Maximum likelihood solution for inclination-only data in paleomagnetism
Maximum likelihood solution for inclination-only data in paleomagnetism
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DOI：
10.1111/j.1365-246x.2010.04671.x
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发表时间：
2010-08-01
影响因子：
2.8
通讯作者：
Levi, S.
中科院分区：
文献类型：
--
作者：
Arason, P.;Levi, S.
我们开发了一种新的鲁棒极大似然方法来估计无偏平均倾角。在古地磁分析中，已知仅倾角数据的算术平均值会引入浅偏置。介绍了几种方法来估计无偏平均倾斜度和分散度的措施。设计了一些只考虑倾角的方法来最大化边际费雪分布的似然函数。然而，最大似然函数的精确解析形式是相当复杂的，所有的方法都需要各种各样的假设和近似，而这些假设和近似往往是不合适的。对于一些陡峭和分散的数据集，这些方法提供的估计明显从似然函数的峰值偏移到系统的较浅倾斜。找到似然函数最大值的问题部分是由于难以准确地评估所有感兴趣值的函数，因为随着精度参数的增加，似然函数的一些元素呈指数增长，导致数值不稳定。在这项研究中，我们成功地从对数似然函数中解析地消去了指数元素，现在我们能够在参数空间的任何地方计算它的值，并且对于任何仅倾角的数据集。此外，我们现在可以以期望的精度计算对数似然函数的偏导数，并找到最大似然，而不需要以前方法所要求的假设。为了评估我们方法的可靠性和准确性，我们生成了大量随机的fisher分布数据集，并计算了平均倾斜度和精度参数。比较表明，我们新的稳健的Arason-Levi极大似然方法是最可靠的，平均倾角估计对浅值的偏差最小。
We have developed a new robust maximum likelihood method for estimating the unbiased mean inclination from inclination-only data. In paleomagnetic analysis, the arithmetic mean of inclination-only data is known to introduce a shallowing bias. Several methods have been introduced to estimate the unbiased mean inclination of inclination-only data together with measures of the dispersion. Some inclination-only methods were designed to maximize the likelihood function of the marginal Fisher distribution. However, the exact analytical form of the maximum likelihood function is fairly complicated, and all the methods require various assumptions and approximations that are often inappropriate. For some steep and dispersed data sets, these methods provide estimates that are significantly displaced from the peak of the likelihood function to systematically shallower inclination. The problem locating the maximum of the likelihood function is partly due to difficulties in accurately evaluating the function for all values of interest, because some elements of the likelihood function increase exponentially as precision parameters increase, leading to numerical instabilities. In this study, we succeeded in analytically cancelling exponential elements from the log-likelihood function, and we are now able to calculate its value anywhere in the parameter space and for any inclination-only data set. Furthermore, we can now calculate the partial derivatives of the log-likelihood function with desired accuracy, and locate the maximum likelihood without the assumptions required by previous methods. To assess the reliability and accuracy of our method, we generated large numbers of random Fisher-distributed data sets, for which we calculated mean inclinations and precision parameters. The comparisons show that our new robust Arason-Levi maximum likelihood method is the most reliable, and the mean inclination estimates are the least biased towards shallow values.