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Constable, C.  1992.  The Bootstrap for Magnetic-Susceptibility Tensors - Reply. Journal of Geophysical Research-Solid Earth. 97:13997-13998.   10.1029/92jb00745   AbstractWebsite
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Constable, C, Tauxe L.  1990.  The Bootstrap for Magnetic Susceptibility Tensors. Journal of Geophysical Research-Solid Earth and Planets. 95:8383-8395.   10.1029/JB095iB06p08383   AbstractWebsite

In studies of the anisotropy of susceptibility or remanence of paleomagnetic samples it is conventional to specify the anisotropy in terms of the parameters of the anisotropy ellipsoids, namely the directions of the principal axes of the ellipsoid and their associated eigenvalues. Confidence intervals for these parameters have in the past often been estimated by using a linearization scheme to propagate the effect of small changes through the eigenvalue decomposition. The validity of these approximations is explored using a Monte-Carlo simulation from measurements that are presumed normally distributed, showing that there are circumstances in which the linearization scheme gives confidence intervals that are much too small. Q-Q plots indicate that the common assumption that the noise in the measurements is Gaussian does not always hold. Because of these shortcomings in the conventional technique we propose using a bootstrap resampling scheme to find empirically the distribution of uncertainties in the results. Confidence intervals for the eigenvalues are found directly from their empirical distributions. For the principal axes, approximate elliptical regions of confidence on the unit sphere are parameterized in terms of the Kent or FB5 distribution. The number of modes observed in the distribution of eigenvalues obtained by bootstrapping is used to classify the shape of the susceptibility ellipsoid as spherical, oblate, prolate or triaxial. The empirical nature of the bootstrap technique allows the extension of the analysis of uncertainties to parameters derived from the principal susceptibilities, such as percentage anisotropy or shape factor.

T
Tauxe, L, Kylstra N, Constable C.  1991.  Bootstrap Statistics for Paleomagnetic Data. Journal of Geophysical Research-Solid Earth and Planets. 96:11723-11740.   10.1029/91jb00572   AbstractWebsite

The power and utility of paleomagnetic analyses stem largely from the ability to quantify such parameters as the degree of rotation of a rock body, or the orientation of an anisotropy axis. Until recently, estimates for uncertainty in these paleomagnetically determined parameters derived from assumptions concerning the underlying parametric distribution functions of the data. In many geologically important situations, the commonly used parametric distribution functions fail to model the data adequately and the uncertainty estimates so obtained are unreliable. Such essentials as the test for common mean require data sets consistent with a spherically symmetric underlying distribution; their application in inappropriate circumstances can result in flawed interpretations. Moreover, the almost universally used approximation for a cone of 95% confidence for the mean of a sample drawn from a Fisher distribution is quite biased even for moderate dispersions (kappa = 25). The availability of inexpensive, powerful computers makes possible the empirical estimation of confidence regions by means of data resampling techniques such as the bootstrap. These resampling schemes replace analytical solutions with repeated simple calculations. We describe a bootstrap approach for the calculation of uncertainties for means or principal directions of paleomagnetic data. The method is tested on means of simulated Fisher distributions with known parameters and is found to be reliable for data sets with more than about 25 elements. Because a Fisher distribution is not assumed, the approach is applicable to a wide range of paleomagnetic data and can be used equally well on directions or associated virtual poles. We also illustrate bootstrap techniques for the discrimination of directions and for the fold test which enable the use of these powerful tests on the wider range of data sets commonly obtained in paleomagnetic investigations.