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Regression

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Mohammad Tawfik
Mohammad Tawfik

How to best-fit a function to a given set of data? Why is this different from interpolation? Do you have more than one independent variable? Learn about regression and least squares method! #WikiCourses #Num001 https://wikicourses.wikispaces.com/Topic+Regression

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Regression Mohammad Tawfik #WikiCourses http://WikiCourses.WikiSpaces.com Regression/Curve Fitting Mohammad Tawfik
Regression Mohammad Tawfik #WikiCourses http://WikiCourses.WikiSpaces.com Objectives • Understanding the 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
Regression Mohammad Tawfik #WikiCourses http://WikiCourses.WikiSpaces.com Measured Data
Regression Mohammad Tawfik #WikiCourses http://WikiCourses.WikiSpaces.com Polynomial Fit!
Regression Mohammad Tawfik #WikiCourses http://WikiCourses.WikiSpaces.com Line Fit!
Regression Mohammad Tawfik #WikiCourses http://WikiCourses.WikiSpaces.com Which is better?
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.
Regression Mohammad Tawfik #WikiCourses http://WikiCourses.WikiSpaces.com Regression
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!
Regression Mohammad Tawfik #WikiCourses http://WikiCourses.WikiSpaces.com Measured Data
Regression Mohammad Tawfik #WikiCourses http://WikiCourses.WikiSpaces.com But, how to get the equation of a line that is “good” for all the data you have!
Regression Mohammad Tawfik #WikiCourses http://WikiCourses.WikiSpaces.com Equation of a 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
Regression Mohammad Tawfik #WikiCourses http://WikiCourses.WikiSpaces.com Solving for the constants! 12 12 1 12 2112 0 & xx yy a xx yxyx a      
Regression Mohammad Tawfik #WikiCourses http://WikiCourses.WikiSpaces.com What if I have more than two points?
Regression Mohammad Tawfik #WikiCourses http://WikiCourses.WikiSpaces.com For every point nn xaay xaay xaay 10 2102 1101                                      1 02 1 2 1 1 1 1 a a x x x y y y nn 
Regression Mohammad Tawfik #WikiCourses http://WikiCourses.WikiSpaces.com So, we may 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 
Regression Mohammad Tawfik #WikiCourses http://WikiCourses.WikiSpaces.com Square the error       aAye                            2 eaAAayAa aAyyyee TTTT TTT  
Regression Mohammad Tawfik #WikiCourses http://WikiCourses.WikiSpaces.com Note: this is a scalar equation!                           2 eaAAayAa aAyyyee TTTT TTT             yAaaAy TTT                  aAAayAayye TTTTT  2 2
Regression Mohammad Tawfik #WikiCourses http://WikiCourses.WikiSpaces.com Note: this is 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
Regression Mohammad Tawfik #WikiCourses http://WikiCourses.WikiSpaces.com Solving the equation 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 
Regression Mohammad Tawfik #WikiCourses http://WikiCourses.WikiSpaces.com Example • If you 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
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
Regression Mohammad Tawfik #WikiCourses http://WikiCourses.WikiSpaces.com Solution                                         14028 287 71 61 51 41 31 21 11 7654321 1111111 AA T
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
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
Regression Mohammad Tawfik #WikiCourses http://WikiCourses.WikiSpaces.com Example • If you 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
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
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
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
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

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