Methods for spherical data analysis and visualization

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Abstract

A systematic analysis of the localization of objects in extra-personal space requires a three-dimensional method of documenting location. In auditory localization studies the location of a sound source is often reduced to a directional vector with constant magnitude with respect to the observer, data being plotted on a unit sphere with the observer at the origin. This is an attractive form of data representation as the relevant spherical statistical and graphical methods are well described. In this paper we collect together a set of spherical plotting and statistical procedures to visualize and summarize these data. We describe methods for visualizing auditory localization data without assuming that the principal components of the data are aligned with the coordinate system. As a means of comparing experimental techniques and having a common set of data for the verification of spherical statistics, the software (implemented in MATLAB) and database described in this paper have been placed in the public domain. Although originally intended for the visualization and summarization of auditory psychophysical data, these routines are sufficiently general to be applied in other situations involving spherical data.

Introduction

While science is arguably based upon observation statement, even relatively unsophisticated analysis of simple data involves assumptions. This is often only implicit in the analytical or statistical methods employed in analysis. Another subtle form of analytical assumption is often buried in the methodology employed in visualizing the data. Although graphical data representation has a long history (Tufte, 1990), the growing availability of inexpensive and powerful computers has resulted in renewed interest in data visualization as a form of analysis for complex or large data sets. One area in neuroscience where data sets can be both large and, from an analytical perspective, fairly complex, is in the representation of extra-personal space and how an animal or individual relates to that space in some meaningful way.

In our laboratory we have been examining how the mammalian nervous system processes auditory information to generate a neural representation of space that results in our perceptions of the auditory world. In its simplest form, we refer to the location of an auditory object (a sound source) in terms of its direction and distance from the observer. As the observer occupies a point in space, the representation and analysis of these data involve at least three spatial dimensions and then a number of other dimensions related to the nature of the signal (e.g. frequency and time) and the experimental manipulation (the independent variables). The most common form of experiment is the placement of a sound source at an unseen location in space and an indication by the subject of the location of the target. The disparity between the indicated and actual location of the auditory target is used as a measure of the localization accuracy of the subject, and we use the term `error' in this paper to describe this value. The analysis of these data generally concentrates on the directional vector of the auditory object and, in most cases, ignores the distance effects. Under this assumption the representation of these data reduces to a more tractable two-dimensional spherical display of the data. Furthermore, the plotting and manipulation of spherical data in real time makes this form of analysis attractive since the complex spatial relations in the data can be easily apprehended as the viewpoint is rotated. Indeed, this is one of the principal advantages of visualization of complex data sets.

Previously, there has been a range of methods applied to the summary description and statistical analysis of these data. For example, Oldfield and Parker (1984)used simple XY plots of error verses target position and shaded contour plots of the errors on an azimuth/elevation grid. Wightman and Kistler (1989)used XY source position versus perceived position plots; and Makous and Middlebrooks (1990)used ellipses drawn on a double pole spherical plot where the size of the major and minor axes were proportional to the signed errors in the vertical or horizontal directions. All of these representations involve separate analysis of the azimuth and elevation components of the data, such as calculating the variance of the azimuth and elevation localization errors for each target location in space.

This approach does not account for some features of these data since azimuth and elevation are likely to covary. We have applied the Kent distribution (Kent, 1982) instead of the commonly used Fisher distribution (described in Fisher et al., 1993) to analyze these spherical data. The Fisher distribution assumes that the data is rotationally symmetric whereas the Kent distribution can be used to model asymmetric data. Such an approach is more likely to expose the coordinate system used by the auditory central nervous system to represent auditory extra-personal space. This is an important methodological step as it allows the comparison of localization performance in individuals or groups to be compared with the spatial variation in their auditory spatial cues to a sound's location. Such an approach provides insights into the processing strategies employed by the auditory system in computing and representing the spatial locations of sound sources (see Carlile, 1996, for review). In considering these issues we have also sought to apply a number of robust statistical methods to the auditory localization data collected in our laboratory and to combine these with a convenient set of visualization tools. We have collected together many of the relevant statistical methods and describe here a library of data manipulation, plotting, summary statistical procedures and routines for hypothesis testing using spherical data. These methods have been developed using MATLAB (The MathWorks, Inc.), a popular data analysis and visualization package and have now been made available as public domain software. This small library, called Spak (for `spherical package'), provides a flexible set of tools for manipulating and processing spherical data (Spak is freely available upon request from the authors, or via the World Wide Web site http://www.physiol.usyd.edu.au/simonc/). It is hoped that, by providing a common resource for the analysis and interpretation of these complex data sets, a greater consistency in approach will be encouraged and thus facilitate more rigorous comparisons between studies in this area. Although these routines were developed to serve the requirements of the research community examining auditory localization, the routines are sufficiently general that they could be employed in other research areas using spherical data (e.g. astronomy, geodesy, geology, geophysics and mathematics; see Fisher et al., 1993).

Section snippets

Spherical coordinate system

The quantitative description of spherical data is dependent on the definition of a particular spherical coordinate system. A number of systems are in general use (Fisher et al., 1993). The two most common in the auditory literature are a single pole system (analogous to the planetary coordinate system), and a double pole system which shares the longitudinal circles of the single pole system (denoting the elevation of a source) but has an orthogonal series of circles centered on the interaural

Results

In this section, we provide examples of the types of visualizations possible using Spak. A feature of this software is that very few lines of code are required to produce visualizations of auditory localization data. All of the code used to generate the plots in Fig. 3, Fig. 4Fig. 5 are given in Appendix A.

Discussion

This paper documents the implementation of routines for data management, visualization and the statistical description of spherical data. In an effort to maximize the utility of these routines we have a exploited a very widely used data visualization and analysis package (MATLAB). The data management features of this package facilitate the efficient manipulation of large amounts of spherical data (>104 data points). Data can be sorted and that associated with particular locations, or a range of

Acknowledgements

This work was supported by the Australian Research Council (AC9330318) and a University of Sydney Research grant. The authors would like to thank Dr Markus Schenkel for reviewing an earlier draft of this paper.

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