Curve Registration
The rigid metric of physical time may not be
directly relevant to the internal dynamics of many
real-life systems.
Rather, there can be a sort of biological or
meteorological time scale that can be nonlinearly
related to physical time…
•
•
•
•
Amplitude Variation vs Phase Variation
Shift Registration
Landmark Registration
More General Registration Techniques
Shift Registration
• The goal is to estimate a δ for each curve such that the criteria
(REGSSE) is minimized
@
HLH
L
D
â
à
n
xi
s + di
-
`
m
s
2
âs
âà@
H
LHL
D
i= 1
n
i= 1
T
T2+di
T1+di
xi
s
-
`
m
s - di
2
â s
• The Procrustes Method involves estimating a delta for each curve.
Then, one reestimates with the newly registered curves the mean
curve and repeats the process.
Numerical method for a simple shift
•
•
•
•
How do we estimate each delta?
Newton-Raphson
1. Choose an initial delta, perhaps using visual landmarks.
2. Modify delta by
di
H
LH
L
n
= di
n- 1
-
a
¶ d i REGSSE
¶ d i 2 REGSSE
• 3. Update the mean and continue Procrustes Method.
• In the text, R & S use this method to improve visual alignment of the
pinch force data.
Landmark Registration
• Mark important structural events (characteristic points).
• Sometimes this is done visually. Other times, events in the derivative
(minima and maxima) may be useful.
• How do we use these events to align the curve?
• Time Warping Function—a strictly increasing transformation of time.
• h(t) where h(0)=0 and h(T)=T for our time in [0,T]
• Must use a reference function (often the mean) and h(t0f)=tif for each
feature f.
• Gasser et al use average times to do this. So, there is no target
function.
• Our registered sample functions will be x(h(t))=x*(t).
• These together will yield the structural average.
Interpolation
•
•
•
•
In R&S, the example uses linear
extrapolation to fit h(t) through
each of the adjusted values for the
handwriting data.
This is data from Ramsay’s “ A
guide to curve registration.” Here a
smooth interpolation is used.
Another example is Gasser et all
using piecewise cubic polynomials
to piece together human growth
curve data.
One problem with curve
registration is the regularity of
landmark features across the
sample. Sometimes there is
ambiguity in maxima and minima
and visual or intuitive methods
must be used.
General Method
• The problem is obtaining a strictly increasing function that
aligns h(0)=0 and h(T)=T. We restrict h to be in this
family and to be twice differentiable.
• Thus,
2
D h = w Dh
H
LA I M
E
H
L
• Which has the solution,
h t = C
D- 1 exp D- 1 w
t
Modeling warping using SDE’s
8H
L
<H
L8H
LH
L
<
• Ramsey and Wang mention that an alternative model would be
d Dh t
= Dh t
w t dt + dz t
• Where everything is as before except z is a stochastic process
• If we take B to be Brownian motion (w.o. drift) there is a solution
H
L A ikH
LH
Ly
E
{
h t = C0 + C1
-1
D
-1
exp D
w t + B t -
s
2
2
t
• This may be “… envisaged as a clock that is running fast or slow from
instant to instant, constantly undergoing a percentage change in rate in
a memoryless chaotic manner.”
Finding w(t)
HÈLàPH
LHH
L
L
T àH
L
• Now, we look for a w(t) that will minimize
Fl
x, y
h =
y t - x h t
2
ât + l
w t
2
ât
• Lambda is a penalty on the second derivative of h(t), which ensures a
smooth h.
• Ramsay and Li recommend an order 1 B-spline to approximate w.
H
Lâ H
L
K
w x =
ck Bk x
k=0
• This allows for a closed form solution of h(t)—which, while not
absolutely necessary, is desirable.
Height Acceleration
This was done with lambda= 0.01 and breakvalues 4,7,10,12,14,16,18.
An Extension
• Alternatively, you could minimize
HÈLâH
LàIQH
L HH
L
L
U
MàH
L
m
Fl
x, y
h =
aj t
j=0
Dj y t - Dj x h t
2
j
ât + l
w t
2
ât
• This allows for differing weights over the intervals, across derivatives,
and over vectors if the function is vector-valued.
An alternate minimizing criteria
• What if the sample functions are multiples of the target function (the
mean function)?
• Minimize the log of the smallest eigenvalue of
H
L
H
L
@
H
L
D
ikÙH
Ù
Ù
y
L@
H
L
DÙ@
H
L
D{
x02 t â t
x0 t xi hi t
x0 t xi hi t
ât
xi 2 hi t
ât
ât
• If these really are multiples, the smallest eigenvalue should be zero.
Other Sources
• Silverman includes the warping function as part of his principal
component analysis. He uses a parametric model for the warping
function.
• Sakoe and Chiba did a much earlier version by minimizing a weighted
distance between curves (with appropriate restrictions on h) using
dynamic programming
• Kneip and Gasser (1992) went through a detailed analysis of the
statistical properties of using warping functions.