Abstract
Functional time series analysis, whether based on time of frequency domain methodology, has traditionally been carried out under the assumption of complete observation of the constituent series of curves, assumed stationary. Nevertheless, as is often the case with independent functional data, it may well happen that the data available to the analyst are not the actual sequence of curves, but relatively few and noisy measurements per curve, potentially at different locations in each curve's domain. Under this sparse sampling regime, neither the established estimators of the time series' dynamics, nor their corresponding theoretical analysis will apply. The subject of this paper is to tackle the problem of estimating the dynamics and of recovering the latent process of smooth curves in the sparse regime. Assuming smoothness of the latent curves, we construct a consistent nonparametric estimator of the series' spectral density operator and use it develop a frequency-domain recovery approach, that predicts the latent curve at a given time by borrowing strength from the (estimated) dynamic correlations in the series across time. Further to predicting the latent curves from their noisy point samples, the method fills in gaps in the sequence (curves nowhere sampled), denoises the data, and serves as a basis for forecasting. Means of providing corresponding confidence bands are also investigated. A simulation study interestingly suggests that sparse observation for a longer time period, may be provide better performance than dense observation for a shorter period, in the presence of smoothness. The methodology is further illustrated by application to an environmental data set on fair-weather atmospheric electricity, which naturally leads to a sparse functional time-series.
Highlights
Functional data analysis constitutes a collection of statistical methods to analyse data comprised of ensembles of random functions: multiple occurrences of random processes evolving continuously in time and/or space, typically over a bounded rectangular domain [33, 11, 19, 41]
Panaretos and Tavakoli [28] introduced the notation of spectral density operators and harmonic principal components, capturing the complete secondorder dynamics in the frequency domain, whereas Panaretos and Tavakoli [29] showed how to estimate the said spectral density operators by smoothing the operator-valued analogue of the periodogram
In this article we address this gap and consider the problem of estimating the complete dynamics, and recovering the latent curves, in a stationary functional time series that is observed sparsely, irregularly, and with measurement errors
Summary
Functional data analysis constitutes a collection of statistical methods to analyse data comprised of ensembles of random functions: multiple occurrences of random processes evolving continuously in time and/or space, typically over a bounded rectangular domain [33, 11, 19, 41]. A step further from the estimation of isolated characteristics such as the mean function and the said long-run covariance operator is to estimate the entire second-order structure of the process, without assuming linearity To this aim, Panaretos and Tavakoli [28] introduced the notation of spectral density operators and harmonic principal components, capturing the complete secondorder dynamics in the frequency domain, whereas Panaretos and Tavakoli [29] showed how to estimate the said spectral density operators by smoothing the operator-valued analogue of the periodogram. Our methodology can be interpreted in a design context: in certain applications, it might be possible for the scientist to choose how to distribute a given fixed budget of measurements over individual curves and over time In this case, one might ask how to better estimate the underlying dynamics: whether it is better to sample a functional time series more densely over shorter time-span, or to record fewer observations per curve but over a longer time-space. In Appendix A we comment on some implementation concerns, the formal proofs are included in Appendix B, and some additional results of the numerical experiments are presented in Appendix C
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