Ted Chang

Evidence for relative motions between the Indian and Australian Plates during the last 20 m.y. from plate tectonic reconstructions: Implications for the deformation of the Indo-Australian Plate

We use plate tectonic reconstructions to establish whether motions between India and Australia occurred since chron 18 (43 Ma). We test the Africa/Antarctica/Australia/India plate circuit closure at chrons 5 (10 Ma), 6 (21 Ma) and 13 (36 Ma) using a compilation of magnetic anomalies and fracture zone traces from the Southeast, Southwest, Central Indian and the Carlsberg ridges. Additional reconstructions at chrons 23 (55 Ma) and 26 (61 Ma) are used to estimate the overall motion between India and Australia. Relative motions between the Indian and Australian plates are estimated using the plate circuit India → Africa → Australia. A new statistical approach, based on spherical regression analyses, is used to assess the uncertainty of the “best-fitting” finite rotations from the uncertainties in the data. The uncertainty in a rotation is described by a covariance matrix directly related to the geometry of the reconstructed plate boundary, to the distribution and estimated errors of the data points along it. Our parameterization of the rotations allows for simple combination of the rotation uncertainties along a plate circuit path. Results for chron 5 are remarkably consistent with present-day kinematics in the Indian Ocean, except that the Arabian and Indian plates are found to be separate plates. Comparisons of the motions between the Indian and African plates across the Carlsberg Ridge with that between the Australian and African plates across the Central Indian Ridge evidence a significant counterclockwise rotation of the Australian plate relative to the Indian plate about a pole located in the Central Indian Basin. The determinations are consistent for chrons 26, 13, 6 and 5. Determination at chron 23 is different but questionable due to the small number of available data. We propose two alternative solutions that both predict convergence within the Wharton and Central Indian basins and extension in the vicinity of the Chagos-Laccadive Ridge. The first solution assumes that all the deformation in the equatorial Indian Ocean started 7 Ma ago as found by Ocean Drilling Program Leg 116. Hence all the determinations at different times represent the total motion between India and Australia. The averaged India/Australia Euler vector (chrons 5, 6, 13, and 26: 11.1°S, 78.0°, ω=3.54°) lies within the Central Indian Basin and yields a N-S contraction of 46±52 km at 85°, and 80±63 km at 90°. However, the difference of the India/Australia Euler vectors at chrons 5 and 6 suggests that the India/Australia convergence started between 10 and 21 Ma, following the continent-continent collision of India with Asia in the Early Miocene. The second averaged solution (chrons 6, 13, and 26: 5.2°S, 74.3°E, ω=5.93°) predicts a total N-S contraction of 123±73 km at 85°E, and 178±91 km at 90°E. Both models are compatible with the deformation pattern observed in the equatorial Indian Ocean.

Estimating the Relative Rotation of Two Tectonic Plates from Boundary Crossings

Abstract Let η1, …, ηs be unknown vectors on the sphere and Ao be an unknown rotation. Suppose that uij are estimates of points lying on the great circle normal to ηi and υik are estimates of points lying on the great circle normal to Aoηi . This article discusses a method to construct a confidence region for Ao . This problem arises in the reconstruction of the relative motion of two tectonic plates on opposing sides of a rift. The boundary on each side is represented by a collection of great circle segments, and Ao is the rotation that takes one boundary into the other. The data consist of measured crossing points uij and υik of the various segments on the opposing boundaries. The analysis completes an analysis of Hellinger (1981). The errors in tectonic data are quite concentrated, and the problem reduces to linear regression. Once this is realized, many interesting problems such as triple junctions or multiple time periods can be examined. The analysis is aided substantially by a parameterization of t...

The Standardized Influence Matrix and its Applications

Abstract In this article we introduce the standardized influence matrix (SIM) of parameter estimators as a conjugate of the sample covariance matrix of standardized influence functions (SIFs) evaluated at the data points. We propose principal component analysis for the SIM and its complement (SIM analysis) as a diagnostic tool for regression or other statistical inference, and provide theoretical insight to the local influence defined by Cook. We show that SIM analysis reveals the multivariate structure of outlying and/or influential points. Specifically, SIM analysis uncovers hidden structures of influence, such as clustering, that cannot be identified by the lengths of the standardized influences. Finally, examples in linear regression show that a diagnostic method using SIM is more effective if robust parameter estimates are used in calculating the sample SIM.

Approximating the matrix Fisher and Bingham distributions: applications to spherical regression and Procrustes analysis

We obtain approximations to the distribution of the exponent in the matrix Fisher distributions on SO(p) and on O(p) whose density with respect to Haar measure is proportional to exp(Tr GX0tX). Similar approximations are found for the distribution of the exponent in the Bingham distribution, with density proportional to exp(xtGx), on the unit sphere Sp-1 in Euclidean p-dimensional space. The matrix Fisher distribution arises as the exact conditional distribution of the maximum likelihood estmate of the unknown orthogonal matrix in the spherical regression model on Sp-1 with Fisher distributed errors. It also arises as the exact conditional distribution of the maximum likelihood estimate of the unknown orthogonal matrix in a model of Procrustes analysis in which location and orientation, but not scale, changes are allowed. These methods allow determination of a confidence region for the unknown rotation for moderate sample sizes with moderate error concentrations when the error concentration parameter is known.

Teaching Survey Sampling Using Simulation

Abstract SURVEY is a computer program that simulates samples drawn from a hypothetical county. We use SURVEY in introductory and advanced sample survey classes to allow students to become involved in all stages of the sampling process, from designing the survey, to analyzing it, to dealing with nonresponse.

Bootstrap Confidence Region Estimation of the Motion of Rigid Bodies

Abstract This article deals with statistical inference of the motion of rigid bodies using bootstrap methodology. We consider two types of motion: motion in p dimensional Euclidean space, and motion on a p-dimensional sphere. This article is motivated by problems of interest to polar scientists understanding the motion of ice pack at the North Pole and those to geoscientists studying the motion of tectonic plates. In addition to obtaining the point estimates for various parameters describing the motion of rigid bodies, we also perform confidence region estimation and testing of certain hypotheses. We consider two types of situations: when homologous data are available, and when the data consists of nonhomologous points. We illustrate the methodology by analyzing two data sets: a data set on the motion of arctic sea ice, and a data set from plate tectonics.

Asymptotic relative Pitman efficiency in group models

The notion of asymptotic efficacy due to Hannan for multivariate statistics in a location problem is reformulated for manifolds. The matrices used in Hannans definition are reformulated as Riemannian metrics on a manifold and hence are seen not to depend upon the particular parameterization of the manifold used to make the calculations. Conditions under which that efficacy does not depend upon basepoint and direction are derived. This leads to the extension of Pitman asymptotic relative efficiency to location parameters in group models. Under stronger conditions, that of a two-point homogeneous space, we introduce a notion of rank and sign and show that, under the null distribution, the sign is uniformly distributed on a suitably defined sphere and that the rank is independent of the sign. This work generalizes previous definitions of Neeman and Chang, Hossjer and Croux. For group models, a definition of a regression group model is given. Unlike the usual linear model, a location model is not a subcase of a regression group model. Nevertheless, it is shown that the Riemannian metrics for the regression model can be derived from those of the location model and hence, in many cases, the asymptotic relative efficiencies coincide for group and location models. As examples, rank score statistics for spherical and Procrustes regressions are derived. The Procrustes regression model arises in problems of image registration.

Priors for ordered conditional variance and vector partial correlation

Let the vector X = [X1, ..., Xp]t have a multivariate normal distribution with unknown population mean vector [mu] and variance-covariance matrix [summation operator]. This paper develops minimally informative priors (in the sense of Bernardo) for use when the parameter of interest is either the vector of ordered conditional variances [delta]i2 = Var[Xi | Xj, j