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Introduction The Model Application
Multivariate Functional Principal Components
Analysis (mFPCA)
Kevin Cummins
Joint Doctoral Program in Interdisciplinary Research in Substance Use
Division of Global Public Health
University of California, San Diego
Department of Social Work
San Diego State University
Wes Thompson
Department of Psychiatry
University of California, San Diego
August 9, 2015
Kevin Cummins JSM 2015v1
Introduction The Model Application
Multivariate Functional Principal Component Analysis
FPCA is a technique for estimating individual (smooth)
trajectories from sparse longitudinal data (James, Hastie, &
Sugar 2000)
Goals:
Estimate smooth mean trajectory,
Determine smooth principal modes of variation of trajectories
around mean levels.
mFPCA is a technique to simultaneously estimation
trajectories of multiple processes.
Added Goals:
Evaluate the association in the modes of variation among the
processes.
Kevin Cummins JSM 2015v1
Introduction The Model Application
Functional PCA model
The response of individual i at time t is multivariate and
modeled as
Yi(tij) = fi(tij) + ij
= µ(tij) + hi(tij) + ij
= φT
(tij)θµ + φT
(tij)Θαi + ij
φ(t) = (φ1(t), φ2(t), . . . , φK (t)): K-dimensional vector of orthogonal
basis functions evaluated at time tij.
θµ: K-dimensional vector of basis coefficients.
Kevin Cummins JSM 2015v1
Introduction The Model Application
Multivariate Functional Principal Component Analysis
(mFPCA)
The response of individual i at time t is multivariate and
modeled as
Yi(t) = ΦT
(t)θµ + ΦT
(t)Θαi + i(t)
Yi(t): P-dimensional observed response at time t
φ(t) = (φ1(t), φ2(t), . . . , φK (t))T : K-dimensional vector of orthogonal
basis functions evaluated at time tij and
ΦT
(t) =




φT
(t) . . . 0T
...
...
...
0T . . . φT
(t)




Θp: K by Qp matrix of spline coefficients subject to ΘT
p Θp = I and
Θ =



Θ1 . . . 0
...
...
...
0T . . . ΘP



Kevin Cummins JSM 2015v1
Introduction The Model Application
Brain Region Trajectories
Kevin Cummins JSM 2015v1
Introduction The Model Application
Modes of Variation
Kevin Cummins JSM 2015v1

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mFPCA

  • 1. Introduction The Model Application Multivariate Functional Principal Components Analysis (mFPCA) Kevin Cummins Joint Doctoral Program in Interdisciplinary Research in Substance Use Division of Global Public Health University of California, San Diego Department of Social Work San Diego State University Wes Thompson Department of Psychiatry University of California, San Diego August 9, 2015 Kevin Cummins JSM 2015v1
  • 2. Introduction The Model Application Multivariate Functional Principal Component Analysis FPCA is a technique for estimating individual (smooth) trajectories from sparse longitudinal data (James, Hastie, & Sugar 2000) Goals: Estimate smooth mean trajectory, Determine smooth principal modes of variation of trajectories around mean levels. mFPCA is a technique to simultaneously estimation trajectories of multiple processes. Added Goals: Evaluate the association in the modes of variation among the processes. Kevin Cummins JSM 2015v1
  • 3. Introduction The Model Application Functional PCA model The response of individual i at time t is multivariate and modeled as Yi(tij) = fi(tij) + ij = µ(tij) + hi(tij) + ij = φT (tij)θµ + φT (tij)Θαi + ij φ(t) = (φ1(t), φ2(t), . . . , φK (t)): K-dimensional vector of orthogonal basis functions evaluated at time tij. θµ: K-dimensional vector of basis coefficients. Kevin Cummins JSM 2015v1
  • 4. Introduction The Model Application Multivariate Functional Principal Component Analysis (mFPCA) The response of individual i at time t is multivariate and modeled as Yi(t) = ΦT (t)θµ + ΦT (t)Θαi + i(t) Yi(t): P-dimensional observed response at time t φ(t) = (φ1(t), φ2(t), . . . , φK (t))T : K-dimensional vector of orthogonal basis functions evaluated at time tij and ΦT (t) =     φT (t) . . . 0T ... ... ... 0T . . . φT (t)     Θp: K by Qp matrix of spline coefficients subject to ΘT p Θp = I and Θ =    Θ1 . . . 0 ... ... ... 0T . . . ΘP    Kevin Cummins JSM 2015v1
  • 5. Introduction The Model Application Brain Region Trajectories Kevin Cummins JSM 2015v1
  • 6. Introduction The Model Application Modes of Variation Kevin Cummins JSM 2015v1