Empirical geodesic graphs and CAT(k) metrics for data analysis

A methodology is developed for data analysis based on empirically constructed geodesic metric spaces. For a probability distribution, the length along a path between two points can be defined as the amount of probability mass accumulated along the path. The geodesic, then, is the shortest such path and defines a geodesic metric. Such metrics are transformed in a number of ways to produce parametrised families of geodesic metric spaces, empirical versions of which allow computation of intrinsic means and associated measures of dispersion. These reveal properties of the data, based on geometry, such as those that are difficult to see from the raw Euclidean distances. Examples of application include clustering and classification. For certain parameter ranges, the spaces become CAT(0) spaces and the intrinsic means are unique. In one case, a minimal spanning tree of a graph based on the data becomes CAT(0). In another, a so-called "metric cone" construction allows extension to CAT($k$) spaces. It is shown how to empirically tune the parameters of the metrics, making it possible to apply them to a number of real cases.Comment: Statistics and Computing, 201

Studies into the detection of buried objects (particularly optical fibres) in saturated sediment. Part 2: design and commissioning of test tank

This report is the second in a series of five, designed to investigate the detection oftargets buried in saturated sediment, primarily through acoustical or acoustics-relatedmethods. Although steel targets are included for comparison, the major interest is intargets (polyethylene cylinders and optical fibres) which have a poor acousticimpedance mismatch with the host sediment. This particular report details theconstruction of a laboratory-scale test facility. This consisted of three maincomponents. Budget constraints were an over-riding consideration in the design.First, there is the design and production of a tank containing saturated sediment. Itwas the intention that the physical and acoustical properties of the laboratory systemshould be similar to those found in a real seafloor environment. Particularconsideration is given to those features of the test system which might affect theacoustic performance, such as reverberation, the presence of gas bubbles in thesediment, or a suspension of particles above it. Sound speed and attenuation wereidentified as being critical parameters, requiring particular attention. Hence, thesewere investigated separately for each component of the acoustic path.Second, there is the design and production of a transducer system. It was the intentionthat this would be suitable for an investigation into the non-invasive acousticdetection of buried objects. A focused reflector is considered to be the most costeffectiveway of achieving a high acoustic power and narrow beamwidth. Acomparison of different reflector sizes suggested that a larger aperture would result inless spherical aberration, thus producing a more uniform sound field. Diffractioneffects are reduced by specifying a tolerance of much less than an acousticwavelength over the reflector surface. The free-field performance of the transducerswas found to be in agreement with the model prediction. Several parameters havebeen determined in this report that pertain to the acoustical characteristics of the waterand sediment in the laboratory tank in the 10 – 100 kHz frequency range.Third, there is the design and production of an automated control system wasdeveloped to simplify the data acquisition process. This was, primarily, a motordrivenposition control system which allowed the transducers to be accuratelypositioned in the two-dimensional plane above the sediment. Thus, it was possible forthe combined signal generation, data acquisition and position control process to be coordinatedfrom a central computer.This series of reports is written in support of the article “The detection by sonar ofxdifficult targets (including centimetre-scale plastic objects and optical fibres) buriedin saturated sediment” by T G Leighton and R C P Evans, written for a Special Issueof Applied Acoustics which contains articles on the topic of the detection of objectsburied in marine sediment. Further support material can be found athttp://www.isvr.soton.ac.uk/FDAG/uaua/target_in_sand.HTM