Consistently recovering the signal from noisy functional data

Hörmann, Siegfried; Jammoul, Fatima

doi:10.1016/j.jmva.2021.104886

Cited by 5 publications

(11 citation statements)

References 21 publications

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“…A theoretical foundation and rigorous investigation of the proposed method is left to a companion paper, see Hörmann and Jammoul (2020).…”

Section: Discussionmentioning

confidence: 99%

Preprocessing noisy functional data: a multivariate perspective

Hörmann¹,

Jammoul²

2020

Preprint

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We consider functional data which are measured on a discrete set of observation points. Often such data are measured with noise, and then the target is to recover the underlying signal. Most commonly, practitioners use some smoothing approach, e.g., kernel smoothing or spline fitting towards this goal. The drawback of such curve fitting techniques is that they act function by function, and don't take into account information from the entire sample. In this paper we argue that signal and noise can be naturally represented as the common and idiosyncratic component, respectively, of a factor model. Accordingly, we propose to an estimation scheme which is based on factor models. The purpose of this paper is to explain the reasoning behind our approach and to compare its performance on simulated and on real data to competing methods.

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“…A theoretical foundation and rigorous investigation of the proposed method is left to a companion paper, see Hörmann and Jammoul (2020).…”

Section: Discussionmentioning

confidence: 99%

Preprocessing noisy functional data: a multivariate perspective

Hörmann¹,

Jammoul²

2020

Preprint

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“…In the formulation of this theorem it is assumed that L is fixed and known. Like in [29] the result can be extended to the cases where L is replaced by a consistent estimator. It is also possible to derive variants of this theorem where L is allowed to diverge with the sample size T .…”

Section: Estimating the Full Curvementioning

confidence: 97%

“…Here b j is the j-th row of B and Y j and U j denote the j-th column of Y and U , respectively. If our data are independent (or satisfy some appropriate weak dependence condition), it holds by the law of large numbers and orthogonality of principal components scores that 1 We have analysed this estimator for the signal in Hörmann and Jammoul [29] and have shown that under mild technical conditions (see Assumptions 2-4 in the Appendix) this estimator converges uniformly, i.e., sup…”

Section: Estimation Approachmentioning

confidence: 99%

“…The two most common techniques for this purpose are either likelihood or PCA based. The PCA technique was shown in our companion paper Hörmann and Jammoul [29] to lead to consistent estimation of the signal under fairly mild assumptions. Here we complement the theoretical results in Hörmann and Jammoul [29] by extending the estimators for X t (s) to consistent estimators for the full curve X t (s) (Section 2.2) and its functional principal components (Section 2.3).…”

Section: Introductionmentioning

confidence: 99%

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Preprocessing noisy functional data: A multivariate perspective

Hörmann

Jammoul

2022

Electron. J. Statist.

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We consider functional data which are measured on a discrete set of observation points. Often such data are measured with additional noise. We explore in this paper the factor structure underlying this type of data. We show that the latent signal can be attributed to the common components of a corresponding factor model and can be estimated accordingly, by borrowing methods from factor model literature. We also show that principal components, which play a key role in functional data analysis, can be accurately estimated by taking such a multivariate instead of a 'functional' perspective. In addition to the estimation problem, we also address testing of the null-hypothesis of iid noise. While this assumption is largely prevailing in the literature, we believe that it is often unrealistic and not supported by a residual analysis.

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“…Although the idea of modeling an FTS by its low-dimensional dynamic component is appealing, only a small number of papers have addressed this topic so far. Hays et al (2012) proposed a functional factor model with a discrete idiosyncratic component, Kowal et al (2017) considered a Bayesian functional dynamic model, Descary and Panaretos (2019) addressed the problem of separate identification of a functional smooth and a rough component, and Hörmann and Jammoul (2021) showed that discretely observed functional data naturally follow some approximate factor model. Other prediction approaches related to factor analysis in the FDA literature focus on algorithms based on functional principal components (FPC) (see Hyndman andUllah 2007 andAue et al 2015).…”

Section: Introductionmentioning

confidence: 99%

Approximate Factor Models for Functional Time Series

Otto¹,

Salish²

2022

Preprint

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A functional dynamic factor model for time-dependent functional data is proposed. We decompose a functional time series into a predictive low-dimensional common component consisting of a finite number of factors and an infinite-dimensional idiosyncratic component that has no predictive power. The conditions under which all model parameters, including the number of factors, become identifiable are discussed. Our identification results lead to a simple-to-use two-stage estimation procedure based on functional principal components. As part of our estimation procedure, we solve the separation problem between the common and idiosyncratic functional components. In particular, we obtain a consistent information criterion that provides joint estimates of the number of factors and dynamic lags of the common component. Finally, we illustrate the applicability of our method in a simulation study and to the problem of modeling and predicting yield curves. In an out-of-sample experiment, we demonstrate that our model performs well compared to the widely used term structure Nelson-Siegel model for yield curves.

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Consistently recovering the signal from noisy functional data

Cited by 5 publications

References 21 publications

Preprocessing noisy functional data: a multivariate perspective

Preprocessing noisy functional data: a multivariate perspective

Preprocessing noisy functional data: A multivariate perspective

Approximate Factor Models for Functional Time Series

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