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MODELING MOTOR LEARNING
USING HETEROSKEDASTIC
FUNCTIONAL PRINCIPAL
COMPONENTS ANALYSIS
JEFF GOLDSMITH, PHD
DEPARTMENT OF BIOSTATISTICS
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Acknowledgements
• Daniel Backenroth, Michelle D. Harran, Juan C. Cortes, John W.
Krakauer, Tomoko Kitago
• Funded by R21EB018917, R01NS097423, and R01HL123407
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Interest is in motor learning
• People get better at motor tasks through practice
• Mean change can be minimal for simple tasks
• Variance change can be relatively large
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Experiment set-up
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Kinematic data
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−20
−10
0
10
20
−20 −10 0 10 20
X
Y
−20
−10
0
10
20
0 10 20
PX(t)
20 20
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Dataset
• 26 right-handed subjects
• 24 motions to each target with each hand
• Target direction was semi-randomized
• Data are [PX
ij
(t), PY
ij
(t)]
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Kinematic data
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−20
−10
0
10
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−20 −10 0 10 20
X
Y
−20
−10
0
10
20
0 10 20 30 40 50
t
PX(t)
−20
−10
0
10
20
0 10 20 30 40 50
t
PY(t)
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−20
−10
0
10
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−20 −10 0 10 20
X
Y
−20
−10
0
10
20
0 10 20 30 40 50
t
PX(t)
−20
−10
0
10
20
0 10 20 30 40 50
t
PY(t)
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Traditional FPCA
Pij(t) = µij(t) + ij(t)
= µij(t) +
K
X
k=1
⇠ijk k(t) + ✏ij(t)
Scores are uncorrelated random variables with and
.
⇠ijk E(⇠ijk) = 0
var(⇠ijk) = k
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Problem
• Distribution of the scores determines the variability curves
• Motion variance can depend on baseline motor control or training
⇠ijk
Constant variance is wrong for our data.
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Solution
• Call the conditional score variance
• Score variance model structure can change depending on FPC k
Let score variance depend on subject and covariates.
var(
⇠ijk
|wijk, zijk, gik) = exp k0 +
q
X
m=1
kmwijkm +
r
X
h=1
gikhzijkh
!
k|wijk,zijk,gik
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Implications I
Conditional covariance
cov[ ij(
s
)
, ij(
t
)
|wijk, zijk, gik] =
K
X
k=1
k|wijk,zijk,gik
k(
s
) k(
t
)
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Implications II
Marginal covariance
cov[ ij(
s
)
, ij(
t
)] =
K
X
k=1
E
⇥
k|wijk,zijk,gik
⇤
k(
s
) k(
t
)
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Assumptions I
• Scores still need to have mean zero
• We model the mean using function-on-scalar regression:
µij(
t
) = 0(
t
) +
p
X
l=1
xijl l(
t
) +
bi(
t
)
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Assumptions II
• FPC basis functions don't depend on subject or covariates
• Maintains a nice interpretation
• Precludes smoothly-varying FPCs but not different expansions across
groups
−0.3
−0.2
−0.1
0.0
0.1
0.2
0 10 20 30 40 50
t
X
−0.3
−0.2
−0.1
0.0
0.1
0.2
0 10 20 30 40 50
t
Y
PC1
PC2
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Alternatives
• Build a model for the scores directly (Chiou et al 2003)
• Implies some covariate dependence in the variance, but mostly
focuses on the mean
• Allow complete covariance to be covariate dependent (Jiang and
Wang 2010)
• Non-constant FPCs are harder to interpret
• Difficult to posit complex models
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Estimation
Several possible approaches:
• Smooth covariance decomposition + score estimation + score
variance model
• Probabilistic / Bayesian FPCA
• Variational Bayes
• MCMC
Advantages and disadvantages for each
We tried each and compared in simulations
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Bayesian model
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Rotations
• Model specification didn't constrain orthogonality
• Needed for scores to be interpretable across iterations
• Quick work-around: SVD step in the sampler / VB algorithm
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Cross sectional simulation
design I
• and are and
• and are and
•
•
•
pi(t) = 0 +
4
X
k=1
⇠ik k(t) + ✏i(t)
t 2 [0, 2⇡]
I 2 {20, 40, 80, 160, 320}
2 = 0.25
1(t) 2(t)
3(t) 4(t)
sin(t)
cos(
t
)
sin(2t)
cos(2
t
)
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Cross sectional simulation
design II
Score Variances FPC 1 FPC 2 FPC 3 FPC 4
Group 1 36 12 6 4
Group 1I 18 24 12 6
pi(t) = 0 +
4
X
k=1
⇠ik k(t) + ✏i(t)
Single binary covariate:
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Simulation results I
FPC1 FPC2 FPC3 FPC4
−0.2
−0.1
0.0
0.1
0.2
0 10 20 30 40 50 0 10 20 30 40 50 0 10 20 30 40 50 0 10 20 30 40 50
t
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FPC1 FPC2 FPC3 FPC4
0.0
0.5
1.0
1.5
2.0
20 40 80 160 320 20 40 80 160 320 20 40 80 160 320 20 40 80 160 320
Number of curves
ISE
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FPC1
Group 1
FPC1
Group 2
FPC2
Group 1
FPC2
Group 2
FPC3
Group 1
FPC3
Group 2
FPC4
Group 1
FPC4
Group 2
−2
−1
0
20 40 80 160320 20 40 80 160320 20 40 80 160320 20 40 80 160320 20 40 80 160320 20 40 80 160320 20 40 80 160320 20 40 80 160320
Number of curves
Relative error
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Simulation results II
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FPC1
Group 1
FPC1
Group 2
FPC2
Group 1
FPC2
Group 2
FPC3
Group 1
FPC3
Group 2
FPC4
Group 1
FPC4
Group 2
−0.50
−0.25
0.00
0.25
VB SC Gibbs VB SC Gibbs VB SC Gibbs VB SC Gibbs VB SC Gibbs VB SC Gibbs VB SC Gibbs VB SC Gibbs
Estimation type
Relative error
Differences show up in computation time. For 80 curves:
• SC takes about 1 second
• VB takes about 20 seconds
• MCMC takes about 10 minutes
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Simulation results III
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Other simulations
• Vary ME variance
• Vary dimension of spline basis
• Vary truncation lag
• Examine multilevel structures
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Analysis of kinematic data
Covariate of interest is repetition number
• Allow separate effects for each target
• Include subject-specific random intercepts and slopes for each target
Prior to analysis, curves are rotated to lie on the positive X axis
• Variation in X direction relates to extent
• Variation in Y direction relates to direction
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Analysis of kinematic data
• Looked for training effects in the mean as well
• Intercepts and slopes are allowed to be correlated within FPCs and
targets
• Final model included 2 FPCs
pij =
8
X
l=1
I(xij0 = l) (⇥ l + ⇥bil) +
K
X
k=1
⇠ijk⇥ k
⇠ijk
⇠
N
"
0,
2
⇠ijk = exp
8
X
l=1
I(xij0 = l) ( lk0 + gilk0 + xij1( lk1 + gilk1))
!#
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Analysis results I
X
0°
X
45°
X
90°
X
135°
X
180°
X
225°
X
270°
X
315°
Y
0°
Y
45°
Y
90°
Y
135°
Y
180°
Y
225°
Y
270°
Y
315°
0
10
20
30
40
50
0
10
20
30
40
50
1 6 12 18 24 1 6 12 18 24 1 6 12 18 24 1 6 12 18 24 1 6 12 18 24 1 6 12 18 24 1 6 12 18 24 1 6 12 18 24
Motion number
Estimated score variance
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Analysis results II
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X Y
−0.06
−0.04
−0.02
0.00
0.02
0° 45° 90° 135° 180° 225° 270° 315° 0° 45° 90° 135° 180° 225° 270° 315°
Target
Estimated slope parameter
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VB
Gibbs
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Analysis results III
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Left Hand Right Hand
−0.5
0.0
0.5
1.0
1.5
Subject
Random intercept
Target
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0°
45°
90°
135°
180°
225°
270°
315°
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Next analyses I
More complexity in healthy controls
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Next analyses I
Stroke recovery
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Discussion
• Useful extension of FPCA framework
• Some more work to do
• Applications to other datasets are ongoing
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Thanks!
• [email protected]
• jeffgoldsmith.com
• github.com/jeff-goldsmith/