1. DETC2008-49101 ASME Computers and Information in Engineering Conference ONE AND TWO DIMENSIONAL DATA ANALYSIS USING BEZIER FUNCTIONS P.Venkataraman Rochester Institute of Technology Department of Mechanical Engineering CTESA: Computational Technologies for Engineering Sciences Applications August 3 - 6, 2008, New York City NY, USA

2. DETC2008-49101 ONE AND TWO DIMENSIONAL DATA ANALYSIS USING BEZIER FUNCTIONS P. Venkataraman Mechanical EngineeringRochester Institute of Technology CTESA: CIE-2-3 Numerical method for modeling Presentation Outline 1. Motivation 2. Bezier Function 3. Data Fitting 4. Computational Resource 5. Example 1 Example 2 Example 4 6. Conclusions 1/20

3. DETC2008-49101 ONE AND TWO DIMENSIONAL DATA ANALYSIS USING BEZIER FUNCTIONS P. Venkataraman Mechanical EngineeringRochester Institute of Technology CTESA: CIE-2-3 Numerical method for modeling Motivation 2/20 Current Status In prior presentations of this technical committee it has been shown that it is possible to obtain explicit solutions in polynomial form for Linear or nonlinear, Single or coupled, Ordinary or partial, Differential equations using Bezier functions

4. DETC2008-49101 ONE AND TWO DIMENSIONAL DATA ANALYSIS USING BEZIER FUNCTIONS P. Venkataraman Mechanical EngineeringRochester Institute of Technology CTESA: CIE-2-3 Numerical method for modeling Motivation 3/20 Inverse Problem One definition of an inverse problem and its solution is Given a sequence of data Establish the differential equation whose solution Is represented by the data

5. DETC2008-49101 ONE AND TWO DIMENSIONAL DATA ANALYSIS USING BEZIER FUNCTIONS P. Venkataraman Mechanical EngineeringRochester Institute of Technology CTESA: CIE-2-3 Numerical method for modeling Motivation 4/20 This Paper This paper approaches the solution of the inverse problem in two steps Given a sequence of data {x i, y i } Find a function that best fits the data – y(x) Then establish the coefficients of the differential system that the function will belong too: This paper addresses only the first step

6. DETC2008-49101 ONE AND TWO DIMENSIONAL DATA ANALYSIS USING BEZIER FUNCTIONS P. Venkataraman Mechanical EngineeringRochester Institute of Technology CTESA: CIE-2-3 Numerical method for modeling Bezier Function 5/20 Description p : parameter Bernstein basis Number of vertices: 5 Order of the function : 4

7. DETC2008-49101 ONE AND TWO DIMENSIONAL DATA ANALYSIS USING BEZIER FUNCTIONS P. Venkataraman Mechanical EngineeringRochester Institute of Technology CTESA: CIE-2-3 Numerical method for modeling Bezier Function 6/20 Matrix Description

8. DETC2008-49101 ONE AND TWO DIMENSIONAL DATA ANALYSIS USING BEZIER FUNCTIONS P. Venkataraman Mechanical EngineeringRochester Institute of Technology CTESA: CIE-2-3 Numerical method for modeling Data Fitting 7/20 Problem Definition For a selected order of the Bezier function (n) Given a set of (m) vector data y a,i, or [Y], find the coefficient matrix, [B] so that the corresponding data set y b,i, [Y B ] produces the least sum of the squared error Minimize FOC:

9. DETC2008-49101 ONE AND TWO DIMENSIONAL DATA ANALYSIS USING BEZIER FUNCTIONS P. Venkataraman Mechanical EngineeringRochester Institute of Technology CTESA: CIE-2-3 Numerical method for modeling Data Fitting 8/20 Data Decoupling The matrix definition for the Bezier function is It can be recognized as And can be decoupled as

10. DETC2008-49101 ONE AND TWO DIMENSIONAL DATA ANALYSIS USING BEZIER FUNCTIONS P. Venkataraman Mechanical EngineeringRochester Institute of Technology CTESA: CIE-2-3 Numerical method for modeling Computational Resources 9/20 Software used: MATLAB 2006b for plots and calculations This work is independent of any language/software/platform 32-bit architecture required limiting the order of the Bezier functions to 20 Standard data statistics is used for comparison

11. DETC2008-49101 ONE AND TWO DIMENSIONAL DATA ANALYSIS USING BEZIER FUNCTIONS P. Venkataraman Mechanical EngineeringRochester Institute of Technology CTESA: CIE-2-3 Numerical method for modeling Examples 10/20 Example 1: Smooth Data at Equidistant Intervals Example 2: Rough Data at Arbitrary Intervals Example 4: Unorganized Data Three of the five examples are presented

12. DETC2008-49101 ONE AND TWO DIMENSIONAL DATA ANALYSIS USING BEZIER FUNCTIONS P. Venkataraman Mechanical EngineeringRochester Institute of Technology CTESA: CIE-2-3 Numerical method for modeling Examples 11/20 Example 1 The data is generated at equidistant intervals of the independent variable (x) The dependent variable (y) values are generated using a smooth function There are 101 data pairs. Best order: 15 Error x: 1.3e-06 Error y: 8.8e-08

13. DETC2008-49101 ONE AND TWO DIMENSIONAL DATA ANALYSIS USING BEZIER FUNCTIONS P. Venkataraman Mechanical EngineeringRochester Institute of Technology CTESA: CIE-2-3 Numerical method for modeling Examples 12/20 Example 1- statistics Comparison of x-dataComparison of y-data

14. DETC2008-49101 ONE AND TWO DIMENSIONAL DATA ANALYSIS USING BEZIER FUNCTIONS P. Venkataraman Mechanical EngineeringRochester Institute of Technology CTESA: CIE-2-3 Numerical method for modeling Examples 13/20 Example 1 - Coefficients Bezier Coefficient Values: The data in Example 1 can be reproduced by a super continuous 15 th order Bezier function, whose derivatives can be easily established by known calculations

15. DETC2008-49101 ONE AND TWO DIMENSIONAL DATA ANALYSIS USING BEZIER FUNCTIONS P. Venkataraman Mechanical EngineeringRochester Institute of Technology CTESA: CIE-2-3 Numerical method for modeling Examples 14/20 Example 1 – Explicit Polynomial

16. DETC2008-49101 ONE AND TWO DIMENSIONAL DATA ANALYSIS USING BEZIER FUNCTIONS P. Venkataraman Mechanical EngineeringRochester Institute of Technology CTESA: CIE-2-3 Numerical method for modeling Examples 15/20 Example 2 Random values between specified limits are used to construct values for x and y. It is then sorted in ascending order. Best order: 13 Range x: Error sum : 38.96 Range y: Error sum: 3.31

17. DETC2008-49101 ONE AND TWO DIMENSIONAL DATA ANALYSIS USING BEZIER FUNCTIONS P. Venkataraman Mechanical EngineeringRochester Institute of Technology CTESA: CIE-2-3 Numerical method for modeling Examples 16/20 Example 2- statistics Data statistics : x Data statistics: y

18. DETC2008-49101 ONE AND TWO DIMENSIONAL DATA ANALYSIS USING BEZIER FUNCTIONS P. Venkataraman Mechanical EngineeringRochester Institute of Technology CTESA: CIE-2-3 Numerical method for modeling Examples 17/20 Example 4 In this example the Adjusted Closing values of the Dow Jones Industrial Average between May 17 and December 18, 2007, is used for the original data Best order: 20 (maximum) Almost all data points are within a standard deviation of the Bezier representation

19. DETC2008-49101 ONE AND TWO DIMENSIONAL DATA ANALYSIS USING BEZIER FUNCTIONS P. Venkataraman Mechanical EngineeringRochester Institute of Technology CTESA: CIE-2-3 Numerical method for modeling Examples 18/20 Example 4- statistics Statistics: y data The Bezier representation preserves the average value of the data

20. DETC2008-49101 ONE AND TWO DIMENSIONAL DATA ANALYSIS USING BEZIER FUNCTIONS P. Venkataraman Mechanical EngineeringRochester Institute of Technology CTESA: CIE-2-3 Numerical method for modeling Conclusions 19/20 Bezier functions are easy to incorporate and can track regular and unpredictable data very well The Bezier functions have excellent blending and smoothing properties High order of functions can be useful in capturing the data content and underlying behavior The mean of the Bezier data is the same as the mean of the original data Bezier functions naturally decouples the independent and the dependent variables

21. DETC2008-49101 ONE AND TWO DIMENSIONAL DATA ANALYSIS USING BEZIER FUNCTIONS P. Venkataraman Mechanical EngineeringRochester Institute of Technology CTESA: CIE-2-3 Numerical method for modeling Conclusions Thank You! Questions?