Electronic Journal of Statistics

Sparsely observed functional time series: estimation and prediction

Tomáš Rubín and Victor M. Panaretos

Full-text: Open access

Abstract

Functional time series analysis, whether based on time or 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 to 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. This new methodology is seen to comprehensively outperform a naive recovery approach that would ignore temporal dependence and use only methodology employed in the i.i.d. setting and hinging on the lag zero covariance. 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 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.

Article information

Source
Electron. J. Statist., Volume 14, Number 1 (2020), 1137-1210.

Dates
Received: February 2019
First available in Project Euclid: 28 February 2020

Permanent link to this document
https://projecteuclid.org/euclid.ejs/1582859034

Digital Object Identifier
doi:10.1214/20-EJS1690

Subjects
Primary: 62M10: Time series, auto-correlation, regression, etc. [See also 91B84]
Secondary: 62M15, 60G10

Keywords
autocovariance operator confidence bands functional data analysis nonparametric regression spectral density operator

Rights
Creative Commons Attribution 4.0 International License.

Citation

Rubín, Tomáš; Panaretos, Victor M. Sparsely observed functional time series: estimation and prediction. Electron. J. Statist. 14 (2020), no. 1, 1137--1210. doi:10.1214/20-EJS1690. https://projecteuclid.org/euclid.ejs/1582859034


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