Solution 4.4 - Chemometrics: Data Analysis for the Laboratory and Chemical Plant

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

1. The centroids of each class are as follows.

Class

x

y

z

A

0.44

0.52 

0.24  

B

0.25

0.325

0.625


2. The Euclidean distances are as follows.

For objects 1 to 5, the distance to the centroid of A is less than that to the distance to the centroid of B, the reverse being true for objects 6 to 9. Hence Euclidean distances to class centroids suggest the original classification is confirmed.

3. The two class distances for the unknown are as follows.

 

Class distance A

Class distance B

unknown

0.236

0.411

Hence the object is most likely to belong to Class A.

4. The nine distances of the unknown are indicated below. The distances are ranked from 1 (closest) to 9 (farthest). The three closest are shaded in the table.

The three nearest neighbours are objects 3 (class A), 1 (class A) and 7 (class B) so it can be concluded that the object belongs to class A. Note that this is the result of a majority vote. However there are not a large number of objects in each class, and a larger training set would probably result in more unambiguous answers. This confirms the conclusion of question 4.

5. Probably object 7 as it has a fairly small distance to class A as well as class B (see question 3). Hence its inclusion in the three closest objects of question 4 may not be very significant.

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