Functional data analysis (FDA) is used to analyze data that changes continuously over time as functions rather than discrete values. The document outlines the need for FDA, the step-by-step procedure including transforming data into functions, smoothing functions, registering functions, and performing analysis. An example is provided showing the use of FDA to analyze knee angle data during walking and how it identifies relationships between knee angle and another variable over different parts of the gait cycle. FDA allows the use of all data information rather than reducing it to single values.

1. FUNCTIONAL
DATA ANALYSIS
-By
Nitisha Agarwal
(Junior Statistical Consultant)
COSTARCH ANALYTICAL CONSULTING P. LTD.

2. CONTENT:
Functional Data
 Need of FDA
 Step by Step Procedure
 Example
 Conclusion
 Reference


3. FUNCTIONAL DATA:
The important feature of functional data is that they consist of
functions and assesses the trend of data that how data changes
continuously over time. This feature differentiate Functional data
analysis from traditional analytics methods.

4. NEED OF FDA:
The fundamental aims of the analysis of functional data are the
same as those of more conventional statistics:
 To formulate the problem at hand in a way amenable to
statistical thinking and analysis.
 To develop ways of presenting the data that highlight interesting
and important features.
 To investigate variability as well as mean characteristics.
 To build models for the data observed, including
those that
allow for dependence of one observation or variable on another.

5. PROCEDURE:
Transforming data into functions,
 Smoothing functions,
 Registering functions,
 Analysis.


6. 1. TRANSFORMING DATA INTO FUNCTIONS
The first step in FDA is transforming the captured time series
data into functions. For cyclical behaviors, it is common to
begin by dividing a long sequence of data into individual
cycles. Each cycle is then transformed into a function.
Therefore, if one cycle was represented by n
observations, the behavior can now be considered a single
functional observation. In FDA, functions are represented as
a linear combination of a set of basis functions.

7. One of important decision that must be made is the type of
basis system to use to represent the data.
 Typical selections are Fourier and spline basis systems, but
others such as polynomials or wavelets can also be used.
 For K basis functions, there will be K corresponding
coefficients. As the number of basis functions increases the
model can produce an excellent fit to the data.


8. 2. SMOOTHING:
Functions can be smoothed by minimizing the number of basis
functions and using regression analysis or by using a roughness
penalty approach.
 The use of a large number of basis functions relative to the
number of data points, tends to over fit the data.
 Conversely, using a small number of function having risk of
losing localized functional features.
 So, regression analysis can not be used for complex cases.


9. Alternative Approach:

10. To create the smooth curve, a multiple of the roughness
penalty is applied to the error sum of squares.
 a smoothing parameter λ is used to specify the degree of
penalty.
 As λ approaches 0, the fit of the function to the observations
improves.


11. 3. REGISTERING FUNCTIONS - CURVE
ALIGNMENT:
An important curve is a mean curve to represent a specific
behavior. The purpose of curve alignment is to reduce
phase variability while preserving the curves shape and
amplitude. A common method for aligning curves used is a
“linear time normalization procedure”.

13. 4. STATISTICAL TESTS:
An important advantage of treating the behavior as a function
rather than n discrete data points is that the data do not have to
be reduced to one or two descriptive statistics, such as an
average, maxima etc. since, this eliminates a large amount of
valuable information that cannot be summarized with one or two
number.

14. Example

17. By considering the data as a sample of observations from a
smooth underlying curve, we will be able to use all of the
information. Before fitting a functional linear model, we must
first fit a smooth curve to the data, we do this using b-spline
basis functions. To avoid over fitting, a roughness penalty is
imposed on the curvature of the fitted function.
The below graph shows the smooth fit for 3 trials-

19. The above graph shows how knee angle affects Y at various
points in the gait cycle. The results of OLS are consistent with
graph above which shows β(t) close to 0 near the times of peak
flexion.
Both the functional model and the ordinary linear model show no
substantial relationship between Y and knee angle in this range
of the gait cycle. The functional linear model does indicate that
knee angle between 0.31 and 0.46, 0.55 and 0.64, as well as 0.85
and 0.95 seem to have an association with Y.

20. CONCLUSION:
Functional Data Analysis is an important analytical method
that can be used for exploratory and hypothesis driven
analyses.
1. A primary advantage to FDA is the ability to assess
continuous data that change over time without having
to reduce the signal into discrete variables.
2. By using FDA and representing each curve as a
function, it is possible to use a functional analogue of
traditional methods without the problem of multicollinearity.

21. Functional data analysis gives a collection of techniques to
model data from dynamic systems• possibly governed by differential equations
• in terms of some set of basis functions

22. REFERENCE:

Elizabeth Crane, David Childers, Geoffrey Gerstner and
Edward Rothman, “Functional Data Analysis for
Biomechanics”.