Recovering covariance from functional fragments

Abstract

We consider nonparametric estimation of a covariance function on the unit square, given a sample of discretely observed fragments of functional data. When each sample path is only observed on a subinterval of length δ < 1, one has no statistical information on the unknown covariance outside a δ-band around the diagonal. The problem seems unidentifiable without parametric assumptions, but we show that nonparametric estimation is feasible under suitable smoothness and rank conditions on the unknown covariance. This remains true even when observation is discrete, and we give precise deterministic conditions on how fine the observation grid needs to be relative to the rank and fragment length for identifiability to hold true. We show that our conditions translate the estimation problem to a low-rank matrix completion problem, construct a nonparametric estimator in this vein, and study its asymptotic properties. We illustrate the numerical performance of our method on real and simulated data.

Nonparametric covariance estimation for mixed longitudinal studies, with applications in midlife women's health

“…It is Table 3. We can see that The fourth simulation compares the performance of the proposed method with the matrix completion method proposed in Descary and Panaretos (2018). The data generating procedure is the same as the one in previous simulations.…”

“…Application to midlife women's working memory study: We downloaded the data from SWAN database (link: http://www.icpsr.umich. is the matrix completion method proposed in Descary and Panaretos (2018), and δ denotes the fraction of domains observed. trials (range, 0-12).…”

“…The analysis of fragmentary functional observations have been studied under other modeling assumptions, for example Delaigle and Hall (2013) and Delaigle and Hall (2016). Very recently, two manuscripts Descary and Panaretos (2018) and Kneip and Liebl (2017) settings since a large portion of covariate pairs will never appear in the same sample (such as the pairs between the earlier and latest observations in the longitudinal studies) by the nature of the design. Bishop and Byron (2014) has studied a similar sequential-aggregation scheme for matrix completion, however, they mainly consider the completion of high-dimensional low rank positive semidefinite matrix in a deterministic setting, while we provide a statistical guarantee for covariance estimation from partially-observed noisy functional data.…”

Inferential procedures for partially observed functional data

“…This is why Condition 1(c) does not require the function ν(s, t) to be bounded away from zero on the entire domain [0, 1] 2 which is needed for the estimation of R, as will be seen in Section 3, Condition 2(a). This means that the theorem applies also in the context of short fragments of curves considered, e.g., by Delaigle and Hall (2016) or Descary and Panaretos (2019), where each curve in the sample is observed on a short interval and no completely observed curves are available.…”

“…Due to new applications recent years have seen the emergence of a new type of observation of functional data, called functional fragments or partially observed functional data. For various examples see Bugni (2012), Delaigle and Hall (2013), Liebl (2013), Gellar et al (2014), Goldberg et al (2014), Kraus (2015), Delaigle and Hall (2016), Gromenko et al (2017), Kneip and Liebl (2017), Dawson and Müller (2018), Mojirsheibani and Shaw (2018), Stefanucci et al (2018), Descary and Panaretos (2019), Kraus and Stefanucci (2019) or Liebl and Rameseder (2019).…”