A functional (lagged) time series regression model involves the regression of scalar response time series on a time series of regressors that consists of a sequence of random functions. In practice, the underlying regressor curve time series are not always directly accessible, but are latent processes observed (sampled) only at discrete measurement locations. In this paper, we consider the so-called sparse observation scenario where only a relatively small number of measurement locations have been observed, possibly different for each curve. The measurements can be further contaminated by additive measurement error. A spectral approach to the estimation of the model dynamics is considered. The spectral density of the regressor time series and the cross-spectral density between the regressors and response time series are estimated by kernel smoothing methods from the sparse observations. The impulse response regression coefficients of the lagged regression model are then estimated by means of ridge regression (Tikhonov regularisation) or PCA regression (spectral truncation). The latent functional time series are then recovered by means of prediction, conditioning on all the observed observed data. The performance and implementation of our methods are illustrated by means of a simulation study and the analysis of meteorological data.

中文翻译：

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具有稀疏噪声观测的函数滞后回归

函数（滞后）时间序列回归模型涉及标量响应时间序列对由一系列随机函数组成的回归量时间序列的回归。在实践中，潜在的回归曲线时间序列并不总是可以直接访问的，而是仅在离散测量位置观察（采样）的潜在过程。在本文中，我们考虑了所谓的稀疏观测场景，其中仅观测到相对少量的测量位置，每条曲线可能不同。测量结果可能会受到附加测量误差的进一步影响。考虑了模型动力学估计的谱方法。回归量时间序列的谱密度以及回归量与响应时间序列之间的交叉谱密度是通过核平滑方法从稀疏观测中估计出来的。然后通过岭回归（Tikhonov 正则化）或 PCA 回归（频谱截断）估计滞后回归模型的脉冲响应回归系数。然后通过预测恢复潜在的功能时间序列，以所有观察到的观察数据为条件。通过模拟研究和气象数据分析来说明我们方法的性能和实施。然后通过岭回归（Tikhonov 正则化）或 PCA 回归（频谱截断）估计滞后回归模型的脉冲响应回归系数。然后通过预测恢复潜在功能时间序列，以所有观察到的观察数据为条件。通过模拟研究和气象数据分析来说明我们方法的性能和实施。然后通过岭回归（Tikhonov 正则化）或 PCA 回归（频谱截断）估计滞后回归模型的脉冲响应回归系数。然后通过预测恢复潜在功能时间序列，以所有观察到的观察数据为条件。通过模拟研究和气象数据分析来说明我们方法的性能和实施。