Ellipsoid fitting with the Cayley transform

We introduce an algorithm, Cayley transform ellipsoid fitting (CTEF), that uses the Cayley transform to fit ellipsoids to noisy data in any dimension. Unlike many ellipsoid fitting methods, CTEF is ellipsoid specific -- meaning it always returns elliptic solutions -- and can fit arbitrary ellipsoids. It also outperforms other fitting methods when data are not uniformly distributed over the surface of an ellipsoid. Inspired by calls for interpretable and reproducible methods in machine learning, we apply CTEF to dimension reduction, data visualization, and clustering. Since CTEF captures global curvature, it is able to extract nonlinear features in data that other methods fail to identify. This is illustrated in the context of dimension reduction on human cell cycle data, and in the context of clustering on classical toy examples. In the latter case, CTEF outperforms 10 popular clustering algorithms

Sequential Gibbs Posteriors with Applications to Principal Component Analysis

Gibbs posteriors are proportional to a prior distribution multiplied by an exponentiated loss function, with a key tuning parameter weighting information in the loss relative to the prior and providing a control of posterior uncertainty. Gibbs posteriors provide a principled framework for likelihood-free Bayesian inference, but in many situations, including a single tuning parameter inevitably leads to poor uncertainty quantification. In particular, regardless of the value of the parameter, credible regions have far from the nominal frequentist coverage even in large samples. We propose a sequential extension to Gibbs posteriors to address this problem. We prove the proposed sequential posterior exhibits concentration and a Bernstein-von Mises theorem, which holds under easy to verify conditions in Euclidean space and on manifolds. As a byproduct, we obtain the first Bernstein-von Mises theorem for traditional likelihood-based Bayesian posteriors on manifolds. All methods are illustrated with an application to principal component analysis

Random Splitting of Fluid Models: Ergodicity and Convergence

We introduce a family of stochastic models motivated by the study of nonequilibrium steady states of fluid equations. These models decompose the deterministic dynamics of interest into fundamental building blocks, i.e., minimal vector fields preserving some fundamental aspects of the original dynamics. Randomness is injected by sequentially following each vector field for a random amount of time. We show under general assumptions that these random dynamics possess a unique invariant measure and converge almost surely to the original, deterministic model in the small noise limit. We apply our construction to the Lorenz-96 equations, often used in studies of chaos and data assimilation, and Galerkin approximations of the 2D Euler and Navier-Stokes equations. An interesting feature of the models developed is that they apply directly to the conservative dynamics and not just those with excitation and dissipation

Cancer Diagnosis using LIBS and Machine Learning Tools: Progress and Challenges

Despite numerous research and development efforts that provide important tools to fight cancer, this disease still poses great challenges to diagnosis and treatment, and it remains one of the leading causes of death worldwide. Early diagnosis is crucial to increase the survival rate and quality of life of cancer patients. Thus, developing non-invasive screening methods would represent a key step towards point-of-care large scale screening and prevention of asymptomatic tumors such as Epithelian Ovarian Cancer (EOC) and others. Our group has developed two experimental strategies to pursue early cancer diagnosis through Laser-Induced Breakdown Spectroscopy (LIBS), a versatile atomic spectroscopy technique whose main advantages are: little or no sample preparation required; real-time multi-elemental response; virtually no limitation about the kind of sample that can be analyzed. The first is a LIBS-based immunoassay (Tag-LIBS), where a cancer biomarker is tagged with a suitably functionalized inorganic microparticles, which are in turn quantitatively and sensitively detected by LIBS. The second is based on the direct analysis of biological fluids through the combined use of LIBS and machine learning algorithms. By combining femtosecond LIBS with unsupervised classification techniques, we have shown that it is possible to discriminate blood samples extracted from healthy and diseased mice with an accuracy that approaches 80%. We will present our most recent results obtained with both approaches, and in particular we will report about the effects of various substrates used for LIBS measurements on the classification accuracy of blood samples extracted from cancerous and healthy mice

Modeling of laser-induced breakdown spectroscopic data analysis by an automatic classifier

Laser-induced breakdown spectroscopy (LIBS) is a multi-elemental and real-time analytical technique with simultaneous detection of all the elements in any type of sample matrix including solid, liquid, gas, and aerosol. LIBS produces vast amount of data which contains information on elemental composition of the material among others. Classification and discrimination of spectra produced during the LIBS process are crucial to analyze the elements for both qualitative and quantitative analysis. This work reports the design and modeling of optimal classifier for LIBS data classification and discrimination using the apparatus of statistical theory of detection. We analyzed the noise sources associated during the LIBS process and created a linear model of an echelle spectrograph system. We validated our model based on assumptions through statistical analysis of “dark signal” and laser-induced breakdown spectra from the database of National Institute of Science and Technology. The results obtained from our model suggested that the quadratic classifier provides optimal performance if the spectroscopy signal and noise can be considered Gaussian

Pulsed injection-seeded optical parametric oscillator with low frequency chirp for high-resolution spectroscopy

This paper was published in Optics Letters, 2003; 28(14):1248-1250 and is made available as an electronic reprint with the permission of OSA. The paper can be found at the following URL on the OSA website: http://www.opticsinfobase.org/ol/abstract.cfm?uri=ol-28-14-1248 Systematic or multiple reproduction or distribution to multiple locations via electronic or other means is prohibited and is subject to penalties under law.Richard T. White, Yabai He, Brian J. Orr, Mitsuhiko Kono, and K. G. H. Baldwi

Transition from single-mode to multimode operation of an injection-seeded pulsed optical parametric oscillator

This paper was published in Optics Express and is made available as an electronic reprint with the permission of OSA. The paper can be found at the following URL on the OSA website: http://www.opticsinfobase.org/oe/abstract.cfm?URI=OPEX-12-23-5655. Systematic or multiple reproduction or distribution to multiple locations via electronic or other means is prohibited and is subject to penalties under law.Richard White, Yabai He, Brian Orr, Mitsuhiko Kono and K. Baldwi