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A relaxed approach to combinatorial problems in robustness and diagnostics

机译：轻松解决健壮性和诊断性组合问题的方法

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摘要

A range of procedures in both robustness and diagnostics require optimisation of a target functional over all subsamples of given size. Whereas such combinatorial problems are extremely difficult to solve exactly, something less than the global optimum can be 'good enough' for many practical purposes, as shown by example. Again, a relaxation strategy embeds these discrete, high-dimensional problems in continuous, low-dimensional ones. Overall, nonlinear optimisation methods can be exploited to provide a single, reasonably fast algorithm to handle a wide variety of problems of this kind, thereby providing a certain unity. Four running examples illustrate the approach. On thernrobustness side, algorithmic approximations to minimum covariance determinant (MCD) and least trimmed squares (LTS) estimation. And, on the diagnostic side, detection of multiple multivariate outliers and global diagnostic use of the likelihood displacement function. This last is developed here as a global complement to Cook's (in J. R. Stat. Soc. 48:133-169, 1986) local analysis. Appropriate convergence of each branch of the algorithm is guaranteed for any target functional whose relaxed form is-in a natural generalisation of concavity, introduced here-'gravitational'. Again, its descent strategy can downweight to zero contaminating cases in the starting position. A simulation study shows that, although not optimised for the LTS problem, our general algorithm holds its own with algorithms that are so optimised. An adapted algorithm relaxes the gravitational condition itself.
机译：鲁棒性和诊断性方面的一系列程序都需要在给定大小的所有子样本上优化目标功能。这样的组合问题非常难以精确解决，但对于许多实际目的而言，小于全局最佳值的东西可能“足够好”，如示例所示。同样，松弛策略将这些离散的高维问题嵌入到连续的低维问题中。总体而言，可以利用非线性优化方法来提供一个单一的，相当快的算法来处理各种各样的此类问题，从而提供一定的统一性。四个运行示例说明了该方法。在稳健性方面，对最小协方差行列式（MCD）和最小修剪平方（LTS）估计的算法近似。并且，在诊断方面，检测多个多元离群值并使用似然位移函数进行整体诊断。这是库克（1986. J. R. Stat。Soc。48：133-169）局部分析的全球补充，在这里得到发展。对于其松弛形式为凹面的自然概括（此处介绍为“引力”）的任何目标函数，均保证了算法各分支的适当收敛。同样，其下降策略可以将起始位置的污染案例减少到零。仿真研究表明，尽管未针对LTS问题进行优化，但我们的通用算法在经​​过如此优化的算法中占有一席之地。一种经过修改的算法可以放松重力条件本身。

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