Regression modelling is a powerful statistical tool often used in biomedical and clinical research. It could be formulated as an inverse problem that measures the discrepancy between the target outcome and the data produced by representation of the modelled predictors. This approach could simultaneously perform variable selection and coefficient estimation. We focus particularly on a linear regression issue, Y ∼ N ( X β , σ I n ) , where β ∈ R p is the parameter of interest and its components are the regression coefficients. The inverse problem finds an estimate for the parameter β , which is mapped by the linear operator ( L : β ⟶ X β ) to the observed outcome data Y = X β + ε . This problem could be conveyed by finding a solution in the affine subspace L - 1 ( Y ) . However, in the presence of collinearity, high-dimensional data and high conditioning number of the related covariance matrix, the solution may not be unique, so the introduction of prior information to reduce the subset L - 1 ( Y ) and regularize the inverse problem is needed. Informed by Huber's robust statistics framework, we propose an optimal regularizer to the regression problem. We compare results of the proposed method and other penalized regression regularization methods: ridge, lasso, adaptive-lasso and elastic-net under different strong hypothesis such as high conditioning number of the covariance matrix and high error amplitude, on both simulated and real data from the South London Stroke Register. The proposed approach can be extended to mixed regression models. Our inverse problem framework coupled with robust statistics methodology offer new insights in statistical regression and learning. It could open a new research development for model fitting and learning.

中文翻译：

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正则化回归模型的逆问题方法及其应用于预测中风后恢复的应用

回归模型是生物医学和临床研究中经常使用的强大统计工具。它可以被表述为一个逆问题，用于衡量目标结果与建模预测变量表示产生的数据之间的差异。该方法可以同时执行变量选择和系数估计。我们特别关注线性回归问题，Y ∼ N ( X β , σ I n ) ，其中 β ∈ R p 是感兴趣的参数，其分量是回归系数。反演问题找到参数 β 的估计值，该估计值由线性算子 ( L : β ⟶ X β ) 映射到观测结果数据 Y = X β + ε 。这个问题可以通过在仿射子空间L-1(Y) 中寻找解来表达。然而，在存在共线性、高维数据和相关协方差矩阵的条件数较高的情况下，解可能不唯一，因此引入先验信息来减少子集L-1(Y)并正则化逆问题是需要的。根据 Huber 强大的统计框架，我们提出了回归问题的最佳正则化器。我们比较了所提出的方法和其他惩罚回归正则化方法的结果：岭，套索，自适应套索和弹性网络在不同的强假设下，例如协方差矩阵的高条件数和高误差幅度，在模拟和真实数据上南伦敦中风登记册。所提出的方法可以扩展到混合回归模型。我们的逆问题框架与强大的统计方法相结合，为统计回归和学习提供了新的见解。它可以为模型拟合和学习开辟新的研究发展。