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Geometric median and its application in the identification of multiple outliers

机译：几何中值及其在多个异常值识别中的应用

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摘要

Geometric mean (GM) is having growing and wider applications in statistical data analysis as a measure of central tendency. It is generally believed that GM is less sensitive to outliers than the arithmetic mean (AM) but we suspect likewise the AM the GM may also suffer a huge set back in the presence of outliers, especially when multiple outliers occur in a data. So far as we know, not much work has been done on the robustness issue of GM. In quest of a simple robust measure of central tendency, we propose the geometric median (GMed) in this paper. We show that the classical GM has only 0% breakdown point while it is 50% for the proposed GMed. Numerical examples also support our claim that the proposed GMed is unaffected in the presence of multiple outliers and can maintain the highest possible 50% breakdown. Later we develop a new method for the identification of multiple outliers based on this proposed GMed. A variety of numerical examples show that the proposed method can successfully identify all potential outliers while the traditional GM fails to do so.
机译：几何平均数（GM）在统计数据分析中具有越来越广泛的应用，作为衡量集中趋势的一种方法。通常认为，GM对异常值的敏感性不如算术平均值（AM），但我们怀疑，如果存在异常值，GM的AM也可能遭受巨大的挫折，尤其是当数据中出现多个异常值时。据我们所知，在通用汽车的鲁棒性问题上还没有做很多工作。为了简单地测量中心趋势，我们在本文中提出了几何中位数（GMed）。我们显示经典GM的击穿点仅为0％，而建议的GMed则为50％。数值示例也支持我们的论点，即提出的GMed在存在多个异常值的情况下不受影响，并且可以保持最高50％的故障率。后来，我们根据提出的GMed开发了一种用于识别多个异常值的新方法。大量的数值例子表明，该方法可以成功地识别出所有潜在的异常值，而传统的GM无法做到。

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