ECE257 Numerical Methods andECE257 Numerical Methods andScientific ComputingScientific ComputingCurve FittingCurve FittingECE 257 Numerical Methods and Scientific ComputingFall 2004Lecture 23John A. ChandyDept. of Electrical and Computer EngineeringUniversity of ConnecticutTodayToday’’s class:s class:••Curve Fitting IntroductionCurve Fitting Introduction••Least-Squares RegressionLeast-Squares Regression––Linear RegressionLinear Regression––Polynomial RegressionPolynomial Regression––Multiple Linear RegressionMultiple Linear RegressionECE 257 Numerical Methods and Scientific ComputingFall 2004Lecture 23John A. ChandyDept. of Electrical and Computer EngineeringUniversity of ConnecticutCurve FittingCurve Fitting••Given a set of discrete points how do you fill in theGiven a set of discrete points how do you fill in thepoints in between to form a continuum?points in between to form a continuum?––Approximation or regressionApproximation or regression••Find a simple function that represents the trend of theFind a simple function that represents the trend of thecurve given that the data may have measurement errorcurve given that the data may have measurement erroror or ““noisenoise””––InterpolationInterpolationThe data is exact, so you need to find a function thatThe data is exact, so you need to find a function thatpasses through all the given pointspasses through all the given pointsECE 257 Numerical Methods and Scientific ComputingFall 2004Lecture 23John A. ChandyDept. of Electrical and Computer EngineeringUniversity of ConnecticutCurve FittingCurve Fittinga)a)Least-Squares RegressionLeast-Squares Regressionb)b)Linear InterpolationLinear Interpolationc)c)Curvilinear InterpolationCurvilinear InterpolationECE 257 Numerical Methods and Scientific ComputingFall 2004Lecture 23John A. ChandyDept. of Electrical and Computer EngineeringUniversity of ConnecticutLeast-Squares RegressionLeast-Squares Regression••Linear RegressionLinear Regression••Polynomial RegressionPolynomial Regression••QuadraticQuadratic••CubicCubic••Multiple Variable Least-SquaresMultiple Variable Least-SquaresECE 257 Numerical Methods and Scientific ComputingFall 2004Lecture 23John A. ChandyDept. of Electrical and Computer EngineeringUniversity of ConnecticutReview of Basic StatisticsReview of Basic Statistics∑==NiiyNy1_1Given a set of data, say, y1, y2,……….yN. € sy=StN −1 where St= yi− y ( )2i=1N∑ € sy2 =St(N −1)€ C.V =syy••Arithmetic Mean:Arithmetic Mean:••Standard Deviation:Standard Deviation:••Variance:Variance:••Coefficient of Variance:Coefficient of Variance:ECE 257 Numerical Methods and Scientific ComputingFall 2004Lecture 23John A. ChandyDept. of Electrical and Computer EngineeringUniversity of ConnecticutBasic StatisticsBasic Statistics••Standard DeviationStandard Deviation€ sy=yi− y ( )2∑N −1=yi2∑− 2y yi∑+ y 2∑N −1=yi2∑− 2y Ny ( )+ Ny 2N −1=yi2∑− Ny 2N −1=yi2∑−yi∑( )2NN −1ECE 257 Numerical Methods and Scientific ComputingFall 2004Lecture 23John A. ChandyDept. of Electrical and Computer EngineeringUniversity of ConnecticutBasic StatisticsBasic StatisticsMeasurement of the coefficient of thermal expansionof structural steel [×10−6 in/(in⋅°F)]535666766706592663366276598649966216445645164036703654266596624673366626721639665646435662565556655647866216543639965526667632564956775655464856....................................Mean, standard deviation, variance, etc.ECE 257 Numerical Methods and Scientific ComputingFall 2004Lecture 23John A. ChandyDept. of Electrical and Computer EngineeringUniversity of Connecticut 1 6.485 0.007173 42.055 2 6.554 0.000246 42.955 3 6.775 0.042150 45.901 4 6.495 0.005579 42.185 5 6.325 0.059875 40.006 6 6.667 0.009468 44.449 7 6.552 0.000313 42.929 8 6.399 0.029137 40.947 9 6.543 0.000713 42.811 10 6.621 0.002632 43.838 11 6.478 0.008408 41.964 12 6.655 0.007277 44.289 13 6.555 0.000216 42.968 14 6.625 0.003059 43.891 15 6.435 0.018143 41.409 16 6.564 0.000032 43.086 17 6.396 0.030170 40.909 18 6.721 0.022893 45.172 19 6.662 0.008520 44.382 20 6.733 0.026669 45.333 21 6.624 0.002949 43.877 22 6.659 0.007975 44.342 23 6.542 0.000767 42.798 24 6.703 0.017770 44.930 25 6.403 0.027787 40.998 26 6.451 0.014088 41.615 27 6.445 0.015549 41.538 28 6.621 0.002632 43.838 29 6.499 0.004998 42.237 30 6.598 0.000801 43.534 31 6.627 0.003284 43.917 32 6.633 0.004008 43.997 33 6.592 0.000498 43.454 34 6.670 0.010061 44.489 35 6.667 0.009468 44.449 36 6.535 0.001204 42.706 ∑ 236.509 0.406514 1554.198( )2i2i i y yyy i −ECE 257 Numerical Methods and Scientific ComputingFall 2004Lecture 23John A. ChandyDept. of Electrical and Computer EngineeringUniversity of ConnecticutCoefficient of Thermal ExpansionCoefficient of Thermal Expansion( )( )%.%..%..../).(./.....64110056976107770100ysvc01161470354065140 353650923619815541nnyys 1077701364065140 1nSs4065140yyS5697636509236nyyy22i2i2yty2iti=×=×===−=−−==−=−==−====∑ ∑∑∑Sum of the square of residualsStandard deviationVarianceCoefficient of variationMeanECE 257 Numerical Methods and Scientific ComputingFall 2004Lecture 23John A. ChandyDept. of Electrical and Computer EngineeringUniversity of Connecticut A histogram used to depict the distribution of data For large data set, the histogram often approaches thenormal distributionNormalDistributionHistogramHistogramECE 257 Numerical Methods and Scientific ComputingFall 2004Lecture 23John A. ChandyDept. of Electrical and Computer EngineeringUniversity of Connecticutxy0yixiLinear Least-SquaresLinear Least-SquaresApproximationApproximationy = a0 + a1x€ e = yi− a0− a1xiECE 257 Numerical Methods and Scientific ComputingFall 2004Lecture 23John A. ChandyDept. of Electrical and Computer EngineeringUniversity of ConnecticutLinear