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Rodney Martin Conducts Seminar with Faculty and PhD Students at Lund University in Sweden
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Rodney Martin Conducts Seminar with Faculty and PhD Students at Lund University in Sweden

On June 27, 2014, Rodney Martin conducted a seminar with faculty and PhD students in mathematical statistics, automatic control, applied mathematics (the image group), and numerical mathematics at Lund University in Sweden. The seminar was on “Extensions and Analysis of a State-Space Approach to Optimal Level-Crossing Prediction for Linear Gaussian Processes.” It was arranged by Georg Lindgren, one of the pioneers of the technique.

Martin presented an overview of recent contributions to the literature on optimal level-crossing prediction for the linear Gaussian processes. There have been many pioneering efforts on this topic from the area of extreme value analysis beginning in the late 70s and early 80s, and a small subset of researchers from that community have continued on with this research track, considering many different applications and extensions of the basic idea. An overview of four different papers on a state-space approach to optimal level-crossing prediction for the linear Gaussian processes was presented, spanning a range of topics from extreme value analysis to providing motivation for a control theoretic perspective. A common theme among all of the papers is advocacy for use of the Area Under (the receiver operator) Curve (AUC), or area under the Receiver Operating Characteristic (ROC) curve to characterize the ability accurately to predict the level-crossing event. More formally, it quantifies the Mann-Whitney-Wilcoxon U test statistic, which is equivalent to the probability of correctly ranking two randomly selected data points, one belonging to the level-crossing event class, the other not. The AUC has been deemed as a theoretically legitimate metric for model selection and algorithmic comparison. This metric of performance is used ubiquitously in the fields of machine learning, medical diagnostics, and signal processing, among others. As such, the insights derived from the findings to be presented here have implications that span all of these fields, both from a theoretical and a practical perspective.

BACKGROUND: Some of the techniques investigated in tandem for this work have their origins in application to legacy NASA platforms. Rudolf E. Kalman found a unique application of his now very well-known Kalman filter for the Apollo program and more broadly to aerospace applications in general, due in part to finding support at NASA Ames Research Center in the mid-1960s. However, practical applications of Kalman filtering for aerospace have largely been relegated to state estimation for guidance, navigation, and control purposes. Although the study of auxiliary failure detection and bad data rejection algorithms have been developed in concert with Kalman filters, the main purpose of those Kalman filters were for state estimation in guidance, navigation, and control systems. Kalman filtering has seen limited practical application dedicated to system reliability and health management as related to exceedance of predetermined failure thresholds in aerospace systems. The difference in the approach taken with this investigation is that the Kalman filter machinery will be implemented for the express purpose of system reliability and health management, invoking more recently available data-mining and machine-learning techniques to develop suitable models.

Almost in parallel with Kalman's breakthrough, a perhaps lesser known study was conducted by Ross Leadbetter and Harald Cramer who are pioneers in the field of the statistics of level crossings and extremes. This study was also funded by NASA and yielded interesting results on the more theoretical aspects of the level-crossing behavior of random processes. The motivation behind the work was as a result of Gertrude Cox's charter to Ross Leadbetter and Harald Cramer at the time to “make comprehensive statistical models for manned space-flight systems.” They ended up supporting a small corner of that effort having to do with the reliability of guidance systems, approaching the problem by modeling the error in a guidance system and declaring failure if it went out of prescribed limits in a mission period, leading to their work on crossings and extremes. All three researchers are legendary, celebrated mathematicians/statisticians in their own right; however, the work was never truly developed to its fullest potential for its originally-intended purpose as with the Kalman filter. Over the years, Leadbetter’s younger Swedish colleagues developed theories that ultimately yielded the idea of optimal alarm systems. This work marries the largely uncultivated portions of Leadbetter’s theory for its intended purpose and the results generated by his younger Swedish colleagues, enabled by none other than the Kalman filter, and thus the unification these two techniques for current NASA challenges. Therefore, with further development and implementation across a broad spectrum of NASA aerospace platforms, this activity also has the potential to generate new knowledge that has evolved from the results of NASA-based legacy programs, which may enhance the potential for future mission success.

NASA PROGRAM FUNDING: System-wide Safety and Assurances Technologies (SSAT) project, Aviation Safety Program (ASP), Aeronautics Research Mission Directorate (ARMD)

Contact: Rodney Martin

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