M7777 Applied Functional Data Analysis
5. From Data to Functions — Constrained
Functions
Jan Koláček (kolacek@math.muni.cz)
Dept. of Mathematics and Statistics, Faculty of Science, Masaryk University, Brno
n Koláček (SCI MUNI)
M7777 Applied FDA
Fall 2019
1/17
Constrained Functions
Constrained Functions
There are some situations in which we want to include known restrictions about x(t).
• x(t) is always positive
• x(t) is always increasing (or decreasing)
• x(t) is a density
Idea: Enforce these conditions by transforming x(t).
Jan Koláček (SCI MUNI)
M7777 Applied FDA
Fall 2019       2 / 17
Constrained Functions
Angular Acceleration for Handwrite Data
-0.04 -0.02 0.00 0.02 0 500 1000 1500 2000
Position [mm] Time [ms]
We know that angular acceleration
a2(t) = [D2x(t)}2 + [D2y(t)}2
must be positive.
Jan Koláček (SCI MUNI)	M7777 Applied FDA		Fall 2019       3 / 17
Constrained Functions
Positive Smoothing of Angular Acceleration for Handwrite Data
1.0e-10
CD
o 2.5e-11 <
0.0e+00
1000 1500 Time [ms]
Jan Koláček (SCI MUNI)
M7777 Applied FDA
Fall 2019       4 / 17
Constrained Functions
Positive Smoothing
• We want to ensure that x(t) > 0.
• Set W(t) = 0*(t)c
• Let us consider the transformation
x(t) = ewW
Now we need to minimize
N 2 r
PENSSEX(W) = J] (y; - e^^)  + A / [LW(t)]2dt,
i=l '
This does not have an explicit formula.
It is convex     there is only one minimum.
Requires numerical optimization, but this is generally fast.
Jan Koláček (SCI MUNI)
M7777 Applied FDA
Fall 2019       5 / 17
Constrained Functions
Monotone Smoothing
Growth of baby's tibia
Baby's tibia Baby's tibia
Growth process is increasing =4> the derivative should be positive!
Jan Koláček (SCI MUNI)	M7777 Applied FDA		Fall 2019       6 / 17
Constrained Functions
Monotone Smoothing
• We need x(t) always increasing, i.e. Dx(t) > 0
• Set again W(t) = 4>*(t)c.
• Let us consider the transformation
r
Dx(t) = ew^     x(t) = a + J ew&ds.
to
We want to minimize
PENSSEX{W) = £[y/-a- ^ e^c/s )   +XJ [LW(t)]2dt.
i=i
to
Still convex problem, numerics work fairly quickly
/_!/!/(£) = D2l/l/(t) suggests that any x(t) = a + e/3t is smooth
Jan Koláček (SCI MUNI)
M7777 Applied FDA
Fall 2019       7 / 17
Constrained Functions
Monotone Smoothing
Estimation with the constraint of monotonity
Baby's tibia Baby's tibia
Jan Koláček (SCI MUNI)
M7777 Applied FDA
Constrained Functions
Density Estimation
St. Johns
10.0-       •;
Days
Jan Koláček (SCI MUNI)
M7777 Applied FDA
Fall 2019       9 / 17
Constrained Functions
Density Estimation
• The function x(t) is a density =4> we need
x(t) > 0 and Jx(t)dt = 1.
Set again W(t) = 4>*(t)c.
Let us consider the transformation
x(t) =
W(t)
feeds'
But we observe only yi,..., y/v (correspond to ii,..., t/v) What would we to minimize?
Jan Koláček (SCI MUNI)
M7777 Applied FDA
Fall 2019       10 / 17
Constrained Functions
Penalized Likelihood
• Likelihood of W(t) is probability of seeing ti,
• We maximize the likelihood function
N N
L(W\tu...,tN) = [[x(ti) = e-i
i=i
• Easier to work with log-likelihood
N
., tisi if W is true.
— A/
/(l/l/|ti,..., t/v) = ^ l/l/(t/) - A/In / ew(s)cfe.
/=i '
Minimize the penalized negative log-likelihood
PENLOGUK^W) = -£ W(«)) + Wl„/.«'C)* + A/[UV(0P*.
/ = 1
Jan Koláček (SCI MUNI)	M7777 Applied FDA		Fall 2019       11 / 17
Constrained Functions
Thinking about Smoothness
• What is an appropriate measure of smoothness for densities?
x(t) = Cew^
Compare to Normal density
m =
-(t-M)2/2a2
V2
TTd
Then W(t) — t2 should be smooth ^> roughness penalty
LW(t) = D3W(t).
Jan Koláček (SCI MUNI)
M7777 Applied FDA
Fall 2019       12 / 17
Constrained Functions
Density Estimation for St. Johns Precipitation
St. Johns
0.2
0.1
0.0
								
	I							
	/							
								
								
								
								
2.5 5.0 7.5 10.0
Precipitation
Used: B-spline basis of order 6 with 29 knots, log A =
Jan Koláček (SCI MUNI)
M7777 Applied FDA
-2
Fall 2019       13 / 17
Problems to solve
O Absorbance Data
• Load the variable absorb from the absorb.RData file and plot it.
• Fit the data using a B-spline basis and a curvature penalty. Try some values of A, do not consider any constraint.
• Consider the monotonicity constraint and fit the data using the same basis. Try some values of A and observe how the "optimal" value changes with the monotonicity constraint. Plot both final fits (see Figure 1).
© Turany Precipitation Data
• Load the variable df .turany.monthly from the turany.RData file. The dataset contains monthly precipitation amounts in Brno-Turany in years 2016 - 2018.
• Fit the temperature density with Fourier bases and the third derivative rougness penalties at a number of values of A (see Figure 2 for
A = 100).
• Use the generic function density to get the density estimate, plot it (see Figure 3) and compare with the previous step result.
0 (optional) Program the CV procedure for monotone smoothing.
Jan Koláček (SCI MUNI)	M7777 Applied FDA		Fall 2019       14 / 17
Problems to solve
Classic
0.4-
o 0.3-o
-Q o
C/5 _Q <
0.2-
0.1
Monotone
Plate
B1 ■ C1 D1 E1
0 5 10 15 20 25   0 5 10 15 20 25
Time [hrs]
Figure 1.
Jan Koláček (SCI MUNI)
M7777 Applied FDA
Fall 2019       15 / 17
Problems to solve
Tu rany
0.03 4
0.02 4
C/5
Q
0.01 4
0.00 4-
25	50	75	100	
	Precipitation			
	Figure 2.			
Jan Koláček (SCI MUNI)	M7777 Applied FDA		Fall 2019	16 / 17
Problems to solve
Tu rany
0.020
0.015
.-ST 0.010
0.005
0.000
					
	/	\			
		\			
					
	1	\			
	7				
					
	1				
	• •	•           • • • • _  _  _ m» _______	• • •		
	•	-         - 7---. • •		•	•
					
0		50		100	
Precipitation
Figure 3.
Jan Koláček (SCI MUNI)
M7777 Applied FDA
Fall 2019       17 / 17