In this paper we introduce a new methodology to determine an optimal coefficient of penalized functional regression. We assume the dependent, independent variables and the regression coefficients are functions of time and error dynamics follow a stochastic differential equation. First we construct our objective function as a time dependent residual sum of square and then minimize it with respect to regression coefficients subject to different error dynamics such as LASSO, group LASSO, fused LASSO and cubic smoothing spline. Then we use Feynman-type path integral approach to determine a Schr\"odinger-type equation which have the entire information of the system. Using first order conditions with respect to these coefficients give us a closed form solution of them.
翻译:在本文中,我们引入了一种新的方法,以确定受惩罚功能回归的最佳系数。我们假设依赖的、独立的变量和回归系数是时间的函数,错误动态则遵循随机差分方程式。首先,我们构建了我们的目标函数,作为有时间依赖的剩余平方块,然后在受不同错误动态影响的回归系数方面将其最小化,如LASSO、LASSO集团、LASSO引信化的LASSO和立方平滑样样样样。然后,我们用费曼型路径的一体化方法来确定具有系统全部信息的Schr\'odinger型方程式。使用有关这些系数的第一顺序条件,我们就可以以封闭的形式解决这些系数。