Chapter 6: Curve Fitting - McMaster University

Chapter 6: Curve FittingTwo types of Curve Fitting Least square regressionGiven data for discrete values, derive a single Curve that represents the generaltrend of the data. When the given data exhibit a significant degree of error or noise. InterpolationGiven data for discrete values, fit a Curve or a series of curves that pass di-rectly through each of the points. When data are very I: Least Square Regression1 Simple linear RegressionFitting a straight line to a set of paired observations(x1, y1),(x2, y2), .. ,(xn, yn).Mathematical expression for the straight line (model)y=a0+a1xwherea0is the intercept, anda1is the ,measured yi,model=yi (a0+a1xi)Criterion for a best fit:minSr= mina0,a1n i=1e2i= mina0,a1n i=1(yi a0 a1xi)2 Finda0anda1:2 Sr a0= 2n i=1(yi a0 a1xi) = 0 (1) Sr a1= 2n i=1[(yi a0 a1xi)xi] = 0 (2)From (1), ni=1yi ni=1a0 ni=1a1xi= 0, orna0+n i=1xia1=n i=1yi(3)From (2), ni=1xiyi ni=1a0xi ni=1a1x2i= 0, orn i=1xia0+n i=1x2ia1=n i=1xiyi(4)(3) and (4) are called normal (3),a0=1nn i=1yi 1nn i=1xia1= y xa1where x=1n ni=1xi, y=1n ni= (4), ni=1xi(1n ni=1yi 1n ni=1xia1) + ni=1x2ia1= ni=1xiyi,a1= ni=1xiyi 1n ni=1xi ni=1yi ni=1x2i 1n( ni=1xi)2ora1=n ni=1xiyi ni=1xi ni=1yin ni=1x2i ( ni=1xi)2 Definitions:Sr=n i=1e2i=n i=1(yi a0 a1xi)2 Standard error of the estimate:Sy/x= Srn 2 Spread around the regression lineStandard deviation of data pointsSy= Stn 1= ni=1(yi y)2n 14whereSt= ni=1(yi y)2.

alent to solving a system of 3 simultaneous linear equations. † In general, to fit an m-th order polynomial y = a0 +a1x1 +a2x 2 +:::+a mx m using least-square regression is equivalent to solving a system of (m + 1) simultaneous linear equations. Standard error: Sy=x = q Sr n¡(m+1) 3 Multiple Linear Regression

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  Linear, Multiple, Multiple linear

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