PCA-based estimation for functional linear regression with functional responses
PCA-based estimation for functional linear regression with functional responses
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DOI：
10.1016/j.jmva.2017.10.001
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发表时间：
2016-09
期刊：
影响因子：
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通讯作者：
M. Imaizumi;Kengo Kato
中科院分区：
文献类型：
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作者：
M. Imaizumi;Kengo Kato
本文研究了一个回归模型，其中预测变量和响应变量都是随机函数。我们考虑一个函数线性模型，其中每个时间点响应变量的条件均值由预测变量的线性函数给出。在本文中，我们感兴趣的是条件期望算子的积分核 b (s, t) 的估计，其中 s 是输出变量，而 t 是与预测变量相互作用的变量。这个问题是一个不适定的反问题，我们考虑两个基于函数主成分分析（PCA）的估计量。我们证明，在适当的规律性条件下，基于单次截断的估计器可以获得积分平方误差的收敛速度，其特征是函数 b (s, t) 在 t 中的平滑度以及协方差算子特征值的衰减率，但该速度并不依赖于 b (s, t) 在 s 中的平滑度。该速率被证明是极小极大最优，因此 s 中 b (s, t) 的平滑度不会影响估计 b 的难度。我们还考虑基于双截断的替代估计器，并提供替代估计器达到最优速率的条件。我们进行模拟来验证基于 PCA 的估计器在有限样本中的性能。最后，我们应用我们的估计器来研究工作时间的寿命模式与总收入之间的关系，以及电力现货价格与风电馈电之间的关系。
This paper studies a regression model where both predictor and response variables are random functions. We consider a functional linear model where the conditional mean of the response variable at each time point is given by a linear functional of the predictor variable. In this paper, we are interested in estimation of the integral kernel b (s, t) of the conditional expectation operator, where s is an output variable while t is a variable that interacts with the predictor variable. This problem is an ill-posed inverse problem, and we consider two estimators based on functional principal component analysis (PCA). We show that under suitable regularity conditions, an estimator based on the single truncation attains the convergence rate for the integrated squared error that is characterized by smoothness of the function b (s, t) in t together with the decay rate of the eigenvalues of the covariance operator, but the rate does not depend on the smoothness of b (s, t) in s. This rate is shown to be minimax optimal, and consequently smoothness of b (s, t) in s does not affect difficulty of estimating b. We also consider an alternative estimator based on the double truncation, and provide conditions under which the alternative estimator attains the optimal rate. We conduct simulations to verify the performance of PCA-based estimators in the finite sample. Finally, we apply our estimators to investigate the relation between the lifetime pattern of working hours and total income, and the relation between the electricity spot price and the wind power in-feed.