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TABLE 15.2 ANOVA TABLE TO EXAMINE DIFFERENCES BETWEEN SIC CODE GROUPS SOURCE OF VARIATION
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Between groups Within groups Total
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1.344158
SS = sum of squares DF = degrees of freedom MS = mean sum of squares
SOLID WASTE CHARACTERIZATION BY BUSINESS ACTIVITIES
15.4 Multivariate Cluster Analysis and Discussion
This section discusses the nal waste grouping process, multivariate cluster analysis, utilized to reduce the 65 SIC code groups further. Cluster analysis is a group of multivariate techniques whose primary purpose is to objectively group objects based on characteristics they possess. Cluster analysis was used to identify SIC code groups that generate similar solid waste material composition percentages. Solid waste stream composition percentage means and variances were calculated for all records gathered from each SIC code group in the previous steps. Based on previous research, a ve-step cluster analysis procedure was applied for this research (Romesburg, 1984):
1 2 3 4 5
Obtain data matrix Standardize the data matrix (z scores) Compute the resemblance matrix Execute the cluster method Report and evaluate the results (statistical testing)
Each step is discussed in the following sections.
15.4.1 STEP 1: OBTAIN THE DATA MATRIX
A data matrix is a table containing the objects and attributes of each object to be grouped. The columns of the matrix represent each object (t total objects) and the rows represent the attributes or properties of each object (n total attributes). For this research, the objects are the 65 SIC code groups and the attributes are the means and variances of each waste material generated by the respective waste group ( and 2). Figure 15.5 displays the canonical form of the data matrix.
Waste Material 1 2 i n
X11 X21 Xi1 Xn1
SIC Code Groups j t 2 j S X12 X22 Xi2 Xn2 X1j X2j Xij Xnj
t X1t X2t Xit Xnt
Figure 15.5 matrix.
Format of cluster analysis data
MULTIVARIATE CLUSTER ANALYSIS AND DISCUSSION
For this research, the objects were each SIC code group for which suf cient data was collected (438 company records covering 65 SIC code groups). Details on the attributes (waste material means and variances) for each object (SIC code group) are listed in the following bullet points. As mentioned, if a material comprised less than 2 percent of total waste for all groups, the material was not included in the analysis for simpli cation and noise reduction (four in total aerosol cans, rags, lamps, batteries). Material composition percentage means and standard deviations of
Biohazard wastes Construction and demolition debris (sand, stone, and concrete) FABRIC and textiles Food waste Glass Metal Old corrugated containers (cardboard) Chemicals, sludges, and used oil Organic wastes (agricultural) Paper (excluding cardboard) Plastic Rubber Wood Yard waste
15.4.2 STEP 2: STANDARDIZE THE DATA MATRIX
This is an optional step that standardizes the data matrix by converting the original attributes into new unit-less attributes. This is important for two reasons (Romesburg, 1984):
1 The original units for measuring attributes can arbitrarily affect the similarities
among objects. 2 Attributes will contribute more equally to the similarities among objects. To standardize the matrix, a standardizing function is selected and applied to normalize the data matrix. The standardizing function (or standard normal form) that is most commonly used in practice and applied for this research (Romesburg, 1984), was X ij X i Si
Zij =
SOLID WASTE CHARACTERIZATION BY BUSINESS ACTIVITIES
where
Xi =
Xij
j =1
1/ 2
t 2 (X ij X i ) j =1 Si = 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 coef cient measures the overall degree of similarity between each pair of objects in the standardized data matrix (Romesburg, 1984). Of the many resemblance coef cients available, the Euclidean distance coef cient was chosen. This coef cient is based on the Pythagorean theorem and used in the following calculation: n 2 (X ij X ik ) d jk = i =1 n
1/ 2
The Euclidean resemblance coef cient is considered a dissimilarity coef cient because the smaller the value the more similar two objects are. Resemblance matrices are square and symmetric; each column identi es the rst object in the pair and each row identi es the second object. The cell formed by the intersection of a column and
Waste Material 1 2 i n
Z11 Z21 Zi1 Zn1
SIC Code Groups j t j S2 Z12 Z22 Zi2 Zn 2 Z1j Z2j Zij Znj
t Z1t Z2t Zit Znt
Figure 15.6 Format of cluster analysis standardized data matrix.
MULTIVARIATE CLUSTER ANALYSIS AND DISCUSSION
First object in pair (SIC code group) 1 Second object in pair (SIC code group) 1 2 3 d21 d31 2 d32 3 n dn1 dn 2 dn3 n
Figure 15.7 Format of cluster analysis resemblance matrix.
row contains the value of the resemblance coef cient for the given pair of objects (Romesburg, 1984). For these reasons, only the lower half of the resemblance matrix contains values. Figure 15.7 provides an example.
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