Course description:
In many practical problems originating from areas like geophysical sciences, astronomy, medicine, etc., the observations often correspond to measurements of a continuous process, which is a function of time and/or space. With rapid advances in technology, such measurements are becoming more and more frequent, thus pushing the boundaries of high-dimensional data analysis. In data science, high dimensional problems are often approached through dimension free analysis by treating an observation as a function. This is one way to circumvent the technical aspects identified with analyzing high-dimensional data, and thus avoiding the curse of dimensionality. This course is meant to introduce the students to various aspects of dimension free analysis, which obviates the technical and computational hurdles associated with high dimensional data. This way of dealing with high dimensional data is an emerging and rapidly developing field that requires understanding both established methods and newly adopted techniques. The primary objective of this course will be to focus on the application of functional data analysis techniques to real world problems, and thus, mathematical rigor is often traded for adaptability to applications. Beginning with the basics of the analysis of data that may be considered to be ``functions'', this course will discuss various visualization and data exploration techniques. Specifically, the course will extensively deal with nonparametric spline smoothing, functional linear models, functional PCA, regularization methods, analysis involving derivatives, registration and nonlinear smoothing. Students are required to work on projects to apply the techniques on real world problems. The preferred software for this course will be `R' and/or Python, however, the students are permitted to use any mathematical software of their liking that have facilities to perform all task in the course (Matlab being one example). Project discussions will enable students to share and compare ideas with each other and to receive specific guidance from the instructors. Efforts will be made to help students to embed real-world problems into mathematical models so that suitable algorithms can be applied with consideration of computational constraints. By surveying special topics, students will be exposed growing range of new methodologies.

Learning outcomes:

• For a passing grade, the student demonstrates
  • an understanding both technical and conceptual aspects of dealing with data treated as samples of functions,
  • knowledge of methods that goes beyond standard multivariate statistical methodology through dimension free approach that accounts for the order in observed variables,
  • an understanding how to represent data as functions, when the data are inherently too highly dimensional to be effectively treated as a vector, and
  • an understanding the role which derivative functions play in construction solutions to practical problems.
  • demonstrate the ability to efficiently handle the data that have an ordering inscribed in them and/or are highly dimensional through functional decomposition,
  • demonstrate understanding of the functional data approach which extends from statistical methodology such as the principal component analysis and multivariate regression, and
  • master a number of algorithmic approaches, with assistance of software packages, both to analyze and visualize functional data.
  • demonstrate available choices for visualization and data exploration in the functional setting,
  • identify the type of approach that is most suitable for the problem at hand, and
  • critically discuss the results of analysis obtained by a particular method.

• Textbooks:
  

  • PRIMARY - Ramsay, J.O., Hooker, G., Graves, S. 2009. ``Functional data analysis with R and MATLAB''. Springer
  • SUPPORTING -
    • Ramsay, J.O. and B. W. Silverman, 2005, "Functional Data Analysis", Springer: New York.
    • Horvath, L., Kokoszka, P. 2012. Inference for functional data with applications. Springer
    • Ferraty, F., Vieu, P. 2006. Nonparametric Functional Data Analysis. New York: Springer, New York
    • Ramsay, J. O. 1982. ``When the data are functions.'' Psychometrika, 47:379—396
    • Kokoszka, P. and Reimherr, M. 2017, ``Introduction to Functional Data Analysis'', Chapman and Hall, CRC: Boca Raton.
  
  

• Supporting computer package:
  All projects and exercises can be completed using

  • Statistical R-package available for free download
  and data sets used throughout the course will be downloadable through suitable links. An alternative tool for the computing coursework is Python.
  • Sreekar Vadlamani , Department of Statistics, e-mail: Sreekar.Vadlamani@stat.lu.se Office Hours: Wednesdays 15:00-16:00 or by appointment.
  • Krzysztof Podgorski , Department of Statistics, e-mail: Krzysztof.Podgorski@stat.lu.se Office Hours: Tuesdays 11:00-12:00 or by appointment.

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  Weakly Timetable

  
  

  Detailed schedule of classes

  
  (sections and chapters refer to the primary textbook)
  

  Week #	Day	Time and Location	Lecture/Lab/Discussion #	Material	Handouts/Reading material
  17	Monday, 23/04/2018	09:15-11:00, E1:369 (Blue Room)	Lecture 1: : The concept of functional data and examples	Overview ,
  		13:15-15:00 E1:369 (Blue Room)	Lecture 2: Representing functional data in functional bases	Orthonormal bases
  	Wednesday, 25/04/2018	09:15-11:00 E1:369 (Blue Room)	Discussion 1: Basis decomposition of a function.	Assignment 1
  18	Wednesday, 02/05/2018	09:15-11:00 E1:369 (Blue Room)	Lecture 3: Locality and smoothing: wavelets and splines	Wavelets and splines,	Code to be used in Lab 1
  		15:15-17:00, Alfa1:0043	Lab 1: Introduction to fda package, working with functional bases and regularization	Topics for Lab 1
  	Friday, 04/05/2018	09:15-11:00 E1:369 (Blue Room)	Discussion 2: Splines and regularization	Assignment 2
  19	Monday, 07/05/2018	09:15-11:00 E1:369 (Blue Room)	Lecture 4: Exploratory data analysis of functional data	Descriptive functional statisitcs
  		13:15-15:00 E1:369 (Blue Room)	Lecture 5: Mathematical model of functional data	FDA model
  	Wednesday, 09/05/2018	09:15-11:00 E1:369 (Blue Room)	Discussion 3: Eigenfunctions	Assignment 3
  20	Monday, 14/05/2018	09:15-11:00 E1:369 (Blue Room)	Lecture 6: Inference for functional data	Statistical Inference	Code to be used in Lab 2
  		13:15-15:00 E1:369 (Blue Room)	Lecture 7: Functional principal component analysis	FPCA
  	Wednesday, 16/05/2018	09:15-11:00	Lab 2: Smoothing and PCA	Lab 2
  21	Monday, 21/05/2018	10:15-12:00 E1:369 (Blue Room)	Lecture 8: Functional linear models	Linear models
  		13:15-15:00 E1:369 (Blue Room)	Discussion 4: Functional regression - properties	Functional Regression
  	Wednesday, 23/05/2018	09:15-11:00	Lab 3: Data Analysis for linear models
  22	Monday, 28/05/2018	09:15-11:00 E1:369 (Blue Room)	Lecture 9: Functional autoregressive models
  		13:15-15:00 E1:369 (Blue Room)	Discussion 5: Functional autoregression
  	Wednesday, 30/05/2018	09:15-11:00	Lab 4: Example with autoregressive functional data
  	Friday, 01/06/2018	09:15-11:00 E1:369 (Blue Room)	Lecture 10: Dynamical functional model
  		09:15-11:00	Lab 5: Example with dynamical functional data

  
  

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• Comprehensive Exam: The work throughout the course will be compounded into one comprehensive examination paper. It will comprise three parts:

  1. Take-home Assignments;
  2. In-class Computer Projects;
  3. Oral Presentations.
  Each part graded on the scale from 0-100 contributes equally to the total score which is computed according to the formula: T=(S1+S2+S3)/3, where S1, S2, S3, are scores for corresponding parts.

  The final grades will be assigned according to the following table:

  Percentage	Grade
  49 - 0	F
  54 - 50	E
  64 - 55	D
  74 - 65	C
  84 - 75	B
  100 - 85	A

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