Low-Rank Covariance Function Estimation for Multidimensional Functional Data

Wang, Jiayi; Wong, Raymond K. W.; Zhang, Xiaoke

doi:10.1080/01621459.2020.1820344

Cited by 16 publications

(13 citation statements)

References 52 publications

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“…Under the multivariate regime, Di et al (2009Di et al ( , 2014; Hasenstab et al (2017) considered the multilevel/multidimensional functional data PCA. Wang et al (2020c) studied the low-rank covariance estimation for multidimensional functional data. Fan et al (2015) considered the functional additive regression method for high-dimensional functional regression.…”

Section: Related Workmentioning

confidence: 99%

“…The works on multivariate functional principal component analysis (Greven et al, 2010;Zipunnikov et al, 2011;Allen, 2013;Chiou et al, 2014;Scheffler et al, 2020;Berrendero et al, 2011;Happ and Greven, 2018) also address dimension reduction of multivariate functional data. Among these literature, Happ and Greven (2018); Wang et al (2020c) are the most relevant to our work. In particular, Wang et al (2020c) focused on i.i.d.…”

Section: Related Workmentioning

confidence: 99%

“…Among these literature, Happ and Greven (2018); Wang et al (2020c) are the most relevant to our work. In particular, Wang et al (2020c) focused on i.i.d. p-dimensional function objects, and they impose tensor low-rankness on the corresponding (2p)-dimensional covariance functions; Happ and Greven (2018) considered the multivariate functional PCA for i.i.d.…”

Section: Related Workmentioning

confidence: 99%

“…The proposed model setting is flexible and adaptive. For example, in the scenario of multivariate functional data analysis, different from many existing literature on (multivariate) functional PCA (Di et al, 2009;Happ and Greven, 2018;Wang et al, 2020c), which assumed the samples are i.i.d. distributed and try to estimate the covariance function, our model allows significant heterogeneity among samples (which is characterized by the singular vector of the sample mode).…”

Section: Introductionmentioning

confidence: 99%

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Guaranteed Functional Tensor Singular Value Decomposition

Han¹,

Shi²,

Zhang³

2021

Preprint

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This paper introduces the functional tensor singular value decomposition (FTSVD), a novel dimension reduction framework for tensors with one functional mode and several tabular modes. The problem is motivated by high-order longitudinal data analysis. Our model assumes the observed data to be a random realization of an approximate CP low-rank functional tensor measured on a discrete time grid. Incorporating tensor algebra and the theory of Reproducing Kernel Hilbert Space (RKHS), we propose a novel RKHS-based constrained power iteration with spectral initialization. Our method can successfully estimate both singular vectors and functions of the low-rank structure in the observed data. With mild assumptions, we establish the non-asymptotic contractive error bounds for the proposed algorithm. The superiority of the proposed framework is demonstrated via extensive experiments on both simulated and real data.

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Section: Related Workmentioning

confidence: 99%

Section: Related Workmentioning

confidence: 99%

Section: Related Workmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Guaranteed Functional Tensor Singular Value Decomposition

Han¹,

Shi²,

Zhang³

2021

Preprint

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“…Xiao (2020) considered estimates with a general weighing scheme via penalized splines and showed that the L 2 rate of convergence for mean function estimation is minimax optimal with properly chosen regularization parameters. Recently, Wang, Wong & Zhang (2020) investigated covariance function estimation for multidimensional functional data and introduced a novel low-rank estimation approach under a reproducing kernel Hilbert space (RKHS) framework.…”

Section: Introductionmentioning

confidence: 99%

Adaptive estimation for functional data: Using a framelet block‐thresholding method

Cheng

Chen

2022

Can J Statistics

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This article considers a framelet block-thresholding method for estimating mean and covariance functions from discretely sampled noisy observations. Estimated convergence rates are established for all types of sampling schemes. In particular, the results reveal a phase transition phenomenon related to the number of observations on each curve. It is shown that the proposed procedures are adaptive in automatically adjusting the smoothness properties of the underlying mean and covariance functions. In contrast, theoretical results for other smoothing methods hold in the setting where smoothness parameters are assumed to be known, since the regularization parameters of estimators that depend on smoothness properties need to be chosen carefully. Simulation studies are provided to offer empirical support for the theoretical results. A comparison with other methods demonstrates that the proposed method outperforms in adaptivity. An application to a real dataset is also provided to illustrate the proposed estimation procedure.

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