Some Examples
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 = 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) + …
Day of the week effect:
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 :
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 variable
Y = daily accident frequency
Independent variables
ANALYSIS OF VARIANCE
REGRESSION
RESIDUAL
SUM OF SQUARES
976292.38
1547102.1
DF
18
3269
MEAN SQUARE
54238.46
473.2646
F RATIO
114.60
VARIABLES IN EQUATION FOR PACC
STD. ERROR
COEFFICIENT
OF COEFF
60.48909 )
0.11107E-02
0.4017E-03
4.99945
1.4272
9.86107
1.4200
9.43565
1.4195
13.84377
1.4195
28.69194
1.4185
21.63193
1.4202
-7.89293
0.5413
-3.41996
0.5385
-3.56763
0.5386
15.40978
0.5384
7.53336
0.5397
-3.67034
0.5399
-1.40299
0.5392
-1.36866
0.5393
32.46759
7.3664
35.95494
7.3516
-18.38942
7.4039
VARIABLE
(Y-INTERCEPT
day
1
D1
9
D2
10
D3
11
D4
12
D5
13
D6
14
S1
15
S2
16
S4
18
C1
21
C2
22
C3
23
C4
24
C5
25
NYE
27
HW
28
T2
33
***** F LEVELS(
4.000,
.
STD REG
COEFF
TOLERANCE
F
TO REMOVE
0.038
0.063
0.124
0.119
0.175
0.363
0.273
-0.201
-0.087
-0.091
0.393
0.192
-0.094
-0.036
-0.035
0.061
0.068
-0.035
0.99005
0.57785
0.58367
0.58311
0.58304
0.58284
0.58352
0.98285
0.99306
0.99276
0.99279
0.98816
0.98722
0.98999
0.98955
0.97171
0.97565
0.96191
7.64
12.27
48.22
44.19
95.11
409.11
232.00
212.65
40.34
43.88
819.12
194.85
46.21
6.77
6.44
19.43
23.92
6.17
.
LEVEL.
.
1 .
1 .
1 .
1 .
1 .
1 .
1 .
1 .
1 .
1 .
1 .
1 .
1 .
1 .
1 .
1 .
1 .
1 .
3.900) OR TOLERANCE INSUFFICIENT FOR FURTHER STEPPING
VARIABLES NOT IN EQUATION
VARIABLE
IACC
Dths
S3
S5
S6
C6
V1
V2
cd
T1
7
8
17
19
20
26
29
30
31
32
PARTIAL
CORR. TOLERANCE
0.49837
0.04788
-0.02761
-0.01625
-0.00489
-0.02856
-0.01331
-0.02555
0.00555
0.00000
0.78647
0.93491
0.99511
0.99348
0.99539
0.98788
0.96168
0.96088
0.97172
0.00000
F
TO ENTER
1079.91
7.51
2.49
0.86
0.08
2.67
0.58
2.13
0.10
0.00
LEVEL
0
0
1
1
1
1
1
1
1
1
Day of the week effects
D1
4.99945
D2
9.86107
D3
9.43565
D4
13.84377
D5
28.69194
D6
21.63193
Day of Week Effect
100.0
80.0
60.0
40.0
20.0
0.0
Mon
Tue
Wed
Thu
Fri
Sat
Sun
Holiday Effects
NYE
HW
T2
32.46759
35.95494
-18.38942
Cyclical Effects
S1
S2
S4
C1
C2
C3
C4
C5
-7.89293
-3.41996
-3.56763
15.40978
7.53336
-3.67034
-1.40299
-1.36866
40
30
20
10
0
0
-10
-20
-30
30
60
90
120
150
180
210
240
270
300
330
360
Example 2
• Data on the reaction rate of the catalytic
isomerization of л-pentane to isopentane
versus the partial pressures of hydrogen, лpentane, and isopentane are reproduced in
the Table on the following slide
Table:
Partial Pressure (psai)
Reaction
Hydrogen Isopentane n-Pentane Rate(hr-1)
(x1)
(x2)
(x3)
(y)
205.8
404.8
209.7
401.6
224.9
402.6
212.7
406.2
133.3
470.9
300.0
301.6
297.3
314.0
305.7
300.1
305.4
305.2
300.1
106.6
417.2
251.0
250.3
145.1
37.1
36.3
49.4
44.9
116.3
128.9
134.4
134.9
87.6
86.9
81.7
101.7
10.5
157.1
86.0
90.2
87.4
87.0
66.4
33.0
32.9
41.5
14.7
50.2
90.9
92.9
174.9
187.2
92.7
102.2
186.9
192.6
140.8
144.2
68.3
214.6
142.2
146.7
142.0
143.7
141.1
141.5
83.0
209.6
83.9
294.4
148.0
291.0
3.541
2.397
6.694
4.722
0.593
0.268
2.797
2.451
3.196
2.021
0.896
5.084
5.686
1.193
2.648
3.303
3.054
3.302
1.271
11.648
2.002
9.604
7.754
11.590
Isomerization is a chemical process in which a
complex chemical is converted into more simple
units, called isomers: catalytic isomerization
employs catalysts to speed the reaction.
The reaction rate depends on various factors,
such as partial pressures of the products and the
concentration of the catalyst.
The differential reaction rate was expressed as
grams of isopentane produced per gram of
catalyst per hour (hr-1), and the instantaneous
partial pressure of a component was
calculated as the mole fraction of the
component times the total pressure, in pounds
per square inch absolute (psia).
A common form of model for the reaction rate
is the Hougen-Watson model:
x3 

13  x2 

1.632


y  f  x1 , x2 , x3 ; 1 ,  2 , 3 ,  4  
1 +  2 x1 + 3 x2 +  4 x3
Fit this model