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