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1 / 11Lesson 16Least SquaresC. L. CoxClemson UniversityC. L. CoxLesson 16, Least Squares
2 / 11MATH 3650-002 Numerical Methods for Engineers, March 27, 2018ObjectivesTo explore:1Curve fitting by linear least squares regression2Curve fitting by transformation to linear least squaresregression3Curve fitting by polynomial regressionC. L. CoxLesson 16, Least Squares
3 / 11MATH 3650-002 Numerical Methods for Engineers, March 27, 2018In Lesson 13, we distinguished betweeninterpolationandcurvefitting, saying that:Curve fitting is a process by which afunction is constructed tobest fit, insome sense, a given set of datapoints.The goal is to capture thetrendof thedata, for example the line in the plotto the rightfitsthe 5 data points.A fitting curve is normally not aninterpolant.xy(x5, y5)(x1, y1)(x2, y2)(x3, y3)(x4, y4)We will quantify the meaning ofbest fit, with the concept ofleast squares regression.Inlinear regression, we fit the data (i.e.(xi, yi), i= 1. . . n)with a linear polynomial of the formy=a0+a1x.C. L. CoxLesson 16, Least Squares
4 / 11MATH 3650-002 Numerical Methods for Engineers, March 27, 2018Inlinear least squares regression(LLSR) the coefficientsa0anda1arechosen to minimize the sum of theerror at each point, i.e.e(a0, a1) =nXi=1[yi-(a0+a1xi)]2Theresidual at theithpointisri=yi-(a0+a1xi)xyy=a1+a0(x5, y5)|r5||r1|(x1, y1)(x2, y2)|r2|(x3, y3)|r3|(x4, y4)|r4|So LLSR minimizes the2

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