A general theory on multivariable sampled-data control systems.
Digital controller design for forward and feedback loops of multivariable sample-data control systems
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Polytopic vector analysis in igneous petrology: Application to lunar petrogenesis
Lunar samples represent a heterogeneous assemblage of rocks with complex inter-relationships that are difficult to decipher using standard petrogenetic approaches. These inter-relationships reflect several distinct petrogenetic trends as well as thermomechanical mixing of distinct components. Additional complications arise from the unequal quality of chemical analyses and from the fact that many samples (e.g., breccia clasts) are too small to be representative of the system from which they derived. Polytopic vector analysis (PVA) is a multi-variate procedure used as a tool for exploratory data analysis. PVA allows the analyst to classify samples and clarifies relationships among heterogenous samples with complex petrogenetic histories. It differs from orthogonal factor analysis in that it uses non-orthogonal multivariate sample vectors to extract sample endmember compositions. The output from a Q-mode (sample based) factor analysis is the initial step in PVA. The Q-mode analysis, using criteria established by Miesch and Klovan and Miesch, is used to determine the number of endmembers in the data system. The second step involves determination of endmembers and mixing proportions with all output expressed in the same geochemical variable as the input. The composition of endmembers is derived by analysis of the variability of the data set. Endmembers need not be present in the data set, nor is it necessary for their composition to be known a priori. A set of any endmembers defines a 'polytope' or classification figure (triangle for a three component system, tetrahedron for a four component system, a 'five-tope' in four dimensions for five component system, et cetera).
Shervais, John W.↗
Simulation techniques for estimating error in the classification of normal patterns
Methods of efficiently generating and classifying samples with specified multivariate normal distributions were discussed. Conservative confidence tables for sample sizes are given for selective sampling. Simulation results are compared with classified training data. Techniques for comparing error and separability measure for two normal patterns are investigated and used to display the relationship between the error and the Chernoff bound.
Whitsitt, S. J.↗
Multivariate Methods for Prediction of Geologic Sample Composition with Laser-Induced Breakdown Spectroscopy
Laser-induced breakdown spectroscopy (LIBS) uses pulses of laser light to ablate a material from the surface of a sample and produce an expanding plasma. The optical emission from the plasma produces a spectrum which can be used to classify target materials and estimate their composition. The ChemCam instrument on the Mars Science Laboratory (MSL) mission will use LIBS to rapidly analyze targets remotely, allowing more resource- and time-intensive in-situ analyses to be reserved for targets of particular interest. ChemCam will also be used to analyze samples that are not reachable by the rover's in-situ instruments. Due to these tactical and scientific roles, it is important that ChemCam-derived sample compositions are as accurate as possible. We have compared the results of partial least squares (PLS), multilayer perceptron (MLP) artificial neural networks (ANNs), and cascade correlation (CC) ANNs to determine which technique yields better estimates of quantitative element abundances in rock and mineral samples. The number of hidden nodes in the MLP ANNs was optimized using a genetic algorithm. The influence of two data preprocessing techniques were also investigated: genetic algorithm feature selection and averaging the spectra for each training sample prior to training the PLS and ANN algorithms. We used a ChemCam-like laboratory stand-off LIBS system to collect spectra of 30 pressed powder geostandards and a diverse suite of 196 geologic slab samples of known bulk composition. We tested the performance of PLS and ANNs on a subset of these samples, choosing to focus on silicate rocks and minerals with a loss on ignition of less than 2 percent. This resulted in a set of 22 pressed powder geostandards and 80 geologic samples. Four of the geostandards were used as a validation set and 18 were used as the training set for the algorithms. We found that PLS typically resulted in the lowest average absolute error in its predictions, but that the optimized MLP ANN and the CC ANN often gave results comparable to PLS. Averaging the spectra for each training sample and/or using feature selection to choose a small subset of wavelengths to use for predictions gave mixed results, with degraded performance in some cases and similar or slightly improved performance in other cases. However, training time was significantly reduced for both PLS and ANN methods by implementing feature selection, making this a potentially appealing method for initial, rapid-turn-around analyses necessary for Chemcam's tactical role on MSL. Choice of training samples has a strong influence on the accuracy of predictions. We are currently investigating the use of clustering algorithms (e.g. k-means, neural gas, etc.) to identify training sets that are spectrally similar to the unknown samples that are being predicted, and therefore result in improved predictions
Morris, Richard↗
The saturated activities of Na-22, Mn-54, and Co-56 and the depth of sampling of soils
A multivariate linear regression study shows that the saturated activities of Na-22, Co-56, and Mn-54 are functions of the chemical composition and the reciprocal of the cube root of the sample weight. A procedure for taking the temporal variation in these short-lived radionuclides into account is described. The regression results are used to estimate the average depths of sampling of soil material. The data indicate that all 28 samples were in the same conditions of exposure when recovered as they had been for the last approximately 10 years. The saturation activity of Mn-54 is a very weak function of time.
Keith, J. E.↗
Salvaging Data Records with Missing Data: Data Imputation using the Multivariate t Distribution
When doing multivariate data analysis, one commonobstacle is the presence of incomplete observations, i.e., observationsfor which one or more key fields are blank. Missing datais often countered by deleting entire observations that containmissing data. The negative effects of deleting entire observationsare multiple: deleting observations reduces sample size andcan also result in biased inferences even if data is missing atrandom. In addition, knowledge contained within incompleteobservations is knowledge lost when they are deleted– and theeffort spent collecting that knowledge is effort wasted. Data imputationmethods, or methods of statistically “filling-in” missingdata, can help combat small sample sizes by using the existinginformation in partially complete observations with the end goalof producing less biased and higher confidence inferences. Whena sample from a multivariate normal population is only partiallycomplete, and the missing data meets appropriate assumptions(missing at random), robust data imputation of the missing datacan be implemented with monotone data augmentation (MDA)using the multivariate t distribution.Missing data imputation is applied to data from the NASA InstrumentCost Model (NICM) using the MDA algorithm underthe assumption of having a multivariate t distribution with fixeddegrees of freedom. A sensitivity analysis to the degrees offreedom parameter is presented to demonstrate robustness ofthe multivariate t distribution when dealing with small samplesas compared to the multivariate normal distribution.
DiNicola, Michael↗
Analysis of Forest Foliage Using a Multivariate Mixture Model
Data with wet chemical measurements and near infrared spectra of ground leaf samples were analyzed to test a multivariate regression technique for estimating component spectra which is based on a linear mixture model for absorbance. The resulting unmixed spectra for carbohydrates, lignin, and protein resemble the spectra of extracted plant starches, cellulose, lignin, and protein. The unmixed protein spectrum has prominent absorption spectra at wavelengths which have been associated with nitrogen bonds.
Hlavka, C. A.↗
Experimental analysis of computer system dependability
This paper reviews an area which has evolved over the past 15 years: experimental analysis of computer system dependability. Methodologies and advances are discussed for three basic approaches used in the area: simulated fault injection, physical fault injection, and measurement-based analysis. The three approaches are suited, respectively, to dependability evaluation in the three phases of a system's life: design phase, prototype phase, and operational phase. Before the discussion of these phases, several statistical techniques used in the area are introduced. For each phase, a classification of research methods or study topics is outlined, followed by discussion of these methods or topics as well as representative studies. The statistical techniques introduced include the estimation of parameters and confidence intervals, probability distribution characterization, and several multivariate analysis methods. Importance sampling, a statistical technique used to accelerate Monte Carlo simulation, is also introduced. The discussion of simulated fault injection covers electrical-level, logic-level, and function-level fault injection methods as well as representative simulation environments such as FOCUS and DEPEND. The discussion of physical fault injection covers hardware, software, and radiation fault injection methods as well as several software and hybrid tools including FIAT, FERARI, HYBRID, and FINE. The discussion of measurement-based analysis covers measurement and data processing techniques, basic error characterization, dependency analysis, Markov reward modeling, software-dependability, and fault diagnosis. The discussion involves several important issues studies in the area, including fault models, fast simulation techniques, workload/failure dependency, correlated failures, and software fault tolerance.
Iyer, Ravishankar, K.↗
A hill-sliding strategy for initialization of Gaussian clusters in the multidimensional space
A hill sliding technique was devised to extract Gaussian clusters from the multivariate probability density estimate of sample data for the first step of iterative unsupervised classification. Each cluster was assumed to posses a unimodal normal distribution. A clustering function proposed distinguished elements of a cluster under formation from the rest in the feature space. Initial clusters were extracted one by one according to the hill sliding tactics. A dimensionless cluster compactness parameter was proposed as a universal measure of cluster goodness and used satisfactorily in test runs with LANDSAT multispectral scanner data. The normalized divergence, defined by the cluster divergence divided by the entropy of the entire sample data, was utilized as a general separability measure between clusters. An overall clustering objective function was set forth in terms of cluster covariance matrices, from which the cluster compactness measure could be deduced. Minimal improvement of initial data partitioning was evaluated by this objective function in eliminating scattered sparse data points. The hill sliding clustering technique developed herein has the potential applicability to decomposition any multivariate mixture distribution into a number of unimodal distributions when an appropriate distribution function to the data set is employed.
Park, J. K.↗
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.↗
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.↗
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.↗
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.↗
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.↗
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.↗
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. Previously announced in STAR as N82-28109
Kalayeh, H. M.↗
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.↗
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.↗