Environmental Engineering Reference
In-Depth Information
Figure 1. Example of triangular simplex region from three-component mixture
experiment.
‗e' is the random error term, represents the combined effects of all
variables not included in the model. For a mixture experiment,
x
1
+
x
2
+
x
3
= 1 therefore, the model can be re-written as:
y = b
1
x
1 +
b
2
x
2
+ b
3
x
3
+ e
(2)
where,
b*
0
= b*
0
.
(x
1
+ x
2
+ x
3
)
This form is known as the Scheffé linear mixture polynomial [2]
Similarly, the quadratic polynomial is as follows:
y = b
0
+ b
*
1
x
1 +
b
*
2
x
2
+ b
*
3
x
3
+ b
*
12
x
1
x
2 +
b
*
13
x
1
x
3
+ b
*
23
x
2
x
3 +
b
*
11
x
1
2
+
b
*
22
x
2
2+
b
33
x
3
2
+
e
(3)
Again this can be represented as:
y = b
1
x
1 +
b
2
x
2
+ b
3
x
3
+ b
12
x
1
x
2 +
b
13
x
1
x
3
+ b
23
x
2
x
3 +
e
where,
x
1
2
= x
1
.(1- x
2
- x
3
)
x
2
2
= x
2
.(1- x
1
- x
3
)
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