7,023 research outputs found
Density of Spherically-Embedded Stiefel and Grassmann Codes
The density of a code is the fraction of the coding space covered by packing
balls centered around the codewords. This paper investigates the density of
codes in the complex Stiefel and Grassmann manifolds equipped with the chordal
distance. The choice of distance enables the treatment of the manifolds as
subspaces of Euclidean hyperspheres. In this geometry, the densest packings are
not necessarily equivalent to maximum-minimum-distance codes. Computing a
code's density follows from computing: i) the normalized volume of a metric
ball and ii) the kissing radius, the radius of the largest balls one can pack
around the codewords without overlapping. First, the normalized volume of a
metric ball is evaluated by asymptotic approximations. The volume of a small
ball can be well-approximated by the volume of a locally-equivalent tangential
ball. In order to properly normalize this approximation, the precise volumes of
the manifolds induced by their spherical embedding are computed. For larger
balls, a hyperspherical cap approximation is used, which is justified by a
volume comparison theorem showing that the normalized volume of a ball in the
Stiefel or Grassmann manifold is asymptotically equal to the normalized volume
of a ball in its embedding sphere as the dimension grows to infinity. Then,
bounds on the kissing radius are derived alongside corresponding bounds on the
density. Unlike spherical codes or codes in flat spaces, the kissing radius of
Grassmann or Stiefel codes cannot be exactly determined from its minimum
distance. It is nonetheless possible to derive bounds on density as functions
of the minimum distance. Stiefel and Grassmann codes have larger density than
their image spherical codes when dimensions tend to infinity. Finally, the
bounds on density lead to refinements of the standard Hamming bounds for
Stiefel and Grassmann codes.Comment: Two-column version (24 pages, 6 figures, 4 tables). To appear in IEEE
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Analyzing big time series data in solar engineering using features and PCA
In solar engineering, we encounter big time series data such as the satellite-derived irradiance data and string-level measurements from a utility-scale photovoltaic (PV) system. While storing and hosting big data are certainly possible using today’s data storage technology, it is challenging to effectively and efficiently visualize and analyze the data. We consider a data analytics algorithm to mitigate some of these challenges in this work. The algorithm computes a set of generic and/or application-specific features to characterize the time series, and subsequently uses principal component analysis to project these features onto a two-dimensional space. As each time series can be represented by features, it can be treated as a single data point in the feature space, allowing many operations to become more amenable. Three applications are discussed within the overall framework, namely (1) the PV system type identification, (2) monitoring network design, and (3) anomalous string detection. The proposed framework can be easily translated to many other solar engineer applications
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