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  1. Applications The General Linear Model

  2. Transformations

  3. 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 X1, X2, ... , Xp . • 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.

  4. Intrinsically Linear (Linearizable) Curves 1Hyperbolas y = x/(ax-b) Linear form: 1/y = a -b (1/x) or Y = b0 + b1 X Transformations: Y = 1/y, X=1/x, b0 = a, b1 = -b

  5. 2.Exponential y = aebx = aBx Linear form: ln y = lna + b x = lna + lnB x or Y = b0 + b1 X Transformations: Y = ln y, X = x, b0 = lna, b1 = b = lnB

  6. 3. Power Functions y = a xb Linear from: ln y = lna + blnx or Y =b0 + b1 X

  7. Logarithmic Functions y = a + b lnx Linear from: y = a + b lnx or Y =b0+ b1X Transformations: Y = y, X = ln x,b0 = a,b1= b

  8. Other special functions y = aeb/x Linear from: ln y = lna + b 1/x or Y =b0 +b1X Transformations: Y = ln y, X = 1/x,b0= lna, b1= b

  9. The Box-Cox Family of Transformations

  10. The Transformation Staircase

  11. Graph of ln(x)

  12. The effect of the transformation

  13. The ln-transformation is a member of the Box-Cox family of transformations with l= 0 • If you decrease the value of lthe effect of the transformation will be greater. • If you increase the value of lthe effect of the transformation will be less.

  14. The effect of the ln transformation • It spreads out values that are close to zero • Compacts values that are large

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

  16. Non-Linear Models Nonlinearizable models

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

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

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

  20. Polynomial Regression models

  21. Polynomial Models y = b0 + b1x + b2x2 + b3x3 Linear form Y = b0 + b1 X1 + b2 X2 + b3 X3 Variables Y = y, X1 = x , X2 = x2, X3 = x3

  22. Suppose that we have two variables • Y – the dependent variable (response variable) • X – the independent variable (explanatory variable, factor)

  23. Assume that we have collected data on two variables X and Y. Let (x1, y1) (x2, y2) (x3, y3) … (xn, yn) denote thepairs of measurements on the on two variables X and Y for n cases in a sample (or population)

  24. The assumption will be made that y1,y2, y3 …, yn are • independent random variables. • Normally distributed. • Have the common variance, s. • The mean of yiis:

  25. Each yi is assumed to be randomly generated from a normal distribution with mean and standard deviation s.

  26. The matrix formulation The Model

  27. The Normal Equations

  28. Example In the following example two quantities are being measured X = amount of an additive to a chemical process Y = the yield of the process

  29. Graph X vs Y

  30. Comment: A cubic polynomial in x can be fitted to y by defining the variables X1 = x, X2 = x2, and X3 = x3 Then fitting the linear model The Model – Cubic polynomial (degree 3)

  31. Response Surface Models Extending polynomial regression models to k independent variables

  32. Dependent variable Y and two independent variables x1 and x2. (These ideas are easily extended to more the two independent variables) Response Surface models (2 independent vars.) The Model (A cubic response surface model) Compare with a linear model:

  33. The response surface model can be put into the form of a linear model : Y = b0 + b1X1 +b2X2 + b3X3 +b4X4 + b5X5 + b6X6 + b7X7 + b8X8 + b9X9+ e by defining

  34. More Generally, consider the random variable Y with 1. E[Y] = g(U1 ,U2 , ... , Uk) = b1f1(U1 ,U2 , ... , Uk) + b2f2(U1 ,U2 , ... , Uk) + ... + bpfp(U1 ,U2 , ... , Uk) = and 2. var(Y) = s2 • where b1, b2 , ... ,bp are unknown parameters • and f1 ,f2 , ... , fp are known functions of the nonrandom variables U1 ,U2 , ... , Uk. • Assume further that Y is normally distributed.

  35. Now suppose that n independent observations of Y, (y1, y2, ..., yn) are made corresponding to n sets of values of (U1 ,U2 , ... , Uk) : (u11 ,u12 , ... , u1k), (u21 ,u22 , ... , u2k), ... (un1 ,un2 , ... , unk). Let xij = fj(ui1 ,ui2 , ... , uik) j =1, 2, ..., p; i =1, 2, ..., n. Then or

  36. Polynomial Regression Model: One variable U. Quadratic Response Surface Model: Two variables U1, U2.

  37. Trigonometric Polynomial Models

  38. Trigonometric Polynomial Models y = b0 + g1cos(2pf1x) + d1sin(2pf1x) + … + gkcos(2pfkx) + dksin(2pfkx) Linear formY = b0 + g1C1 + d1S1 + … + gk Ck+ dk Sk Variables Y = y, C1 = cos(2pf1x) , S2 = sin(2pf1x) , … Ck = cos(2pfkx) , Sk = sin(2pfkx)

  39. The Normal equations: given data General set of models

  40. Polynomial Models Two important Special Cases Trig-polynomial Models

  41. Orthogonal Polynomial Models

  42. Definition Consider the values x0, x1, … , xnand the polynomials are orthogonal relative to x0, x1, … , xnif: If in addition , they are called orthonormal

  43. This is equivalent to a polynomial model. Rather than the basis for this model being The basis is ,polynomials of degree 0, 1, 2, 3, etc Consider the model

  44. The Normal Equations given the data

  45. Derivation of Orthogonal Polynomials With equally spaced data points

  46. Suppose x0 = a, x1 = a + b, x2 = a + 2b, … , xn = a + nb