6.6 - 6.6 Applications of Least Squares Many applications...

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6.6 Applications of Least Squares Many applications in science and engineering, e.g. for statistical data analysis Instead of notation Avectorx = vector b , write X vector β = vectory X : design matrix vector β : parameter vector vectory : observation vector L.S. curve fitting Basic idea: Given data points ( x 1 , y 1 ) , · · · , ( x n , y n ), “fit” the data with a simple curve simplest case is to fit data with straight line | | | x 1 x j x n Data Point ( x j , y j ) ( x j , β 0 + β 1 x j ) Residual y = β 0 + β 1 x Point on line Figure 1: Fitting a line to experimental data find constants β 0 , β 1 , so line y = β 0 + β 1 x fits data in the sense that at each point x j predicted value β 0 + β 1 x j y j , y j observed value. 1
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L.S. Line y = β 0 + β 1 x minimizes sum of squares of residuals minimizes ( β 0 + β 1 x 1 y 1 ) 2 + · · · + ( β 0 + β 1 x n y n ) 2 β 0 + β 1 x called line of regression of y on x β 0 , β 1 called linear regression coefficients. solve β 0 + β 1 x 1 = y 1 β 0 + β 1 x 2 = y 2 . . . β 0 + β 1 x n = y n or X vector β = vectory X = 1 x 1 1 x 2 . . . . . . 1 x n , vector β = bracketleftBigg β 0 β 1 bracketrightBigg , vector y = y 1 . . . y n Since columns of X are linearly independent (for x i ’s distinct), can solve normal equations X T X vector β = X T vectory This minimizes distance between X vector β and vectory , bardbl X vector β vector y bardbl 2
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EX: Find the equation y = β 0 + β 1 x of the least-
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