Solution 2.3 - Chemometrics: Data Analysis for the Laboratory and Chemical Plant

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  • Published: Jan 1, 2000
  • Channels: Chemometrics & Informatics

1. The design matrix is as follows.

  1. x1

    x2

    x3

    x1x2

    x1x3

    x2x3

    1

    0

    0

    0

    0

    0

    0

    1

    0

    0

    0

    0

    0

    0

    1

    0

    0

    0

    0.5

    0.5

    0

    0.25

    0

    0

    0.5

    0

    0.5

    0

    0.25

    0

    0

    0.5

    0.5

    0

    0

    0.25

    The six coefficients are as follows. This model is commonly called the Sheffé model.

     

    b1

    41

     

    b2

    12

     

    b3

    18

     

    b12

    10

     

    b13

    -22

     

    b23

    8

2. The design matrix using the alternative (Cox) model is, together with the coefficients are as follows.

  1. x0

    x1

    x2

    x1x2

      

    1

    1

    0

    1

    0

    0

      

    1

    0

    1

    0

    1

    0

      

    1

    0

    0

    0

    0

    0

      

    1

    0.5

    0.5

    0.25

    0.25

    0.25

      

    1

    0.5

    0

    0.25

    0

    0

      

    1

    0

    0.5

    0

    0.25

    0

      

    Coefficients

     

    a0

    18

     

    a1

    1

     

    a2

    2

     

    a11

    22

     

    a22

    -8

     

    a12

    24

     

3. The algebra is as follows.


The equivalence of the coefficients is, therefore,

              

a0

b3
                         

a1

b1 - b3 + b13
 

a2

b2 - b3 + b23
 

a11

-b13
 

a22

-b23
 

a12

b12 - b13 - b23


This is easy to show numerically, using the models in questions 1 and 2. For example, a0 for the second model equals 18, which is the same as b3 for the first model, and so on.

The two models are equivalent. The model of question 1 is more common and more symmetrical when there are several factors. Note that it is impossible to include both intercept and all three single factor mixture terms in one model, because the proportion of the third component is dependent on the proportions of the first two.

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