Download Presentation
## Applications

- - - - - - - - - - - - - - - - - - - - - - - - - - - E N D - - - - - - - - - - - - - - - - - - - - - - - - - - -

**Applications**The General Linear Model**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.**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**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**3. Power Functions**y = a xb Linear from: ln y = lna + blnx or Y =b0 + b1 X**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**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**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.**The effect of the ln transformation**• It spreads out values that are close to zero • Compacts values that are large**y up**y up x down x up x down x up y down y down The Bulging Rule**Non-Linear Models**Nonlinearizable models**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”=**The Logistic Growth Model**Equation: or (ignoring e) “rate of increase in Y”=**The Gompertz Growth Model:**Equation: or (ignoring e)“rate of increase in Y”=**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**Suppose that we have two variables**• Y – the dependent variable (response variable) • X – the independent variable (explanatory variable, factor)**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)**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:**Each yi is assumed to be randomly generated from a normal**distribution with mean and standard deviation s.**The matrix formulation**The Model**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**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)**Response Surface Models**Extending polynomial regression models to k independent variables**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:**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**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.**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**Polynomial Regression Model: One variable U.**Quadratic Response Surface Model: Two variables U1, U2.**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)**The Normal equations: given data**General set of models**Polynomial Models**Two important Special Cases Trig-polynomial Models**Definition**Consider the values x0, x1, … , xnand the polynomials are orthogonal relative to x0, x1, … , xnif: If in addition , they are called orthonormal**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**Derivation of Orthogonal Polynomials**With equally spaced data points