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Publication# Sparsely Observed Functional Time Series: Theory and Applications

Résumé

Functional time series is a temporally ordered sequence of not necessarily independent random curves. While the statistical analysis of such data has been traditionally carried out under the assumption of completely observed functional data, it may well happen that the statistician only has access to a relatively low number of sparse measurements for each random curve. These discrete measurements may be moreover irregularly scattered in each curve's domain, missing altogether for some curves, and be contaminated by measurement noise. This sparse sampling protocol escapes from the reach of established estimators in functional time series analysis and therefore requires development of a novel methodology.

The core objective of this thesis is development of a non-parametric statistical toolbox for analysis of sparsely observed functional time series data. Assuming smoothness of the latent curves, we construct a local-polynomial-smoother based estimator of the spectral density operator producing a consistent estimator of the complete second order structure of the data. Moreover, the spectral domain recovery approach allows for prediction of latent curve data at a given time by borrowing strength from the estimated dynamic correlations in the entire time 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.

A classical non-parametric apparatus for encoding the dependence between a pair of or among a multiple functional time series, whether sparsely or fully observed, is the functional lagged regression model. This consists of a linear filter between the regressors time series and the response. We show how to tailor the smoother based estimators for the estimation of the cross-spectral density operators and the cross-covariance operators and, by means of spectral truncation and Tikhonov regularisation techniques, how to estimate the lagged regression filter and predict the response process.

The simulation studies revealed the following findings: (i) if one has freedom to design a sampling scheme with a fixed number of measurements, it is advantageous to sparsely distribute these measurements in a longer time horizon rather than concentrating over a shorter time horizon to achieve dense measurements in order to diminish the spectral density estimation error, (ii) the developed functional recovery predictor surpasses the static predictor not exploiting the temporal dependence, (iii) neither of the two considered regularisation techniques can, in general, dominate the other for the estimation in functional lagged regression models. The new methodologies are illustrated by applications to real data: the meteorological data revolving around the fair-weather atmospheric electricity measured in Tashkent, Uzbekistan, and at Wank mountain, Germany; and a case study analysing the dependence of the US Treasury yield curve on macroeconomic variables.

As a secondary contribution, we present a novel simulation method for general stationary functional time series defined through their spectral properties. A simulation study shows universality of such approach and superiority of the spectral domain simulation over the temporal domain in some situations.

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Victor Panaretos, Shahin Tavakoli

We develop the basic building blocks of a frequency domain framework for drawing statistical inferences on the second-order structure of a stationary sequence of functional data. The key element in such a context is the spectral density operator, which generalises the notion of a spectral density matrix to the functional setting, and characterises the second-order dynamics of the process. Our main tool is the functional Discrete Fourier Transform (fDFT). We derive an asymptotic Gaussian representation of the fDFT, thus allowing the transformation of the original collection of dependent random functions into a collection of approximately independent complex-valued Gaussian random functions. Our results are then employed in order to construct estimators of the spectral density operator based on smoothed versions of the periodogram kernel, the functional generalisation of the periodogram matrix. The consistency and asymptotic law of these estimators are studied in detail. As immediate consequences, we obtain central limit theorems for the mean and the long-run covariance operator of a stationary functional time series. Our results do not depend on structural modelling assumptions, but only functional versions of classical cumulant mixing conditions, and are shown to be stable under discrete observation of the individual curves.

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.

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 article, 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 regularization) or principal component analysis (PCA) regression (spectral truncation). The latent functional time series are then recovered by means of prediction, conditioning on all the observed data. The performance and implementation of our methods are illustrated by means of a simulation study and the analysis of meteorological data.