M7777 Applied Functional Data Analysis
1. Introduction
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 1/22
Course Outline and Requirements
Outline
1. Introduction
2. Basis Systems
3. Basis Smoothing
4. Smoothing Penalties
5. Constrained Smoothing
6. Exploratory Data Analysis, FPCA
7. Scalar-on-function Regression
8. Functional Data Simulation
9. Function-on-scalar Regression
10. Function-on-function Regression
11. Registration
12. Sparse FDA
Jan Koláček (SCI MUNI)	M7777 Applied FDA		Fall 2019 2/22
Course Outline and Requirements
Assumed Knowledge
This class will focus on the application of functional data analysis techniques to real-world problems and is not intended to be mathematically technical. However, we will make use of linear algebra and I assume a background in applied statistics on the level of M5120.
Computing Software
The course will be taught using the fda library in <®. I do not assume knowledge of©, but some programming experience will be helpful, ©is freely available from www.r-project.org.
Assessment
• Attendance
• Homework
• Final Project
Students are expected to work individually on homework. The project may be undertaken in small groups.
Jan Koláček (SCI MUNI)
M7777 Applied FDA
Fall 2019 3/22
What is Functional Data?
What are the most obvious features of these data?
Jan Koláček (SCI MUNI)
M7777 Applied FDA
Fall 2019 4/22
What is Functional Data?
What are the most obvious features of these data?
Time [ms]
Jan Koláček (SCI MUNI)
M7777 Applied FDA
Fall 2019 4/22
What is Functional Data?
What are the most obvious features of these data?
sample
Time [ms]
quantity
frequency (resolution) similar trends
Jan Koláček (SCI MUNI)
M7777 Applied FDA
Fall 2019 4/22
What is Functional Data?
What are the most obvious features of these data?
sample
quantity
frequency (resolution)
similar trends
the same domain (not necessary)
Time [ms]
Jan Koláček (SCI MUNI)
M7777 Applied FDA
Fall 2019 4/22
What is Functional Data?
What are the most obvious features of these data?
sample
quantity
frequency (resolution)
similar trends
the same domain (not necessary)
smoothness
Time [ms]
Jan Koláček (SCI MUNI)
M7777 Applied FDA
Fall 2019 4/22
What is Functional Data?
6 replications, 200 observations within replications
FDA involves repeated measures of the same process
3 100 200 300
Time [ms]
Jan Koláček (SCI MUNI)
M7777 Applied FDA
Fall 2019 5/22
What is Functional Data?
6 replications, 200 observations within replications
				„.••:::m,",,!»££*			
						..„ "::s..	
							
							
	•V						
	.••vy"						
							
sample
FDA involves repeated measures of the same process
1 observation = 1 function
Time [ms]
Jan Koláček (SCI MUNI)
M7777 Applied FDA
Fall 2019 5/22
What is Functional Data?
6 replications, 200 observations within replications
				„.••:::m,",,!»££*			
						..„ "::s..	
							
							
	•V						
	.••vy"						
							
sample
FDA involves repeated measures of the same process
1 observation = 1 function
FDA = Analysis of data that are functions
Time [ms]
Jan Koláček (SCI MUNI)
M7777 Applied FDA
Fall 2019 5/22
What is Functional Data?
6 replications, 1401 observations within replications
Functional data is often com plicated:
• not easily described by mathematical formulae
1000 1500
Time [ms]
Jan Koláček (SCI MUNI)
M7777 Applied FDA
Fall 2019 6/22
What is Functional Data?
6 replications, 1401 observations within replications
Functional data is often complicated:
• not easily described by mathematical formulae
• variation between replications even harder to describe
Time [ms]
Jan Koláček (SCI MUNI)
M7777 Applied FDA
Fall 2019 6/22
What is Functional Data?
6 replications, 1401 observations within replications, 2 dimensions
^-0.04-.2 0.04-
Functional data is often complex:
• often a large number of related quantities
Time [ms]
Jan Koláček (SCI MUNI)
M7777 Applied FDA
Fall 2019 7/22
What is Functional Data?
6 replications, 1401 observations within replications, 2 dimensions
^-0.04-.2 0.04-
Time [ms]
Functional data is often complex:
• often a large number of related quantities
• viewing each replication as a single observation can make the data easier to think about
What are these data? Let us plot one component against another!
Jan Koláček (SCI MUNI)
M7777 Applied FDA
Fall 2019 7/22
What is Functional Data?
Measures of position of nib of a pen writing "fda". 6 replications,
measurements taken at 200 hertz.
0.04 H
-0.04 -0.02 0.00 0.02
Position [mm]
Jan Koláček (SCI MUNI)
M7777 Applied FDA
Fall 2019 8/22
What is Functional Data?
Data may be measured more noisily
St. Johns
				-A	^—1--- ./ <			
				-v. / •	• • •	>		
			Mr	V* m_/ • •		•	•	
		♦	%r * \				.•v	
	•	• / Ar*'* •					%\ •	• >ř ••
	• *     • • < *	• •						
0 100 200 300
Days
Jan Koláček (SCI MUNI)
M7777 Applied FDA
Fall 2019 9/22
What is Functional Data?
Data may be measured more noisily
St. Johns
10.0-						•				
									•	
7.5-		•	•						•	
		• • • • •	• • •	t	• •		• • •	• • •	•	
Precipitation b	<	• • • •   • : • •		•	•	• • •	• • • • • •  •  •          " • •• •  •         J     * •• • • • • • • • • • •		• • •	
		—a»^sJ •••: • • •	• •	• • • •• • •    • • • • •/ •	• •• • •	• • • • • • • -	•	* • Mi •           • •	•	
2.5-		•• •	•   • • • • • •    • *	» •               • • "	• •	• • • /•• •	• • • • •	M •	• • • •	
		• •	•• • •	• • • •	• • • ...    . •	*    •     " •• > • • • • •          • • •			•	
				•						
(		) 1(		10                                                         200 300 Days						
Jan Koláček (SCI MUNI)							Fall 2019 1			0/22
What is Functional Data?
Data may be measured more sparsely
Berkeley Growth Data
200
175
150
X
125
100
75
					•		•
					• •                   .   •   •   T * .   *           .   • '		
				: ,		. •    : « i »	
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			• .'"ti	• • i « : i • • ! *	i >		
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		• t : :	• ■ •				
	• 4 4 • :	t s > »					
• • ■ * • •• • •	• • • :						
							
•							
10
15
sample
• 1
• 2
• 3
• 4
• 5
• 6
Age
Jan Koláček (SCI MUNI)
M7777 Applied FDA
Fall 2019 11/22
What is Functional Data?
Longitudinal Data
6 patients
1500-
-17   -15   -13   -11    -9    -7    -5    -3    -1      1      3      5      7      9     11     13     15     17    19    21     23    25    27    29    31     33    35 37
Time since seroconversion
Jan Koláček (SCI MUNI)
M7777 Applied FDA
Fall 2019
12/22
What is Functional Data?
We may not have repeated measurements
year
• 1964
• 1965
• 1966
	■  i   i   i   i   i  i  i  i  i  i i 1    2    3    4    5    6    7    8    9   10   11 12		
	Time		
Jan Koláček (SCI MUNI)	M7777 Applied FDA	Fall 2019	13/22
What is Functional Data?
From discrete to functional data - intuition
• The term functional in reference to observed data refers to the intrinsic structure of the data being functional; i.e. there is an underlying function that gives rise to the observed data.
• Advantages of representing the data as a smooth function:
• allows evaluation at any time point
• allows evaluation of rates of change of the underlying curve
• allows registration to a common time-scale
Main idea in FDA: treat the observed data functions as single entities, rather than sequence of individual observations.
Jan Koláček (SCI MUNI)	M7777 Applied FDA		Fall 2019 14/22
References
• Ferraty, F.f Vieu, P., 2006. Nonparametric functional data analysis: theory and practice. Springer Science & Business Media.
• Ramsay, J. 0., Silverman, B. W., 2005. Functional data analysis, 2nd Edition. Springer, New York.
• Ramsay, J. 0., Silverman, B. W., 2007. Applied functional data analysis: methods and case studies. Springer.
• Ramsay, J. 0., Wickham, H., Graves, S., Hooker, G., 2019. fda: Functional Data Analysis. R package version 2.4.8.
https:// CRAN. R-project.org/package=fda
• Giles Hooker's course BTRY 6150 http:// faculty bscb. Cornell, edu/^ hooker/
• Kokoszka, P., Reimherr, M., 2017. Introduction to functional data analysis. Taylor & Francis Group.
Jan Koláček (SCI MUNI)	M7777 Applied FDA		Fall 2019 15/22
Problems to solve
O Install the fda package. © Berkeley Growth Data
• load the variable growth from the fda package
• plot the first 6 samples for boys and girls separately (see Figure 1)
• plot the first 6 samples for boys and girls into one plot (see Figure 2)
© Canadian Weather Data
• load the variable Canadian Weather from the fda package
• plot temperatures measured in Edmonton, Halifax, Montreal and Ottawa (see Figure 3)
• plot precipitations observed in Edmonton, Halifax, Montreal and Ottawa (see Figure 4)
• plot temperatures for all the places in dependence on regions; into one plot and separately (see Figures 5 and 6)
O (optional) Plot other data from this presentation.
Jan Koláček (SCI MUNI)
M7777 Applied FDA
Fall 2019 16/22
Problems to solve
Berkeley Growth Data
200
175
150
E o
125
100
75
boys								girls							
					•<										
					• •										
					11									!!!!!!	ü
															
			•	......											
		•									• •				
	<	• i							1		•				
	• ! • ! :	! •							• 1 • < •						
•	i • • •							J	! *						
								/							
•								W							
sample
• 1
• 2
• 3
• 4
• 5
• 6
10
15
10
15
Age [yrs]
Figure 1
Jan Koláček (SCI MUNI)
M7777 Applied FDA
Fall 2019 17/22
Problems to solve
Berkeley Growth Data
200
175
150
e ü
CD
X 125
100
75
					•		
					• •	• • • •:	
				•		Hi !!!	
			i			• • • • • i	
			• 1 •:; i •      •      A 1	!!!'"			
		S »	• ill! •	! 1			
	I						
	•						
ll							
iÜ!!							
i	[		i	0	1	5	
Age [yrs]
Figure 2.
Sex
• boys
• girls
Jan Koláček (SCI MUNI)
M7777 Applied FDA
Fall 2019 18/22
Problems to solve
Figure 3.
Jan Koláček (SCI MUNI)	M7777 Applied FDA		Fall 2019 19/22
Problems to solve
Canadian Weather Data
Place
• Edmonton
• Halifax Montreal
• Ottawa
0	100	200 Days Figure 4.	300		
Jan Koláček (SCI MUNI)		M7777 Applied FDA		Fall 2019	20/22
Problems to solve
Jan Koláček (SCI MUNI)
M7777 Applied FDA
Fall 2019 21/22
Problems to solve
Canadian Weather Data
20
-20
CD
C\5 CD
Arctic
Atlantic
& 20
-20
Continental
Pacific
Region
• Arctic
• Atlantic
• Continental
• Pacific
100 200
300
100 200
300
Days
Figure 6.
Jan Koláček (SCI MUNI)	M7777 Applied FDA		Fall 2019 22/22