M7777 Applied Functional Data Analysis 9. Functional Response with Scalar Covariate
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 1/31
Functional Response Models
Functional ANOVA
Just as in the standard ANOVA, let be K (K > 3) groups and
K
Xij(t)... /-th curve in j-th group, / = 1,..., ny, n — nj
7=1
An over-all mean
m =
,    K nj
*(0 = -EEx</(0
7=1 i = l
Effects for each group
An error process
"j(o = ^Emo-*(0)
£«(0 = xí/(0 - "/(o - x(0
Jan Koláček (SCI MUNI)	M7777 Applied FDA		Fall 2019       2 / 31
Functional Response Models
Functional ANOVA model
Xij(t) = ii(t) + aj(t) + eij(t) Suppose we observe scalar covariates z/i,..., z,x
Zij =
1 if x,y(t) belongs to j-th group 0 otherwise
Model (1) can be rewritten as a functional linear model
K
y/(t) = A)(t) + ^2ßj{t)zij + Ei{t)
7=1
K
with conditions ^ ßj{t) = 0, Ee;(t) = 0
7=1
(1)
Jan Koláček (SCI MUNI)	M7777 Applied FDA		Fall 2019       3 / 31
Functional Response Models
Canadian Weather
Divide all 35 locations to 4 regions: Atlantic, Continental, Pacific, Arctic
□ays
We will study the effect of geographic region on the shape of the temperature curves.
Jan Koláček (SCI MUNI)	M7777 Applied FDA		Fall 2019       4 / 31
Functional Response Models
Let us denote model parameters
ßo(t) ... Canada ßi(t) ß3(t) ... Pacific /34(t)
03
m
Atlantic ß2(t) Arctic
Continental
Estimates of ßj(t)
Canada										Atlantic								
																		
																		
																		
																		
																		
																		
																		
																		
Pacific										Arctic								
																		
																		
																		
																		
																		
																		
																		
(	) 100			200		300		(			)	100		200		300		
Continental						
						
						
						
						
						
						
						
						
100
200
300
Jan Koláček (SCI MUNI)
M7777 Applied FDA
Fall 2019       5 / 31
Functional Response Models
Prediction of the temperature curve in j-th group
yj(t) = ß0(t) + ßj(t),      7 = 1,..., 4.
20
10
w 0
"O 0)
o
TD
-20
-30
Predicted temperatures in each region
region
^ Arctic
™ Atlantic
^ Continental
^m Pacific
100
200
300
Days
Jan Koláček (SCI MUNI)
M7777 Applied FDA
Fall 2019       6 / 31
Functional Response Models
Confidence Intervals
Consider a basis representation
(3j(t) = *j(t)cj set b = (cq, ..., cK) =4> /5/(t) depends on b.
Let t = (ti,..., t/v), y; = (y/(ti), • • • ,yi(ti\i))f, generally, we minimize penalized least squares and get the estimate
b = y2cMapy.
Estimate the covariance matrix
1 "
Z =    _    22 where ^' = y' ~~ z'^(*)-
n - K
;=i
Thus (formally)
Var&(t) = 0/(t)y2cMapiy2cMap'0/(t)/.
Jan Koláček (SCI MUNI)	M7777 Applied FDA		Fall 2019       7 / 31
Functional Response Models
Estimates of ßj{t)
20
10
CD CD
-10
-20
20
10
Canada								
								
					■ ^ —			
			4		- \\	A		
			T 1/ l/l		V			
	----	1 11	l/l l/l Ml-- (1 1	—	----	s V	—	
		'/' '/>'					\\ <	
		/ / / /					\\	
	✓							
								
Atlantic								
								
								
								
	"■■*-■	. — -				✓		
	: - - -		-«. _			^ ^	~" N» >	—
								
								
								
								
Continental								
								
								
								
								
-	^ -			—			N	-
		✓					\ S	
								
								
								
-10
-20
Pacific								
								
1                      — "		x \ \						
		*^X N	S N				✓	
		N	X x	X		>		
			X ^ X	x    **" —				
								
								
								
								
Arctic								
								
								
								
								
				s — ~ "				
	™ X X		✓			X X		-
	X	> ^ ,	✓ /	✓		■««. x X. X		
	""* *«.		✓				X ^	—
	X	^ ^						
100
200
300
100
200
300
100
200
Days
300
Jan Koláček (SCI MUNI)
M7777 Applied FDA
Fall 2019
8/31
Functional Response Models
Jan Koláček (SCI MUNI)
M7777 Applied FDA
Fall 2019       9 / 31
Functional Response Models
Assessing the fit of the fANOVA
Residuals
15
10
CO "O
s. 0
-10
place	
— Arvida	- Quebec
- Bagottville	- Regina
— Calgary	- Resolute
— Dawson -	— Sherbrooke
— Edmonton -	Scheffervll
- Fredericton -	- St. Johns
- Halifax	- Sydney
- Charlottvl	- The Pas
- Churchill	— Thunder Bay
— Inuvik	— Toronto
- Iqaluit	- Uranium City
- Kamloops	Vancouver
— London -	- Victoria
— Montreal -	— Whitehorse
- Ottawa	— Winnipeg
- Pr. Albert	— Yarmouth
- Pr. George -	— Yellowknife
- Pr. Rupert	
100
200
Days
Jan Koláček (SCI MUNI)
M7777 Applied FDA
Fall 2019       10 / 31
Functional Response Models
Jan Koláček (SCI MUNI)
M7777 Applied FDA
Fall 2019       11 / 31
Functional Response Models
F-statistic
To test significance, we can define a pointwise F-statistic
Var(y(t))
F(t) = —
i=l
indicates where there is a large amount of signal relative to variance. Test over-all regression significance based on
F* = maxF(t).
Jan Koláček (SCI MUNI)
M7777 Applied FDA
Fall 2019       12 / 31
Functional Response Models
Permutation Test
We would like to test the null hypothesis
H0:Ey(t) = 0 Vte[tutN]
Do b times
O Permute indexes 1,..., n to get /'i,..., in, leaving the design unchanged.
© Define yf{t) = y/,(t).
© Estimate the model using yb(t) as the response.
O Measure FT and set
if F* > F if F* < F
Then p-value for the test
b=l
Jan Koláček (SCI MUNI)
Functional Response Models
Canadian Weather
b = 200, pb = 0
■H2 w
as
I
100
200
300
Observed statistic pointwise 0.05 critical value maximum 0.05 critical value
Days
Jan Koláček (SCI MUNI)	M7777 Applied FDA		Fall 2019       14 / 31
Functional Response Models
Canadian Weather
detailed test results
Observed statistic pointwise 0.05 critical value Sample statistic (B=200)
Days
Jan Koláček (SCI MUNI)
M7777 Applied FDA
Fall 2019       15 / 31
Functional Response Models
Canadian Weather
F* = 3.41, F0*95 = 0.747
Ll_ 2
X
								
								
								
								
								
				'--—11 J	M________			
	I				I i	11	Ii	
1			Wll			m	Al	•
Observed maximum F* maximum 0.05 critical value Sample maximum F*
1 1 0 50	100 150 Sample	200		
Jan Koláček (SCI MUNI)	M7777 Applied FDA		Fall 2019	16 / 31
Functional Response Models
Functional ř-test
Just 2 groups of curves (x,y(t), x/2(0) significant?
Is the difference statistically
Example. Berkeley Growth Study (39 boys, 54 girls)
200
150
CD
X
100
— boys
— girls
5	10 Age [years]		15
Jan Koláček (SCI MUNI)	M7777 Applied FDA		Fall 2019       17 / 31
Functional Response Models
Functional t-statistic
To test significance, we can define a pointwise ř-statistic
T(t) =
xi(t)-x2(t)|
^£Var[xi(t)] + ^Var[x2(t)]
indicates where there is a large mean difference relative to variance Test over-all significance based on
7* = max T(t).
Jan Koláček (SCI MUNI)
M7777 Applied FDA
Fall 2019       18 / 31
Functional Response Models
Permutation Test
We would like to test the null hypothesis
H0 : Exi(t) = Ex2(t)   Vt G [ti, t/v]
Do 6 times O Randomly shuffle the labels of the curves. © Calculate the ^-statistic Tf,(£) with the new labels.
1
0 Measure T£ and set 1^ = Then p-value for the test
0
if T* > T* if T* < T*
Pb =
Jan Koláček (SCI MUNI)
M7777 Applied FDA
Fall 2019       19 / 31
Functional Response Models
Berkeley Growth Study
b = 200, pb = 0
Functional Response Models
Berkeley Growth Study
detailed test results
Observed statistic pointwise 0.05 critical value Sample statistic (B=200)
Jan Koláček (SCI MUNI)
M7777 Applied FDA
Fall 2019       21 / 31
Functional Response Models
Berkeley Growth Study
7* = 10.3, T0*95 = 2.61
10.0
7.5
'x 5.0
2.5
0.0
Observed maximum T* maximum 0.05 critical value Sample maximum T*
0 50	100 150 Sample	200		
Jan Koláček (SCI MUNI)	M7777 Applied FDA		Fall 2019	22 / 31
Problems to solve
O Sound Intensity Data
Load the variable rat3 from the rat3.RData file. The variable rat3 contains observations of a rat neural activity evoked by sound intensity. The evoked potential (EPI) was measured in dependence on 19 sound intensities for 5 days. The dataset contains 79 repetitions for each day.
• Smooth the data by B-spline bases with second-derivative penalties and plot the result with color-day specification (see Figure 1).
• Conduct a study of the effect of the day on the shape of the EPI curves. Consider the fANOVA model with days as covariates. Plot estimated parameters with its pointwise confidence bands (see Figure 2).
• Plot predictions for each day with its pointwise confidence bands (see Figure 3).
• Plot functional R2 of the model (see Figure 4) and interpret it.
• Asses the model by the permutation test for F-statistic and plot the result (see Figure 5).
Jan Koláček (SCI MUNI)	M7777 Applied FDA		Fall 2019       23 / 31
Problems to solve
© Sound Intensity Data
• Consider just days SS4 and SS5 and plot the EPI estimates with color-day specification (see Figure 6).
• Is the difference between days statistically significant? Conduct the functional t-test.
• Asses the model by the permutation test for t-statistic, plot the result (see Figure 7) and interpret it.
Jan Koláček (SCI MUNI)	M7777 Applied FDA		Fall 2019       24 / 31
Problems to solve
Problems to solve
Figure 2.
Jan Koláček (SCI MUNI)
M7777 Applied FDA
Fall 2019       26 / 31
Problems to solve
Jan Koláček (SCI MUNI)
M7777 Applied FDA
Fall 2019       27 / 31
Problems to solve
0.6
0.4
0.2
0.0
								
								
								
								
								
								
								
--------								
		1 )                                                             25 5						
Intensity
Figure 4.
Jan Koláček (SCI MUNI)	M7777 Applied FDA		Fall 2019       28 / 31
Problems to solve
Observed statistic pointwise 0.05 critical value maximum 0.05 critical value
1 0	25 50 Intensity Figure 5.	i 75		
Jan Koláček (SCI MUNI)	M7777 Applied FDA		Fall 2019	29 / 31
Problems to solve
10(H
u-1- 0	25 Intensity Figure 6.	50	75	
Jan Koläcek (SCI MUNI)	M7777 Applied FDA		Fall 2019	30 / 31
Problems to solve
Observed statistic pointwise 0.05 critical value maximum 0.05 critical value
1 0	25 50 Intensity Figure 7.	i 75		
Jan Koláček (SCI MUNI)	M7777 Applied FDA		Fall 2019	31 / 31