Transformations

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.

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

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 

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

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

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

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

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

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)

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

The Box-Cox Family of Transformations

The Transformation Staircase

The Bulging Rule x up y up y down x down

Non-Linear Models Nonlinearizable models

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” =

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

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

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

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) + …

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 :

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( *i*day). Ci=cos( *i*day). Dependent variable Y = daily accident frequency

Independent variables ANALYSIS OF VARIANCE SUM OF SQUARES DF MEAN SQUARE F RATIO REGRESSION RESIDUAL 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 ). day E E IACC D Dths D S D S D S D C D V S V S cd S T C C C C C NYE HW T ***** F LEVELS( 4.000, 3.900) OR TOLERANCE INSUFFICIENT FOR FURTHER STEPPING

D D D D D D Day of the week effects

NYE HW T Holiday Effects

S S S C C C C C Cyclical Effects

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.

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

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

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

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

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

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

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

Trigonometric Polynomials

 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:

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

The Box-Cox Family of Transformations

The Transformation Staircase

The Bulging Rule x up y up y down x down