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# Functional Data Analysis

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A Brief Presentation about Functional Data Analysis.

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### Functional Data Analysis

1. 1. FUNCTIONAL DATA ANALYSIS -By Nitisha Agarwal (Junior Statistical Consultant) COSTARCH ANALYTICAL CONSULTING P. LTD.
2. 2. CONTENT: Functional Data  Need of FDA  Step by Step Procedure  Example  Conclusion  Reference 
3. 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. 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. 5. PROCEDURE: Transforming data into functions,  Smoothing functions,  Registering functions,  Analysis. 
6. 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. 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. 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. 9. Alternative Approach:
10. 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. 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”.
12. 12. 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.
13. 13. Example
14. 14. 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-
15. 15. 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.
16. 16. 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.
17. 17. 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
18. 18. REFERENCE:  Elizabeth Crane, David Childers, Geoffrey Gerstner and Edward Rothman, “Functional Data Analysis for Biomechanics”.