1,720,960 research outputs found
Towards a Common Dimensionality Reduction Approach; Unifying PCA, tSNE, and UMAP through a Cohesive Framework
Get PDFDimensionality reduction is a widely studied field that is used to visualize data, cluster samples, and extract insights from high-dimensional distributions. The classical approaches such as PCA, Isomap, and Laplacian eigenmaps rely on clear optimization strategies while more modern approaches such at tSNE and UMAP define gradient descent search spaces through disparities between the high- and low-dimensional datasets. In this work, we notice that all of these approaches can be interpreted as minimizing the difference between two kernel functions – one for the high dimensional space and one for the low dimensional space. In particular, once we abstract the kernel functions, we can develop a common framework for any dimensionality reduction problem. Namely, one needs to identify their high-dimensional distance kernel, the low-dimensional distance kernel, and the method used for minimization. With this in mind, we identify the relevant general framework and then proceed to discuss the ways in which PCA, tSNE, and UMAP all fit into it. For each, we discuss insights that were obtained during the process. We lastly highlight next steps and directions for future work
Towards Metric Measure Space Learning
Get PDFDensity estimation is pivotal in manifold learning and data analysis, providing insightinto the underlying geometric structure of data. In the first part of this thesis, we link density estimation to the concept of weight from metric geometry. Specifically, we show that the weight function estimates the reciprocal of the data density function on an embedded Riemannian manifold. This leads to a novel normalization method for estimating the Laplace-Beltrami operator. In the second part, we develop the theory to generalize density estimation to metric measure spaces beyond smooth manifolds. By extending the Lebesgue Differentiation Theorem to kernel-weighted integrals, we accommodate a broader class of kernel functions than previously used in manifold learning. The final chapter shows that many aspects of our work fit into enriched category theory. We demonstrate that generalized metric spaces form an enriched category, following Lawvere’s results. This perspective allows us to define generalized kernel spaces and show that the outer measure defines a functor. This section is speculative and aims to inspire future research directions in data science
Higher Order Kalman Filtering for Nonlinear Systems
Get PDFWe seek to improve upon and generalize the Ensemble Kalman Filter (EnKF) by defining a Higher Order Kalman Filter. The Kalman filter consists of two steps: forecast and assimilation. In this thesis we develop the forecast step of our desired Higher Order Kalman Filter with the higher order unscented transform (HOUT). The HOUT is a quadrature rule that estimates the expected value of the first four moments of a distribution, i.e. the mean, covariance, skewness and kurtosis. We then discuss how to generalize the assimilation step. The original Kalman Filter can be derived in three ways: the Bayesian approach, the Minimum Mean-Square Estimate (MMSE) approach and the Closure approach. Each derivation provides a different avenue for us to derive the Higher Order Kalman Filter. In order to generalize the Bayesian approach to the first four moments, instead of using a Gaussian likelihood and prior, we use exponentials with a quartic polynomial as the exponent. In order to generalize the MMSE approach we consider deriving optimal quadratic filters. Finally we may generalize the closure approach by deriving the ordinary differential equations for the skewness and kurtosis and instead of assuming that the skewness is zero, we seek new closures for the first four moments rather than just the first two
Towards a theory of Self Calibrating Sensors
Get PDFWe examine the impact of inference models on uncertainty when using continuous wave Optically Detected Magnetic Resonance (ODMR) measurements to infer temperature. Our approach employs a probabilistic feedforward model designed to maximize the likelihood of observed ODMR spectra through automatic differentiation. This model achieves a state-of-the-art prediction uncertainty of ±1 K across a temperature range of 243 K to 323 K. We show that, for out-of-sample data within the training temperature range, data-driven methods can reduce uncertainty by up to 0.67 K without incorporating expert knowledge of the spectroscopic-temperature relationship. However, the probabilistic model demonstrates superior robustness and generalizability and in contrast, data-driven methods exhibit up to ten times greater uncertainty when tasked with extrapolating beyond their training data range. The Probabilistic feedforward model motivated our development of a mathematical theory of self-calibrating sensors. We operate under the implicit assumption that sensors inevitably introduce slight disturbances while measuring the very phenomena they are designed to monitor. We perturb the dynamics of the system, proposing that these minor perturbations, such as those introduced by the sensor itself, can enhance observability. In a system with two states and a single observation, we show that, under certain conditions, the uncertainty in estimating the system's states is inversely proportional to the size of a small perturbation applied to the system. The one-dimensional observation case guided us to go to higher dimensions, and when the number of observations is only one fewer than the number of states we find similar results. However, for even fewer observations we illustrate significant additional challenges. Ultimately, this dissertation shows that self-calibrating sensors provide a robust solution for maintaining accuracy and reliability in the face of hidden state uncertainty and environmental fluctuations
Going Beyond Counting First Authors in Author Co-citation Analysis
Get PDFThe present study examines one of the fundamental aspects of author co-citation analysis (ACA) - the way co-citation
counts are defined. Co-citation counting provides the data on which all subsequent statistical analyses and mappings
are based, and we compare ACA results based on two different types of co-citation counting - the traditional type that
only counts the first one among a cited work's authors on the one hand and a non-traditional type that takes into
account the first 5 authors of a cited work on the other hand. Results indicate that the picture produced through this non-traditional author co-citation counting contains more coherent author groups and is therefore considerably clearer. However, this picture represents fewer specialties in the research field being studied than that produced through the traditional first-author co-citation counting when the same number of top-ranked authors is selected and analyzed. Reasons for these effects are discussed
Model Free Techniques for Reduction of High-Dimensional Dynamics
Get PDFThere is a growing need in science and engineering to extract information about complex phenomena from large data sets. A rapidly developing approach to building a model from data is manifold learning, and analysis of such a model may allow isolation of the desired features of the data. By introducing an additional geometric structure, the techniques of differential geometry become available for analyzing the model. In this dissertation we extend previous methods of analyzing the geometry of data. Our key contribution is the theory of local kernels, which generalizes previous nonparametric techniques such as Laplacian eigenmaps and diffusion maps. We show that every geometry can be represented by a local kernel in the limit of large data. Moreover, using the discrete exterior calculus (DEC) we show that a local kernel can be used to introduce a discrete Hodge star operator on a data set. This shows that local kernels introduce a discrete geometry on a data set without the need for an explicit simplicial complex
Variations on the Author
Get PDF“Variations on the Author” discusses two of Eduardo Coutinho’s recent films (Um Dia na Vida, from 2010, and Últimas Conversas, posthumously released in 2015) and their contribution to the general question of documentary authorship. The director’s filmography is characterized by a consistent yet self-effacing form of authorial self-inscription: Coutinho often features as an interviewer that rather than express opinions propels discourses; an interviewer that is good at listening. This mode of self-inscription characterizes him as an author who is not expressive but who is nonetheless markedly present on the screen. In Um Dia na Vida, however, Coutinho is completely absent form the image, while Últimas Conversas, on the contrary, includes a confessional prologue that moves the director from the margins to the center of his films. This article examines the ways in which these works stand out in the filmography of a director who offers new insights into the notion of cinematic authorship
Appropriate Similarity Measures for Author Cocitation Analysis
Get PDFWe provide a number of new insights into the methodological discussion about author cocitation analysis. We first argue that the use of the Pearson correlation for measuring the similarity between authors’ cocitation profiles is not very satisfactory. We then discuss what kind of similarity measures may be used as an alternative to the Pearson correlation. We consider three similarity measures in particular. One is the well-known cosine. The other two similarity measures have not been used before in the bibliometric literature. Finally, we show by means of an example that our findings have a high practical relevance.information science;Pearson correlation;cosine;similarity measure;author cocitation analysis
Dispelling the Myths Behind First-author Citation Counts
Get PDFWe conducted a full-scale evaluative citation analysis study of scholars in the XML research field to explore just how different from each other author rankings resulting from different citation counting methods actually are, and to demonstrate the capability of emerging data and tools on the Web in supporting more realistic citation counting methods. Our results contest some common arguments for the continued
use of first-author citation counts in the evaluation of scholars, such as high correlations between author rankings by first-author citation counts and other citation
counting methods, and high costs of using more realistic citation counting methods that are not well-supported by the ISI databases. It is argued that increasingly available digital full text research papers make it possible for citation analysis studies to go beyond what the ISI databases have directly supported and to employ more
sophisticated methods
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