We introduce the partial martingale difference correlation, a scalar-valued measure of conditional mean dependence of $Y$ given $X$, adjusting for the nonlinear dependence on $Z$, where $X$, $Y$ and $Z$ are random vectors of arbitrary dimensions. At the population level, partial martingale difference correlation is a natural extension of partial distance correlation developed recently by Székely and Rizzo [14], which characterizes the dependence of $Y$ and $X$, after controlling for the nonlinear effect of $Z$. It extends the martingale difference correlation first introduced in Shao and Zhang [10] just as partial distance correlation extends the distance correlation in Székely, Rizzo and Bakirov [13]. Sample partial martingale difference correlation is also defined building on some new results on equivalent expressions of sample martingale difference correlation. Numerical results demonstrate the effectiveness of these new dependence measures in the context of variable selection and dependence testing.
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机译：我们介绍了部分mar差异相关性，即给定$ X $时对条件$ Y $的条件均值依赖的标量度量，并调整了对$ Z $的非线性依赖，其中$ X $，$ Y $和$ Z $是随机的任意维度的向量。在人口水平上，部分ting差异相关性是Székely和Rizzo最近开发的部分距离相关性的自然扩展[14]，在控制了$ Z $的非线性效应之后，它表征了$ Y $和$ X $的依赖性。 。正如部分距离相关扩展了Székely，Rizzo和Bakirov [13]中的距离相关一样，它扩展了Shao和Zhang [10]中首次引入的mar差相关性。还基于样本mar差异相关的等价表达式的一些新结果，定义了样本部分difference差异相关。数值结果表明，在变量选择和依赖测试的背景下，这些新的依赖度量的有效性。
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