M7777 Applied Functional Data Analysis
11. Registration
Jan Koláček (kolacek@math.muni.cz)
Dept. of Mathematics and Statistics, Faculty of Science, Masaryk University, Brno
Jan Koláček (SCI MUNI)
M7777 Applied FDA
Fall 2019 1/17
Registration
Berkeley Growth Data
• Heights of 54 girls taken from ages 0 through 18.
• Growth process easier to visualize in terms of acceleration (2nd derivative).
• Peaks in acceleration = start of growth spurts.
<
^ -2
5 10 Age [years]	15 5 Sample of 10 girls.	10 Age [years]	15	
Jan Koláček (SCI MUNI)	M7777 Applied FDA		Fall 2019	2/17
Registration
The Registration Problem
• Most analyzes only account for variation in amplitude.
• Frequently, observed data exhibit features that vary in time - phase variation.
Phase variation										
										
										
										
				Av \ \ \						
		S At	V /	* \ \ \ \ \						
					V  \ \ \\ \ \	/ /*				
						/ J				
										
										
										
Amplitude variation								
								
								
								
								
								
								
								
								
								
								
-5.0
-2.5
0.0
2.5
5.0 -5.0
-2.5
0.0
2.5
5.0
• Mean of unregistered curves (dashed) has smaller peaks than any individual curve.
• Aligning the curves reduces variation.
Jan Koláček (SCI MUNI)	M7777 Applied FDA		Fall 2019       3 / 17
Registration
Berkeley Growth Data
Observed
Sample of 10 girls
Jan Koláček (SCI MUNI)
M7777 Applied FDA
Registration
Defining a Warping Function
Requires a transformation of time.
• time-warping function is a continuous function h,(t) defined on [0, T], which is strictly increasing and /7/(0) = 0, h,(T) — T
• aligning function /7/~1(t) is the functional inverse of h;(t), i.e.
hr\h,{t)) = t
• the registered curves
X?(t)=Xi(h7\t))
Jan Koláček (SCI MUNI)
M7777 Applied FDA
Fall 2019       5 / 17
Registration
Landmark Registration
• For each curve x,-(t) we choose points t/i,..., t,x
• We define a reference (usually one of the curves or mean etc.)
*di? • • • ? *bK
• We define constraints
• Finally we find a constrained smooth function. Berkeley Growth example
• Just one reference point to = H-7 .. . average time of maximal pubertal growth spurt (acceleration crosses 0)
• t\ ... maximal pubertal growth spurt for /-th curve
• Thus the constraints
•_-i ■_-i
/    J.5... 5 A75       J.5...
/»/(11.7) = t,-
/' = 1,...,54.
Jan Koláček (SCI MUNI)
M7777 Applied FDA
Fall 2019       6 / 17
Registration
	girl 3									girl 7							
								Result — 0 .... A									
										aserved gned							
0-					íi \ \\ i ■ i 1 i									- Vif -			
					1 '• 1      1 • \      1 *i \  1 V										\    \ / /		
					\   1 /' \   1 / 1												
					1 1 1 1 1												
					1												
			1		0 1		5					1		0 1		5	
Age [years]
15
5
girl 3								girl 7						
						/								
					y y/ y /								>y y y	
				y y y								y y		
			■ y /								y			
		y / y								y y y				
	y / /								/ y ' y y					
/								1    7 y / * f y						
								/y /y /-						
5                             10 1					5                                                      5 1						0 1		5	
Age [years]
Aligned data with warping functions.
Jan Koláček (SCI MUNI)	M7777 Applied FDA		Fall 2019       7 / 17
Registration
Age [years] Age [years]
Aligned data for 10 girls with their warping functions.
Jan Koláček (SCI MUNI)	M7777 Applied FDA		Fall 2019       8 / 17
Registration
Identifying Landmarks
Major landmarks of interest:
• where curve x,-(t) crosses some value
• location of peaks or valleys
• location of inflections
Almost all are points at which some derivative of x,-(t) crosses zero.
In practise, zero-crossings can be found automatically, but usually still require manual checking.
For landmark registration in R, the procedure landmarkreg is used.
Jan Koláček (SCI MUNI)
M7777 Applied FDA
Fall 2019       9 / 17
Registration
Continuous Registration
• Let xo(t) be a reference curve (dashed)
Phase variation									
									
									
									
			' '/ /	\ \ \					
			// /	l   \M  \ \ A   \ \\   \ \					-
				\  \  V \ \ \   \   \\ \ V \ \ \N\/		s  f  r / / /' / /			
						t i*			
									
									
									
Amplitude variation									
									
									
									
									
									-
									
									
									
									
									
-5.0 -2.5 0.0 2.5 5.0    -5.0 -2.5 0.0 2.5 5.0
• Phase variation: curves are less correlated with xq(£),
FPCA     the first 3 PC's explain 55%, 39% and 5% of the variation
• Amplitude variation: curves are high correlated with xq(£), FPCA     just the first PC explains 100% of the variation
Main idea: Find h(t) to maximize correlation with xo(t)
Jan Koláček (SCI MUNI)	M7777 Applied FDA		Fall 2019       10 / 17
Registration
Practical Using O Do the preprocessing of data by landmark registration © Do the continuous registration (we can repeat).
as
as
2
<D O O <
£ -
O) <D X
0--
Age [years]
For continuous registration in R, the procedure registered is used.
Jan Koláček (SCI MUNI) EE
Fall 2019       11 / 17
M7777 Applied FDA
Registration
Assessing the process of Registration
n T
• The total MSE: MSEtota, = ai"1 £ / [x/(t) - x(t)]2 eft
;=i o
• The amplitude MSE: MSEamp = Cn~x £ / [x*(t) - x*(t)]2 c/t
;=i o
T T
• The phase MSE: MSEphase = C Jx*2(t)c/t + /x2(t)dt,
o o
where x,-(t) ... unregistered curves, x*(t) .. . registered version It can be shown
MSEtota! = MSEamp + MSEphase Thus we can define squared multiple correlation index
" MSEtota!
Berkeley Growth Data:   unregistered x landmark: R2 = 0.7
landmark x continuous: R2 = —0.06
Jan Koláček (SCI MUNI)	M7777 Applied FDA		Fall 2019       12 / 17
Problems to solve
O Simulation
• Conduct 3 steps of continuous registration on simulated data from example (use the attached script). Do it for the case of Phase variation (see Figure 1). Does it have any sense to do it for the case of Amplitude variation?
© Pinch-Force Data
• Load the variable pinchraw from the f da package (see Lecture 6 for data details).
• Smooth data by B-spline basis of order 5 (with GCV-optimal A) and plot the first derivatives of curves (see Figure 2).
• Compute a mid point for landmark registration (see the location of the maximum of each curve) and align curves. Plot unregistered curves together with aligned curves (see Figure 3).
• Conduct the continuous registration on aligned curves and compare results (see Figure 4).
Jan Koláček (SCI MUNI)	M7777 Applied FDA		Fall 2019       13 / 17
Registration
Simulation
Figure 1.
Jan Koláček (SCI MUNI)
M7777 Applied FDA
Fall 2019       14 / 17
Registration
Jan Koláček (SCI MUNI)
Figure 2.
M7777 Applied FDA
Fall 2019       15 / 17
Registration
Pinch-Force Data
12.54
10.0 4
CD Ü
0.3 0.0
Seconds
Figure 3.
Jan Koláček (SCI MUNI)
M7777 Applied FDA
Registration
Pinch-Force Data
12.54
10.0 4
CD Ü
0.0 0.1	0.2	0.3 0.0	0.1	0.2 0.3
		Seconds		
		Figure 4.		
Jan Koláček (SCI MUNI)		M7777 Applied FDA		Fall 2019       17 / 17