Sparsely observed functional time series: estimation and prediction

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.

Functional lagged regression with sparse noisy observations

“…The forecasting of the response process is then implemented by first predicting the latent functional regressor data (using their estimated spectral characteristics) and then plugging-in these predictions into the estimated lagged regression model. Sparsely observed functional time series have only recently received attention (Kowal et al 2017a,b;Sen and Klüppelberg, 2019;Rubín and Panaretos, 2020), and our results appear to be the first in the context of the lagged regression model where the regressor process is functional. A related problem of dynamic function-on-scalar regression was studied by Kowal (2018) by means of Bayesian factor models.…”

“…Here we employ cumulant mixing conditions which have been successfully used in functional time series before (Panaretos and Tavakoli, 2013;Rubín and Panaretos, 2020). Alternatives to the cumulant assumptions are L p -m-approximability (Hörmann and Kokoszka, 2010;Hörmann et al, 2015a,b) and strong mixing conditions (Rubín and Panaretos, 2020). Bellow we list the assumptions that we will make use of to study the asymptotics of the spectral density estimator {f X } ∈[− , ] and the cross-spectral density estimator…”

“…The non-parametric estimation in functional models given sparse noisy measurements has been frequently based on local-polynomial kernel smoothers (Yao et al, 2005a;Hall et al, 2006;Li and Hsing, 2010;Rubín and Panaretos, 2020). Throughout the article we use the Epanechnikov kernel…”

“…for a sequence of point processes (x t1 , … , x tN t ), t = 1, … , T, independent in time, and a white noise measurement error sequence. This observation scheme is illustrated in Figure 1 and is exhibited, for example, when dealing with fair weather atmospheric electricity data (Tammet, 2009;Rubín and Panaretos, 2020).…”

“…In the dense case, one can typically behave as if the true latent function were observed. The sparse case, however, needs new tools in Yao et al (2005a,b), Hall et al (2006) and Li and Hsing (2010), and extended to the time series case in Rubín and Panaretos (2020). Once these are estimated, one obtains estimating equations whose solution yields the estimators of the filter coefficients.…”