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Regression
The document discusses the concepts of regression and interpolation, highlighting their differences and applications. It explains how to fit a polynomial to data, particularly focusing on the best-fit lines and parabolas, and addresses the factors influencing the choice between interpolation and regression based on data accuracy. Additionally, it provides step-by-step examples and mathematical formulations for determining the best-fit equations.
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Regression
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- 2. Regression Mohammad Tawfik #WikiCourses http://WikiCourses.WikiSpaces.com Objectives • Understandingthe difference between regression and interpolation • Knowing how to “best fit” a polynomial into a set of data • Knowing how to use a polynomial to interpolate data
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- 7. Regression Mohammad Tawfik #WikiCourses http://WikiCourses.WikiSpaces.com Curve Fitting •If the data measured is of high accuracy and it is required to estimate the values of the function between the given points, then, polynomial interpolation is the best choice. • If the measurements are expected to be of low accuracy, or the number of measured points is too large, regression would be the best choice.
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- 9. Regression Mohammad Tawfik #WikiCourses http://WikiCourses.WikiSpaces.com Why Regression? •Measurements that we get from real situations are not usually consistent! • The number of “pieces” of information that we can get about a certain project is HUGE • You can NEVER measure exact values!
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- 11. Regression Mohammad Tawfik #WikiCourses http://WikiCourses.WikiSpaces.com But, howto get the equation of a line that is “good” for all the data you have!
- 12. Regression Mohammad Tawfik #WikiCourses http://WikiCourses.WikiSpaces.com Equation ofa Line: Revision xaay 10 If you have two points 1101 xaay 2102 xaay 2 1 1 0 2 1 1 1 y y a a x x
- 13. Regression Mohammad Tawfik #WikiCourses http://WikiCourses.WikiSpaces.com Solving forthe constants! 12 12 1 12 2112 0 & xx yy a xx yxyx a
- 14. Regression Mohammad Tawfik #WikiCourses http://WikiCourses.WikiSpaces.com What ifI have more than two points?
- 15. Regression Mohammad Tawfik #WikiCourses http://WikiCourses.WikiSpaces.com For everypoint nn xaay xaay xaay 10 2102 1101 1 02 1 2 1 1 1 1 a a x x x y y y nn
- 16. Regression Mohammad Tawfik #WikiCourses http://WikiCourses.WikiSpaces.com So, wemay write the error vector 1 02 1 2 1 2 1 1 1 1 a a x x x y y y e e e nnn 1*22*1*1* aAye nnn
- 17. Regression Mohammad Tawfik #WikiCourses http://WikiCourses.WikiSpaces.com Square theerror aAye 2 eaAAayAa aAyyyee TTTT TTT
- 18. Regression Mohammad Tawfik #WikiCourses http://WikiCourses.WikiSpaces.com Note: thisis a scalar equation! 2 eaAAayAa aAyyyee TTTT TTT yAaaAy TTT aAAayAayye TTTTT 2 2
- 19. Regression Mohammad Tawfik #WikiCourses http://WikiCourses.WikiSpaces.com Note: thisis a quadratic equation in {a}!!! aAAayAayye TTTTT 2 2 To minimize the error in the above equation, we need to differentiate with respect to the parameters 022 2 aAAyA ad ed TT
- 20. Regression Mohammad Tawfik #WikiCourses http://WikiCourses.WikiSpaces.com Solving theequation We get: 022 2 aAAyA ad ed TT 1**21*22**2 n T nn T n yAaAA 1*21*22*2 yaA yAa 1
- 21. Regression Mohammad Tawfik #WikiCourses http://WikiCourses.WikiSpaces.com Example • Ifyou are given the data. • Find the equation of the “best-fit” line. y=a1+a2x yx 0.51 2.52 23 44 3.55 66 5.57
- 22. Regression Mohammad Tawfik #WikiCourses http://WikiCourses.WikiSpaces.com Solution 5.5 6 5.3 4 2 5.2 5.0 71 61 51 41 31 21 11 1 0 a a 5.5 6 5.3 4 2 5.2 5.0 & 71 61 51 41 31 21 11 yA
- 23. Regression Mohammad Tawfik #WikiCourses http://WikiCourses.WikiSpaces.com Solution 14028 287 71 61 51 41 31 21 11 7654321 1111111 AA T
- 24. Regression Mohammad Tawfik #WikiCourses http://WikiCourses.WikiSpaces.com Solution 5.119 24 5.5 6 5.3 4 2 5.2 5.0 7654321 1111111 yA T
- 25. Regression Mohammad Tawfik #WikiCourses http://WikiCourses.WikiSpaces.com Solution 5.119 24 14028 287 1 0 a a yAaAA TT 8393.0 0714.0 1 0 a a 0714.08393.0 xy
- 26. Regression Mohammad Tawfik #WikiCourses http://WikiCourses.WikiSpaces.com Example • Ifyou are given the data. • Find the equation of the “best-fit” parabola. y=a0+a1x+a2x2 yx 0.51 2.52 23 44 3.55 66 5.57
- 27. Regression Mohammad Tawfik #WikiCourses http://WikiCourses.WikiSpaces.com Solution 5.5 6 5.3 4 2 5.2 5.0 49 36 25 16 9 4 1 71 61 51 41 31 21 11 2 1 0 a a a 5.5 6 5.3 4 2 5.2 5.0 & 49 36 25 16 9 4 1 71 61 51 41 31 21 11 yA
- 28. Regression Mohammad Tawfik #WikiCourses http://WikiCourses.WikiSpaces.com Solution 4676784140 78414028 140287 49 36 25 16 9 4 1 71 61 51 41 31 21 11 49362516941 7654321 1111111 AA T
- 29. Regression Mohammad Tawfik #WikiCourses http://WikiCourses.WikiSpaces.com Solution 5.665 5.119 24 5.5 6 5.3 4 2 5.2 5.0 49362516941 7654321 1111111 yA T
- 30. Regression Mohammad Tawfik #WikiCourses http://WikiCourses.WikiSpaces.com Solution yAaAA TT 0298.0 0774.1 2857.0 2 1 0 a a a 2857.00774.10298.0 2 xxy