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One of the basic challenges of understanding hyperspectral data arises from the fact that it is intrinsically 3-dimensional. A diverse range of algorithms have been developed to help visualize hyperspectral data trichromatically in 2-dimensions. In this paper we take a different approach and show how virtual reality provides a way of visualizing a hyperspectral data cube without collapsing the spectral dimension. Using several different real datasets, we show that it is straightforward to find signals of interest and make them more visible by exploiting the immersive, interactive environment of virtual reality. This enables signals to be seen which would be hard to detect if we were simply examining hyperspectral data band by band. 

Dimensionality-reduction methods are a fundamental tool in the analysis of large datasets. These algorithms work on the assumption that the "intrinsic dimension" of the data is generally much smaller than the ambient dimension in which it is collected. Alongside their usual purpose of mapping data into a smaller-dimensional space with minimal information loss, dimensionality-reduction techniques implicitly or explicitly provide information about the dimension of the dataset.In this paper, we propose a new statistic that we call the kappa-profile for analysis of large datasets. The kappa-profile arises from a dimensionality-reduction optimization problem: namely that of finding a projection that optimally preserves the secants between points in the dataset. From this optimal projection we extract kappa, the norm of the shortest projected secant from among the set of all normalized secants. This kappa can be computed for any dimension k; thus the tuple of kappa values (indexed by dimension) becomes a kappa-profile. Algorithms such as the Secant-Avoidance Projection algorithm and the Hierarchical Secant-Avoidance Projection algorithm provide a computationally feasible means of estimating the kappa-profile for large datasets, and thus a method of understanding and monitoring their behavior. As we demonstrate in this paper, the kappa-profile serves as a useful statistic in several representative settings: weather data, soundscape data, and dynamical systems data. 

A fundamental question in many data analysis settings is the problem of discerning the ``natural'' dimension of a data set. That is, when a data set is drawn from a manifold (possibly with noise), a meaningful aspect of the data is the dimension of that manifold. Various approaches exist for estimating this dimension, such as the method of Secant-Avoidance Projection (SAP). Intuitively, the SAP algorithm seeks to determine a projection which best preserves the lengths of all secants between points in a data set; by applying the algorithm to find the best projections to vector spaces of various dimensions, one may infer the dimension of the manifold of origination. That is, one may learn the dimension at which it is possible to construct a diffeomorphic copy of the data in a lower-dimensional Euclidean space. Using Whitney's embedding theorem, we can relate this information to the natural dimension of the data. A drawback of the SAP algorithm is that a data set with $$n$$ points has $n(n-1)/2$ secants, making the computation and storage of all secants infeasible for very large data sets. In this paper, we propose a novel algorithm that generalizes the SAP algorithm with an emphasis on addressing this issue. That is, we propose a hierarchical secant-based dimensionality-reduction method, which can be employed for data sets where explicitly calculating all secants is not feasible. 

New depth sensors, like the Microsoft Kinect, produce streams of human pose data. These discrete pose streams can be viewed as noisy samples of an underlying continuous ideal curve that describes a trajectory through high-dimensional pose space. This paper introduces a technique for generalized curvature analysis (GCA) that determines features along the trajectory which can be used to characterize change and segment motion. Tools are developed for approximating generalized curvatures at mean points along a curve in terms of the singular values of local mean-centered data balls. The features of the GCA algorithm are illustrated on both synthetic and real examples, including data collected from a Kinect II sensor. We also applied GCA to the Carnegie Mellon University Motion Capture (MoCaP) database. Given that GCA scales linearly with the length of the time series we are able to analyze large data sets without down sampling. It is demonstrated that the generalized curvature approximations can be used to segment pose streams into motions and transitions between motions. The GCA algorithm can identify 94.2% of the transitions between motions without knowing the set of possible motions in advance, even though the subjects do not stop or pause between motions. 

A fundamental question in many data analysis settings is the problem of discerning the “natural” dimension of a data set. That is, when a data set is drawn from a manifold (possibly with noise), a meaningful aspect of the data is the dimension of that manifold. Various approaches exist for estimating this dimension, such as the method of Secant-Avoidance Projection (SAP). Intuitively, the SAP algorithm seeks to determine a projection which best preserves the lengths of all secants between points in a data set; by applying the algorithm to find the best projections to vector spaces of various dimensions, one may infer the dimension of the manifold of origination. That is, one may learn the dimension at which it is possible to construct a diffeomorphic copy of the data in a lower-dimensional Euclidean space. Using Whitney's embedding theorem, we can relate this information to the natural dimension of the data. A drawback of the SAP algorithm is that a data set with T points has O(T 2 ) secants, making the computation and storage of all secants infeasible for very large data sets. In this paper, we propose a novel algorithm that generalizes the SAP algorithm with an emphasis on addressing this issue. That is, we propose a hierarchical secant-based dimensionality-reduction method, which can be employed for data sets where explicitly calculating all secants is not feasible.