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NASA NTRS1982

Mixture densities, maximum likelihood, and the EM algorithm

The problem of estimating the parameters which determine a mixture density is reviewed as well as maximum likelihood estimation for it. A particular iterative procedure for numerically approximating maximum likelihood estimates for mixture density problems is considered. This EM algorithm, is a specialization to the mixture density context of a general algorithm of the same name used to approximate maximum likelihood estimates for incomplete data problems. The formulation and theoretical and practical properties of the EM algorithm for mixture densities are discussed focussing in particular on mixtures of densities from exponential families.

Redner, R. A.↗

NASA NTRS2004

Symmetric normal mixtures

We consider mixture density estimation under the symmetry constraint x = Az for an orthogonal matrix A. This distributional constraint implies a corresponding constraint on the mixture parameters. Focusing on the gaussian case, we derive an expectation-maximization (EM) algorithm to enforce the constraint and show results for modeling of image feature vectors.

symmetry constraint↗

NASA NTRS1994

Software for Data Analysis with Graphical Models

Probabilistic graphical models are being used widely in artificial intelligence and statistics, for instance, in diagnosis and expert systems, as a framework for representing and reasoning with probabilities and independencies. They come with corresponding algorithms for performing statistical inference. This offers a unifying framework for prototyping and/or generating data analysis algorithms from graphical specifications. This paper illustrates the framework with an example and then presents some basic techniques for the task: problem decomposition and the calculation of exact Bayes factors. Other tools already developed, such as automatic differentiation, Gibbs sampling, and use of the EM algorithm, make this a broad basis for the generation of data analysis software.

Buntine, Wray L.↗

NASA NTRS1994

Data Analysis with Graphical Models: Software Tools

Probabilistic graphical models (directed and undirected Markov fields, and combined in chain graphs) are used widely in expert systems, image processing and other areas as a framework for representing and reasoning with probabilities. They come with corresponding algorithms for performing probabilistic inference. This paper discusses an extension to these models by Spiegelhalter and Gilks, plates, used to graphically model the notion of a sample. This offers a graphical specification language for representing data analysis problems. When combined with general methods for statistical inference, this also offers a unifying framework for prototyping and/or generating data analysis algorithms from graphical specifications. This paper outlines the framework and then presents some basic tools for the task: a graphical version of the Pitman-Koopman Theorem for the exponential family, problem decomposition, and the calculation of exact Bayes factors. Other tools already developed, such as automatic differentiation, Gibbs sampling, and use of the EM algorithm, make this a broad basis for the generation of data analysis software.

Buntine, Wray L.↗

NASA NTRS1986

Statistical methods for astronomical data with upper limits. II - Correlation and regression

Statistical methods for calculating correlations and regressions in bivariate censored data where the dependent variable can have upper or lower limits are presented. Cox's regression and the generalization of Kendall's rank correlation coefficient provide significant levels of correlations, and the EM algorithm, under the assumption of normally distributed errors, and its nonparametric analog using the Kaplan-Meier estimator, give estimates for the slope of a regression line. Monte Carlo simulations demonstrate that survival analysis is reliable in determining correlations between luminosities at different bands. Survival analysis is applied to CO emission in infrared galaxies, X-ray emission in radio galaxies, H-alpha emission in cooling cluster cores, and radio emission in Seyfert galaxies.

Isobe, T.↗

NASA NTRS2003

Density Estimation with Mercer Kernels

We present a new method for density estimation based on Mercer kernels. The density estimate can be understood as the density induced on a data manifold by a mixture of Gaussians fit in a feature space. As is usual, the feature space and data manifold are defined with any suitable positive-definite kernel function. We modify the standard EM algorithm for mixtures of Gaussians to infer the parameters of the density. One benefit of the approach is it's conceptual simplicity, and uniform applicability over many different types of data. Preliminary results are presented for a number of simple problems.

Macready, William G.↗

NASA NTRS2012

Particle Filtering for Model-Based Anomaly Detection in Sensor Networks

A novel technique has been developed for anomaly detection of rocket engine test stand (RETS) data. The objective was to develop a system that postprocesses a csv file containing the sensor readings and activities (time-series) from a rocket engine test, and detects any anomalies that might have occurred during the test. The output consists of the names of the sensors that show anomalous behavior, and the start and end time of each anomaly. In order to reduce the involvement of domain experts significantly, several data-driven approaches have been proposed where models are automatically acquired from the data, thus bypassing the cost and effort of building system models. Many supervised learning methods can efficiently learn operational and fault models, given large amounts of both nominal and fault data. However, for domains such as RETS data, the amount of anomalous data that is actually available is relatively small, making most supervised learning methods rather ineffective, and in general met with limited success in anomaly detection. The fundamental problem with existing approaches is that they assume that the data are iid, i.e., independent and identically distributed, which is violated in typical RETS data. None of these techniques naturally exploit the temporal information inherent in time series data from the sensor networks. There are correlations among the sensor readings, not only at the same time, but also across time. However, these approaches have not explicitly identified and exploited such correlations. Given these limitations of model-free methods, there has been renewed interest in model-based methods, specifically graphical methods that explicitly reason temporally. The Gaussian Mixture Model (GMM) in a Linear Dynamic System approach assumes that the multi-dimensional test data is a mixture of multi-variate Gaussians, and fits a given number of Gaussian clusters with the help of the wellknown Expectation Maximization (EM) algorithm. The parameters thus learned are used for calculating the joint distribution of the observations. However, this GMM assumption is essentially an approximation and signals the potential viability of non-parametric density estimators. This is the key idea underlying the new approach.

Solano, Wanda↗

NASA NTRS2020

Shape Estimation for Elongated Deformable Object using B-spline Chained Multiple Random Matrices Model

In this paper, a B-spline chained multiple random matrix models (RMMs) representation is proposed to model geometric characteristics of an elongated deformable object. The hyper degrees of freedom structure of the elongated deformable object make its shape estimation challenging. Based on the likelihood function of the proposed B-spline chained multiple RMMs, an expectation-maximization (EM) method is derived to estimate the shape of the elongated deformable object. A split and merge method based on the Euclidean minimum spanning tree (EMST) is proposed to provide initialization for the EM algorithm. The proposed algorithm is evaluated for the shape estimation of the elongated deformable objects in scenarios, such as the static rope with various configurations (including configurations with intersection), the continuous manipulation of a rope and a plastic tube, and the assembly of two plastic tubes. The execution time is computed and the accuracy of the shape estimation results is evaluated based on the comparisons between the estimated width values and its ground-truth, and the intersection over union (IoU) metric.

Gang Yao↗

NASA NTRS2018

Sensor Data Quality and Angular Rate Down-Selection Algorithms on SLS EM-1

The NASA Space Launch System Block 1 launch vehicle is equipped with an Inertial Navigation System (INS) and multiple Rate Gyro Assemblies (RGA) that are used in the Guidance, Navigation, and Control (GN&C) algorithms. The INS provides the inertial position, velocity, and attitude of the vehicle along with both angular rate and specific force measurements. Additionally, multiple sets of co-located rate gyros supply angular rate data. The collection of angular rate data, taken along the launch vehicle, is used to separate out vehicle motion from flexible body dynamics. Since the system architecture uses redundant sensors, the capability was developed to evaluate the health (or validity) of the independent measurements. A suite of Sensor Data Quality (SDQ) algorithms is responsible for assessing the angular rate data from the redundant sensors. When failures are detected, SDQ will take the appropriate action and disqualify or remove faulted sensors from forward processing. Additionally, the SDQ algorithms contain logic for down-selecting the angular rate data used by the GNC software from the set of healthy measurements. This paper explores the trades and analyses that were performed in selecting a set of robust fault-detection algorithms included in the GN&C flight software. These trades included both an assessment of hardware-provided health and status data as well as an evaluation of different algorithms based on time-to-detection, type of failures detected, and probability of detecting false positives. We then provide an overview of the algorithms used for both fault-detection and measurement down selection. We next discuss the role of trajectory design, flexible-body models, and vehicle response to off-nominal conditions in setting the detection thresholds. Lastly, we present lessons learned from software integration and hardware-in-the-loop testing.

Park, Thomas↗

NASA NTRS2018

Sensor Data Quality and Angular Rate Down-Selection Algorithms on SLS EM-1

Park, Thomas↗

DOE OSTI2026

Computationally efficient Bayesian estimation of graphical networks for omics data

Graphical networks are useful, widely-used modeling approaches to represent complex biological processes with biological measurements generated by platforms such as mass spectrometry. Bayesian analyses of graphical networks for omics data have several advantages over their frequentist counterparts, such as the inclusion of prior knowledge in the estimation of models. However, Bayesian approaches to date have only been feasible for data with a couple hundred biomolecules due to prohibitive computational time, but omics data often contains tens of thousands of biomolecules. Here, we present and illustrate a more computationally efficient approach named BPlane (Bayesian PseudoLikelihood-based Algorithm for Network Estimation) to extend Bayesian modeling capabilities for larger-sized datasets, such as most untargeted proteomics data. Via simulation, we demonstrate that BPlane produces substantial computational savings over a current state-of-the-art Bayesian algorithm while maintaining competitive edge detection accuracy. On a SARS-CoV2 proteomics data with 7000 proteins, the competing algorithm takes three times as long to complete the first iteration as BPlane takes to converge after over 100 iterations.

EM algorithm↗

NASA NTRS2004

Automatic Generation of Algorithms for the Statistical Analysis of Planetary Nebulae Images

Analyzing data sets collected in experiments or by observations is a Core scientific activity. Typically, experimentd and observational data are &aught with uncertainty, and the analysis is based on a statistical model of the conjectured underlying processes, The large data volumes collected by modern instruments make computer support indispensible for this. Consequently, scientists spend significant amounts of their time with the development and refinement of the data analysis programs. AutoBayes [GF+02, FS03] is a fully automatic synthesis system for generating statistical data analysis programs. Externally, it looks like a compiler: it takes an abstract problem specification and translates it into executable code. Its input is a concise description of a data analysis problem in the form of a statistical model as shown in Figure 1; its output is optimized and fully documented C/C++ code which can be linked dynamically into the Matlab and Octave environments. Internally, however, it is quite different: AutoBayes derives a customized algorithm implementing the given model using a schema-based process, and then further refines and optimizes the algorithm into code. A schema is a parameterized code template with associated semantic constraints which define and restrict the template s applicability. The schema parameters are instantiated in a problem-specific way during synthesis as AutoBayes checks the constraints against the original model or, recursively, against emerging sub-problems. AutoBayes schema library contains problem decomposition operators (which are justified by theorems in a formal logic in the domain of Bayesian networks) as well as machine learning algorithms (e.g., EM, k-Means) and nu- meric optimization methods (e.g., Nelder-Mead simplex, conjugate gradient). AutoBayes augments this schema-based approach by symbolic computation to derive closed-form solutions whenever possible. This is a major advantage over other statistical data analysis systems which use numerical approximations even in cases where closed-form solutions exist. AutoBayes is implemented in Prolog and comprises approximately 75.000 lines of code. In this paper, we take one typical scientific data analysis problem-analyzing planetary nebulae images taken by the Hubble Space Telescope-and show how AutoBayes can be used to automate the implementation of the necessary anal- ysis programs. We initially follow the analysis described by Knuth and Hajian [KHO2] and use AutoBayes to derive code for the published models. We show the details of the code derivation process, including the symbolic computations and automatic integration of library procedures, and compare the results of the automatically generated and manually implemented code. We then go beyond the original analysis and use AutoBayes to derive code for a simple image segmentation procedure based on a mixture model which can be used to automate a manual preproceesing step. Finally, we combine the original approach with the simple segmentation which yields a more detailed analysis. This also demonstrates that AutoBayes makes it easy to combine different aspects of data analysis.

Fischer, Bernd↗

NASA NTRS1996

Software For Multivariate Bayesian Classification

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.

Saul, Ronald↗

NASA NTRS1992

Antenna analysis using neural networks

Conventional computing schemes have long been used to analyze problems in electromagnetics (EM). The vast majority of EM applications require computationally intensive algorithms involving numerical integration and solutions to large systems of equations. The feasibility of using neural network computing algorithms for antenna analysis is investigated. The ultimate goal is to use a trained neural network algorithm to reduce the computational demands of existing reflector surface error compensation techniques. Neural networks are computational algorithms based on neurobiological systems. Neural nets consist of massively parallel interconnected nonlinear computational elements. They are often employed in pattern recognition and image processing problems. Recently, neural network analysis has been applied in the electromagnetics area for the design of frequency selective surfaces and beam forming networks. The backpropagation training algorithm was employed to simulate classical antenna array synthesis techniques. The Woodward-Lawson (W-L) and Dolph-Chebyshev (D-C) array pattern synthesis techniques were used to train the neural network. The inputs to the network were samples of the desired synthesis pattern. The outputs are the array element excitations required to synthesize the desired pattern. Once trained, the network is used to simulate the W-L or D-C techniques. Various sector patterns and cosecant-type patterns (27 total) generated using W-L synthesis were used to train the network. Desired pattern samples were then fed to the neural network. The outputs of the network were the simulated W-L excitations. A 20 element linear array was used. There were 41 input pattern samples with 40 output excitations (20 real parts, 20 imaginary). A comparison between the simulated and actual W-L techniques is shown for a triangular-shaped pattern. Dolph-Chebyshev is a different class of synthesis technique in that D-C is used for side lobe control as opposed to pattern shaping. The interesting thing about D-C synthesis is that the side lobes have the same amplitude. Five-element arrays were used. Again, 41 pattern samples were used for the input. Nine actual D-C patterns ranging from -10 dB to -30 dB side lobe levels were used to train the network. A comparison between simulated and actual D-C techniques for a pattern with -22 dB side lobe level is shown. The goal for this research was to evaluate the performance of neural network computing with antennas. Future applications will employ the backpropagation training algorithm to drastically reduce the computational complexity involved in performing EM compensation for surface errors in large space reflector antennas.

Smith, William T.↗

NASA NTRS2004

Putting Priors in Mixture Density Mercer Kernels

This paper presents a new methodology for automatic knowledge driven data mining based on the theory of Mercer Kernels, which are highly nonlinear symmetric positive definite mappings from the original image space to a very high, possibly infinite dimensional feature space. We describe a new method called Mixture Density Mercer Kernels to learn kernel function directly from data, rather than using predefined kernels. These data adaptive kernels can en- code prior knowledge in the kernel using a Bayesian formulation, thus allowing for physical information to be encoded in the model. We compare the results with existing algorithms on data from the Sloan Digital Sky Survey (SDSS). The code for these experiments has been generated with the AUTOBAYES tool, which automatically generates efficient and documented C/C++ code from abstract statistical model specifications. The core of the system is a schema library which contains template for learning and knowledge discovery algorithms like different versions of EM, or numeric optimization methods like conjugate gradient methods. The template instantiation is supported by symbolic- algebraic computations, which allows AUTOBAYES to find closed-form solutions and, where possible, to integrate them into the code. The results show that the Mixture Density Mercer-Kernel described here outperforms tree-based classification in distinguishing high-redshift galaxies from low- redshift galaxies by approximately 16% on test data, bagged trees by approximately 7%, and bagged trees built on a much larger sample of data by approximately 2%.

Srivastava, Ashok N.↗

NASA NTRS2005

An Ensemble Approach to Building Mercer Kernels with Prior Information

This paper presents a new methodology for automatic knowledge driven data mining based on the theory of Mercer Kernels, which are highly nonlinear symmetric positive definite mappings from the original image space to a very high, possibly dimensional feature space. we describe a new method called Mixture Density Mercer Kernels to learn kernel function directly from data, rather than using pre-defined kernels. These data adaptive kernels can encode prior knowledge in the kernel using a Bayesian formulation, thus allowing for physical information to be encoded in the model. Specifically, we demonstrate the use of the algorithm in situations with extremely small samples of data. We compare the results with existing algorithms on data from the Sloan Digital Sky Survey (SDSS) and demonstrate the method's superior performance against standard methods. The code for these experiments has been generated with the AUTOBAYES tool, which automatically generates efficient and documented C/C++ code from abstract statistical model specifications. The core of the system is a schema library which contains templates for learning and knowledge discovery algorithms like different versions of EM, or numeric optimization methods like conjugate gradient methods. The template instantiation is supported by symbolic-algebraic computations, which allows AUTOBAYES to find closed-form solutions and, where possible, to integrate them into the code.

Srivastava, Ashok N.↗

NASA NTRS2013

Orion MPCV GN and C End-to-End Phasing Tests

End-to-end integration tests are critical risk reduction efforts for any complex vehicle. Phasing tests are an end-to-end integrated test that validates system directional phasing (polarity) from sensor measurement through software algorithms to end effector response. Phasing tests are typically performed on a fully integrated and assembled flight vehicle where sensors are stimulated by moving the vehicle and the effectors are observed for proper polarity. Orion Multi-Purpose Crew Vehicle (MPCV) Pad Abort 1 (PA-1) Phasing Test was conducted from inertial measurement to Launch Abort System (LAS). Orion Exploration Flight Test 1 (EFT-1) has two end-to-end phasing tests planned. The first test from inertial measurement to Crew Module (CM) reaction control system thrusters uses navigation and flight control system software algorithms to process commands. The second test from inertial measurement to CM S-Band Phased Array Antenna (PAA) uses navigation and communication system software algorithms to process commands. Future Orion flights include Ascent Abort Flight Test 2 (AA-2) and Exploration Mission 1 (EM-1). These flights will include additional or updated sensors, software algorithms and effectors. This paper will explore the implementation of end-to-end phasing tests on a flight vehicle which has many constraints, trade-offs and compromises. Orion PA-1 Phasing Test was conducted at White Sands Missile Range (WSMR) from March 4-6, 2010. This test decreased the risk of mission failure by demonstrating proper flight control system polarity. Demonstration was achieved by stimulating the primary navigation sensor, processing sensor data to commands and viewing propulsion response. PA-1 primary navigation sensor was a Space Integrated Inertial Navigation System (INS) and Global Positioning System (GPS) (SIGI) which has onboard processing, INS (3 accelerometers and 3 rate gyros) and no GPS receiver. SIGI data was processed by GN&C software into thrust magnitude and direction commands. The processing changes through three phases of powered flight: pitchover, downrange and reorientation. The primary inputs to GN&C are attitude position, attitude rates, angle of attack (AOA) and angle of sideslip (AOS). Pitch and yaw attitude and attitude rate responses were verified by using a flight spare SIGI mounted to a 2-axis rate table. AOA and AOS responses were verified by using a data recorded from SIGI movements on a robotic arm located at NASA Johnson Space Center. The data was consolidated and used in an open-loop data input to the SIGI. Propulsion was the Launch Abort System (LAS) Attitude Control Motor (ACM) which consisted of a solid motor with 8 nozzles. Each nozzle has active thrust control by varying throat area with a pintle. LAS ACM pintles are observable through optically transparent nozzle covers. SIGI movements on robot arm, SIGI rate table movements and LAS ACM pintle responses were video recorded as test artifacts for analysis and evaluation. The PA-1 Phasing Test design was determined based on test performance requirements, operational restrictions and EGSE capabilities. This development progressed during different stages. For convenience these development stages are initial, working group, tiger team, Engineering Review Team (ERT) and final.

Neumann, Brian C.↗

NASA NTRS2018

PEG Enhancement for EM1 and EM2+ Missions

NASA is currently building the Space Launch System (SLS) Block-1 launch vehicle for the Exploration Mission 1 (EM-1) test flight. The next evolution of SLS, the Block-1B Exploration Mission 2 (EM-2), is currently being designed. The Block-1 and Block-1B vehicles will use the Powered Explicit Guidance (PEG) algorithm. Due to the relatively low thrust-to-weight ratio of the Exploration Upper Stage (EUS), certain enhancements to the Block-1 PEG algorithm are needed to perform Block-1B missions. In order to accommodate mission design for EM-2 and beyond, PEG has been significantly improved since its use on the Space Shuttle program. The current version of PEG has the ability to switch to different targets during Core Stage (CS) or EUS flight, and can automatically reconfigure for a single Engine Out (EO) scenario, loss of communication with the Launch Abort System (LAS), and Inertial Navigation System (INS) failure. The Thrust Factor (TF) algorithm uses measured state information in addition to a priori parameters, providing PEG with an improved estimate of propulsion information. This provides robustness against unknown or undetected engine failures. A loft parameter input allows LAS jettison while maximizing payload mass. The current PEG algorithm is now able to handle various classes of missions with burn arcs much longer than were seen in the shuttle program. These missions include targeting a circular LEO orbit with a low-thrust, long-burn-duration upper stage, targeting a highly eccentric Trans-Lunar Injection (TLI) orbit, targeting a disposal orbit using the low-thrust Reaction Control System (RCS), and targeting a hyperbolic orbit. This paper will describe the design and implementation of the TF algorithm, the strategy to handle EO in various flight regimes, algorithms to cover off-nominal conditions, and other enhancements to the Block-1 PEG algorithm. This paper illustrates challenges posed by the Block-1B vehicle, and results show that the improved PEG algorithm is capable for use on the SLS Block 1-B vehicle as part of the Guidance, Navigation, and Control System.

Von der Porten, Paul↗