Transformations
description
Transcript of 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 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 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
b/a
1/a
positive curvature b>0
y=x/(ax-b)
y=x/(ax-b)
negative curvature b< 0
1/a
b/a
2. Exponential
y = ex = 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
2100
5
Exponential (B > 1)
x
y aB
a
2100
1
2
Exponential (B < 1)
x
y
a
aB
3. Power Functions
y = a xb
Linear from: ln y = lna + blnx or Y = 0 + 1 X
Power functionsb>0
b > 1
b = 1
0 < b < 1
Power functionsb < 0
b < -1b = -1
-1 < b < 0
Logarithmic Functionsy = a + b lnx
Linear from: y = a + b lnx or Y = 0 + 1 X
Transformations: Y = y, X = ln x, 0 = a, 1 = b
b > 0b < 0
Other special functionsy = 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
b > 0 b < 0
Polynomial Models
y = 0 + 1x + 2x2 + 3x
3
Linear form Y = 0 + 1 X1 + 2 X2 + 3 X3
Variables Y = y, X1 = x , X2 = x2, X3 = x3
0 0.5 1 1.5 2 2.5 3
0.5
1
1.5
2
2.5
3
Exponential Models with a polynomial exponent
y e x x 0 1 44
Linear form lny = 0 + 1 X1 + 2 X2 + 3 X3+ 4 X4
Y = lny, X1 = x , X2 = x2, X3 = x3, X4 = x4
0 5 10 15 20 25 30
0.25
0.5
0.75
1
1.25
1.5
1.75
2
Trigonometric Polynomial Models
y = 0 + 1cos(2f1x) + 1sin(2f1x) + … +
kcos(2fkx) + ksin(2fkx)
Linear form Y = 0 + 1 C1 + 1 S1 + … + k Ck + k Sk
Variables Y = y, C1 = cos(2f1x) , S2 = sin(2f1x) , …
Ck = cos(2fkx) , Sk = sin(2fkx)
-20
-10
0
10
20
30
0 1
Response Surface modelsDependent 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 = 0 + 1 X1 + 2 X2 + 3 X3 + 4 X4 + 5 X5 + 6 X6 + 7 X7 + 8 X8 + 9 X9+
where
21421322110 xxxxxY
319
22182
217
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225 xxxxxxx
, , , , , 225214
2132211 xXxxXxXxXxX
319
22182
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Mechanistic Growth Model
Non-Linear Growth models • many models cannot be transformed into a linear model
The Mechanistic Growth Model
Equation: kxeY 1
or (ignoring ) “rate of increase in Y” = Ykdx
dY
The Logistic Growth Model
or (ignoring ) “rate of increase in Y” = YkY
dx
dY
Equation:
kxeY
1
10864200.0
0.5
1.0
Logistic Growth Model
x
y
k=1/4
k=1/2k=1k=2
k=4
The Gompertz Growth Model:
or (ignoring ) “rate of increase in Y” =
YkY
dx
dY ln
Equation: kxeeY
10864200.0
0.2
0.4
0.6
0.8
1.0
Gompertz Growth Model
x
y
k = 1
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
TrendThis will be modeled by a Linear function :
Y = 0 +1 X
(more generally a polynomial)
Y = 0 +1 X +2 X2 + 3 X3 + ….
Yearly Cyclical Trend
This will be modeled by a Trig Polynomial – Sin and Cos functions with differing frequencies(periods) :
Y = 1 sin(2f1X) + 1 cos(2f2X) 1 sin(2f2X)
+ 2 cos(2f2X) + …
Day of the week effect:This will be modeled using “dummy”variables :
1 D1 + 2 D2 + 3 D3 + 4 D4 + 5 D5 + 6 D6
Di = (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(0.017202423838959*i*day). Ci=cos(0.017202423838959*i*day).
Dependent variableY = daily accident frequency
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
S1 -7.89293
S2 -3.41996
S4 -3.56763
C1 15.40978
C2 7.53336
C3 -3.67034
C4 -1.40299
C5 -1.36866
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 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 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
b/a
1/a
positive curvature b>0
y=x/(ax-b)
y=x/(ax-b)
negative curvature b< 0
1/a
b/a
2. Exponential
y = a ebx = aBx
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
2100
5
Exponential (B > 1)
x
y aB
a
2100
1
2
Exponential (B < 1)
x
y
a
aB
3. Power Functions
y = a xb
Linear from: ln y = lna + blnx or Y = 0 + 1 X
Power functionsb>0
b > 1
b = 1
0 < b < 1
Power functionsb < 0
b < -1b = -1
-1 < b < 0
Logarithmic Functionsy = a + b lnx
Linear from: y = a + b lnx or Y = 0 + 1 X
Transformations: Y = y, X = ln x, 0 = a, 1 = b
b > 0b < 0
Other special functionsy = 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
b > 0 b < 0
Polynomial Models
y = 0 + 1x + 2x2 + 3x
3
Linear form Y = 0 + 1 X1 + 2 X2 + 3 X3
Variables Y = y, X1 = x , X2 = x2, X3 = x3
0 0.5 1 1.5 2 2.5 3
0.5
1
1.5
2
2.5
3
Exponential Models with a polynomial exponent
y e x x 0 1 44
Linear form lny = 0 + 1 X1 + 2 X2 + 3 X3+ 4 X4
Y = lny, X1 = x , X2 = x2, X3 = x3, X4 = x4
0 5 10 15 20 25 30
0.25
0.5
0.75
1
1.25
1.5
1.75
2
0
1
2
3
4
5
6
7
8
9
0 0.5 1 1.5 2
Trigonometric Polynomials
0 1 1 1 1sin 2 cos 2Y
sin 2 cos 2k k k k
• 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:
0 1 1 1 1 k k k kS C S C
sin 2 cos 2k k k k
0 1 1 1 1sin 2 cos 2Y
where sin 2 and cos 2k k k kS C
Response Surface modelsDependent 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 = 0 + 1 X1 + 2 X2 + 3 X3 + 4 X4 + 5 X5 + 6 X6 + 7 X7 + 8 X8 + 9 X9+
where
21421322110 xxxxxY
319
22182
217
316
225 xxxxxxx
, , , , , 225214
2132211 xXxxXxXxXxX
319
22182
217
316 and , , xXxxXxxXxX