Estimating Truncated Functional Linear Models with a Nested Group Bridge Approach

We study a scalar-on-function truncated linear regression model which assumes that the functional predictor does not influence the response when the time passes a certain cutoff point. We approach this problem from the perspective of locally sparse modeling, where a function is locally sparse if it is zero on a substantial portion of its defining domain. In the truncated linear model, the slope function is exactly a locally sparse function that is zero beyond the cutoff time. A locally sparse estimate then gives rise to an estimate of the cutoff time. We propose a nested group bridge penalty that is able to specifically shrink the tail of a function. Combined with the B-spline basis expansion and penalized least squares, the nested group bridge approach can identify the cutoff time and produce a smooth estimate of the slope function simultaneously. The proposed nested group bridge estimator is shown to be consistent, while its numerical performance is illustrated by simulation studies. The proposed nested group bridge method is demonstrated with an application of determining the effect of the past engine acceleration on the current particulate matter emission. This article has online supplementary material.

本文研究一类标量对函数截断线性回归模型，该模型假设当时间超过某一截断点时，函数型预测变量不再对响应变量产生影响。本文从局部稀疏建模的视角处理该问题：若某函数在其定义域的绝大部分区域内取值为零，则称该函数为局部稀疏函数。在该截断线性模型中，斜率函数恰好是一类局部稀疏函数——在截断时间之后其取值恒为零。通过局部稀疏估计，可同时得到截断时间的估计值。本文提出一种嵌套群桥惩罚项（nested group bridge penalty），其可针对性地对函数的尾部区域进行收缩估计。将该方法与B样条基展开（B-spline basis expansion）及惩罚最小二乘法（penalized least squares）相结合，可同时实现截断时间的识别与斜率函数的平滑估计。理论上证明所提嵌套群桥估计量具有相合性，同时通过模拟实验验证了该方法的数值表现。本文将所提嵌套群桥方法应用于探究过往发动机加速度对当前颗粒物排放的影响，以此验证方法的实用性。本文附带在线补充材料。