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 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.

3. Intrinsically Linear (Linearizable) Curves
1 Hyperbolas
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

4. 2. Exponential
y = a ebx = 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

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

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

7. Other special functions
y = a e b/x
Linear from: ln y = lna + b 1/x or Y = b0 + b1 X
Transformations: Y = ln y, X = 1/x, b0 = lna, b1 = b

8. 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

9. Exponential Models with a polynomial exponent
Linear form lny = b0 + b1 X1 + b2 X2 + b3 X3+ b4 X4
Y = lny, X1 = x , X2 = x2, X3 = x3, X4 = x4

10. Trigonometric Polynomials

11. b0, d1, g1, … , dk, gk are parameters that have to be estimated,
n1, n2, n3, … , nk are known constants (the frequencies in the trig polynomial.
Note:

12. Trigonometric Polynomial Models
y = b0 + g1cos(2pn1x) + d1sin(2pn1x) + … +
gkcos(2pnkx) + dksin(2pnkx)
Linear form Y = b0 + g1 C1 + d1 S1 + … + gk Ck + dk Sk
Variables Y = y, C1 = cos(2pn1x) , S2 = sin(2pn1x) , …
Ck = cos(2pnkx) , Sk = sin(2pnkx)

13. Response Surface models
Dependent variable Y and two independent variables x1 and x2. (These ideas are easily extended to more the two independent variables)
The Model (A cubic response surface model) or
Y = b0 + b1 X1 + b2 X2 + b3 X3 + b4 X4 + b5 X5 + b6 X6 + b7 X7 + b8 X8 + b9 X9+ e
where

15. The Box-Cox Family of Transformations

16. The Transformation Staircase

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

18. Nonlinearizable models
Non-Linear Models
Nonlinearizable models

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

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

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

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

23. Trend Yearly Cyclical Trend
This will be modeled by a Linear function :
Y = b0 +b1 X
(more generally a polynomial)
Y = b0 +b1 X +b2 X2 + b3 X3 + ….
Yearly Cyclical Trend
This will be modeled by a Trig Polynomial – Sin and Cos functions with differing frequencies(periods) :
Y = d1 sin(2pf1X) + g1 cos(2pf2X) + d1 sin(2pf2X)
+ g2 cos(2pf2X) + …

24. Day of the week effect: Holiday Effects
This will be modeled using “dummy”variables :
a1 D1 + a2 D2 + a3 D3 + a4 D4 + a5 D5 + a6 D6
Di = (1 if day of week = i, 0 otherwise)
Holiday Effects
Also will be modeled using “dummy”variables :

25. 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

26. 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( , ) OR TOLERANCE INSUFFICIENT FOR FURTHER STEPPING

27. Day of the week effects D1 4.99945 D2 9.86107 D3 9.43565 D4 13.84377 D6

29. Holiday Effects
NYE HW T2

30. Cyclical Effects S1 -7.89293 S2 -3.41996 S4 -3.56763 C1 15.40978 C2 C3 C4 C5