Computation of High Breakdown Nonlinear Regression Parameters
高崩溃点非线性回归参数的计算AI 解读 ↓
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高崩溃点非线性回归参数的计算
- 研究问题
- 非线性回归中遍历大量基本子集拟合的成本很高，如何减少计算并获得高崩溃点估计及可用的标准误？
- 方法与思路
- 作者提出只需少量基本子集最小二乘拟合的算法，用于 LMS 与 MM 估计，并比较自助法、蒙特卡洛及渐近标准误。
- 材料陈述的贡献
- 摘要称，所考察设定中较少拟合即可达到相近表现；同时发现异常比例上升时，自助法及部分渐近标准误会变得不可靠。
- 阅读关注点
- 应核对子集选择和初始化、算法能否找到目标解、污染比例的设定，以及高崩溃点点估计与标准误可靠性的区别。
- 阅读边界
- 仅据 OpenAlex 索引摘要，稿件版本未注明；计算节省与标准误失效均来自特定设定，不能推断所有非线性模型均有相同表现。
原文摘要
Most algorithms for estimating high breakdown regression estimators in linear regression rely on finding the least squares fit to many p-point elemental sets, where p is the dimension of the X matrix. Such an approach is computationally infeasible in nonlinear regression. This article presents a new algorithm for computing high breakdown estimates in nonlinear regression that requires only a small number of least squares fits to p points. The algorithm is used to compute Rousseeuw's least median of squares (LMS) estimate and Yohai's MM estimate in both simulations and examples. It is also used to compute bootstrapped and Monte Carlo Standard error estimates for MM estimates, which are compared with asymptotic Standard errors (ASE's). Using the PROGRESS algorithm for a two-parameter nonlinear model with sample size 30 would require finding the least squares fit to 435 two-point subsets of the data. In the settings considered in this article, the proposed algorithm performs just as well with 25 as with 435 least squares fits, thus substantially reducing computation time. Using the new algorithm to compare Standard error estimates for MM estimates reveals that. at least in the setting considered here, bootstrapped and to a lesser extent ASE's become increasingly unreliable as the percentage of outliers increases. In one example presented, the high breakdown estimates are useful in determining that a point appearing to be an outlier in the least squares analysis probably should not be considered an outlier. In the other example, the least squares analysis reveals no outliers and suggests that the model is inappropriate for the data. The residual plots using high breakdown estimates reveal that two points are outliers, and the model provides a reasonable fit to the remainder of the data.