1. Transformations

2. Transformations to Linearity Many non-linear curves can be put into a linear form by appropriate transformations of the either – the dependent variable Y or –some (or all) of the independent variables X 1, X 2,..., X p. This leads to the wide utility of the Linear model. We have seen that through the use of dummy variables, categorical independent variables can be incorporated into a Linear Model. We will now see that through the technique of variable transformation that many examples of non-linear behaviour can also be converted to linear behaviour.

3. Intrinsically Linear (Linearizable) Curves 1 Hyperbolas y = x/(ax-b) Linear form: 1/y = a -b (1/x) or Y =  0 +  1 X Transformations: Y = 1/y, X=1/x,  0 = a,  1 = -b

4. 2. Exponential y =  e  x =  x Linear form: ln y = ln  +  x = ln  + ln  x or Y =  0 +  1 X Transformations: Y = ln y, X = x,  0 = ln ,  1 =  = ln 

5. 3. Power Functions y = a x b Linear from: ln y = lna + blnx or Y =  0 +  1 X

6. Logarithmic Functions y = a + b lnx Linear from: y = a + b lnx or Y =  0 +  1 X Transformations: Y = y, X = ln x,  0 = a,  1 = b

7. Other special functions y = a e b/x Linear from: ln y = lna + b 1/x or Y =  0 +  1 X Transformations: Y = ln y, X = 1/x,  0 = lna,  1 = b

8. Polynomial Models y =  0 +  1 x +  2 x 2 +  3 x 3 Linear form Y =  0 +  1 X 1 +  2 X 2 +  3 X 3 Variables Y = y, X 1 = x, X 2 = x 2, X 3 = x 3

9. Exponential Models with a polynomial exponent Linear form lny =  0 +  1 X 1 +  2 X 2 +  3 X 3 +  4 X 4 Y = lny, X 1 = x, X 2 = x 2, X 3 = x 3, X 4 = x 4

10. Trigonometric Polynomial Models y =  0 +  1 cos(2  f 1 x) +  1 sin(2  f 1 x) + … +  k cos(2  f k x) +  k sin(2  f k x) Linear form Y =  0 +  1 C 1 +  1 S 1 + … +  k C k +  k S k Variables Y = y, C 1 = cos(2  f 1 x), S 2 = sin(2  f 1 x), … C k = cos(2  f k x), S k = sin(2  f k x)

11. Response Surface models Dependent variable Y and two independent variables x 1 and x 2. (These ideas are easily extended to more the two independent variables) The Model (A cubic response surface model) or Y =  0 +  1 X 1 +  2 X 2 +  3 X 3 +  4 X 4 +  5 X 5 +  6 X 6 +  7 X 7 +  8 X 8 +  9 X 9 +  where

13. The Box-Cox Family of Transformations

14. The Transformation Staircase

15. The Bulging Rule x up y up y down x down

16. Non-Linear Models Nonlinearizable models

17. Non-Linear Growth models many models cannot be transformed into a linear model The Mechanistic Growth Model Equation: or (ignoring  ) “rate of increase in Y” =

18. The Logistic Growth Model or (ignoring  ) “rate of increase in Y” = Equation:

19. The Gompertz Growth Model: or (ignoring  ) “rate of increase in Y” = Equation:

20. Example: daily auto accidents in Saskatchewan to 1984 to 1992 Data collected: 1.Date 2.Number of Accidents Factors we want to consider: 1.Trend 2.Yearly Cyclical Effect 3.Day of the week effect 4.Holiday effects

21. Trend This will be modeled by a Linear function : Y =  0 +  1 X (more generally a polynomial) Y =  0 +  1 X +  2 X 2 +  3 X 3 + …. Yearly Cyclical Trend This will be modeled by a Trig Polynomial – Sin and Cos functions with differing frequencies(periods) : Y =  1 sin(2  f 1 X) +  1 cos(2  f 2 X)  1 sin(2  f 2 X) +  2 cos(2  f 2 X) + …

22. Day of the week effect: This will be modeled using “dummy”variables :  1 D 1 +  2 D 2 +  3 D 3 +  4 D 4 +  5 D 5 +  6 D 6 D i = (1 if day of week = i, 0 otherwise) Holiday Effects Also will be modeled using “dummy”variables :

23. Independent variables X = day,D1,D2,D3,D4,D5,D6,S1,S2,S3,S4,S5, S6,C1,C2,C3,C4,C5,C6,NYE,HW,V1,V2,cd,T1, T2. Si=sin(0.017202423838959*i*day). Ci=cos(0.017202423838959*i*day). Dependent variable Y = daily accident frequency

24. Independent variables ANALYSIS OF VARIANCE SUM OF SQUARES DF MEAN SQUARE F RATIO REGRESSION 976292.38 18 54238.46 114.60 RESIDUAL 1547102.1 3269 473.2646 VARIABLES IN EQUATION FOR PACC. VARIABLES NOT IN EQUATION STD. ERROR STD REG F. PARTIAL F VARIABLE COEFFICIENT OF COEFF COEFF TOLERANCE TO REMOVE LEVEL. VARIABLE CORR. TOLERANCE TO ENTER LEVEL (Y-INTERCEPT 60.48909 ). day 1 0.11107E-02 0.4017E-03 0.038 0.99005 7.64 1. IACC 7 0.49837 0.78647 1079.91 0 D1 9 4.99945 1.4272 0.063 0.57785 12.27 1. Dths 8 0.04788 0.93491 7.51 0 D2 10 9.86107 1.4200 0.124 0.58367 48.22 1. S3 17 -0.02761 0.99511 2.49 1 D3 11 9.43565 1.4195 0.119 0.58311 44.19 1. S5 19 -0.01625 0.99348 0.86 1 D4 12 13.84377 1.4195 0.175 0.58304 95.11 1. S6 20 -0.00489 0.99539 0.08 1 D5 13 28.69194 1.4185 0.363 0.58284 409.11 1. C6 26 -0.02856 0.98788 2.67 1 D6 14 21.63193 1.4202 0.273 0.58352 232.00 1. V1 29 -0.01331 0.96168 0.58 1 S1 15 -7.89293 0.5413 -0.201 0.98285 212.65 1. V2 30 -0.02555 0.96088 2.13 1 S2 16 -3.41996 0.5385 -0.087 0.99306 40.34 1. cd 31 0.00555 0.97172 0.10 1 S4 18 -3.56763 0.5386 -0.091 0.99276 43.88 1. T1 32 0.00000 0.00000 0.00 1 C1 21 15.40978 0.5384 0.393 0.99279 819.12 1. C2 22 7.53336 0.5397 0.192 0.98816 194.85 1. C3 23 -3.67034 0.5399 -0.094 0.98722 46.21 1. C4 24 -1.40299 0.5392 -0.036 0.98999 6.77 1. C5 25 -1.36866 0.5393 -0.035 0.98955 6.44 1. NYE 27 32.46759 7.3664 0.061 0.97171 19.43 1. HW 28 35.95494 7.3516 0.068 0.97565 23.92 1. T2 33 -18.38942 7.4039 -0.035 0.96191 6.17 1. ***** F LEVELS( 4.000, 3.900) OR TOLERANCE INSUFFICIENT FOR FURTHER STEPPING

25. D14.99945 D29.86107 D39.43565 D413.84377 D528.69194 D621.63193 Day of the week effects

27. NYE32.46759 HW35.95494 T2-18.38942 Holiday Effects

28. S1-7.89293 S2-3.41996 S4-3.56763 C115.40978 C27.53336 C3-3.67034 C4-1.40299 C5-1.36866 Cyclical Effects

30. Transformations to Linearity Many non-linear curves can be put into a linear form by appropriate transformations of the either – the dependent variable Y or –some (or all) of the independent variables X 1, X 2,..., X p. This leads to the wide utility of the Linear model. We have seen that through the use of dummy variables, categorical independent variables can be incorporated into a Linear Model. We will now see that through the technique of variable transformation that many examples of non-linear behaviour can also be converted to linear behaviour.

31. Intrinsically Linear (Linearizable) Curves 1 Hyperbolas y = x/(ax-b) Linear form: 1/y = a -b (1/x) or Y =  0 +  1 X Transformations: Y = 1/y, X=1/x,  0 = a,  1 = -b

32. 2. Exponential y = a e bx = aB x Linear form: ln y = lna + b x = lna + lnB x or Y =  0 +  1 X Transformations: Y = ln y, X = x,  0 = lna,  1 = b = lnB

33. 3. Power Functions y = a x b Linear from: ln y = lna + blnx or Y =  0 +  1 X

34. Logarithmic Functions y = a + b lnx Linear from: y = a + b lnx or Y =  0 +  1 X Transformations: Y = y, X = ln x,  0 = a,  1 = b

35. Other special functions y = a e b/x Linear from: ln y = lna + b 1/x or Y =  0 +  1 X Transformations: Y = ln y, X = 1/x,  0 = lna,  1 = b

36. Polynomial Models y =  0 +  1 x +  2 x 2 +  3 x 3 Linear form Y =  0 +  1 X 1 +  2 X 2 +  3 X 3 Variables Y = y, X 1 = x, X 2 = x 2, X 3 = x 3

37. Exponential Models with a polynomial exponent Linear form lny =  0 +  1 X 1 +  2 X 2 +  3 X 3 +  4 X 4 Y = lny, X 1 = x, X 2 = x 2, X 3 = x 3, X 4 = x 4

38. Trigonometric Polynomials

39.  0,  1,  1, …,  k,  k are parameters that have to be estimated, 1, 2, 3, …, k are known constants (the frequencies in the trig polynomial. Note:

40. Response Surface models Dependent variable Y and two independent variables x 1 and x 2. (These ideas are easily extended to more the two independent variables) The Model (A cubic response surface model) or Y =  0 +  1 X 1 +  2 X 2 +  3 X 3 +  4 X 4 +  5 X 5 +  6 X 6 +  7 X 7 +  8 X 8 +  9 X 9 +  where

42. The Box-Cox Family of Transformations

43. The Transformation Staircase

44. The Bulging Rule x up y up y down x down