Fast covariance estimation for sparse functional data

Xiao, Luo; Li, Cai; Checkley, William; Crainiceanu, Ciprian M.

doi:10.1007/s11222-017-9744-8

Cited by 60 publications

(72 citation statements)

References 35 publications

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“…While the estimation procedure described in this paper is similar to that proposed in Cederbaum et al, the procedure has not been implemented in any peer‐reviewed publication. Moreover, our covariance estimation procedure is fast, accurate, and was designed specifically for sparse functional covariance estimation.…”

Section: Discussionmentioning

confidence: 99%

“…This results in estimates

{\overset{μ}{true}}_{z} false(t false)

Ĉ_{z} false(s, t false)

, and

{\overset{σ}{true}}_{ϵ z}^{2}

. Then, conditional on the observed data

Z_{i, obs} = {false{Z_{i, obs} false(t_{i 1} false), ⋯, Z_{i, obs} false(t_{i m_{i}} false) false}}^{'}

, we predict Z i ( t ) by its best linear unbiased predictor (BLUP),

{\overset{Z}{true}}_{i} false(t false) = {\overset{μ}{true}}_{z} false(t false) + Ĉ_{z} {false(t, t_{i} false)}^{'} {({}, Ĉ_{z} false(t_{i}, t_{i} false) + {\overset{σ}{true}}_{ϵ z}^{2} I_{m_{i}})}^{- 1} false{Z_{i, obs} - {\overset{μ}{true}}_{z} false(t_{i} false) false}

, where

t_{i} = {false(t_{i 1}, ⋯, t_{i m_{i}} false)}^{'} \in R^{m_{i}}

Ĉ_{z} false(t, t_{i} false) = {false{C_{z} false(t, t_{i 1} false), ⋯, C_{z} false(t, t_{i m_{i}} false) false}}^{'} \in R^{m_{i}}

{\overset{μ}{true}}_{z} false(

…”

Section: Modelmentioning

confidence: 99%

“…We use the R package face for estimating C ( s , t ) and

σ_{ϵ}^{2}

using the residuals e i ,1≤ i ≤ N . The face package is based on Xiao et al, where C ( s , t ) is decomposed as ∑ 1≤ k , ℓ ≤ c B k ( s ) B k ( t ) γ k ℓ .…”

Section: Modelmentioning

confidence: 99%

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Dynamic prediction in functional concurrent regression with an application to child growth

Leroux

Xiao

Crainiceanu

2017

Statistics in Medicine

Self Cite

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In many studies, it is of interest to predict the future trajectory of subjects based on their historical data, referred to as dynamic prediction. Mixed effects models have traditionally been used for dynamic prediction. However, the commonly used random intercept and slope model is often not sufficiently flexible for modeling subject‐specific trajectories. In addition, there may be useful exposures/predictors of interest that are measured concurrently with the outcome, complicating dynamic prediction. To address these problems, we propose a dynamic functional concurrent regression model to handle the case where both the functional response and the functional predictors are irregularly measured. Currently, such a model cannot be fit by existing software. We apply the model to dynamically predict children's length conditional on prior length, weight, and baseline covariates. Inference on model parameters and subject‐specific trajectories is conducted using the mixed effects representation of the proposed model. An extensive simulation study shows that the dynamic functional regression model provides more accurate estimation and inference than existing methods. Methods are supported by fast, flexible, open source software that uses heavily tested smoothing techniques.

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Section: Discussionmentioning

confidence: 99%

“…This results in estimates

{\overset{μ}{true}}_{z} false(t false)

Ĉ_{z} false(s, t false)

, and

{\overset{σ}{true}}_{ϵ z}^{2}

. Then, conditional on the observed data

Z_{i, obs} = {false{Z_{i, obs} false(t_{i 1} false), ⋯, Z_{i, obs} false(t_{i m_{i}} false) false}}^{'}

, we predict Z i ( t ) by its best linear unbiased predictor (BLUP),

{\overset{Z}{true}}_{i} false(t false) = {\overset{μ}{true}}_{z} false(t false) + Ĉ_{z} {false(t, t_{i} false)}^{'} {({}, Ĉ_{z} false(t_{i}, t_{i} false) + {\overset{σ}{true}}_{ϵ z}^{2} I_{m_{i}})}^{- 1} false{Z_{i, obs} - {\overset{μ}{true}}_{z} false(t_{i} false) false}

, where

t_{i} = {false(t_{i 1}, ⋯, t_{i m_{i}} false)}^{'} \in R^{m_{i}}

Ĉ_{z} false(t, t_{i} false) = {false{C_{z} false(t, t_{i 1} false), ⋯, C_{z} false(t, t_{i m_{i}} false) false}}^{'} \in R^{m_{i}}

{\overset{μ}{true}}_{z} false(

…”

Section: Modelmentioning

confidence: 99%

See 1 more Smart Citation

Dynamic prediction in functional concurrent regression with an application to child growth

Leroux

Xiao

Crainiceanu

2017

Statistics in Medicine

Self Cite

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“…CD4 counts have been extensively analyzed using longitudinal data methods, e.g., semiparametric and linear random effects models (Taylor et al, 1994;Zeger and Diggle, 1994;Fan and Zhang, 2000). Recently, functional data methods have also been applied to this data (Yao et al, 2005;Goldsmith et al, 2013;Xiao et al, 2018). While the nonparametric functional data methods are highly flexible and better adapt to subject-specific patterns, they are more difficult to implement and interpret compared to the parametric approaches.…”

Section: Introductionmentioning

confidence: 99%

“…For functional methods, the covariance within a subject is assumed to be smooth with an unknown nonparametric form. The covariance can be estimated by smoothing the sample covariance (Besse and Ramsay, 1986;Yao et al, 2005;Xiao et al, 2018) or constructing a reduced rank approximation by estimating basis functions from smoothed sample curves (James et al, 2000;Peng and Paul, 2009). In contrast, longitudinal data approaches typically assume a simple parametric covariance structure with a few parameters, such as autoregressive or exponential (see Diggle et al (2002) for an overview), or induced by a random effects model (Laird and Ware, 1982).…”

Section: Introductionmentioning

confidence: 99%

A Smoothing-Based Goodness-of-Fit Test of Covariance for Functional Data

2018

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Summary: Functional data methods are often applied to longitudinal data as they provide a more flexible way to capture dependence across repeated observations. However, there is no formal testing procedure to determine if functional methods are actually necessary. We propose a goodness-of-fit test for comparing parametric covariance functions against general nonparametric alternatives for both irregularly observed longitudinal data and densely observed functional data. We consider a smoothing-based test statistic and approximate its null distribution using a bootstrap procedure. We focus on testing a quadratic polynomial covariance induced by a linear mixed effects model and the method can be used to test any smooth parametric covariance function. Performance and versatility of the proposed test is illustrated through a simulation study and three data applications.

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Functional clustering methods for resistance spot welding process data in the automotive industry

Capezza

Centofanti

Lepore

et al. 2021

Appl Stoch Models Bus & Ind

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In the automotive industry, quality assessment of resistance spot welding (RSW) joints of metal sheets is typically based on costly and lengthy offline tests, which are unfeasible in the full‐scale production on a large scale. However, the massive industrial digitalization triggered by the Industry 4.0 framework makes online measurements of RSW process parameters available for every joint produced. Among these, the so‐called dynamic resistance curve (DRC) is recognized as the full technological signature of the spot welds. Motivated by this context, this article intends to show the potentiality and practical applicability of clustering methods to data represented by curves, and, in general, to functional data. In this way, the task of separating DRCs into homogeneous groups pertaining to spot welds with common mechanical and metallurgical properties can be performed without the need for arbitrary and problem‐specific feature extraction. We provide a hands‐on overview of the most promising functional clustering methods, and apply them to DRCs collected during RSW lab tests at Centro Ricerche Fiat. The identified groups of DRCs emerge to be strictly linked with the wear status of the electrodes, which, in turn, is conjectured to impact the RSW joint final quality. The analysis code, developed in the software environment R, accompanied by an essential tutorial and the ICOSAF project data set containing DRC measurements, are openly available online at https://github.com/unina-sfere/funclustRSW/.

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Fast covariance estimation for sparse functional data

Cited by 60 publications

References 35 publications

Dynamic prediction in functional concurrent regression with an application to child growth

Dynamic prediction in functional concurrent regression with an application to child growth

A Smoothing-Based Goodness-of-Fit Test of Covariance for Functional Data

Functional clustering methods for resistance spot welding process data in the automotive industry

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