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Transcript of mFPCA
Introduction The Model Application
Multivariate Functional Principal ComponentsAnalysis (mFPCA)
Kevin CumminsJoint Doctoral Program in Interdisciplinary Research in Substance Use
Division of Global Public HealthUniversity of California, San Diego
Department of Social WorkSan Diego State University
Wes ThompsonDepartment 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 trajectoriesaround mean levels.
mFPCA is a technique to simultaneously estimationtrajectories of multiple processes.
Added Goals:
Evaluate the association in the modes of variation among theprocesses.
Kevin Cummins JSM 2015v1
Introduction The Model Application
Functional PCA model
The response of individual i at time t is multivariate andmodeled as
Yi(tij) = fi(tij) + εij
= µ(tij) + hi(tij) + εij
= φT (tij)θµ + φT (tij)Θαi + εij
φ(t) = (φ1(t), φ2(t), . . . , φK(t)): K-dimensional vector of orthogonalbasis 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 andmodeled as
Yi(t) = ΦT (t)θµ + ΦT (t)Θαi + εi(t)
Yi(t): P -dimensional observed response at time t
φ(t) = (φ1(t), φ2(t), . . . , φK(t))T : K-dimensional vector of orthogonalbasis functions evaluated at time tij and
ΦT (t) =
φT (t) . . . 0T
.... . .
...0T . . . φT (t)
Θp: K by Qp matrix of spline coefficients subject to ΘTp 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