Statistical analysis via the curvature of data space
نویسندگان
چکیده
It has been known that the curvature of data spaces plays a role in data analysis. For example, the Frechet mean (intrinsic mean) always exists uniquely for a probability measure on a non-positively curved metric space. In this paper, we use the curvature of data spaces in a novel manner. A methodology is developed for data analysis based on empirically constructed geodesic metric spaces. The population version defines distance by the amount of probability mass accumulated on travelling between two points and geodesic metric arises from the shortest path version. Such metrics are then transformed in a number of ways to produce families of geodesic metric spaces. Empirical versions of the geodesics allow computation of intrinsic means and associated measures of dispersion. A version of the empirical geodesic is introduced based on some metric graphs computed from the sample points. For certain parameter ranges the spaces become CAT(0) spaces and the intrinsic means are unique. In the graph case a minimal spanning tree obtained as a limiting case is CAT(0). In other cases the aggregate squared distance from a test point provides local minima which yield information about clusters. This is particularly relevant for metrics based on so-called metric cones which allow extensions to CAT(κ) spaces. We show how our methods work by using some actual data. This paper is a summary of a longer version [5]. See it for proof of theorems and details.
منابع مشابه
RICCI CURVATURE OF SUBMANIFOLDS OF A SASAKIAN SPACE FORM
Involving the Ricci curvature and the squared mean curvature, we obtain basic inequalities for different kind of submaniforlds of a Sasakian space form tangent to the structure vector field of the ambient manifold. Contrary to already known results, we find a different necessary and sufficient condition for the equality for Ricci curvature of C-totally real submanifolds of a Sasakian space form...
متن کاملInterpretation of gravity anomalies via terracing method of the profile curvature
One of the main goals of interpretation of gravity data is to detect location and edges of the anomalies. Edge detection of gravity anomalies is carried out by different methods. Terracing of the data is one of the approaches that help the interpreter to achieve appropriate results of edge detection. This goal becomes a complex task when the gravity anomalies have smooth borders due to gradual ...
متن کاملProbing the Probabilistic Effects of Imperfections on the Load Carrying Capacity of Flat Double-Layer Space Structures
Load carrying capacity of flat double-layer space structures majorly depends on the structures' imperfections. Imperfections in initial curvature, length, and residual stress of members are all innately random and can affect the load-bearing capacity of the members and consequently that of the structure. The double-layer space trusses are susceptible to progressive collapse due to sudden buckli...
متن کاملSpacelike hypersurfaces with constant $S$ or $K$ in de Sitter space or anti-de Sitter space
Let $M^n$ be an $n(ngeq 3)$-dimensional complete connected and oriented spacelike hypersurface in a de Sitter space or an anti-de Sitter space, $S$ and $K$ be the squared norm of the second fundamental form and Gauss-Kronecker curvature of $M^n$. If $S$ or $K$ is constant, nonzero and $M^n$ has two distinct principal curvatures one of which is simple, we obtain some charact...
متن کاملLandforms identification using neural network-self organizing map and SRTM data
During an 11 days mission in February 2000 the Shuttle Radar Topography Mission (SRTM) collected data over 80% of the Earth's land surface, for all areas between 60 degrees N and 56 degrees S latitude. Since SRTM data became available, many studies utilized them for application in topography and morphometric landscape analysis. Exploiting SRTM data for recognition and extraction of topographic ...
متن کامل