Functional Lagged Regression with Sparse Noisy Observations
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by
Tomáš Rubín, Victor M. Panaretos
2020
Abstract
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.
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