1. 1 Curve Fitting ( 曲線適配 )

2. Function Approximation 領域技術  Trend analysis  A process of using the pattern of data to make predictions  Function approximation (Chapters 13, 14)  Given data of points have noises, find the trend represented by data Method of least squares Minimize the residuals  Function interpolation (Chapters 15, 16)  Given precise data of points, find data between these points  Approximating function match the given data exactly 2

3. 3 Curve Fitting: Fitting a Straight Line

4. Least Square Regression  Regression analysis ( 迴歸分析 ) : Statistical technique for estimating the relationships among variables  本單元應習知技巧  Curve fitting  Statistics review  Linear least square regression  Linearization of nonlinear relationships  MATLAB 內建指令 4

5. Wind Tunnel Experiment ( 風洞實驗 ) 5 Measure air resistance as a function of velocity Curve fitting

6. Regression and Interpolation 6 (a)Least-squares regression ( 最小平方廻歸 ) (b) Linear interpolation ( 線性內插 ) (c) Curvilinear interpolation ( 曲線內插 ) Curve fitting

7. Least-squares Fit of a Straight Line v, m/s F, N 風洞實驗中，力 (N) 與速度 (m/s) 的觀測數據

8. 8  Mean ( 平均、均數 ), arithmetic mean (or average)  例： Given numbers 10, 12, 18, 6, 4, 5, and 15  Variance ( 變異數，常以  2 表示 )  Var(X )= 上例之變異數為：  Standard deviation ( 標準差，常記為  )  稱為 X 的標準差  Var(X ) 表示 X 的分散程度。 Var(X) 越小 ( 即  愈小 ) 則 X 越集 中於平均值 E(X) 。反之， Var(X) 越大 ( 即  愈大 ) 則 X 越散開 基本統計 (1/3)

9. 9  自一大小為 N 的母體抽出一組隨機樣本 則 樣本平均數 本身亦為隨機變數， 有其機率分配  Sample variance ( 樣本變異數 ) ( 真正的  2 未知時可用 )  Unbiased estimate of  2 基本統計 (2/3) ：樣本平均數的抽樣分配 抽自無限母體抽自有限母體

10. 基本統計 (3/3) ：書例 10 Measurement of the coefficient of thermal expansion of structural steel [  10  6 in/(in  F)] Mean, standard deviation, variance, etc.

11. Basic Statistics  Arithmetic mean  Standard deviation about the mean  Variance (spread)  Coefficient of variation (c.v.) 11

12. 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

13. Coefficient of Thermal Expansion Sum of the square of residuals Standard deviation Variance Coefficient of variation Mean

14. Histogram ( 直方圖 )  A histogram depicts the distribution of data  For large data set, the histogram often approaches the normal distribution 14 Normal Distribution

15. Histogram  書 Sec.14.1.3 15

16. Regression and Residual

17. Linear Regression  Fitting a straight line to observations Small residual errors Large residual errors e i : residual or error

18. Least Squares ( 最小平方 ) Approximation  Minimizing residuals (errors)  minimum average error  minimum absolute error  minimax error (minimizing the maximum error)  least squares (linear, quadratic, ….) 18

19. Minimize the maximum error Minimize sum of errors Minimize sum of absolute errors

20. Linear Least Squares  Given  Minimize total square-error  Straight line approximation  Not likely to pass all points if n > 2 20

21. Linear Least Squares  Given  Total square-error function: Sum of the squares of the residuals  Minimizing square-error S r (a 0, a 1 ) 21 Solve for (a 0, a 1 )

22.  Minimize  Normal equation y = a 0 + a 1 x Linear Least Squares

23. Advantage of Least Squares  Positive differences do not cancel negative differences  Differentiation is straightforward  Weighted differences  Small differences become smaller and large differences are magnified 23

24. Linear Least Squares  Use sum( ) in MATLAB 24

25. 中文書範例 14.4  求最配適的直線  ( 請同學現場操作 ) 25

26. Correlation Coefficient ( 相關係數 )  Sum of squares of the residuals with respect to the mean  Sum of squares of the residuals with respect to the regression line  Coefficient of determination  Correlation coefficient 26 完美配適： S r = 0, r 2 = 1 ( 此直線能 100% 解釋數據的變異性 ) 若 r 2 =0 則 S t = S r ，代表配適未改善

27. Correlation Coefficient  Alternative formulation of correlation coefficient  More convenient for computer implementation 27

28. Standard Error of the Estimate  If the data spread about the line is normal  “Standard deviation” for the regression line Standard error of the estimate No error if n = 2 (a 0 and a 1 ) 下標 y/x 代表此誤差是針對某一 x 的預測值所對應的 y

29. Linear Regression Reduce the Spread of Data 29 Spread of data around the mean Spread of data around the best-fit line Normal distributions

30. Standard Deviation for Regression Line 30 S y/x SySy S y : Spread around the mean S y/x : Spread around the regression line

31. 中文書範例 14.5  針對前面的範例， 計算總標準差、標 準誤差估計以及相 關係數  ( 請同學現場操作 )  r 值為何？代表的物理意義為？ 31

32. 另例 (1/2)

33. Standard deviation about the mean Standard error of the estimate Correlation coefficient 另例 (2/2)

34. function [a, r2] = linregr(x,y) % linregr: linear regression curve fitting % [a, r2] = linregr(x,y): Least squares fit of straight % line to data by solving the normal equations % input: % x = independent variable % y = dependent variable % output: % a = vector of slope, a(1), and intercept, a(2) % r2 = coefficient of determination n = length(x); if length(y)~=n, error('x and y must be same length'); end x = x(:); y = y(:); % convert to column vectors sx = sum(x); sy = sum(y); sx2 = sum(x.*x); sxy = sum(x.*y); sy2 = sum(y.*y); a(1) = (n*sxy-sx*sy)/(n*sx2-sx^2); a(2) = sy/n-a(1)*sx/n; r2 = ((n*sxy-sx*sy)/sqrt(n*sx2-sx^2)/sqrt(n*sy2-sy^2))^2; % create plot of data and best fit line xp = linspace(min(x),max(x),2); yp = a(1)*xp+a(2); plot(x,y,'o',xp,yp) grid on

35. Modified MATLAB M-File 35

36. » x=1:7 x = 1 2 3 4 5 6 7 » y=[0.5 2.5 2.0 4.0 3.5 6.0 5.5] y = 0.5000 2.5000 2.0000 4.0000 3.5000 6.0000 5.5000 » s=linear_LS(x,y) a0 = 0.0714 a1 = 0.8393 x y (a0+a1*x) (y-a0-a1*x) 1.0000 0.5000 0.9107 -0.4107 2.0000 2.5000 1.7500 0.7500 3.0000 2.0000 2.5893 -0.5893 4.0000 4.0000 3.4286 0.5714 5.0000 3.5000 4.2679 -0.7679 6.0000 6.0000 5.1071 0.8929 7.0000 5.5000 5.9464 -0.4464 err = 2.9911 Syx = 0.7734 r = 0.9318 s = 0.0714 0.8393 y =0.0714 + 0.8393 x Sum of squares of residuals S r Standard error of the estimate Correlation coefficient

37. » x=0:1:7; y=[0.5 2.5 2 4 3.5 6.0 5.5]; Linear regression y = 0.0714+0.8393x Error : S r = 2.9911 correlation coefficient : r = 0.9318

38. function [x,y] = example1 x = [ 1 2 3 4 5 6 7 8 9 10]; y = [2.9 0.5 -0.2 -3.8 -5.4 -4.3 -7.8 -13.8 -10.4 -13.9]; » [x,y]=example1; » s=Linear_LS(x,y) a0 = 4.5933 a1 = -1.8570 x y (a0+a1*x) (y-a0-a1*x) 1.0000 2.9000 2.7364 0.1636 2.0000 0.5000 0.8794 -0.3794 3.0000 -0.2000 -0.9776 0.7776 4.0000 -3.8000 -2.8345 -0.9655 5.0000 -5.4000 -4.6915 -0.7085 6.0000 -4.3000 -6.5485 2.2485 7.0000 -7.8000 -8.4055 0.6055 8.0000 -13.8000 -10.2624 -3.5376 9.0000 -10.4000 -12.1194 1.7194 10.0000 -13.9000 -13.9764 0.0764 err = 23.1082 Syx = 1.6996 r = 0.9617 s = 4.5933 -1.8570 y = 4.5933  1.8570 x r = 0.9617

39. Linear Least Square y = 4.5933  1.8570 x Error S r = 23.1082 Correlation Coefficient r = 0.9617

40. » [x,y]=example2 x = Columns 1 through 7 -2.5000 3.0000 1.7000 -4.9000 0.6000 -0.5000 4.0000 Columns 8 through 10 -2.2000 -4.3000 -0.2000 y = Columns 1 through 7 -20.1000 -21.8000 -6.0000 -65.4000 0.2000 0.6000 -41.3000 Columns 8 through 10 -15.4000 -56.1000 0.5000 » s=Linear_LS(x,y) a0 = -20.5717 a1 = 3.6005 x y (a0+a1*x) (y-a0-a1*x) -2.5000 -20.1000 -29.5730 9.4730 3.0000 -21.8000 -9.7702 -12.0298 1.7000 -6.0000 -14.4509 8.4509 -4.9000 -65.4000 -38.2142 -27.1858 0.6000 0.2000 -18.4114 18.6114 -0.5000 0.6000 -22.3720 22.9720 4.0000 -41.3000 -6.1697 -35.1303 -2.2000 -15.4000 -28.4929 13.0929 -4.3000 -56.1000 -36.0539 -20.0461 -0.2000 0.5000 -21.2918 21.7918 err = 4.2013e+003 Syx = 22.9165 r = 0.4434 s = -20.5717 3.6005 Correlation coefficient r = 0.4434 Linear Least Square: y =  20.5717 + 3.6005x Data in arbitrary order Large errors !!

41. Linear regression y =  20.5717 +3.6005x Error S r = 4201.3 Correlation r = 0.4434 !!

42. Linearization of Nonlinear Relationships  非線性關係的線性化  線性迴歸分析是為數據配適最佳直線的常用技巧，但前提是應變 數與自變數之間的關係為線性  使用迴歸分析之前，應先將數據繪出，以目測決定是否可運用線 性模型 下一單元將介紹多項式迴歸技巧  一些情境可透過 “ 轉換 ” 的方式，把數據表示成可以匹配線性迴歸 的形式 42

43. Linearization of Nonlinear Relationships

44. Untransformed power equation x vs. y Transformed data log x vs. log y

45. log : Base-10  Exponential equation  Power equation Linearization of Nonlinear Relationships

46.  Saturation-growth-rate equation  Rational function Linearization of Nonlinear Relationships

47. Example: Power Equation 47 Power equation fit along with the data x vs. y Transformed Data log x i vs. log y i y =  2 x  2

48. 中文書範例 14.6  使用對數轉換，求配適方程式 關係數  ( 請同學現場操作 ) 48

49. >> x=[10 20 30 40 50 60 70 80]; >> y = [25 70 380 550 610 1220 830 1450]; >> [a, r2] = linregr(x,y) a = 19.4702 -234.2857 r2 = 0.8805 y = 19.4702x  234.2857

50. >> x=[10 20 30 40 50 60 70 80]; >> y = [25 70 380 550 610 1220 830 1450]; >> linregr(log10(x),log10(y)) r2 = 0.9481 ans = 1.9842 -0.5620 log x vs. log y log y = 1.9842 log x – 0.5620 y = (10 –0.5620 )x 1.9842 = 0.2742 x 1.9842

51.  Least-square fit of nth-order polynomial  p = polyfit(x,y,n)  Evaluate the value of polynomial using  y = polyval(p,x) MATLAB Functions