bar{{\text {b}{}{}}}{}{}} $ final state is presented, where H is the standard model (SM) Higgs boson. The search uses an event sample of proton-proton collisions corresponding to an integrated luminosity of 133$\,\text {fb}^{-1}$ collected at a center-of-mass energy of 13$\,\text {Te}\hspace{-.08em}\text {V}$ with the CMS detector at the CERN LHC. The analysis introduces several novel techniques for deriving and validating a multi-dimensional background model based on control samples in data. A multiclass multivariate classifier customized for the ${\text {b}{}{}} {\bar{{\text {b}{}{}}}{}{}} {\text {b}{}{}} {\bar{{\text {b}{}{}}}{}{}} $ final state is developed to derive the background model and extract the signal. The data are found to be consistent, within uncertainties, with the SM predictions. The observed (expected) upper limits at 95% confidence level are found to be 3.8 (3.8) and 5.0 (2.9) times the SM prediction for the ${\text {Z}{}{}} {\text {Z}{}{}} $ and ${\text {Z}{}{}} {\text {H}{}{}} $ production cross sections, respectively.

72 PHYSICS OF ELEMENTARY PARTICLES AND FIELDS↗

DOE OSTI2019

Polynomial chaos expansions for dependent random variables

Polynomial chaos expansions (PCE) are well-suited to quantifying uncertainty in models parameterized by independent random variables. The assumption of independence leads to simple strategies for building multivariate orthonormal bases and for sampling strategies to evaluate PCE coefficients. In contrast, the application of PCE to models of dependent variables is much more challenging. Three approaches can be used to construct PCE of models of dependent variables. The first approach uses mapping methods where measure transformations, such as the Nataf and Rosenblatt transformation, can be used to map dependent random variables to independent ones; however we show that this can significantly degrade performance since the Jacobian of the map must be approximated. A second strategy is the class of dominating support methods. In these approaches a PCE is built using independent random variables whose distributional support dominates the support of the true dependent joint density; we provide evidence that this approach appears to produce approximations with suboptimal accuracy. A third approach, the novel method proposed here, uses Gram-Schmidt orthogonalization (GSO) to numerically compute orthonormal polynomials for the dependent random variables. This approach has been used successfully when solving differential equations using the intrusive stochastic Galerkin method, and in this paper we use GSO to build PCE using a non-intrusive stochastic collocation method. The stochastic collocation method treats the model as a black box and builds approximations of the input-output map from a set of samples. Building PCE from samples can introduce ill-conditioning which does not plague stochastic Galerkin methods. To mitigate this ill-conditioning we generate weighted Leja sequences, which are nested sample sets, to build accurate polynomial interpolants. We show that our proposed approach, GSO with weighted Leja sequences, produces PCE which are orders of magnitude more accurate than PCE constructed using mapping or dominating support methods.

97 MATHEMATICS AND COMPUTING↗

NASA NTRS1990

Gaussian windows: A tool for exploring multivariate data

Presented here is a method for interactively exploring a large set of quantitative multivariate data, in order to estimate the shape of the underlying density function. It is assumed that the density function is more or less smooth, but no other specific assumptions are made concerning its structure. The local structure of the data in a given region may be examined by viewing the data through a Gaussian window, whose location and shape are chosen by the user. A Gaussian window is defined by giving each data point a weight based on a multivariate Gaussian function. The weighted sample mean and sample covariance matrix are then computed, using the weights attached to the data points. These quantities are used to compute an estimate of the shape of the density function in the window region. The local structure of the data is described by a method similar to the method of principal components. By taking many such local views of the data, we can form an idea of the structure of the data set. The method is applicable in any number of dimensions. The method can be used to find and describe simple structural features such as peaks, valleys, and saddle points in the density function, and also extended structures in higher dimensions. With some practice, we can apply our geometrical intuition to these structural features in any number of dimensions, so that we can think about and describe the structure of the data. Since the computations involved are relatively simple, the method can easily be implemented on a small computer.

Jaeckel, Louis A.↗

DOE OSTI2025

Use of Fisher's Ratio assisted multivariate curve resolution- alternating least squares for discovery-based analysis using ultrahigh pressure liquid chromatography-high resolution mass spectrometry

Non-targeted analysis of complex chemical mixtures can be difficult considering the convoluted nature of the matrix and the potential unknown chemical differences between samples or classes of samples. Ultrahigh pressure liquid chromatography coupled to quadrupole time-of-flight mass spectrometry (UHPLC-QTOF) is an ideal technique to probe chemical differences for a wide variety of samples. While UHPLC-QTOF can discover minute chemical differences down to low part per billion (ppb) concentrations with a high degree of confidence, the application of high-resolution mass spectrometry can yield massive amounts of information (∼ 10 gb per sample) that cannot be analyzed manually. Therefore, the application of chemometric techniques is mandatory for the interrogation of complex samples. Fisher's ratio (FR) assisted multivariate curve resolution-alternating least squares (MCR-ALS) was used to the discover and identify the chemical differences between two classes of materials: 1) a pond water matrix and 2) the matrix spiked with a pharmaceutical standard mix containing 17 compounds. Thirteen of the seventeen spiked compounds were discovered using FR analysis, and then five were successfully deconvoluted using MCR-ALS wherein the number of curves chosen were automatically determined using singular value decomposition (SVD). In conclusion, the use of an automated FR assisted MCR-ALS will aid in discovering trace levels of chemical components without the need for the researcher to provide potentially biased input which will aid in non-targeted workflow.

37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CH↗

NASA NTRS2011

A Comparison of Multivariate and Pre-Processing Methods for Quantitative Laser-Induced Breakdown Spectroscopy of Geologic Samples

The ChemCam instrument selected for the Curiosity rover is capable of remote laser-induced breakdown spectroscopy (LIBS).[1] We used a remote LIBS instrument similar to ChemCam to analyze 197 geologic slab samples and 32 pressed-powder geostandards. The slab samples are well-characterized and have been used to validate the calibration of previous instruments on Mars missions, including CRISM [2], OMEGA [3], the MER Pancam [4], Mini-TES [5], and Moessbauer [6] instruments and the Phoenix SSI [7]. The resulting dataset was used to compare multivariate methods for quantitative LIBS and to determine the effect of grain size on calculations. Three multivariate methods - partial least squares (PLS), multilayer perceptron artificial neural networks (MLP ANNs) and cascade correlation (CC) ANNs - were used to generate models and extract the quantitative composition of unknown samples. PLS can be used to predict one element (PLS1) or multiple elements (PLS2) at a time, as can the neural network methods. Although MLP and CC ANNs were successful in some cases, PLS generally produced the most accurate and precise results.

Anderson, R. B.↗

DOE OSTI2020

On the quantification and efficient propagation of imprecise probabilities with copula dependence

This paper addresses the problem of quantification and propagation of uncertainties associated with dependence modeling when data for characterizing probability models are limited. Practically, the system inputs are often assumed to be mutually independent or correlated by a multivariate Gaussian distribution. However, this subjective assumption may introduce bias in the response estimate if the real dependence structure deviates from this assumption. In this work, we overcome this limitation by introducing a flexible copula dependence model to capture complex dependencies. Here, a hierarchical Bayesian multimodel approach is proposed to quantify uncertainty in dependence model-form and model parameters that result from small data sets. This approach begins by identifying, through Bayesian multimodel inference, a set of candidate marginal models and their corresponding model probabilities, and then estimating the uncertainty in the copula-based dependence structure, which is conditional on the marginals and their parameters. The overall uncertainties integrating marginals and copulas are probabilistically represented by an ensemble of multivariate candidate densities. A novel importance sampling reweighting approach is proposed to efficiently propagate the overall uncertainties through a computational model. Through an example studying the influence of constituent properties on the out-of-plane properties of transversely isotropic E-glass fiber composites, we show that the composite property with copula-based dependence model converges to the true estimate as data set size increases, while an independence or arbitrary Gaussian correlation assumption leads to a biased estimate.

72 PHYSICS OF ELEMENTARY PARTICLES AND FIELDS↗

NASA NTRS1982

Predicting the required number of training samples

A criterion which measures the quality of the estimate of the covariance matrix of a multivariate normal distribution is developed. Based on this criterion, the necessary number of training samples is predicted. Experimental results which are used as a guide for determining the number of training samples are included.

Kalayeh, H. M.↗

NASA NTRS1983

Predicting the required number of training samples

Kalayeh, H. M.↗

NASA NTRS1995

A multiparametric analysis of the Einstein sample of early-type galaxies. 1: Luminosity and ISM parameters

We have conducted bivariate and multivariate statistical analysis of data measuring the luminosity and interstellar medium of the Einstein sample of early-type galaxies (presented by Fabbiano, Kim, & Trinchieri 1992). We find a strong nonlinear correlation between L(sub B) and L(sub X), with a power-law slope of 1.8 +/- 0.1, steepening to 2.0 +/- if we do not consider the Local Group dwarf galaxies M32 and NGC 205. Considering only galaxies with log L(sub X) less than or equal to 40.5, we instead find a slope of 1.0 +/- 0.2 (with or without the Local Group dwarfs). Although E and S0 galaxies have consistent slopes for their L(sub B)-L(sub X) relationships, the mean values of the distribution functions of both L(sub X) and L(sub X)/L(sub B) for the S0 galaxies are lower than those for the E galaxies at the 2.8 sigma and 3.5 sigma levels, respectively. We find clear evidence for a correlation between L(sub X) and the X-ray color C(sub 21), defined by Kim, Fabbiano, & Trinchieri (1992b), which indicates that X-ray luminosity is correlated with the spectral shape below 1 keV in the sense that low-L(sub X) systems have relatively large contributions from a soft component compared with high-L(sub X) systems. We find evidence from our analysis of the 12 micron IRAS data for our sample that our S0 sample has excess 12 micron emission compared with the E sample, scaled by their optical luminosities. This may be due to emission from dust heated in star-forming regions in S0 disks. This interpretation is reinforced by the existence of a strong L(sub 12)-L(sub 100) correlation for our S0 sample that is not found for the E galaxies, and by an analysis of optical-IR colors. We find steep slopes for power-law relationships between radio luminosity and optical, X-ray, and far-IR (FIR) properties. This last point argues that the presence of an FIR-emitting interstellar medium (ISM) in early-type galaxies is coupled to their ability to generate nonthermal radio continuum, as previously argued by, e.g., Walsh et al. (1989). We also find that, for a given L(sub 100), galaxies with larger L(sub X)/L(sub B) tend to be stronger nonthermal radio sources, as originally suggested by Kim & Fabbiano (1990). We note that, while L(sub B) is most strongly correlated with L(sub 6), the total radio luminosity, both L(sub X) and L(sub X)/L(sub B) are more strongly correlated with L(sub 6 CO), the core radio luminosity. These points support the argument (proposed by Fabbiano, Gioia, & Trinchieri 1989) that radio cores in early-type galaxies are fueled by the hot ISM.

Eskridge, Paul B.↗

NASA NTRS2006

Diurnal Variability of Vertical Structure from a TRMM Passive Microwave "Virtual Radar" Retrieval

Robust description of the diurnal cycle from TRMM observations is complicated by the limitations of Low Earth Orbit (LEO) sampling; from a 'climatological' perspective, sufficient sampling must exist to control for both spatial and seasonal variability, before tackling an additional diurnal component (e.g., with 8 additional 3-hourly or 24 1-hourly bins). For documentation of vertical structure, the narrow sample swath of the TRMM Precipitation Radar limits the resolution of any of these components. A neural-network based 'virtual radar" retrieval has been trained and internally validated, using multifrequency / multipolarization passive microwave(TM1) brightness temperatures and textures parameters and lightning (LIS) observations, as inputs, and PR volumetric reflectivity as targets (outputs). By training the algorithms (essentially highly multivariate, nonlinear regressions) on a very large sample of high-quality co-located data from the center of the TRMM swath, 3D radar reflectivity and derived parameters (VIL, IWC, Echo Tops, etc.) can be retrieved across the entire TMI swath, good to 8-9% over the dynamic range of parameters. As a step in the retrieval (and as an output of the process), each TMI multifrequency pixel (at 85 GHz resolution) is classified into one of the 25 archetypal radar profile vertical structure "types", previously identified using cluster analysis. The dynamic range of retrieved vertical structure appears to have higher fidelity than the current (Version 6) experimental GPROF hydrometeor vertical structure retrievals. This is attributable to correct representation of the prior probabilities of vertical structure variability in the neural network training data, unlike the GPROF cloud-resolving model training dataset used in the V6 algorithms. The LIS lightning inputs are supplementary inputs, and a separate offline neural network has been trained to impute (predict) LIS lightning from passive-microwave-only data. The virtual radar retrieval is thus, in principle, extensible to Aqua/AMSR-E and NPOESS/CMIS passive microwave instruments. The virtual radar approach yields a threefold increase in effective sampling from the mission, albeit of lower-quality "retrieved" data, reducing the variance of local estimates by one third (or the standard deviation by-0.57). In this talk, the variance reduction is leveraged to more finely resolve global diurnal variability in both space and time (local hour).

Boccippio, Dennis J.↗

NASA NTRS1994

Noise and drift analysis of non-equally spaced timing data

Generally, it is possible to obtain equally spaced timing data from oscillators. The measurement of the drifts and noises affecting oscillators is then performed by using a variance (Allan variance, modified Allan variance, or time variance) or a system of several variances (multivariance method). However, in some cases, several samples, or even several sets of samples, are missing. In the case of millisecond pulsar timing data, for instance, observations are quite irregularly spaced in time. Nevertheless, since some observations are very close together (one minute) and since the timing data sequence is very long (more than ten years), information on both short-term and long-term stability is available. Unfortunately, a direct variance analysis is not possible without interpolating missing data. Different interpolation algorithms (linear interpolation, cubic spline) are used to calculate variances in order to verify that they neither lose information nor add erroneous information. A comparison of the results of the different algorithms is given. Finally, the multivariance method was adapted to the measurement sequence of the millisecond pulsar timing data: the responses of each variance of the system are calculated for each type of noise and drift, with the same missing samples as in the pulsar timing sequence. An estimation of precision, dynamics, and separability of this method is given.

Vernotte, F.↗

NASA NTRS1982

Accuracy of remotely sensed data: Sampling and analysis procedures

A review and update of the discrete multivariate analysis techniques used for accuracy assessment is given. A listing of the computer program written to implement these techniques is given. New work on evaluating accuracy assessment using Monte Carlo simulation with different sampling schemes is given. The results of matrices from the mapping effort of the San Juan National Forest is given. A method for estimating the sample size requirements for implementing the accuracy assessment procedures is given. A proposed method for determining the reliability of change detection between two maps of the same area produced at different times is given.

Congalton, R. G.↗

DOE OSTI2025

Chemistry imaging and distribution analysis of rare earth elements in coal using LIBS and LA-ICP-MS instruments

Currently, demand for rare earth elements (REEs) increased significantly. Coal is actively evaluated as potential economic sources for extraction of REEs. Here, in this work, laser-induced breakdown spectroscopy (LIBS) was evaluated for rapid estimation of REEs content and their distribution in the natural coal samples. The results were compared with similar laser ablation–inductively coupled plasma–mass spectrometry (LA-ICP-MS) measurements. Thirteen coal samples (nine standard samples and five natural samples) were used in this study. Powder samples were pressed into pellets while coal chunks were directly ablated for data recording. Pellets of the powder standard samples were used to optimize the data acquisition system and then data recorded with this optimized system was used to identify the proper data acquisition and analysis models. After establishing the proper data acquisition system and analysis model using the standard samples, natural coal samples in powder form and their chunks were utilized to record LIBS and LA-ICP-MS spectra. Multivariate calibration models were developed using four of the natural samples, which were evaluated by predicting the REE content in the fifth sample. Principal component analysis was performed on the LIBS data obtained from the natural samples and it classified all the samples with high accuracy. Two-dimensional (2D) elemental mapping on coal chunk samples was also performed using both LIBS and LA-ICP-MS to study the distribution of REEs in the samples. The resulting elemental images and their correlations can be used to infer mineral distributions.

01 COAL, LIGNITE, AND PEAT↗

NASA NTRS1995

A diagnostic analysis of the VVP single-doppler retrieval technique

A diagnostic analysis of the VVP (volume velocity processing) retrieval method is presented, with emphasis on understanding the technique as a linear, multivariate regression. Similarities and differences to the velocity-azimuth display and extended velocity-azimuth display retrieval techniques are discussed, using this framework. Conventional regression diagnostics are then employed to quantitatively determine situations in which the VVP technique is likely to fail. An algorithm for preparation and analysis of a robust VVP retrieval is developed and applied to synthetic and actual datasets with high temporal and spatial resolution. A fundamental (but quantifiable) limitation to some forms of VVP analysis is inadequate sampling dispersion in the n space of the multivariate regression, manifest as a collinearity between the basis functions of some fitted parameters. Such collinearity may be present either in the definition of these basis functions or in their realization in a given sampling configuration. This nonorthogonality may cause numerical instability, variance inflation (decrease in robustness), and increased sensitivity to bias from neglected wind components. It is shown that these effects prevent the application of VVP to small azimuthal sectors of data. The behavior of the VVP regression is further diagnosed over a wide range of sampling constraints, and reasonable sector limits are established.

Boccippio, Dennis J.↗