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Unformatted text preview: Next Previous Section 4.3: Linear Regression Section 4.3: Linear Regression 11 5 10 15 20 25 30 35 40 45 50 4 6 8 10 12 14 x y x y 5.1 0.6 5.9 2.5 6.5 13 8 21 9.5 25 10.5 31 12 45 data Minimize the sum of the distances between best fit line and the data. Section 4.3: Linear Regression (Curve Fitting) Next Previous Section 4.3: Linear Regression Section 4.3: Linear Regression 22 Paired Data x1,y 1 d 1 d 2 d 3 y = mx + b d 1 = y1  y(x1) = y1 – ( mx1 + b) Note that difference between actual and predicted y value at point 1 is given by: Or in general, for all points, i, we can write: d i = yi  y(xi) = yi  (mxi + b) We now sum the squares of all the differences di (assume n pairs of data) Straight line to be “fit” to the data y x Straight line & y = mx + b Next Previous Section 4.3: Linear Regression Section 4.3: Linear Regression 33 Minimizing differences: least squares regression x1,y 1 d 1 d 2 d 3 y=mx + b y x Regression of y upon x ∑ + + + = i 2 i 2 i i i 2 i 2 i x m bmx 2 y mx 2 b by 2 y b m f ) ( ) , ( ® all xi and yi are known, ® only m and b are unknown ® minimize f(m,b) with respect to m and b Solving eqns. 1 and 2 simultaneously for m and b , gives: ∑ ∑ ∑ ∑ = i 2 i i i i x x x x y y x m ) ( x m y b = n y y i ∑ = n x x i ∑ = where, Line of regression of y upon x & y = mx + b n = number of data pairs ( 29 ( 29 x m 2 x b 2 y x 2 m b m f m b m f 2 i i i i = + + = ∂ ∂ ⇒ = ∂ ∂ ∑ ∑ ∑ ) ( , , eqn. 1 ( 29 ( 29 x m 2 nb 2 y 2 b b m f b b m f i i = + + = ∂ ∂ ⇒ = ∂ ∂ ∑ ∑ , , eqn. 2 Next Previous Section 4.3: Linear Regression Section 4.3: Linear Regression 44 ∑ ∑ = 2 i i i x x y y x x m ) ( ) )( ( x m y b = Both sets of equations give the same results. Alternate regression equations In addition, by substituting our expression for b, back into the equation y = mx + b, we can write the best fit equation as: ( 29 x x m y y = An alternate set of equations for calculating m and b, which may be used equally well are: where x and y are the averages of the x and y data given previously. Next...
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This note was uploaded on 05/19/2008 for the course ENGR 25 taught by Professor All during the Spring '08 term at Lehigh University .
 Spring '08
 ALL

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