Kevin Cummins Joint Doctoral Program in Interdisciplinary Research on Substance Use University of California, San Diego JSM Digital Poster Presentation, August 2015

1. Background
Functional principal component analysis (FPCA)
is a technique for estimating individual smooth
trajectories from sparse longitudinal data
(James, Hastie & Sugar 2000).
FPCA provides:
• Smooth mean trajectory,
• Smooth principal modes of variation of
trajectories around mean levels.
The current methodological work is designed
to extend FPCA to a multivariate application.
Multivariate FPCA additionally provides:
• A measure of the developmental
connectedness of various domains,
• A means to evaluate the association of
the modes of variation with outcomes or
exposures.
Application
The prospective longitudinal study was
developed to document adolescent
neurodevelopment and how developmental
variation is associated with substance use.
One of five investigated pairs of brain regions
predicted to be develop in syncrony is reported
here (Walhovd et al. 2014).
We describe patterns of development in those
two regions:
• Evaluate the association of the modes of
variation for the two domains,
• Evaluate the association between the
modes of variation and clinical outcomes.
Methods
Participants:
• Healthy 12-14 year-olds (54% female) at
recruitment (N=295),
• All non-substance users at intake.
Sampling Design:
• Participants followed for 12 years,
• Users and a matched control scanned
at annual visits after substance use
experience,
• 272 (92%) participants observed at ≥ 2
times, 75 (25%) at ≥ 3 times.
Substance Use Measures:
• Substance use histories are assessed with
the Timeline Follow-back.
mFPCA Model; panel 1 of 4
A Bayesian Model for Multivariate
Functional Principal Components Analysis
Outline Functional Principal Components Analysis (FPCA) Multivariate Functional Principal Component Analysis (mFPCA)
Multivariate Sparse Longitudinal Data
Outcomes on ith subject at time point tij
Yi(tij) = fi(tij) + ✏ij
i = 1, . . . , N; (subjects)
j = 1, . . . , Mi; (number of time points)
Yi(tij): P-dimensional observed fMRI response at time tij
fi(tij): P-dimensional smooth response at time tij
✏ij(t) ⇠ iid NP (0, diag(σ2
✏,p))
Kevin Cummins
Division of Global Public Health
University of California, San Diego
Wesley Thompson
Department of Psychiatry
University of California, San Diego
References
James, GM, Hastie T & Sugar
C. (2000). Principal component
models for sparse functional data.
Biometrika 87(3) 587-602.
Walhovd, KB, Tamnes, CK,
Bjørnerud, A, Due-Tønnessen P,
Holland D, Dale AM, & Fjell AM
(2014). Maturation of Cortico-
Subcortical Structural Networks—
Segregation and Overlap of Medial
Temporal and Fronto-Striatal
Systems in Development. Cerebral
Cortex, 25(7), 1835-1841.
Results
Dorsolateral prefrontal cortex (DLPFC) and parietal lobe (PL)
thickness decreased slightly between ages 12 and 21 (Fig.
1). The first functional PC for both brain regions generally
−2−10123
12 15 18 21 24 27
Age (yrs)
DLPFCThickness(µm)
Age (yrs)
−3−2−10123
ParietalThickness(µm)
12 15 18 21 24 27
−2−10123
12 15 18 21 24 27
Age (yrs)
DLPFCThickness(µm)
Age (yrs)
−3−2−10123
ParietalThickness(µm)
12 15 18 21 24 27
+1 SD
-1 SD
Mean
represented a parallel shift
(Fig. 2). These modes of
variation in the two regions
are highly correlated (r=0.79,
95% credibility interval=0.69,
0.87). The proportion of years
that participants engage in
binge drinking is associated
with the first PC (DLPFC: b=
-0.53, SEb=0.16; PL: b= -0.46,
SEb=0.14).
Conclusion
Development trajectories
of the parietal lobe and
dorsolateral prefrontal cortex
modes of variation are linked.
Their modes of variations
are relatively simple; mFPCA
could have detected complex
modes of variation. Binge
drinking is associated with
more depressed trajectories
of DLPFC and PL thinkness.
mFPCA is capable of
identifying and reducing
associations among complex
modes of variation from
multiple domains.
Figure 2. Modes of variation for DLPFC and PL. Mean trajectory ±1 st. dev. for the first PC.
Figure 1. Observed trajectories of DLPFC and PL thickness.

2. Background
Functional principal component analysis (FPCA)
is a technique for estimating individual smooth
trajectories from sparse longitudinal data
(James, Hastie & Sugar 2000).
FPCA provides:
• Smooth mean trajectory,
• Smooth principal modes of variation of
trajectories around mean levels.
The current methodological work is designed
to extend FPCA to a multivariate application.
Multivariate FPCA additionally provides:
• A measure of the developmental
connectedness of various domains,
• A means to evaluate the association of
the modes of variation with outcomes or
exposures.
Application
The prospective longitudinal study was
developed to document adolescent
neurodevelopment and how developmental
variation is associated with substance use.
One of five investigated pairs of brain regions
predicted to be develop in syncrony is reported
here (Walhovd et al. 2014).
We describe patterns of development in those
two regions:
• Evaluate the association of the modes of
variation for the two domains,
• Evaluate the association between the
modes of variation and clinical outcomes.
Methods
Participants:
• Healthy 12-14 year-olds (54% female) at
recruitment (N=295),
• All non-substance users at intake.
Sampling Design:
• Participants followed for 12 years,
• Users and a matched control scanned
at annual visits after substance use
experience,
• 272 (92%) participants observed at ≥ 2
times, 75 (25%) at ≥ 3 times.
Substance Use Measures:
• Substance use histories are assessed with
the Timeline Follow-back.
mFPCA Model; panel 2 of 4
A Bayesian Model for Multivariate
Functional Principal Components Analysis
Outline Functional Principal Components Analysis (FPCA) Multivariate Functional Principal Component Analysis (mFPCA)
mFPCA
The smooth responses are modeled as
fi(tij) = µ(tij) + hi(tij)
= ΦT
(tij)✓µ + ΦT
(tij)⇥↵i
µi(t) = ΦT
(t)✓µ: overall mean functions at time t
hi(t) = ΦT
(t)⇥↵i: the smoothed deviation of the ith subject from the
mean curves.
FPC functions ΦT
(t)⇥ characterize the major modes of variation in all P
outcome variables.
↵i ⇠ N(0, ⌃↵).
Kevin Cummins
Division of Global Public Health
University of California, San Diego
Wesley Thompson
Department of Psychiatry
University of California, San Diego
References
James, GM, Hastie T & Sugar
C. (2000). Principal component
models for sparse functional data.
Biometrika 87(3) 587-602.
Walhovd, KB, Tamnes, CK,
Bjørnerud, A, Due-Tønnessen P,
Holland D, Dale AM, & Fjell AM
(2014). Maturation of Cortico-
Subcortical Structural Networks—
Segregation and Overlap of Medial
Temporal and Fronto-Striatal
Systems in Development. Cerebral
Cortex, 25(7), 1835-1841.
Results
Dorsolateral prefrontal cortex (DLPFC) and parietal lobe (PL)
thickness decreased slightly between ages 12 and 21 (Fig.
1). The first functional PC for both brain regions generally
−2−10123
12 15 18 21 24 27
Age (yrs)
DLPFCThickness(µm)
Age (yrs)
−3−2−10123
ParietalThickness(µm)
12 15 18 21 24 27
−2−10123
12 15 18 21 24 27
Age (yrs)
DLPFCThickness(µm)
Age (yrs)
−3−2−10123
ParietalThickness(µm)
12 15 18 21 24 27
+1 SD
-1 SD
Mean
represented a parallel shift
(Fig. 2). These modes of
variation in the two regions
are highly correlated (r=0.79,
95% credibility interval=0.69,
0.87). The proportion of years
that participants engage in
binge drinking is associated
with the first PC (DLPFC: b=
-0.53, SEb=0.16; PL: b= -0.46,
SEb=0.14).
Conclusion
Development trajectories
of the parietal lobe and
dorsolateral prefrontal cortex
modes of variation are linked.
Their modes of variations
are relatively simple; mFPCA
could have detected complex
modes of variation. Binge
drinking is associated with
more depressed trajectories
of DLPFC and PL thinkness.
mFPCA is capable of
identifying and reducing
associations among complex
modes of variation from
multiple domains.
Figure 2. Modes of variation for DLPFC and PL. Mean trajectory ±1 st. dev. for the first PC.
Figure 1. Observed trajectories of DLPFC and PL thickness.

3. Background
Functional principal component analysis (FPCA)
is a technique for estimating individual smooth
trajectories from sparse longitudinal data
(James, Hastie & Sugar 2000).
FPCA provides:
• Smooth mean trajectory,
• Smooth principal modes of variation of
trajectories around mean levels.
The current methodological work is designed
to extend FPCA to a multivariate application.
Multivariate FPCA additionally provides:
• A measure of the developmental
connectedness of various domains,
• A means to evaluate the association of
the modes of variation with outcomes or
exposures.
Application
The prospective longitudinal study was
developed to document adolescent
neurodevelopment and how developmental
variation is associated with substance use.
One of five investigated pairs of brain regions
predicted to be develop in syncrony is reported
here (Walhovd et al. 2014).
We describe patterns of development in those
two regions:
• Evaluate the association of the modes of
variation for the two domains,
• Evaluate the association between the
modes of variation and clinical outcomes.
Methods
Participants:
• Healthy 12-14 year-olds (54% female) at
recruitment (N=295),
• All non-substance users at intake.
Sampling Design:
• Participants followed for 12 years,
• Users and a matched control scanned
at annual visits after substance use
experience,
• 272 (92%) participants observed at ≥ 2
times, 75 (25%) at ≥ 3 times.
Substance Use Measures:
• Substance use histories are assessed with
the Timeline Follow-back.
mFPCA Model; panel 3 of 4
A Bayesian Model for Multivariate
Functional Principal Components Analysis
Outline Functional Principal Components Analysis (FPCA) Multivariate Functional Principal Component Analysis (mFPCA)
mFPCA
The response of individual i at time t is multivariate and
modeled as
Yi(t) = ΦT
(t)✓µ + ΦT
(t)⇥↵i + ✏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) =
2
6
6
4
φT
(t) . . . 0T
.
..
...
.
..
0T . . . φT
(t)
3
7
7
5
⇥p: K by Qp matrix of spline coefficients subject to ⇥T
p ⇥p = I and
⇥ =
2
6
4
⇥1 . . . 0
...
...
...
0T . . . ⇥P
3
7
5
Kevin Cummins
Division of Global Public Health
University of California, San Diego
Wesley Thompson
Department of Psychiatry
University of California, San Diego
References
James, GM, Hastie T & Sugar
C. (2000). Principal component
models for sparse functional data.
Biometrika 87(3) 587-602.
Walhovd, KB, Tamnes, CK,
Bjørnerud, A, Due-Tønnessen P,
Holland D, Dale AM, & Fjell AM
(2014). Maturation of Cortico-
Subcortical Structural Networks—
Segregation and Overlap of Medial
Temporal and Fronto-Striatal
Systems in Development. Cerebral
Cortex, 25(7), 1835-1841.
Results
Dorsolateral prefrontal cortex (DLPFC) and parietal lobe (PL)
thickness decreased slightly between ages 12 and 21 (Fig.
1). The first functional PC for both brain regions generally
−2−10123
12 15 18 21 24 27
Age (yrs)
DLPFCThickness(µm)
Age (yrs)
−3−2−10123
ParietalThickness(µm)
12 15 18 21 24 27
−2−10123
12 15 18 21 24 27
Age (yrs)
DLPFCThickness(µm)
Age (yrs)
−3−2−10123
ParietalThickness(µm)
12 15 18 21 24 27
+1 SD
-1 SD
Mean
represented a parallel shift
(Fig. 2). These modes of
variation in the two regions
are highly correlated (r=0.79,
95% credibility interval=0.69,
0.87). The proportion of years
that participants engage in
binge drinking is associated
with the first PC (DLPFC: b=
-0.53, SEb=0.16; PL: b= -0.46,
SEb=0.14).
Conclusion
Development trajectories
of the parietal lobe and
dorsolateral prefrontal cortex
modes of variation are linked.
Their modes of variations
are relatively simple; mFPCA
could have detected complex
modes of variation. Binge
drinking is associated with
more depressed trajectories
of DLPFC and PL thinkness.
mFPCA is capable of
identifying and reducing
associations among complex
modes of variation from
multiple domains.
Figure 2. Modes of variation for DLPFC and PL. Mean trajectory ±1 st. dev. for the first PC.
Figure 1. Observed trajectories of DLPFC and PL thickness.

4. Background
Functional principal component analysis (FPCA)
is a technique for estimating individual smooth
trajectories from sparse longitudinal data
(James, Hastie & Sugar 2000).
FPCA provides:
• Smooth mean trajectory,
• Smooth principal modes of variation of
trajectories around mean levels.
The current methodological work is designed
to extend FPCA to a multivariate application.
Multivariate FPCA additionally provides:
• A measure of the developmental
connectedness of various domains,
• A means to evaluate the association of
the modes of variation with outcomes or
exposures.
Application
The prospective longitudinal study was
developed to document adolescent
neurodevelopment and how developmental
variation is associated with substance use.
One of five investigated pairs of brain regions
predicted to be develop in syncrony is reported
here (Walhovd et al. 2014).
We describe patterns of development in those
two regions:
• Evaluate the association of the modes of
variation for the two domains,
• Evaluate the association between the
modes of variation and clinical outcomes.
Methods
Participants:
• Healthy 12-14 year-olds (54% female) at
recruitment (N=295),
• All non-substance users at intake.
Sampling Design:
• Participants followed for 12 years,
• Users and a matched control scanned
at annual visits after substance use
experience,
• 272 (92%) participants observed at ≥ 2
times, 75 (25%) at ≥ 3 times.
Substance Use Measures:
• Substance use histories are assessed with
the Timeline Follow-back.
mFPCA Model; panel 4 of 4
A Bayesian Model for Multivariate
Functional Principal Components Analysis
Outline Functional Principal Components Analysis (FPCA) Multivariate Functional Principal Component Analysis (mFPCA)
mFPCA
The strength of association among variables is modeled on the
smoothed response level via correlations between PC scores:
↵i ⇠ N(0, ⌃↵) and ⌃↵ is restricted to the form
0
B
B
B
@
⌃1 C12 · · · C1P
CT
12 ⌃2 · · · C2P
.
..
.
..
...
.
..
CT
1P CT
2P · · · ⌃P
1
C
C
C
A
⌃1, ⌃2, . . . , ⌃P are diagonal matrices
⌃↵ can also be written as ⌃↵ = D↵R↵D↵ where D↵ is diagonal and R↵
is restricted to the form as
0
B
B
B
@
IQ1
R12 · · · R1P
RT
12 IQ2
· · · R2P
...
...
...
...
RT
1P RT
2P · · · IQP
1
C
C
C
A
Kevin Cummins
Division of Global Public Health
University of California, San Diego
Wesley Thompson
Department of Psychiatry
University of California, San Diego
References
James, GM, Hastie T & Sugar
C. (2000). Principal component
models for sparse functional data.
Biometrika 87(3) 587-602.
Walhovd, KB, Tamnes, CK,
Bjørnerud, A, Due-Tønnessen P,
Holland D, Dale AM, & Fjell AM
(2014). Maturation of Cortico-
Subcortical Structural Networks—
Segregation and Overlap of Medial
Temporal and Fronto-Striatal
Systems in Development. Cerebral
Cortex, 25(7), 1835-1841.
Results
Dorsolateral prefrontal cortex (DLPFC) and parietal lobe (PL)
thickness decreased slightly between ages 12 and 21 (Fig.
1). The first functional PC for both brain regions generally
−2−10123
12 15 18 21 24 27
Age (yrs)
DLPFCThickness(µm)
Age (yrs)
−3−2−10123
ParietalThickness(µm)
12 15 18 21 24 27
−2−10123
12 15 18 21 24 27
Age (yrs)
DLPFCThickness(µm)
Age (yrs)
−3−2−10123
ParietalThickness(µm)
12 15 18 21 24 27
+1 SD
-1 SD
Mean
represented a parallel shift
(Fig. 2). These modes of
variation in the two regions
are highly correlated (r=0.79,
95% credibility interval=0.69,
0.87). The proportion of years
that participants engage in
binge drinking is associated
with the first PC (DLPFC: b=
-0.53, SEb=0.16; PL: b= -0.46,
SEb=0.14).
Conclusion
Development trajectories
of the parietal lobe and
dorsolateral prefrontal cortex
modes of variation are linked.
Their modes of variations
are relatively simple; mFPCA
could have detected complex
modes of variation. Binge
drinking is associated with
more depressed trajectories
of DLPFC and PL thinkness.
mFPCA is capable of
identifying and reducing
associations among complex
modes of variation from
multiple domains.
Figure 2. Modes of variation for DLPFC and PL. Mean trajectory ±1 st. dev. for the first PC.
Figure 1. Observed trajectories of DLPFC and PL thickness.