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Unformatted text preview: Leastsquares line Polynomial leastsquares General linear models General Linear LeastSquares Dhavide Aruliah UOIT MATH 2070U c D. Aruliah (UOIT) General Linear LeastSquares MATH 2070U 1 / 34 Leastsquares line Polynomial leastsquares General linear models General Linear LeastSquares 1 Leastsquares fitting with straight line (leastsquares line) 2 Leastsquares fitting with polynomials (polynomial leastsquares) 3 Leastsquares fitting of general linear models c D. Aruliah (UOIT) General Linear LeastSquares MATH 2070U 2 / 34 Leastsquares line Polynomial leastsquares General linear models Data approximation problem Data approximation problem Given n data points { ( x 1 , y 1 ) , ( x 2 , y 2 ) , . . . , ( x n , y n ) } , determine a function e f that approximates the data, i.e., e f ( x k ) ’ y k ( k = 1: n ) .54321 1 2 3 4 5321 1 2 3 c D. Aruliah (UOIT) General Linear LeastSquares MATH 2070U 4 / 34 Leastsquares line Polynomial leastsquares General linear models Leastsquares fit with straight line (cont.) 1 2 3 4 5 6 7 8 9 1021 1 2 3 4 5 Least squares line fit c D. Aruliah (UOIT) General Linear LeastSquares MATH 2070U 5 / 34 Leastsquares line Polynomial leastsquares General linear models Leastsquares fit with straight line Let e f ( x ) = a + a 1 x Given { ( x k , y k ) } n k = 1 , exact interpolation conditions are a + a 1 x k = y k ( k = 1: n ) Write system of equations to fit in matrix form 1 · a + x 1 · a 1 = y 1 1 · a + x 2 · a 1 = y 2 . . . 1 · a + x n · a 1 = y n ⇒ 1 x 1 1 x 2 1 x 3 . . . . . . 1 x n  {z } [ V ] a a 1  {z } { a } = y 1 y 2 y 3 . . . y n  {z } { y } [ V ] ∈ R n × 2 { a } ∈ R 2 × 1 { y } ∈ R n × 1 Overdetermined system to approximate solution is [ V ] { a } = { y } c D. Aruliah (UOIT) General Linear LeastSquares MATH 2070U 6 / 34 Leastsquares line Polynomial leastsquares General linear models Normal equations for leastsquares line From before, normal equations are n ∑ k = 1 [ a + a 1 x k y k ] = n ∑ k = 1 a x k + a 1 x 2 k y k x k = Normal equations in matrix form: [ V ] T [ V ] { a } = [ V ] T { y } In practice, leastsquares approximation found directly from [ V ] { a } = { y } c D. Aruliah (UOIT) General Linear LeastSquares MATH 2070U 7 / 34 Leastsquares line Polynomial leastsquares General linear models Solution of the normal equations For overdetermined linear system of equations [ V ] { a } = { y } , normal equations [ V ] T [ V ] { a } = [ V ] T { y } solved directly using backslash in MATLAB c D. Aruliah (UOIT) General Linear LeastSquares MATH 2070U 8 / 34 Leastsquares line Polynomial leastsquares General linear models Example: biomechanical data k σ k e k k σ k e k 1 0.00 0.00 5 0.31 0.23 2 0.06 0.08 6 0.47 0.25 3 0.14 0.14 7 0.60 0.28 4 0.25 0.20 8 0.70 0.29 c.f., Problem 3.3, Figure 3.16 in Quarteroni & Saleric....
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 Winter '10
 aruliahdhavidhe
 Math, Least Squares, Regression Analysis, Ordinary least squares, Linear least squares, D. Aruliah

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