M7777 Applied Functional Data Analysis 8. Functional Data Simulation
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 1/19
Functional Data Simulation
1. Wiener process (limit of a Random Walk)
0 2000 4000 6000 8000 10000 0 2000 4000 6000 8000
Combinations with Wiener process
xi(tk) = m(tk) + Sk
Jan Koláček (SCI MUNI)	M7777 Applied FDA		Fall 2019       2 / 19
Functional Data Simulation
2. Regression model
• Simulate N x M measurements
Xj(tk) = m(tk) + eik,       iid eik ~ A/(0, a2),
m{t)... any regression function, / = 1,..., /V, k = 1,..., M Smooth the data by FDA     x/(t), / = 1,..., N
Regression
Smoothed regression
Jan Koláček (SCI MUNI)
M7777 Applied FDA
Fall 2019       3 / 19
Functional Data Simulation
3. Basis expansion
k
7=1
0*(t) = ..., <t>k(t)) ■ ■ ■ a g'ven basis system
Cjj . .. iid random basis coefficients for /-th curve, j = 1,..., K
Jan Koláček (SCI MUNI)
M7777 Applied FDA
Fall 2019       4 / 19
Functional Data Simulation
4. Gaussian process
Let us consider a regression model
Xi(tk) = m(tk) + eik
with a covariance function c(r,s), i.e. Cov(e/r,£/s) = c(tr,ts), usually
c(u, v) = a2 exp (~^{u - v):
2/2
Set m = (m(ti),..., m(tN))', 53 = (c(t;, tj))^.=1, x, = (x,(ti),..., x/(t/v))' Thus x/ ~ A//v(m, 53) and x,-(t) = lim/v-^ x,-.
Gaussian process
Jan Koláček (SCI MUNI)	M7777 Applied FDA		Fall 2019       5 / 19
Functional Data Simulation
5. Random Gaussian process
Let be given (tj>yD • • •, (^,y^), L < M, and suppose
xi(t*k)=y*k+e*k, ** ~ NL(0,a2JL).
Then
x/|y*~A/M(m*,S*)
with
m* = £tt* (St*t* +^l/.)"1y*J
= Stt - Ett* (St*t* + cr*'/.) 1 £t*t- where £ab = Cov(a, b).
Random Gaussian process
0 100 200 300
Jan Koláček (SCI MUNI)	M7777 Applied FDA		Fall 2019       6 / 19
Functional Data Simulation
Regression Simulation
O Generate y; with known a,       x,-(t) and     / = 1,..., 30
/\
0 Get estimates /3(t),y by considered methods
a) Estimation through a basis expansion . .. /3se(£), ybe
b) Estimation with a roughness penalty .. . /3pp(£),ypp
c) Regression on functional principal components . .. /3pc(t), ypc
d) Nonparametric regression ... /3/v/?(t), y/v/?
✓V s\ s\ s\
© Compare @be, @rp, fipc, @nr with known /3
O Compare estimates 9be, 9rp, 9pc, 9nr with known model fits.
Jan Koláček (SCI MUNI)
M7777 Applied FDA
Fall 2019       7 / 19
Functional Data Simulation
Regression Simulation 1
Let (ti,..., tyf) = (1,2,... 365), we will simulate 30 regression curves x,-(t) as the Gaussian process with
m{t) = sin(t/365),    c(u, v) = 0.01 exp (-^^(u - vf The regression model takes the form
	365	
	-10+ / ß(t)xi(t)dt + ei l	
with ß(t) = l + 2t/365 -	(t/365)2 and e,- - A/(0,5).	
Jan Kolaiek (SCI MUNI)	M7777 Applied FDA	Fall 2019       8 / 19
Functional Data Simulation
Simulated Data
Jan Koláček (SCI MUNI)
M7777 Applied FDA
Fall 2019       9 / 19
Functional Data Simulation
Jan Koláček (SCI MUNI)
M7777 Applied FDA
Fall 2019       10 / 19
Functional Data Simulation
Comparison of fits
Type
• 1. Basis expansion
• 2. Roughness Penalty 3. Functional PCA
• 4. Nonparametric
Jan Koláček (SCI MUNI)
M7777 Applied FDA
Fall 2019       11 / 19
Functional Data Simulation
Regression Simulation 2
Let (ti,..., tyf) = (0, 0.01,... 1), we will simulate 30 regression curves x,-(t) as the Gaussian process with
m(t) = exp(t/27r),    c(u, v) = 0.5 exp (—10(iv - \/)2) .
The regression model takes the form
l
y; = 5 + ^ P(t)xi(t)dt + ei o
with       = sin(27rt) and e\ - A/(0,0.1).
Jan Koláček (SCI MUNI)	M7777 Applied FDA		Fall 2019       12 / 19
Functional Data Simulation
Simulated Data
Jan Koláček (SCI MUNI)
M7777 Applied FDA
Fall 2019       13 / 19
Functional Data Simulation
Jan Koláček (SCI MUNI)
M7777 Applied FDA
Fall 2019       14 / 19
Functional Data Simulation
Jan Koláček (SCI MUNI)
M7777 Applied FDA
Fall 2019       15 / 19
Problems to solve
O Conduct the following simulation (Kokoszka and Reimherr, 2017).
• Generate 1000 random functions
X{tJ) = Ztj + U + r1{tj) + e{tj),
where tj are 101 equidistantly distributed points on [0,1],
rj(tj) ~ A/(0,1), Z - A/(l, 0.22), U ~ UNIF(0, 5) and the random
curves e(t) will be generated as
10 1
e(t) = ^2t {Zik sin(27rt/c) + Z2/c cos(27rt/c)}
K
k=l
with independent standard normal Zi/c,Z2/c.
• Consider a regression model of the form
l
y; = 0.01 J /3(t)Xi(t)dt + ei o
with fi(t) = -fi{t) + 3f2(t) + f3(t) and e-, - A/(0, 0.4), where /i, f2, f3 are normal densities A/(0.2, 0.032), A/(0.5, 0.042), A/(0.75, 0.052), respectively. _
Jan Koláček (SCI MUNI)	M7777 Applied FDA		Fall 2019       16 / 19
Problems to solve
• Try all regression approaches studied in the previous lesson, i.e.
• estimation through a basis expansion,
• estimation with a roughness penalty and
• regression on FPCA.
Plot the estimates @(t) and compare it with the original @(t) (see Figure 1).
• Conduct the nonparametric regression. Plot estimated values y; against the simulated y; for all cases (see Figure 2).
Jan Koláček (SCI MUNI)
M7777 Applied FDA
Fall 2019       17 / 19
Problems to solve
Jan Koláček (SCI MUNI)
M7777 Applied FDA
Fall 2019       18 / 19
Problems to solve
Jan Koláček (SCI MUNI)
Figure 2.
M7777 Applied FDA
Fall 2019       19 / 19