Page 270 - Solid Waste Analysis and Minimization a Systems Approach

248 SOLID WASTE CHARACTERIZATION BY BUSINESS ACTIVITIES

where

t
∑ X ij

X = j=1
i
t

/
⎡ t ⎤ 12
⎢∑ ( X − X ) 2 ⎥
ij
i
S = j ⎢ =1 ⎥
i ⎢ t −1 ⎥
⎢ ⎥
⎢ ⎥
⎦
⎣
Figure 15.6 displays the canonical form of the standardized data matrix.

15.4.3 STEP 3: COMPUTE THE RESEMBLANCE MATRIX

A resemblance coefficient measures the overall degree of similarity between each
pair of objects in the standardized data matrix (Romesburg, 1984). Of the many resem-
blance coefficients available, the Euclidean distance coefficient was chosen. This coef-
ficient is based on the Pythagorean theorem and used in the following calculation:

/
⎡ n ⎤ 12
⎢∑ ( X − X ) 2 ⎥
ik
ij
d = i ⎢ =1 ⎥
jk ⎢ n ⎥
⎢ ⎥
⎣ ⎦

The Euclidean resemblance coefficient is considered a dissimilarity coefficient
because the smaller the value the more similar two objects are. Resemblance matrices
are square and symmetric; each column identifies the first object in the pair and each
row identifies the second object. The cell formed by the intersection of a column and

SIC Code Groups
Waste 1 j t
Material μ S 2 … j … t
1 Z 11 Z 12 … Z 1j … Z 1t
2 Z 21 Z 22 … Z 2j … Z 2t
… … … … …
i Z i1 Z i2 … Z ij … Z it
… … … … …
n Z n1 Z n2 … Z nj … Z nt

Figure 15.6 Format of cluster analysis
standardized data matrix.