Multivariate 

Cluster ng sed 

on Entropy 

U.S. GEOLOGICAL SURVEY BULL-EJIN 1893 q 
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Multivariate 

Clustering Based 

on Entropy 

By JOSEPH MOSES BOTBOL 

Entropy-derived degrees of association proved as effective in 

describing degrees of association between geochemical 

variables as correlation coefficients of the raw data or their 

logarithms 

U.S. GEOLOGICAL SURVEY BULLETIN 1893 
DEPARTMENT OF THE INTERIOR 

MANUEL LUJAN, JR., Secretary 

U.S. GEOLOGICAL SURVEY 

Dallas L. Peck, Director 

Any use of trade, product, or firm names in this publication is for descriptive purposes 

only and does not imply endorsement by the U.S. Government 

UNITED STATES GOVERNMENT PRINTING OFFICE: 1989 

For sale by the 

Books and Open-File Reports Section 

U.S. Geological Survey 

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Denver, CO 80225 

Library of Congress Cataloging in Publication Data 

Botbol, Joseph Moses. 

Multivariate clustering based on entropy I by Joseph Moses Botbol. 

p. cm.-(U.S. Geological Survey bulletin ; 1893) 

Supt. of Docs. no.: 119.3:1893 

1. Manganese nodules-North Pacific Ocean-Composition. 2. Cluster 

analysis. I. Title. II. Series. 

QE390.2.M35B67 1990 89-600183 

553.4'629-dc20 CIP 
CONTENTS 

Abstract 1 

Introduction 1 

Acknowledgments 1 

Relevant Theoretical Aspects of Entropy 1 

Data Description 3 

Entropy Computations 3 

Cobalt 3 

Nickel 3 

Manganese 3 

Iron 4 

Copper 4 

Zinc 4 

Lead 4 

Aluminum 4 

Silicon 4 

Depth 4 

Cluster Analysis 4 

Correlation Coefficients of Raw Data 5 

Correlation Coefficients of Logs of Raw Data 5 

Uncertainty Coefficients of Elemental Enrichments 5 

Uncertainty Coefficients of Raw Data 5 

Conclusions 5 

References Cited 30 

Figures 

1. Scatter plots of cobalt versus nine variables 6 

2. Perspective plots showing uncertainties of cobalt versus nine variables 7 

3. Scatter plots of nickel versus nine variables 8 

4. Perspective plots showing uncertainties of nickel versus nine variables 9 

5. Scatter plots of manganese versus nine variables 10 

6. Perspective plots showing uncertainties of manganese versus nine 

variables 11 

7. Scatter plots of iron versus nine variables 12 

8. Perspective plots showing uncertainties of iron versus nine variables 13 

9. Scatter plots of copper versus nine variables 14 

10. Perspective plots showing uncertainties of copper versus nine variables 15 

11. Scatter plots of zinc versus nine variables 16 

12. Perspective plots showing uncertainties of zinc versus nine variables 17 

13. Scatter plots of lead versus nine variables 18 

14. Perspective plots showing uncertainties of lead versus nine variables 19 

15. Scatter plots of aluminum versus nine variables 20 

16. Perspective plots showing uncertainties of aluminum versus nine variables 21 

17. Scatter plots of silicon versus nine variables 22 

18. Perspective plots showing uncertainties of silicon versus nine variables 23 

19. Scatter plots of depth versus nine variables 24 

Contents Ill 
IV Contents 

20. Perspective plots showing uncertainties of depth versus nine variables 25 

21-24. Dendritic diagrams showing clusters of variables based on: 

21. Linear correlation coefficients of raw data 26 

22. Linear correlation coefficients of logarithms of the raw data 27 

23. Uncertainty coefficients of elemental enrichments 28 

24. Uncertainty coefficients of the raw data 29 

Tables 

1-10. Entropy computations for: 

1. Cobalt versus nine variables 7 

2. Nickel versus nine variables 9 

3. Manganese versus nine variables 11 

4. Iron versus nine variables 13 

5. Copper versus nine variables 15 

6. Zinc versus nine variables 17 

7. Lead versus nine variables 19 

8. Aluminum versus nine variables 21 

9. Silicon versus nine variables 23 

10. Depth versus nine variables 25 

11. Correlation coefficients of raw data 26 

12. Correlation coefficients of logarithms of raw data 27 

13. Symmetrical uncertainties of elemental enrichments 28 

14. Symmetrical uncertainties of raw data 29 

15. Cluster analysis of linear correlation coefficients of raw data and their logarithmic transforms and uncertainty coefficients of elemental enrichments and 

raw data 30 
Multivariate Clustering Based on Entropy 

By Joseph Moses Botbol 

Abstract 

To test the efficacy of entropy as a metric for degrees 

of association between pairs of variables, bivariate scatter 

plots of the sample depth and concentration of 9 elements 

within 1,200 "ferromanganese" nodules of the northern 

Pacific Ocean were classified into two-way contingency 

tables. Entropy attributes were computed for each table. 

The uncertainty coefficients (measures of the degree of 

association computed from other entropy attributes) of 

the raw data, data transformed to enrichments, linear 

correlation coefficients of the raw data, and raw data 

transformed to logarithms were input as similarity matrices to cluster analysis to determine the differences 

between the clusters of variables in each of the four input 

data groups. 

Four similarity matrices were used as input to cluster 

analysis. Clusters of all input sources fell into three major 

groups of variables, with aluminum-silicon-depth, cobaltlead, and nickel-copper-manganese-zinc common to their 

respective clusters in all four of the analyzed groups. The 

association of manganese-iron occurs only in the analysis 

of the raw data uncertainties. 

To the extent of this study, entropy proved effective 

in describing degrees of association between variables in 

chemical analyses. Similar studies can be used as effective 

tools in the multivariate analysis of geochemical data. 

INTRODUCTION 

The various industrial, academic, and governmental 

agencies that are involved in marine geology or marine 

mineral resource studies are compiling and updating an 

ever-increasing number of marine geologic data bases. As 

the size and scope of these data bases increase, modem 

comprehensive geologic data analysis must embrace far 

more data and variables than ever before, thus resulting in 

a need to develop and use appropriate multivariate analytical methods to determine sensible and meaningful relationships between or among variables. 

In the course of developing a hard-mineral resource 

model of the northern Pacific Ocean (Botbol and Evenden, 

1988), chemical analyses of 9 elements for each of approxManuscript approved for publication April 20, 1989. 

imately 1,250 oxide-rich-phase marine nodules (also 

referred to as ferromanganese nodules) were evaluated 

statistically. Intermediate products of the analyses were 

scatter plots and two-way contingency tables of all pairs of 

variables. Linear correlation coefficients were initially computed for all pairs of variables, but these correlation 

coefficients were unacceptable as a final product because 

they were too easily corrupted by outliers. Ultimately, the 

relations between variables as portrayed in the scatter plots 

and contingency tables were visually classified as weakly, 

moderately, or strongly linear. 

This report presents the results of a study undertaken 

to develop a technique for assessing the information content 

between pairs of variables so that their interrelations can be 

assessed for subsequent multivariate analysis. The primary 

inputs to this procedure are scatter plots of all pairs of 

variables. These are considered as order-disorder information diagrams, and entropy, as used in information theory, 

is the major metric used to describe relations between the 

variables in the diagrams. 

Acknowledgments 

The author gratefully acknowledges R. SindingLarsen for suggestions regarding the use of entropy to 

generate similarity matrices, L. L. Montgomery for suggestions on various aspects of information theory, and G.I. 

Evenden for assistance with computing. A.L. Clark and 

J.R. Hein reviewed this manuscript. 

RELEVANT THEORETICAL ASPECTS OF 

ENTROPY 

Entropy is used to measure the degree of order or 

disorder of the components within a system. The systems 

may be as varied as the spatial distribution of electrons 

around the nucleus of an atom, the degree to which an 

evolving river system has come to equilibrium, or the 

amount of information and degree of association between 

variables in a two-way contingency table. 

This study deals with the entropy characteristics of 

two-way contingency tables derived directly from bivariate 

Relevant Theoretical Aspects of Entropy 
scatter plots of the chemical contents of marine ferromanganese nodules. A contingency table is composed of rows 

and columns whose intersections form cells that are analogous to the squares on a checkerboard. The row-column 

(X,Y) values of each datum determine the cell to which it 

belongs. The frequencies of points in the cells of their 

respective contingency tables constitute the basic data 

required to compute entropy. Equations 1 through 8 are 

from Press and others (1988). 

The entropy of the whole of each contingency table is 

not used in computing the degrees of association of the 

variables, but it is a major entropy attribute useful in 

comparing scatter plots. It is calculated as follows: 

where 

I J 

H=-I I P;J ln Pi.J 

i=l j= 1 

H = total entropy of the contingency table, 

P. . = the proportion of total points in cell i ,j, 1,) 

I = number of columns, and 

J = number of rows. 

(1) 

The logarithm is used to keep the proportional components 

of a given contingency table additive. The negation is used 

to keep the sum of the logarithms of fractions as a positive 

number. H is highest when all cells have an equal number 

of points, and His lowest when all points are in one cell. 

The higher the disorder in the data, the higher the entropy, 

and conversely. This means that high degrees of association 

or dependence between variables are associated with low 

entropy values. [Singh (1966) refers to L. Brillouin who 

coined the term "negentropy" because of the inverse relation between entropy and information content.] 

Four computation procedures are used to generate 

degrees of association between a pair of variables: (1) the 

individual entropy of each variable of the pair that composes the two-way contingency table (equations 2 and 3), 

(2) the entropy of each variable, taking into consideration 

the influence of the other (equations 4 and 5), (3) the 

coefficient of uncertainty for each variable, taking into 

consideration the influence of the other (equations 6 and 7), 

and ( 4) the weighted average of the coefficients of uncertainty (equation 8). 

The individual entropies of X and Y in a contingency 

table of X versus Yare computed as follows: 

J 

H(Y)=-I P1 ln P1 

j=l 

I 

H(X)=-I P; ln P; 

i=l 

2 Multivariate Clustering Based on Entropy 

(2) 

(3) 

where 

H(Y) = entropy of Y, 

H(X) = entropy of X, 

P1 = proportion of frequency of each cell relative 

to the frequency of its host row in the array, 

Pi = proportion of frequency of each cell relative 

to the frequency of its host column in the 

array, 

I = number of columns, and 

J = number of rows. 

Entropies for Y given X and for X given Y are computed as 

follows: 

where 

~ ~ P; 1 ln P;,J H(Y I X)= L L _. __ 

p. i=l j=l l 

~ ~ P;J ln P;,1 

H(X I Y)= L L 

p. 

j=l i=l J 

H(Y I X) = entropy of Y given X, 

H(X I Y) = entropy of X given Y, 

P = proportion of points in cell, 

I = number of columns, and 

J = number of rows. 

(4) 

(5) 

The computations that quantify the degrees of intervariable association are derived by computing the uncertainty coefficient, U, for each variable, taking into consideration the effect of the other variable of the pair. U 

indicates how much entropy is lost by one variable (that is, 

information that is gained) because of the presence of the 

other. 

The formulas for U(X I Y) and U(Y I X) are as 

follows: 

where 

U(Y I X) H(Y)-H(Y I X) 

H(Y) 

U(X I Y) H(X)-H(X I Y) 

H(X) 

U(Y I X) = uncertainty of Y given X, 

U(X I Y) = uncertainty of X given Y, 

H(Y) = entropy of Y, 

H(X) = entropy of X, 

H(Y I X) = entropy of Y given X, and 

H(X I Y) = entropy of X given Y. 

(6) 

(7) 

U ranges from 0 to 1 where zero indicates no relation 

between X and Y, and 1 indicates complete predictability of 

one variable by the other. 
To represent both U(X I Y) and U(Y I X) with a single 

number, the formula for symmetrical uncertainty, U(X,Y), 

is computed as follows: 

H(X) U(X I Y) + H(Y) U(Y I X) (8) 

U(X,Y) H(X)+H(Y) 

U(X, Y) is the weighted average of both U(X I Y) and 

U(Y I X). 

Entropy calculations are based on the information 

content of the data and do not preserve the sign of relations 

in the data. For example, two mirror image distributions 

having linear correlation coefficients of 0. 7 and -0. 7, 

respectively, will have the same entropy. 

DATA DESCRIPTION 

A northern Pacific Ocean oxide-rich-phase (ferromanganese) nodule subset of the Scripps Institute of Oceanography Sediment Data Bank (Frazer and Fisk, 1980) was 

used for this study. These data are those described by 

McKelvey and others (1983). The study area is bounded by 

0° N., 40° N., 120° E., and 100° W. Each record in the data 

base is composed of multiple fields that describe one ocean 

bottom sample. The subset retrieved for this study is 

composed of the following variables: water depth above 

sample site, geographic coordinates, and concentration (in 

weight percent) of Co, Ni, Mn, Fe, Cu, Zn, Pb, AI, and Si. 

Not all variables are present in all samples. 

Scatter plots were constructed for all pairs of variables, and each scatter plot was partitioned into a two-way 

contingency table. In each of the contingency tables, the 

number of classes for each variable was determined according to Sturges' Rule (from Huntsberger, 1963): 

where 

k= 1 +3.3 log n 

k = number of classes and 

n = population size. 

(9) 

All class boundaries for the frequency distribution of each 

variable were arbitrarily set at equal intervals. 

ENTROPY COMPUTATIONS 

By using two-way contingency tables as input, 

entropy computations were made for all pairs of variables. 

The components and results of the entropy computations are 

presented herein for each variable and include the original 

scatter plots of each variable versus each of the other 

variables, a tabular summary of all entropy computations, 

and a perspective plot of the uncertainty coefficients 

U(X I Y), U(Y I X), and U(X,Y). 

The scatter plots present a synopsis of the orderdisorder that exists in the raw data. Although the plots have 

been classified into two-way contingency tables for computation of entropy attributes, correlations can be visually 

estimated from the scatter plots for comparison with computed entropy values. 

Tables 1 through 10 provide the complete range of 

entropy attributes computed for this study. Parenthetical 

table entries indicate the entropy calculation normalized by 

1/log n. In the tables, "Percent data content" refers to the 

percentage of rows, columns, and cells for which there are 

data. For example, in table 1, cobalt versus nickel, every 

column and row of the contingency table contains at least 

one datum, whereas only 50 percent of the individual cells 

contain data. In all figures and tables, "de" stands for depth 

of water above sample site. 

Perspective plots spatially portray the uncertainty 

coefficient of the subject variable as the dependent variable 

versus the other variables as independent variables 

( U(X I Y)), the subject variable as the independent variable 

versus the other variables as dependent variables 

(U(Y I X)), and the symmetrical uncertainty (U(X,Y)) of the 

subject variable versus the symmetrical uncertainties of the 

others. These diagrams provide a rapid review of the salient 

entropy relations of each subject variable with all of the 

other variables. 

Cobalt 

In figure 1 , the scatter plots of cobalt versus lead and 

iron appear to be the strongest visually assessable linear 

correlations. The uncertainty plot in figure 2 demonstrates 

unequivocally that the association of cobalt and lead is the 

strongest of all the associations with cobalt. Although the 

average uncertainty coefficient of aluminum is numerically 

the next value lower than lead, it is clustered with the other 

variables. Zinc appears to be slightly removed from the 

major group of variables, but its unique position is in the 

low value side of the plot. 

Nickel 

Nickel scatter plots (fig. 3) show that nickel has a 

strong linear correlation with manganese, iron, copper, and 

zinc. These strong relations are also true of the uncertainties 

shown in figure 4, where copper has the highest degree of 

association with nickel. Associations with manganese, iron, 

and zinc are well defined but of lower value. Other 

variables are grouped near the origin of the figure (0,0) and 

show no significant degree of association. 

Manganese 

The three strong, visually assessable, linear correlations in figure 5 are those of manganese versus zinc, nickel, 

Entropy Computations 3 
and copper. These correlations are borne out by the uncertainty coefficients in figure 6, which set these variables 

apart from the other elements. Other variables are all 

grouped near the origin of the figure (0,0). 

Iron 

Figure 7 shows strong linear correlation of iron with 

nickel and copper and its moderate correlation with zinc and 

lead. These relations are clearly depicted in figure 8. The 

degrees of association implied by the uncertainties of the 

other variables are distinctly lower. 

Copper 

Copper has a very strong linear relation to nickel and 

moderate to weak correlations with manganese, iron, and 

zinc (fig. 9). Figure 10 shows an exceptionally high degree 

of association between nickel and copper. Although lower 

than the copper-nickel uncertainty coefficient, those of 

copper to iron, manganese, and zinc are also high. 

Zinc 

Figure 11 shows moderate to strong linear correlation 

between zinc and nickel, manganese, and copper. These 

variables also have the highest degrees of association with 

zinc (fig. 12). In figure 12, the almost horizontal alignment 

of the variables is due to the overall high entropy in the Y 

direction, which is seen in figure 11 where a narrow range 

of zinc concentrations occurs with a very broad range of 

data values of the other elements. 

Lead 

Figure 13 shows a weak to moderate correlation 

between lead and cobalt, iron, copper, and zinc. Uncertainty coefficients in figure 14 show that lead has a high 

degree of association with zinc, iron, copper, and cobalt. 

Other variables show no significant relations. 

Aluminum 

Aluminum's only strong correlation is with silicon 

(fig. 15). This correlation is well emphasized in the plot of 

the pair's high uncertainty coefficient (fig. 16). At the scale 

of figure 16, the high value of the uncertainty coefficient of 

the aluminum and silicon pair seriously diminishes the 

graphic portrayal of any other significant relations existing 

between the other variables and aluminum. 

4 Multivariate Clustering Based on Entropy 

Silicon 

Silicon is very significantly associated with aluminum in both the scatter plots of figure 17 and the plot of the 

uncertainty coefficients in figure 18. As with aluminum, the 

relation between silicon and aluminum dominates any 

relations between silicon and the other variables. 

Depth 

In figure 19, depth shows very weak correlation with 

copper and almost no correlation with any of the other 

variables. Also, there appears to be a vestige of a second 

population in each of the scatter plots. The uncertainties in 

figure 20 show three distinct clusters of variables associated 

with depth: (1) copper, (2) iron, aluminum, nickel, lead, 

and silicon, and (3) zinc, cobalt, and manganese. 

CLUSTER ANALYSIS 

Up to this point in the analysis, it has been shown that 

uncertainty coefficients of the variables more or less agree 

with a visual estimate of their linear correlations and, when 

displayed as perspective pole diagrams, show a striking 

visual representation of information content. The next 

logical step in the comparison of entropy-derived similarities with those derived by traditional methods is to cluster 

the variables. 

Cluster analysis is a widely accepted numerical taxonomy method of partitioning the variables that compose an 

input similarity matrix to classify the variables into groups 

of maximum similarity. If the groups of variables determined from the entropy-based similarity matrices sensibly 

simulate those of accepted similarity matrices, such as 

matrices of correlation coefficients, similarity between the 

input matrices is implicit, and further methodological development and utilization are justified. 

Four similarity matrices were used as input to cluster 

analysis: (1) product-moment linear correlation coefficients 

between all pairs of variables in the raw data; (2) productmoment linear correlation coefficients between all pairs of 

variables in the raw data as transformed to their logarithms; 

(3) uncertainty coefficients of all pairs of variables in the 

raw data as transformed to their elemental enrichments 

(Botbol, 1987); and (4) uncertainty coefficients of all pairs 

of variables in the raw data. These four matrices provide a 

basis for comparison between the entropy-derived similarities and those derived by traditional methods. The results 

of this analysis are presented in four dendritic diagrams, 

figures 21 through 24, derived from the source data presented in tables 11 through 14, respectively. Final summary 

of the data clustering is presented in table 15, where the 

component variables of each cluster for each input similarity matrix are grouped. 
Included in the cluster analysis is a statement of the 

cophenetic correlation. This is a measure of the degree of 

distortion introduced into the dendritic diagram. If the value 

is less than 0.8, the cluster analysis is subject to question 

(Davis, 1973, p. 466). All cluster analyses were performed 

by using the algorithms presented in the CLUSTR program 

(Davis, 1973, p. 467). 

Correlation coefficients of raw data 

Table 11 is the matrix of product-moment linear 

correlation coefficients of the raw data used in the cluster 

analysis procedure. Intuitive partitioning of the dendritic 

diagram in figure 21 shows three major clusters of variables: (1) cobalt, lead, and iron; (2) nickel, copper, manganese, and zinc; and (3) aluminum, silicon, and depth. 

Correlation coefficients of logs of raw data 

Table 12, the second cluster analysis similarity 

matrix, is composed of the product-moment linear correlation coefficients of the logarithms of the raw data. Figure 22 

shows three major clusters of variables that are the same as 

those in the first analysis. Interestingly, the ordinal aspects 

of the clusters and the cluster component variables for the 

first two analyses are identical. 

Uncertainty coefficients of elemental 

enrichments 

The third cluster analysis (fig. 23) is based on the 

similarity matrix of uncertainty coefficients of elemental 

enrichments shown in table 13. The same three major 

clusters of variables occur as in the analysis of the raw data. 

Note that, although the order of some of the variables has 

changed, the variables within the clusters are the same. 

Uncertainty coefficients of raw data 

The fourth and final cluster analysis uses as input the 

uncertainty coefficient matrix for the raw data shown in 

table 14. Three distinct clusters result: (1) cobalt and lead; 

(2) nickel, copper, iron, manganese, and zinc; and (3) 

aluminum, silicon, and depth (fig. 24). 

CONCLUSIONS 

Table 15 summarizes the four data groups that were 

subjected to cluster analysis. There are three distinct clusters in each of the analyses. In the table, variables for each 

group are arranged so that the leftmost variable has the 

highest value. 

The most revealing aspect of table 15 is the shifting of 

iron from cluster I of the correlation coefficients of the raw 

data, their logarithmic transforms, and elemental enrichment uncertainty coefficients partitions to cluster II of the 

raw data uncertainty coefficients. This is the only partition 

in which iron is grouped with manganese. Interestingly, the 

most fundamental attribute of the data used in this study is 

the association of the two dominant elements that compose 

ferromanganese nodules. The uncertainty coefficients of the 

raw data compose the only input similarity matrix that 

reflects this association. 

The strongest and most pervasive cluster in the entire 

analysis is cluster III, which groups aluminum, silicon, and 

depth This composition undoubtedly reflects the geologic 

environment of the deep sea nodules in terms of nodule core 

material and in terms of the geologic distribution of the 

aluminosilicate fraction in the north Pacific. All three 

components are equivalent at an ordinal level and show 

exceptionally strong degrees of association in the plots of 

uncertainty coefficients. 

Two sets of variables that occurred within the same 

clusters during all of the analyses are cobalt and lead in 

cluster I and nickel, copper, manganese, and zinc in cluster 

II. All show strong degrees of association in the plots of the 

uncertainty coefficients. 

Another significant aspect of the results of this study 

is that the uncertainty coefficients of the raw data ordinally 

correspond almost exactly with the visual estimation of their 

linear correlation coefficients. In and of itself, this correspondence is not categorical proof of the efficacy of 

uncertainty coefficients as replacements for linear correlation coefficients, but it does establish significant credibility 

in entropy attributes as viable metrics of degrees of association between variables. 

With respect to establishing credibility of multivariate 

methods that use entropy-based attributes of data, this study 

has shown that entropy attributes of the raw data compare 

favorably with data attributes that are determined by linear 

correlation of both raw data and their log transforms and 

with the entropy-derived attributes of enrichment transforms. 

Conclusions 5 
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1.6 

1.6 

Figure 1. Scatter plots of Co content versus content of Ni, Mn, Fe, Cu, Zn, Pb, AI, and Si and depth of water above the 

sample site. 

6 Multivariate Clustering Based on Entropy 
Table 1. Entropy computations for cobalt versus nine variables 

[Parenthetical entries indicate entropy calculation normalized by 1/log n. de, depth of water above the sample site] 

Ni Mn Fe Cu Zn Pb 

Entropy of X 1.32 1.32 1.32 1.31 1.21 1.33 

(.55) (.55) (.55) (.54) (.50) (.55) 

Entropy of Y 2.18 1.74 1.49 2.15 0.76 1.45 

(.91) (.72) (.62) (.89) (.33) (.63) 

Entropy of (X ,Y) 3.36 2.93 2.66 3.28 1.92 2.54 

(.70) (.61) (.55) (.68) (.40) (.54) 

Entropy of (Y I X) 2.03 1.61 1.34 1.97 0.70 1.20 

(.84) (.67) (.55) (.82) (.30) (.52) 

Entropy of (X I Y) 1.17 1.19 1.17 1.13 1.15 1.09 

(.49) (.49) (.48) (.47) (.48) (.45) 

Uncertainty of (Y IX) 0.06 0.07 0.10 0.08 0.08 0.16 

Uncertainty of (X I Y) .11 .09 .11 .13 .05 .18 

Uncertainty of (X ,Y) .08 .08 .10 .10 .06 .17 

No. of points 1,164 1,153 1,146 1,158 688 624 

Percent data content * 

Columns 100 90 100 100 90 100 

Rows 100 90 72 90 50 100 

Cells so 40 38 39 18 42 

*Percentage of rows, columns, and cells for which there are data. 

Pb 

0. 1 8 ,--------1 

U(YIX) de 

0,0 U(XIY) 0.18 

Figure 2. Perspective plot of uncertainties of Co 

versus uncertainties of Ni, Mn, Fe, Cu, Zn, Pb, AI, 

Si, and depth of water above the sample site (de). 

U(Y I X), U(X I Y), and U(X, Y) are uncertainty 

coefficients. 

AI Si de 

1.47 138 1.35 

(.61) (.57) (.56) 

1.66 1.44 1.93 

(.75) (.65) (.80) 

2.94 2.69 3.16 

(.64) (.58) (.66) 

1.47 1.31 1.81 

(.67) (.59) (.75) 

1.28 1.24 1.23 

(.53) (.52) (.51) 

0.11 0.09 0.06 

.12 .09 .08 

.11 .09 .07 

301 258 1,058 

90 81 100 

100 77 100 

37 30 55 

Conclusions 7 
1.6 

'? 1.2 

Q) 

~ Q) 

.e 0.8 ~ 

0 

..0 

0 

u 0.4 

,........1.5 

-

c 

Q) 

~ Q) 

a. 

-1... 

Q) 

a. 

a. 

8 0.5 

7-

-...... 

c 6 -· Q) 

0 

1... 

Q) 5- a. 

- I 

E 4-, 

. . ·. .. : ... ,;•.;. 

0.5 1 1.5 

Nickel (percent) 

... 

0.5 1 1.5 

Nickel (percent) 

I I I 

.. ~ . . .. ... '~ . ::J ,~· • • •• • • • .i c 3 ~·. • ;-:_; I ' ~ .:f' .. -. :."-• • ~ .E ·~ ··.vw• ••• :.~~~:..,:,,. ••• • ::J ··;.·: ·' ... . ... ~::.~·. 

< 2 ~ • • ••••••• . ···• ·~.-. . . . . .. . ' 

1 1- • •• • r •I• • ·~ o' 

I! , .. r •• ~ ... I ,• I 

0.5 1 1.5 

Nickel (percent) 

.. 

-

-

-

-

0. -

-

-

,..... 

...... 

c 

Q) 

0 

1... 

Q) 

a. 

-Q) 

II) 

Q) 

c 

0 

0'1 

c 

0 

~ 

-...... 

c 

Q) 

0 

50 

40 

30 

20 

10 

0.31-

0 • 

0.5 1 1.5 

Nickel (percent) 

I I I 

-

... . ~ 0.2 t- ·. ; .... ·.: . - .... • ·'·.c. .~ • 

-0 

c 

N 

-...... 

c 

Q) 

0 

1... 

Q) 

a. 

-c 

0 

.~ 

Vi 

-

• • : ,ir-Y~.J· : 

o o oo I :~·~.ik_~ ... o 

.. · : .. :·. ! ~,y..,-;:~t 

0.1 t- 0 •• ·:.·~ ~·-:···r;;,'i~" .. ...... ,~~-· .. ~~,~-·.·.· . •' " '•I • l ' '1 

0.5 1 1.5 

Nickel (percent) 

I I I 

25 1- -

20 -· -

15 1- -

.... .. .. 

10; 1 :'· 

1

1

1 

~I 11 ·.:~~~ tflll I 

! ...... •. • ·.~ :, .: ,, ·... . 

o0t ... : 

0

•: 

0 .1:f1~o.:l•~, 'lo 0 0 

5 --· ••••• <:: ... ··: _ •• , •• ~· • •• .. . ·:: i':: "' .. . .. 

-

-

-• •' j" : I I 

0.5 1 1.5 

Nickel (percent) 

,..... 

...... 

c 

Q) 

0 

1... 

Q) 

a. 

-c 

0 

~ 

,..... 

....... 

c 

Q) 

0 

1... 

40 

35 

30 

25 

20 

10 

5 

o .. :n0.5 1 1.5 

Nickel (percent) 

I I I 

-

8_ 0.2 ~ -

-'0 

0 

j 

-II) 

1... 

Q) 

...... 

Q) 

E 

..2 

:.i 

-

L. 

....... 

a. 

Q) 

0 

0.1 f; 

... -·-·· . ........ .. ·- . . . . ..... . . . . . ... .. . ..... . .. -··· ... . . . ... .. . ·- ..... -· ... ··- --·. --... . ······---······ .. --- --·· --- ..... . . . . --.... ---·-·-. 

-

• •• ": -: l.:.: 0 : ... .1":'=-:;:::;..~:·: • • 

0.5 1 1.5 

Nickel (percent) 

0.5 1 1.5 

Nickel (percent) 

Figure 3. Scatter plots of Ni content versus content of Co, Mn, Fe, Cu, Zn, Pb, AI, and Si and depth of water above the 

sample site. 

8 Multivariate Clustering Based on Entropy 
Table 2. Entropy computations for nickel versus nine variables 

[Parenthetical entries indicate entropy calculation normalized by 1/log n. de, depth of water above the sample site] 

Co Mn Fe Cu Zn Pb 

Entropy of X 2.18 2.18 2.17 2.18 2.13 2.19 

(.91) (.90) (.90) (.91) (.89) (.91) 

Entropy of Y 1.32 1.73 1.49 2.15 0.76 1.44 

(.55) (.72) (.62) (.89) (.33) (.62) 

Entropy of (X ,Y) 3.36 3.39 3.23 3.54 2.57 3.41 

(.70) (.70) (.67) (.73) (.54) (.72) 

Entropy of (Y IX) 1.17 1.21 1.05 1.35 0.43 1.22 

(.49) (.50) (.44) (.56) (.19) (.53) 

Entropy of (X I Y) 2.03 1.65 1.73 1.38 1.80 1.96 

(.84) (.69) (.72) (.57) (.75) (.82) 

Uncertainty of (Y IX) 0.11 0.30 0.29 0.37 0.42 0.15 

Uncertainty of (X I Y) .06 .24 .20 .36 .15 .10 

Uncertainty of (X ,Y) .08 .26 .23 .36 .22 .12 

No. of points 1,164 1,215 1,208 1,158 693 628 

Percent data content * 

Columns 100 100 100 100 90 100 

Rows 100 90 72 90 50 100 

Cells 50 50 45 55 23 49 

* Percentage of rows, columns, and cells for which there are data. 

Cu 

Zn 

Pb 

SiU(YIX)I CJLdc \\J~ 0,0 U(XIY) 0.44 

Figure 4. Perspective plot of uncertainties of Ni 

versus uncertainties of Co, Mn, Fe, Cu, Zn, Pb, 

AI, Si, and depth of water above the sample site 

(de). U(Y I X), U(X I Y), and U(X, Y) are uncertainty coefficients. 

AI Si de 

2.17 2.18 2.18 

(.90) (.91) (.91) 

1.67 1.45 1.96 

(.76) (.66) (.81) 

3.67 3.46 3.89 

(.80) (.75) (.81) 

1.49 1.27 1.71 

(.68) (.58) (.71) 

2.00 2.00 1.93 

(.83) (.83) (.80) 

0.10 0.12 0.12 

.07 .08 .11 

.09 .09 .12 

310 266 1,118 

100 100 100 

100 77 100 

56 48 67 

Conclusions 9 
1.6 

'? 1.2 Cl) 

~ Cl) 

.e 0.8 

~ 

c 

..c 

0 

u 0.4 

-1.5 

...... 

c 

Cl) 

~ CIJ 

Cl. ........, 

I... 

Cl) 

Cl. 

Cl. 

8 0.5 

7 

-...... 

c 6 Cl) 

0 

~ 

Cl) 5 Cl. ........, 

E 4 

::3 

c: 3 .E 

:l 

< 2 

.... : '· ... .··· :.. ·.: 

10 20 30 40 

Manganese (percent) 

10 20 30 40 

Manganese (percent) 

.. 

. . . . ...... ·~: .·· . _., ... . ::-

... ~..;·~~ ~ · .. ).:~~, .. . . . . r .. ·.·t'-: .. ;.~·.;: .. -:._ .. ,. • • I 

. ;:_....· .. · .. . ... : .... ~ 

10 20 30 40 

Manganese (percent) 

50 

50 

50 

,.......1.5 

-

c: 

Cl) 

u 

I... 

Cl) 

Cl. ........., 

Q) 

.:::£ 

.~ 

z 

--

c: 

Cl) 

u 

0.5 

0.3 

~ 0.2 ........., 

u 

c: 

N 

-...... 

c: 

Cl) 

u 

I... 

Cl) 

Cl. ........., 

c: 

0 

.~ 

(i) 

0.1 

25-

20 ,..." 

15-

10..: 

s.: 

.·:· 

10 20 30 40 

Manganese (percent) 

· .. 

10 20 30 40 

Manganese (percent) 

I I I I 

·-·:· ·,. ··. . . ··::~ .•..•. ,_.·. • ••• r .... ,.;~~· ... : .).\~ ... ~.Aft, • • .. 0::-:\ •• •• .. .. ,;:.. :~.. . ,. . .. . .. .. I ·.1 • .:. ~·· .I 

10 20 30 40 

Manganese (percent) 

50 

50 

-

-

-

-

-

50 

--

c: 

Cl) 

u 

I... 

Cl) 

Cl. ........., 

c: 

0 

-= 

-...... 

c: 

CIJ 

~ 

40 

35 

30 

25 

20 

15 

10 

5 

0.3 

~ 0.2 ........, 

"'C 

c 

Cl) 

.....1 

-en 

I... 

Cl) 

...... 

Cl) 

E 

..Q 

........, 

~ 

.c. 

...... 

Cl. 

Cl) 

c 

0.1 

7 

0 

10 20 30 40 

Manganese (percent) 

. . .. . . . ...... . ···-· .. . .. .. . _. --· .... . . .. . ··-·-· .. ··---···· ... ·-··-···· 

···--· ···---· 

10 20 30 40 

Manganese (percent) 

. . . 

:.· . . ~ :• .· 

.· .... ~.· . .. .,·~··: . I • • • • ·~ 

..... ... :,:1 :,· ,• 

. .•. ·--'=-=·. : .... ::: 

10 20 30 40 

Manganese (percent) 

50 

50 

50 

Figure 5. Scatter plots of Mn content versus content of Co, Ni, Fe, Cu, Zn, Pb, AI, and Si and depth of water above the 

sample site. 

10 Multivariate Clustering Based on Entropy 
Table 3. Entropy computations for manganese versus nine variables 

[Parenthetical entries indicate entropy calculation normalized by 1/log n. de, depth of water above the sample site] 

Co Ni Fe Cu Zn Pb AI 

Entropy of X 1.74 1.73 1.73 1.73 1.63 1.75 1.78 

(.72) (.72) (.72) (.72) (.68) (.73) (.74) 

Entropy of Y 1.32 2.18 1.49 2.15 0.76 1.44 1.67 

(.55) (.90) (.62) (.89) (.33) (.62) (.76) 

Entropy of (X ,Y) 2.93 3.39 3.01 3.50 2.12 3.04 3.27 

(.61) (.70) (.62) (.73) (.45) (.64) (.71) 

Entropy of (Y I X) 1.19 1.65 1.27 1.76 0.48 1.29 1.49 

(.49) (.69) (.53) (.73) (.21) (.56) (.68) 

Entropy of (X I Y) 1.61 1.21 1.51 1.34 1.35 1.60 1.60 

(.67) (.50) (.63) (.56) (.56) (.66) (.67) 

Uncertainty (Y I X) 0.09 0.24 0.14 0.18 0.36 0.10 0.10 

Uncertainty (X I Y) .07 .30 .12 .22 .17 .08 .09 

Uncertainty (X ,Y) .08 .26 .13 .19 .23 .09 .10 

No. of points 1,153 1,215 1,208 1,205 693 627 311 

Percent data content * 

Columns 90 90 90 90 81 90 

Rows 90 100 72 90 50 100 

Cells 40 50 44 52 18 49 

* Percentage of rows, columns, and cells for which there are data. 

~li 

Cu 

Fe 

0,0 U(X I Y) 0.38 

Figure 6. Perspective plot of uncertainties of Mn 

versus uncertainties of Co, Ni, Fe, Cu, Zn, Pb, AI, Si, 

and depth of water above the sample site (de). 

U(Y I X), U(X I Y), and U(X, Y) are uncertainty 

coefficients. 

81 

100 

48 

Si de 

1.71 1.76 

(.71) (.73) 

1.45 1.96 

(.66) (.81) 

2.98 3.59 

(.64) (.74) 

1.26 1.83 

(.57) (.76) 

1.52 1.63 

(.63) (.68) 

0.13 0.06 

.11 .07 

.12 .06 

267 1,108 

81 90 

77 100 

35 69 

Conclusions 11 
1.6 

~ 1.2 

(I) 

~ (I) 

.e 0.8 ..... 

0 

..c 

0 

(.) 0.4 

.-.1.5 

..... 

c: 

(I) 

~ (I) 

Q. 

-I... 

(I) 

Q. 

Q. 

8 0.5 

7 

5 10 15 20 25 30 35 40 

Iron (percent) 

5 10 15 20 25 30 35 40 

Iron (percent) 

5 1 0 1 5 20 25 30 

Iron (percent) 

35 40 

,......._1.5 

-c: 

(I) 

u 

.... 

(I) 

Q. 

-q:; 

..::L. 

u 

z 0.5 

0.3 

--r:: 

(I) 

u 

~ 0.2 

-u 

c 

N 

-

0.1 

25 

-c: 20 (I) 

~ 

8. 15 

-

5 10 15 20 25 30 35 40 

Iron (percent) 

5 10 15 20 25 30 35 40 

Iron (percent) 

5 1 0 15 20 25 30 35 40 

Iron (percent) 

-1: 40 (I) 

u 

I... 

(I) 

Q. 

-(I) 

en 

~ 20 c 

01 

c 

~ 10 

-..... 

r:: 

(I) 

~ 

0.3 

8. 0.2 

-

"C 

c 

(I) 

_J 

0.1 

5 1 0 15 20 25 30 35 40 

Iron (percent) 

- ..... . . ··- .. ... _, __ ... . -. --·· .. ··--.. ----··. ---·- . .. -·--· 

5 1 0 15 20 25 30 35 40 

Iron (percent) 

61\. ·,~· .. =. 'Cii' • \.~··. ~ 5 c: ... ..... . .. ,. Cl) I \-. I 

E 4 • ·,: • ..2 • ;-·~ 

. . . . '· .. '· ~ 3 •••••• ··=-t-- . . .. · : .. -· :'· ~- . ... ". ··. . ·· .. • ...... ,. -·" .,., -~~ .. . ·. . ., . 

5 10 15 20 25 30 35 40 

Iron (percent) 

Figure 7. Scatter plots of Fe content versus content of Co, Ni, Mn, Cu, Zn, Pb, AI, and Si and depth of water above the 

sample site. 

12 Multivariate Clustering Based on Entropy 
Table 4. Entropy computations for iron versus nine variables 

[Parenthetical entries indicate entropy calculation normalized by 1/log n. de, depth of water above the sample site] 

Co Ni Mn Cu Zn Pb AI 

Entropy of X 1.49 1.49 1.49 1.49 1.38 1.48 1.57 

(.62) (.62) (.62) (.62) (.57) (.62) (.65) 

Entropy of Y 1.32 2.17 1.73 2.15 0.767 1.44 1.67 

(.55) (.90) (.72) (.89) (.33) (.62) (.76) 

Entropy of (X ,Y) 2.66 3.23 3.01 3.16 1.98 2.66 3.10 

(.55) (.67) (.62) (.66) (.42) (.56) (.67) 

Entropy of (Y I X) 1.17 1.73 1.51 1.67 0.609 1.17 1.53 

(.48) (.72) (.63) (.70) (.26) (.51) (.69) 

Entropy of (X I Y) 1.34 1.05 1.27 1.01 1.22 1.22 1.43 

(.55) (.44) (.53) (.42) (.50) (.50) (.59) 

Uncertainty (Y IX) 0.11 0.201 0.12 0.22 0.21 0.18 0.08 

Uncertainty (X I Y) .10 .295 .14 .31 .12 .18 .08 

Uncertainty (X ,Y) .10 .238 .13 .264 .15 .18 .08 

No. of points 1,146 1,208 1,208 1,198 694 626 310 

Percent data content * 

columns 72 72 72 72 63 72 72 

rows 100 100 90 90 50 100 100 

cells 38 45 44 38 15 34 42 

* Percentage of rows, columns, and cells for which there are data. 

Cu 

0 1 ~, 

~r I 

I I ,jp :>::" U(YIX) I C c> ;><~ 

Si AI ::) 

0,0 U(XI Y) 0.32 

Figure 8. Perspective plot of uncertainties of Fe 

versus uncertainties of Co, Ni, Mn, Cu, Zn, Pb, AI, 

Si, and depth of water above the sample site (de). 

U(Y I X), U(X I Y), and U(X, Y) are uncertainty 

coefficients. 

Si de 

1.44 1.50 

(.60) (.62) 

1.45 1.96 

(.66) (.81) 

2.79 3.24 

(.60) (.67) 

1.34 1.73 

(.61) (.72) 

1.33 1.27 

(.55) (.53) 

0.07 0.11 

.07 .15 

.07 .13 

266 1,100 

63 72 

77 100 

31 53 

Conclusions 13 
1.6 

~ 

0 

.!l 

0 

u 

0.5 1 1.5 

Copper (percent) 

40 

35 

- 30 -

c: 

Q) 25 0 

L. 

Q) 

c.. 20 ......... 

c 15 _g 

0.5 1 1.5 

Copper (percent) 

I I I 

7- -

--

c: 6-' - Q) 

0 

L. 

Q) 5- .. - c.. 

......... ·. 

4,. • • :.~ •• • • E ... - ::I • ···;:::, : • < •• c 3 -·. ~ :.~.· ':· • ..... ' ., :a .: • ·: .E '· - ·'·· .~·-~J·~:::'··:·:~ .. ~·:. ::I -.. , ...... "'.: ... ~ 

< 2 ;.l : • . • ,,,... .• ~ . - .. . . :· 

1 •: .. ~ .. - ~··. I I\ I 

0.5 1 1.5 

Copper (percent) 

.... -1.5 

-

c 

Q) 

0 

I.. 

Q) 

c.. ......... 

Q) 

..::L. 

0 

z 0.5 

0.3 

--

c 

Q) 

0 

0.5 1 1.5 

Copper (percent) 

. .. I . ~ 0.2 ......... 

0 

c 

N 

-

1: 

Q) 

0 

I.. 

Q) 

c.. 

......... 

c: 

0 

.~ 

Vi 

0.1 

0.5 1 1.5 

Copper (percent) 

I I I 

25 ~ 

20 ~ 

15 ~ 

. · . .. ·. "· •• :.r ....... .· 

10... • 00 "•" 

0 

I • 0 000 0 

~ ~\ '·, .. ~~ · ... : . ; :. . .. ..... • I j,JI .!1.:.: : ... • •• • --=• • I ·"' .. • \ I • I .. :v "'" • •• • 5 ~i~ ·.:;:t-' .. . : .· : ··. .· ~ ,. . .. •:' • I I I 

0.5 1 1.5 

Copper (percent) 

-

-

-

-

-

50 

--

c: 

Q) 

0 

L. 

Q) 

c.. 30 ......... 

Q) 

en 

Q) 

c: 20 0 

0' 

c 

0 

~ 10 

0.5 1 1.5 

Copper (percent) 

0.3 

--

c 

Q) 

~ 8. 0.2 ......... 

"0 

0 

~ 

0.1 •• :-:·.·:..!""" 

-· ·--···· ... . -·· ---·· ...... . 

0.5 1 1.5 

Copper (percent) 

6 ••••••• ••• ... ,.,: -~ .. :•; ·-:.· .. : .. · ...... . I ~ ·~···• I"' . \ . <~ • ~· r~:,. lf; .. " ·····:·::jlii. ~ : .. ·.~·- \.·~ . .,~ .. : .. ·· ...... .. . 4 :~.,~·: .. • .......... ..,. ... . . ... . . , . 

.. . .. 

0.5 1 1.5 

Copper (percent) 

Figure 9. Scatter plots of Cu content versus content of Co, Ni, Mn, Fe, Zn, Pb, AI, and Si and depth of water above the 

sample site. 

14 Multivariate Clustering Based on Entropy 
Table 5. Entropy computations for copper versus nine variables 

[Parenthetical entries indicate entropy calculation normalized by 1/log n. de, depth of water above the sample site] 

Co Ni Mn Fe Zn Pb 

Entropy of X 2.15 2.15 2.15 2.15 2.17 2.16 

(.89) (.89) (.89) (.89) (.90) (.90) 

Entropy of Y 1.31 2.18 1.73 1.49 0.7691 1.45 

(.54) (.91) (.72) (.62) (.33) (.62) 

Entropy of (X ,Y) 3.28 3.54 3.50 3.16 2.68 3.31 

(.68) (.73) (.73) (.66) (.57) (.70) 

Entropy of (Y I X) 1.13 1.38 1.34 1.01 0.51 1.15 

(.47) (.57) (.56) (.42) (.22) (.50) 

Entropy of (X I Y) 1.97 1.35 1.76 1.67 1.91 1.86 

(.82) (.56) (.73) (.70) (.79) (.77) 

Uncertainty(YIX) 0.13 0.36 0.22 0.31 0.33 0.20 

Uncertainty (X I Y) .08 .37 .18 .22 .11 .13 

Uncertainty (X ,Y) .10 .36 .19 .26 .17 .16 

No. of points 1,158 1,158 1,205 1,198 692 625 

Percent data content * 

columns 90 90 90 90 90 90 

rows 100 100 90 72 50 100 

cells 39 55 52 38 25 42 

* Percentage of rows, columns, and cells for which there are data. 

Ni 

Zn 

Pb 

0.38 

LA de > U(YIX) 

Si ~ ::s 

0,0 U(XIY) 0.38 

Figure 10. Perspective plot of uncertainties of Cu 

versus uncertainties of Co, Ni, Mn, Fe, Zn, Pb, AI, 

Si, and depth of water above the sample site (de). 

U(Y I X), U(X I Y), and U(X, Y) are uncertainty 

coefficients. 

AI Si de 

2.05 2.15 2.15 

(.85) (.89) (.89) 

1.66 1.45 1.95 

(.75) (.66) (.81) 

3.47 3.45 3.72 

(.75) (.75) (.77) 

1.41 1.29 1.56 

(.64) (.59) (.65) 

1.80 1.99 1.76 

(.75) (.83) (.73) 

0.15 0.11 0.19 

.12 .07 .17 

.13 .08 .18 

305 261 1,107 

90 90 90 

100 77 100 

47 46 52 

Conclusions 15 
1.6-

? 1.2- c 

Q) 

~ ... &. : ~ 0.8~ ·=. ~ '!"•" 

I 

.0 • ) • 

0 ··:t : .. 

I 

.. ~ . 

I 

u 0.4~ ........ iA~:· .: .. . 

·a;. . '~'··~·. L:.."!-!i.'''i • ' I • • 0 I 

? 

c 

Q) 

~ Q) 

Q. 

~ 

c 

_g 

--

c 

Q) 

~ Q) 

c. 

~ 

E 

::J 

c .E 

::J 

< 

7 

6 

5 

3 

2 

0.1 0.2 O.J 

Zinc (percent) 

0.1 0.2 O.J 

Zinc (percent) 

.. 

.. .. .. ... :~'·.: .. .,,1"'.: : .... ,f.,~l'···.· ~~ • 

·~.· ···.:.·~ . • I• • • • . ta• I 

.. , ... 

0.1 0.2 0.3 

Zinc {percent) 

-

-

-

-

_1.5~. 

-

c 

Q) 

0 

'"-

I I 

,• 

... • ···f'~t~: trJ. • I • • 

=~~;}_"'..:"· 

Q) 

c. 

~ 

··~~l!f·:::.. : . ,. ,;:.,\ . . I II ...... • • • 

1 ~· ··1~·-· ···~ . . ~f . ~ o.s~·.~.f: ~ . .,.:.: . -~--~ . 

I 

~ •• •:: I I I 

0.1 0.2 0.3 

Zinc {percent) 

,.....,.1.5 

-c: 

Q) 

0 

'"- Q) 

c. 

~ 

'"-Q) 

Q. 

0 

u 

--c 

Q) 

0 

'"-

Q) 

c. 

~ 

c 

0 

~ 

iii 

I I I 

25-

20-

151-

' ,,.. ·· .. 10 ~ •• =· 

: ~:'J' : , ·. . ····~· ....... . . .. 51-- •, .• ,:. • •• I a 

1- ·~:.· • • • • 

: I I I 

0.1 0.2 0.3 

Zinc {percent) 

-

-

-

-

-

-

-

-

50----~------~------r-----

--

c 

Q) 

0 

'"- Q) 

c. 

~ 

Q) 

Ul 

Q) 

c 

0 

0'1 

c 

0 

~ 

--

c 

Q) 

0 

40b 

... ·;,;.·· .. ·.: JOf; . • .• : '--l- ••• ·j . -~. · ... 20 ~·~ .• • ·,! ... ~ ... rl 

. ' ...... ,:J. .... . 

···:··· .. '-- . 10 ~ ·~')'( : . ,., . I 

•"• I I I 

0.1 0.2 O.J 

Zinc (percent) 

I I I 

0.3 I- ": 

._ .. : &. 0.2 ~ .:.."" ........., 

"0 

0 

Q) 

...J 

-Ul 

'"- Q) 

-

Q) 

E 

.2 

~ ~ 

..c 

-

c. 

Q) 

0 

0.1 f.;:~~::.~-

.. .. --·--··· ------.. .. - • "'"":'::::;_ • .:_;;, •• • I 

0.1 0.2 0.3 

Zinc (percent) 

7 I I I 

._,. . ~· ~·:~, ....... • ~ ... --..= 

5- • • • • .. ., .... ~ ..... ... ·· .. -."':*:·"i~\1'· :-'· 4~.:{·~ ••• ;( ......... ....... ;·~·~·. 3-··t!." • .. ,. . .~ ... 2 ~ -~'' 

rt .. \•· •• ··~,. 

1 ~·;·. 

0 '" I I I 

0.1 0.2 0.3 

Zinc (percent} 

-

-

-

-

-

-

-

-

-

-

-

-

-

Figure 11. Scatter plots of Zn content versus content of Co, Ni, Mn, Fe, Cu, Pb, AI, and Si and depth of water above the 

sample site. 

16 Multivariate Clustering Based on Entropy 
Table 6. Entropy computations for zinc versu:, r•1ne variables 

[Parenthetical entries indicate entropy calculation normalized by 1/log n. de, depth of water above the sample site] 

Co Ni Mn Fe Cu Pb 

Entropy of X 0.76 0.76 0.76 0.76 0.76 0.76 

(.33) (.33) (.33) (.33) (.33) (.33) 

Entropy of Y 1.21 2.13 1.63 1.38 2.17 1.38 

(.50) (.89) (.68) (.57) (.90) (.60) 

Entropy of (X ,Y) 1.9 2.57 2.12 1.98 2.68 1.96 

(.40) (.54) (.45) (.42) (.57) (.42) 

Entropy of (Y I X) 1.15 1.80 1.35 1.22 1.91 1.20 

(.48) (.75) (.56) (.50) (.79) (.52) 

Entropy of (X I Y) 0.70 0.43 0.48 0.60 0.51 0.57 

(.30) (.19) (.21) (.26) (.22) (.24) 

Uncertainty (Y I X) 0.05 0.15 0.17 0.12 0.11 0.13 

Uncertainty (X I Y) .08 .42 .36 .21 .33 .24 

Uncertainty (X ,Y) .06 .22 .23 .15 .17 .17 

No. of points 688 693 693 694 692 532 

Percent data content * 

columns 50 50 50 50 50 50 

rows 90 90 81 63 90 80 

cells 18 23 18 15 25 16 

• Percentage of rows, columns, and cells for which there are data. 

0,0 U(XI Y) 0.44 

Figure 12. Perspective plot of uncertainties of Zn 

versus uncertainties of Co, Ni, Mn, Fe, Cu, Pb, AI, 

Si, and depth of water above the sample site (de). 

U(Y I X), U(X I Y), and U(X, Y) are uncertainty 

coefficients. 

Al Si de 

0.77 0.63 0.76 

(.33) (.27) (.33) 

1.59 1.43 1.96 

(.72) (.65) (.81) 

2.26 2.02 2.60 

(.50) (.45) (.55) 

1.49 1.39 1.84 

(.67) (.63) (.76) 

0.67 0.58 0.64 

(.29) (.25) (.27) 

0.06 0.03 0.06 

.13 .06 .15 

.08 .04 .08 

178 144 646 

50 40 50 

88 77 100 

20 16 23 

Conclusions 17 
1.6 1- I I I 

'? 1.2 foCI) 

~ Cl) 

.e o.s r-a •• • 

=5 • • •••••• .0 •• •• 

0 • • .: : •• ··=· 

(.) 0.4 i 'II' I: I h i! i!: I!!.. . .. 

I I o •• , :I" •II: i ... I 

tlo ,I :' : • I I 

'"' 

-

c 

Cl) 

0 

.... 

Cl) 

a.. 

'-" 

c 

0 

.!:: 

'"' 

-c: 

Cl) 

~ Cl) 

a.. 

'-" 

0.1 0.2 0.3 

Lead (percent) 

40 

35 

30 

25 

. . . 20 • • • I •: • • • • • 

15 

10 

• • I I• I 'I• IIJI:: • • • ... ·I' i I'' : : .... = :: •'I II· i ... i: ·= . ·Ill...... ... . . .. l':i·= 

.. I • • 

0.1 0.2 0.3 

Lead (percent) 

7 

6 =· 

.. :!. :, ..• • I • • 

3 ·•1 ::: ~· ·. I '• :-, 'a· .. • I .1. • 1 ••" 2 • : ~: • • • • 

I •: I 

I .. 

0.1 0.2 O.J 

Lead (percent) 

-

-

-

-

I I I 

~ 1.5 ~l'i:::· ~ I •I····· ~ lll·i' ; . a.. 1 ,.. :.1. : ':::: : I· I: .. : . . Q) • I i •. I :. • • • 

~ .... ···:···· 

z 05 l:llll:.·!·· .• ii .:= .: • o..: ••• t•·'·:·l•· : • . : ·•I: : : .. • • • 'I 

1

: i •! 1 : I •:: 1 : I 

: • • • •! • I I I 

0.1 0.2 0.3 

'"' 1.5 

-c: 

G) 

~ G) 

a.. 

'-" 

.... 

G) 

a.. 

a.. 

8 0.5 

Lead (percent) 

0.1 0.2 0.3 

Lead (percent) 

25-

'"' 1: 20- G) 

~ 

8. 15- '-" 

c •• 

I 

~ 101-:.~. (i) •: .: •: 'I 1• i Ill" 0 I 

0 0 0 • 

5 -·· ••••• •• • I I • I: • • 

I 

. ..

.. 

I 

• I 1• I 

0.1 0.2 0.3 

Lead (percent) 

-

-

-

-

-

-

-

-

so----~,------~----~,-----

'"' 1: 40r; • Cl) • 

0 I • .... . . 8..Jo~· 

'i' "· 1·1:::· .. :' .. .. . .:. ~ 20 ~ ll.l'lil~h:; . . . 0 I •:•J •: ·= 0'1 i"••t''···· c ··: •••• :· o 10 I, •""• • • ~ :J ••• : .. : 

1 I I I 

'"' 

-

c 

Cl) 

O.J t0.1 0.2 0.3 

Lead (percent) 

I I I 

e .•. Cl) 0.2 t-t=· 

~ ll'i~ .· 

N t"ll•l"• • ••••• 

0.1 r: I "II··· ·.. .. . =II ... p.·::·· :1' .p; •• :·l•i:•==· I • el ,e.J • • • • •I 

•• "I • I I 

0.1 0.2 0.3 

Lead (percent) 

..r::. 

0. .. . .. Cl) 

c 

0.1 0.2 0.3 

Lead (percent) 

-

-

-

-

-

-

-

Figure 13. Scatter plots of Pb content versus content of Co, Ni, Mn, Fe, Cu, Zn, AI, and Si and depth of water above the 

sample site. 

18 Multivariate Clustering Based on Entropy 
Table 7. Entropy computations for lead versus nine variables 

[Parenthetical entries indicate entropy calculation normalized by 1/log n. de, depth of water above the sample site] 

Co Ni Mn Fe Cu Zn AI 

Entropy of X 1.45 1.44 1.44 1.44 1.45 1.38 1.67 

(.63) (.62) (.62) (.62) (.62) (.60) (.72) 

Entropy of Y 1.33 2.19 1.75 1.48 2.16 0.76 1.69 

(.55) (.91) (.73) (.62) (.90) (.33) (.76) 

Entropy of (X ,Y) 2.54 3.41 3.04 2.66 3.31 1.96 3.17 

(.54) (.72) (.64) (.56) (.70) (.42) (.70) 

Entropy of (Y I X) 1.09 1.96 1.60 1.22 1.86 0.57 1.50 

(.45) (.82) (.66) (.50) (.77) (.24) (.68) 

Entropy of (X I Y) 1.20 1.22 1.29 1.17 1.15 1.20 1.48 

(.52) (.53) (.56) (.51) (.50) (.52) (.64) 

Uncertainty (Y I X) 0.18 0.10 0.08 0.18 0.13 0.24 0.11 

Uncertainty(XIY) .16 .15 .10 .18 .20 .13 .11 

Uncertainty (X ,Y) .17 .12 .09 .18 .16 .17 .11 

No. of points 624 628 627 626 625 532 203 

Percent data content • 

columns 100 100 100 100 100 80 100 

rows 100 100 90 72 90 50 88 

cells 42 49 49 34 42 16 42 

• Percentage of rows, columns, and cells for which there are data. 

Zn 

0.26.----~ 

U(YIX) Mn 

0,0 U(X I Y) 0.26 

Figure 14. Perspective plot of uncertainties of Pb versus 

uncertainties of Co, Ni, Mn, Fe, Cu, Zn, AI, Si, and 

depth of water above the sample site (de). U (Y I X), 

U(X I Y), and U(X, Y) are uncertainty coefficients. 

Si de 

1.71 1.46 

(.74) (.63) 

1.51 1.98 

(.68) (.82) 

3.07 3.22 

(.68) (.68) 

1.35 1.76 

(.61) (.73) 

1.56 1.24 

(.67) (.54) 

0.10 0.10 

.09 .14 

.09 .12 

156 616 

90 100 

77 100 

41 58 

Conclusions 19 
1.6- I I I I I I I -

-:;::;- 1.2- c 

Q,) 

~ .. · ... Q,) •• 

~ 0.8 _.: • 

~ 

0 

..0 

0 ••••••• 

() 0.4 -:I • • ... ,.~~.: • ' ...... • 'iJ .. ,y 

~ :·~: ~~!;..~~::, .: . ' ;·.· ~';'1:11: .. ~~· "' ~ ' I ' , I 

""""' +' 

c: 

Q,) 

0 

IQ,) 

a. 

~ 

c: 

_g 

........... 

+' 

c 

Q,) 

0 

I40 

35 

30 

25 

20 

15 

10 

5 

1 2 3 4 5 6 7 

Aluminum (percent) 

I •'•, fl : '•,:' • 

.. '"·l ~ '"''I ,: .... ' . ····~ .. , ... .: .• .... . ~- .. ,- ... . . :t:••"\• ... . ' ..... lA . ..... . ·.· ... 

2 3 4 5 6 7 

Aluminum (percent) 

I ,I I I I I I 

0 . .3-

:g_ 0.2 ~: ~ 

"0 I- : 0 ... Q,) .. _J : . .. 0.1 f-

.. ..... .......... - .. .. .. 

·. ·-· -- ... I ·,:-·:.:~. J . i _j_ I 

2 3 4 5 6 7 

Aluminum (percent) 

-

-

-

-

-

-

""""' 1.5 +' 

c: 

Q,) 

0 

1-

Q,) 

Q. 

~ 

. . ... '. :~::~·.-== 

:c(~~~·:· .. .· ;' .. c· .v• \.' }·~ ... • • =· 

• • r: • • :· Q) ·.:i•,. .::L :. •• • •••• , 

0 'I 0 I :4 I•"' 

z 0.5 '• .~ :: ·1 ~:' ; .. 

,.....,1.5 

+' 

c 

Q,) 

0 

1-

Q,) 

a. 

~ 

1-

Q,) 

a. 

Q. 

8 0.5 

25 

........... 

+' 

c 20 Q,) 

0 

ICl) 

Q. 15 ~ 

c: 

0 

~ 10 

(i) 

5 

,) . . .. . ...... .. •.•• .'J. •• 

· .. ··:·•; .. ! . 

2 3 4 5 6 

Aluminum (percent) 

..... · ... • ... ;.f •• •• •·:c· .... .-:. :. 

···-· 

"-··----

, .. ·:~: • ... y ... 

. ,·.a..,... . . . .. ... . . . ... ····· .. .. .·.;'!"':~~·.· .. . . . . .... • • .;. 1:j.~ _,., •• •• . . . . ... , .. ~ ..•.. · '•' : ... . ·. . .. . ... 2 3 4 5 6 

Aluminum (percent) 

... : ••1•. r ... • 

• • ·,:~L.,: :: • 

•. 'fi'j!_ '•"'" ,","\:" ...... ... : ,· .. ~ ... ,, ... , ... . ·' . . . ... .... 

2 3 4 5 6 

Aluminum (percent) 

7 

7 

........... 

+' 

c 

Q,) 

0 

IQ,) 

a. 

~ 

Q,) 

Ul 

Q,) 

c: 

0 

0'1 

c: 

0 

:::t 

........... 

...... 

c: 

II.) 

0 

1-

50 

40 

... · . :~:..1"\ . • .. ~~ . .. I . .... . ., ... - .. ••• ~~'.· ~.t •• 20 .... .. .. · .. ·' ,. , ... '$ •• ~· •• •. . 

10 

0.3-

.... :.,. ':' .lf, e.:,.·· . . . . ' , .. •• 'll:· . \ \ . ':,• .. 

2 3 4 5 6 

Aluminum (percent) 

7 

I I I I I I I 

·.· 

8. 0.2- .. ~ 

0 

c: 

N 

""""' Ul 

IQ,) 

+' Q,) 

E 

.E 

:.i 

~ 

.L:. 

+' Q. 

Cl) 

Cl 

:£ ": •• . . , . ·&:: .. •• 

0.1- • ~·. 

... :·.J~·.=· .. 

7 

2 

:... .. : ···.A .. I.,.'·: I , .. . .... . .. · .. I \ .t ' I ·~ •I 

0 

I I 

1 2 3 4 5 6 

Aluminum (percent) 

. .. ' . . ··,:. . ·'·'· . I •' I:..& 1'"' • o • 

l ": .! :~ :·: ). .. . .. J.·~·· .... . . . ·· .. ·· .:-. . . ·. .. ~.-: ...... · .. \ .,. ·:!'· J ... . . 

... .. 

2 3 4 5 6 

Aluminum (percent) 

I 

7 

7 

-

-

-

Figure 15. Scatter plots of AI content versus content of Co, Ni, Mn, Fe, Cu, Zn, Pb, and Si and depth of water above the 

sample site. 

20 Multivariate Clustering Based on Entropy 
Table 8. Entropy computations for aluminum versus nine variables 

[Parenthetical entries indicate entropy calculation normalized by 1/log n. de, depth of water above the sample site] 

Co Ni Mn Fe Cu Zn 

Entropy of X 1.66 1.67 1.67 1.67 1.66 1.59 

(.75) (.76) (.76) (.76) (.75) (.72) 

Entropy of Y 1.47 2.17 1.78 1.57 2.05 0.77 

(.61) (.90) (.74) (.65) (.85) (.33) 

Entropy of (X ,Y) 2.94 3.67 3.27 3.10 3.47 2.26 

(.64) (.80) (.71) (.67) (.75) (.50) 

Entropy of (Y I X) 1.28 2.00 1.60 1.43 1.80 0.67 

(.53) (.83) (.67) (.59) (.75) (.29) 

Entropy of (X I Y) 1.47 1.49 1.49 1.53 1.41 1.49 

(.67) (.68) (.68) (.69) (.64) (.67) 

Uncertainty (Y IX) 0.12 0.07 0.09 0.08 0.12 0.13 

Uncertainty(XIY) .11 .10 .10 .08 .15 .06 

Uncertainty (X ,Y) .11 .09 .10 .08 .13 .08 

No. of points 301 310 311 310 305 178 

Percent data content * 

Columns 100 100 100 100 100 88 

Rows 90 100 81 72 90 50 

Cells 37 56 48 42 47 20 

* Percentage of rows, columns, and cells for which there are data. 

Pb 

Mn 

0.3 

Zc ~ \ 

\ 

U(YIX) $:' 

Fe l•li ~ :::5 

0,0 U(XI Y) 0.3 

Figure 16. Perspective plot of uncertainties of AI 

versus uncertainties of Co, Ni, Mn, Fe, Cu, Zn, Pb, 

Si, and depth of water above the sample site (de). 

U(Y I X), U(X I Y), and U(X, Y) are uncertainty 

coefficients. 

Pb Si de 

1.69 1.65 1.66 

(.76) (.75) (.75) 

1.67 1.43 1.95 

(.72) (.65) (.81) 

3.17 2.66 3.38 

(.70) (.60) (.73) 

1.48 1.00 1.72 

(.64) (.45) (.71) 

1.50 1.22 1.42 

(.68) (.55) (.64) 

0.11 0.29 0.12 

.11 .25 .14 

.11 .27 .13 

203 211 303 

88 88 100 

100 77 100 

42 35 55 

Conclusions 21 
1.6 r- I I I I I -

'? 1.2 r- - CI) 

u •• ~ I 

8. ....... 0.8fo". - +" 0 •• ..c 

0 ••• • •• 

u 0.4 ~ ·:.r;;~ . .:. ~· ~ ·. . .1,' ...... •• .. ~: . -)' ~· · ..... ~ r: I • j • • 0

1 I I 

-

5 10 15 20 25 

Silicon (percent) 

40 

35 

- 30 +" 

c 

Cl) 

u 25 .... 

Cl) 

0. 20 

- .... "' . c 0 0 • 

0 15 . ~:i.:::··l- . •. :: .. -= . • •irr( .. ·: ••• =,·. • 

10 .. :-.-;e:.' .. .. . .. . «··· . 5 •• •' I I I I • I • J' 

5 10 15 20 25 

Silicon (percent) 

I I I I I 

0.3 r- : -

-

+" 

c 

Cl) 

u 

~ 8. 0.2 ; ·: :0 - -"0 

c .. - Cl) .. ...J . 0.1 r- -· .. . .. - .. . ~ ..... 

• ·~··:.:: .L. I I I 

5 10 15 20 25 

Silicon (percent) 

I I 

·: ·:(:: .. . .. : ~ .. .. ~.ey.-...,· - 1.5r-

-

c 

Q) 

0 

.... 

Q) 

0. 1 r- •• '.f;-·:i .... . · .• :·~.· .. -

"'i) .. -t~ •• \ 0 0 

~ • ';c.;'" .,1 ... 0 • 

z 0.5 f-·;;,.. •..•. .-: : 

T 

... • • • • I ' ' ,. .... 0 •• .,. 0 ••• • 

. .. . . ~ . I 

T 

.. 

p ~ L

0 I o o I 1° 

5 10 15 20 

Silicon (percent) 

r T I I 

• 0 

. -1.5-

...... . ! .,. c 

Q) 

0 . ··t.~· .. .... 

Q) .. :c.;: : ·: 0. 1 -

- ......... ; .... .... Q) .. fi 0 . a. • • ••• .: _.t.· •• 0. 8 0.5- I : ;J•,; .:. :I I I ,• 

........ .. .. ' .. ·~·:· .~.-:,· '- . _. '-· . . .,<~·.:11::.. ·- ~ 0 ·~ ,. .. 5 10 15 20 

Silicon (percent) 

7 

-

+" 

c 6 Q) 

0 

.... 

Q) 5 a. .. 0 -

E 4 • : ·:,· 0 

;. ••• 0 •• 

:l 

c . -...i·· ... : :· .E 3 .. ,. fl'l • •••• 

:::l . . ~~-.. ~~. . < 2 .. ;..!:;: ·.~:~· .. .. , .. 5 10 15 20 

Silicon (percent} 

I 

-

-

-

I 

25 

I 

-

-

-

I 

25 

25 

50----r-~--~~--~~--~~--~~~~ 

-

+" 

c 

Q) 

0 

~ 

Q) 

0. 

-Q) 

fl) 

Q) 

c 

c 

C'l 

c 

c 

.:i! 

-

+" 

c 

Q) 

0 

.... 

.. 

30 r-·· ~··~; ~ • ~.. ·~ .... .: .... ~~-=~ . 20 r- -~ ..... ~; ~ •• ··.~~-x~·.,.~ .... · 

10 r-

/, • •• • Y. :' •• :". • I o 

0 0 

. . I" I • I I" 

5 10 15 20 

Silicon (percent) 

I I I I 

0.3 r-

. .. 

8. 0.2- ... -0 

c ·. N : ... , 

0.1 - .... -~-

.. ··~ :. • ·:·.:·~; :·.: .,.; fl' • •• 

. • I • I • 

I I : • • I ,· 

5 10 15 20 

Silicon (percent) 

7 

- ... :.., . . . fl) • •J•• .: • r • •• .... 

+" 

Q) ·' ... ·~··' .,;..:.. .:. ... .. Q) .. "'"""" ... · ... E .... ,..;:: ..Q ,; .. ~ 

-,£; 

+" 

0. i Cl) .... .... c 

0 

5 10 15 20 

Silicon (percent) 

-

-

-

-

I 

25 

I 

-

-

-

I 

25 

25 

Figure 17. Scatter plots of Si content versus content of Co, Ni, Mn, Fe, Cu, Zn, Pb, and AI and depth of water above the 

sample site. 

22 Multivariate Clustering Based on Entropy 
Table 9. Entropy computations for silicon versus nine variables 

[Parenthetical entries indicate entropy calculation normalized by 1/log n. de. depth of water above the sample site] 

Co Ni Mn Fe Cu Zn Pb 

Entropy of X 1.44 1.45 1.45 1.45 1.45 1.43 1.51 

(.65) (.66) (.66) (.66) (.66) (.65) (.68) 

Entropy of Y 1.38 2.18 1.71 1.44 2.15 0.63 1.71 

(.57) (.91) (.71) (.60) (.89) (.27) (.74) 

Entropy of (X ,Y) 2.69 3.46 2.98 2.79 3.45 2.02 3.07 

(.58) (.75) (.64) (.60) (.75) (.45) (.68) 

Entropy of (Y IX) 1.24 2.00 1.52 1.33 1.99 0.58 1.56 

(.52) (.83) (.63) (.55) (.83) (.25) (.67) 

Entropy of (X I Y) 1.31 1.27 1.26 1.34 1.29 1.39 1.35 

(.59) (.58) (.57) (.61) (.59) (.63) (.61) 

Uncertainty(YIX) 0.09 0.08 0.11 0.07 0.07 0.06 0.09 

Uncertainty (X I Y) .09 .12 .13 .07 .11 .03 .10 

Uncertainty (X ,Y) .09 .09 .12 .07 .08 .04 .09 

No. of points 258 266 267 266 261 144 156 

Percent data content * 

Columns 77 77 77 77 77 77 

Rows 81 100 81 63 90 40 

Cells 30 48 35 31 46 16 

* Percentage of rows, columns, and cells for which there are data. 

AI 

Mn 

Co de I 0.3 l Fe F 

Zn 

~ I ~ \ ~ $:' U(YIX) Uu ___ Ni 

~ Pb Cu :.::5 

0,0 U(XI Y) 0.3 

Figure 18. Perspective plot of uncertainties of Si 

versus uncertainties of Co, Ni, Mn, Fe, Cu, Zn, Pb, 

AI, and depth of water above the sample site (de). 

U(Y I X), U(X I Y), and U(X, Y) are uncertainty 

coefficients. 

77 

90 

41 

AI de 

1.43 1.45 

(.65) (.66) 

1.65 1.81 

(.75) (.75) 

2.66 3.09 

(.60) (.67) 

1.22 1.64 

(.55) (.68) 

1.00 1.28 

(.45) (.58) 

0.25 0.09 

.29 .11 

.27 .10 

211 254 

77 77 

88 100 

35 45 

Conclusions 23 
1.6 

'? 1.2 Q) 

0 

.... 

Q) 

.e 0.8 ...... 

0 

~ 

0 

(.) 0.4 

40 

35 

.z-30 

c 

~ 25 

8. 20 

-c 15 .e 

10 

0 

5 

-...... 

c: 

Q) 

0 

.... X. 0.2 

-"0 

0 

~ 0.1 

0 

0 

2 3 4 5 6 

Depth (kilometers) 

. ... . .... ,· ... · ... ). ,_.. . : .. . t ...... · ..... ·~ ·.··. : . .. ... ' . ,.. . "•':tt-..··':',;,'".'-:,,).:L .. ~..:!.,I. '• • • • ·.' : :·· I .v I 

o '',i.:: 0 fl I o'':,, o ~ I, 

= • • ~: ·.. :r I I '•,a '• o I 

2 3 4 5 6 

Depth (kilometers) 

·. 

: I I I I I 

. . ... . . -·-· ....... -.. . . . . . ··- .. . . . . . . . -- .. ··----· .... ---· .. . ·-·-·-· . .··--- .. ··----· ·- . 2 3 4 5 6 

Depth (kilometers) 

7 

7 

7 

-1.5-

..... 

c: 

I I I lo I I 

.. r:. .: .. - Q) 

0 

L.. 

8. 1-

.:--'-~. j

•:• .~ .... oo Jo ~~). I 

. . . .. ..•. .·:~·, - . . . (, ·~ _.;i~: . -Q) •• ':. • • • • • 10.:~.,.~ 

.::L ~ •• '•7•:c.-~. 0 : t ... : • • • ....... ;&...·~" .... Z 0 5 • .,., • ' • ~ • •• • P!4l~r . ~ . , .. ,.,... ..····· ····· . - ·, :: a.l. :,· ·~: •.. ..~ ... ) : '-l_, #. • 

,........1.5 

..... 

c: 

Q) 

0 

L.. 

Q) 

Cl. 

-L.. 

Q) 

a. 

Cl. 

8 0.5 

7 

'"" ..... 

c 6 Q) 

0 

.... 

Q) 5 a. 

-

E 4 

;::) 

c 3 .E 

;::) 

2 < 

~ ··~··· ~~··~-:. ·: .:·.·.t:;· . . . . . "·· ·. ' ·. ·.. . 'I ' I ' '• I •, j • ,' ,J • : .I •Jo 1 

0 1 2 3 4 5 6 

Depth (kilometers) 

0 

.. 

0 

2 3 4 5 6 

Depth (kilometers) 

'• 

. \. . . .... .\ ·. I I I I ,_ 

. =~ . .:· .. :~..·.··· .. . --~~"1' . .... o l'"o ', I ofl' lf!'iJI • ~}······" .,J' •;:-3- • ~~ I • I. I ..... 

, . . . ..... 

. - 2 3 4 5 6 

Depth (kilometers) 

7 

7 

7 

so--~--~----~ ----r- --~ 

-c 4Dh, Q) 

0 

L.. 

g_ 30 r-

-Q) 

Cll 

~ 20- 0 

Ol 

c: 

~ 10 ,_ 

-..... 

c 

Q) 

0 

.... 

0 

g_ 0.2 

-0 

c: 

N 

-...... 

c 

Q) 

0 

.... 

Q) 

Cl. 

-c 

0 

~ 

Vi 

0.1 

0 

25-

20 .... 

15-

10-

- ... 

... :.~~. :~ - ... ;:(:: .: :~·;Jr.fi· ~- ...... • .. ... !!f. • ..:·•-.. '~ • -

•. "·.:-· ... =·:.· :c . J I I; -: .. I ~ •:•. ~~ .: I I' I I . . -·.. ··: '=" . ·-=~· .. .· .. .:: · .. ·. ... . - II I : I~ '• 

. . .. 'I ''1 I I' '.1"' 'I 

1 2 3 4 5 6 7 

Depth (kilometers) 

: : 

7 

I I I I I I 

-

. - .. .. - I . · .. I .. . . .......... 

0 t· I - .. l:'•tff: . . i.'"j',. .. : { •• 5-:. :- · ... I I 01 ~r=''o'!" - lo I . .. :- . 'I I I I• I I I 1 

0 1 2 3 4 El 6 7 

Depth (kilometers) 

Figure 19. Scatter plots of depth of water above the sample site versus content of Co, Ni, Mn, Fe, Cu, Zn, Pb, AI, and Si. 

24 Multivariate Clustering Based on Entropy 
Table 10. Entropy computations for depth versus nine variables 

[Parenthetical entries indicate entropy calculation normalized by 1/log n. de, depth of water above the sample site] 

Co Ni Mn Fe Cu Zn 

Entropy of X 1.93 1.96 1.96 1.96 1.95 1.96 

(.80) (.81 (.81) (.81) (.81) (.81) 

Entropy of y 1.35 2.18 1.76 1.50 2.15 0.76 

(.56) (.91 (.73) (.62) (.89) (.33) 

Entropy of (x ,y) 3.16 3.89 3.59 3.24 3.72 2.60 

(.66) (.81 (.74) (.67) (.77) (.55) 

Entropy of (y I x) 1.23 1.93 1.63 1.27 1.76 0.64 

(.51) (.80 (.68) (.53) (.73) (.27) 

Entropy of (x I y) 1.81 1.71 1.83 1.73 1.56 1.84 

(.75) (.71 (.76) (.72) (.65) (.76) 

Uncertainty (y I x) 0.08 0.11 0.07 0.15 0.17 0.15 

Uncertainty (x I y ) .06 .12 .06 .11 .19 .06 

Uncertainty (x ;y ) .07 .12 .06 .13 .18 .08 

No of points 1,058 1,118 1,108 1,100 1,107 646 

Percent data content * 

Colunms 100 100 100 100 100 100 

Rows 100 100 90 72 90 50 

Cells 55 67 69 53 52 23 

* Percentage of rows, columns, and cells for which there are data. 

Cu 

0,0 U(XI Y) 0.2 

Figure 20. Perspective plot of uncertainties of depth 

of water above the sample site (de) versus uncertainties of Co, Ni, Mn, Fe, Cu, Zn, Pb, AI, and Si. 

U(Y I X), U(X I Y), and U(X, Y) are uncertainty 

coefficients. 

Pb AI Si 

1.98 1.95 1.81 

(.82) (.81) (.75) 

1.46 1.66 1.45 

(.63) (.75) (.66) 

3.22 3.38 3.09 

(.68) (.73) (.67) 

1.24 1.42 1.28 

(.54) (.64) (.58) 

1.76 1.72 1.64 

(.73) (.71) (.68) 

0.14 0.14 0.11 

.10 .12 .09 

.12 .13 .10 

616 303 254 

100 100 100 

100 100 77 

58 55 45 

Conclusions 25 
Table 11. Correlation coefficients of raw data 

[Cophenetic correlation coefficient = 0.87. See p. 5 for discussion] 

Co Ni MR Pe Cu Zn Pb AI 

Co 1.00 ..0.11 0.11 0.31 -0.33 -0.11 0.60 -0.42 

Ni -.11 1.10 .71 ~.66 .19 .04 -.42 -.07 

Mn .11 .71 1.00 -. .41 .60 .11 -.12 -.45 

Fe .3& .... -. .41 1.01 -.70 -.11 .S4 -.21 

Cu -.33 .IJ .60 -.70 1.00 .OS -.53 .04 

Zn -.11 .04 .11 -.11 .OS 1.00 -.13 -.15 

Pb .60 -. .42 -.12 .54 -.53 -.13 1.00 -.19 

AI -• .42 -Jl7 -.45 -.21 .04 -.15 -.19 1.00 

Si -.31 -.IJ -~ -.C17 -.06 -.17 -.11 .70 

Dll6 -.40 :J.7 -.02 ··" . .42 -.11 -.44 .33 

I I I I I I I I I I I I I .I I I I I I I I 

I I I I I I I I I I I I I I I I· I I I I I 

-1 -0.5 0 0.5 1 

Si Depth 

-0.31 -0.-40 

-.~ .27 

-.45 -.02 

-.07 -34 

-.06 . .42 

-.17 -.11 

-.11 -.44 

.70 .33 

1.00 .11 

.11 1.00 

Cobalt 

Lead 

Iron 

Nickel 

Copper 

~anganese 

Zinc 

Aluminum 

Silicon 

Depth 

Fpe 21. Dendritic diagram showing clusters of variables 

based on linear correlation coefficients of raw data. 
Table 12. Correlation coefficients of logarithms of raw data 

[Cophenetic correlation coefficient = 0.83. Seep. 5 for discussion] 

Co 

Co 10.00 

Ni .18 

~ .31 

Fe .37 

Cu -.11 

1.a -.17 

Pb .51 

AI -.36 

Si -.1, 

Depth -.12 

Ni Mn Fe Cu 1.a ,.. AI Si .Deptll 

0.18 0.31 0.37 -0.11 -0.17 0.51 -0.36 -0.15 -0.12 

1.00 .70 -.42 .80 .59 -.29 .OS .11 .36 

.70 1.00 -.22 .45 .so -.07 -.32 -.38 .OS 

-.42 -.22 1.00 -.41 -.51 .51 -.12 .09 -.19 

.10 .45 -.48 1.00 .S4 -.41 .ll .22 .63 

.59 .so -.51 .s• 1.00 -.51 -.14 -.28 .10 

-.29 -.07 .51 -.41 •. ,1 1.00 -.19 -.06 -.32 

.OS -.32 -.12 .28 -.1. -.19 1.00 .71 .46 

.11 -.38 .09 .22 -.28 -.06 .71 1.00 .44 

.36 .OS -.19 .63 .10 -32 .46 .44 1.00 

I I I I I I I I I I I I I I I I I I I I I 

Cobalt 

Lead 

Iron 

Nickel 

Copper 

Manganese 

Zinc 

Aluminum 

Silicon 

Depth 

I I I I I I I I I I I I I I I I I I I I I 

-1 -0.5 0 0.5 1 

Figure 22. Dendritic diagram showing clusters of variables 

based on linear correlation coefficients of logarithms of the raw 

data. 

Conclusions 27 
Table 13. Symmetrical uncertainties of elemental enrichments 

[Cophenetic correlation coefficient = 0. 86. See p. 5 for discussion] 

Co Ni Mn Fe Cu Zn Pb Al 

Co 1.00 0.06 0.09 0.09 0.09 0.04 0.15 0.09 

Ni .06 1.00 .18 .12 .27 .12 .08 .07 

Mn .09 .18 1.00 .09 .14 .11 .05 .12 

Fe .09 .12 .09 1.00 .13 .11 .17 .06 

Cu .09 .27 .14 .13 1.00 .10 .11 .10 

Zn .04 .12 .11 .11 .10 1.00 .09 .06 

Pb .15 .08 .05 .17 .11 .09 1.00 .08 

Al .09 .07 .12 .06 .10 .06 .08 1.00 

Si .07 .08 .12 .06 .08 .04 .09 .23 

Depth .03 .05 .02 .03 .04 .02 .03 .09 

Si 

0.07 

.08 

.12 

.06 

.08 

.04 

.09 

.23 

1.00 

.08 

------1~...--__ _ 

I 

0 

,....--

I 

I 

I 

0.5 

Depth 

0.03 

.05 

.02 

.03 

.04 

.02 

.03 

.09 

.08 

1.00 

Cobalt 

Iron 

Lead 

Nickel 

Copper 

Manganese 

Zinc 

Aluminum 

Silicon 

Depth 

Figure 23. Dendritic diagram showing clusters of variables 

based on uncertainty coefficients of elemental enrichments. 

28 Multivariate Clustering Based on Entropy 
Table 14. Symmetrical uncertainties of raw data 

[Cophenetic correlation coefficient = 0.88. See p. 5 for discussion] 

Co 

Co 1.00 

Ni .08 

Mn .08 

Fe .10 

Cu .10 

Zn .06 

Pb .17 

Al .11 

Si .09 

Depth .07 

Ni 

0.08 

1.00 

.26 

.23 

.36 

.22 

.12 

.09 

.09 

.12 

I 

0 

Mn 

0.08 

.26 

1.00 

.13 

.19 

.23 

.09 

.10 

.12 

.06 

Fe Cu 

0.10 0.10 

.23 .36 

.13 .19 

1.00 .26 

.26 1.00 

.15 .17 

.18 .16 

.08 .13 

.07 .08 

.13 .18 

Zn 

0.06 

.22 

.23 

.15 

.17 

1.00 

I 

0.5 

.17 

.08 

.04 

.08 

Pb 

0.17 

.12 

.09 

.18 

.16 

.17 

1.00 

.11 

.09 

.12 

Al Si Depth 

0.11 0.09 0.07 

.09 .09 .12 

.10 .12 .06 

.08 .07 .13 

.13 .08 .18 

.08 .04 .08 

.11 .09 .12 

1.00 .27 .13 

.27 1.00 .10 

.13 .10 1.00 

Cobalt 

Lead 

Nickel 

Copper 

Iron 

Manganese 

Zinc 

Aluminum 

Silicon 

Depth 

Figure 24. Dendritic diagram showing clusters of variables 

based on uncertainty coefficients of the raw data. 

Conclusions 29 
Table 15. Cluster analysis of linear correlation coefficients of raw data and their 

logarithmic transforms and uncertainty coefficients of elemental enrichments and 

raw data 

Cluster Correlation coefficient 

Raw data Log raw data 

Cobalt Cobalt 

I Lead Lead 

Iron Iron 

Nickel Nickel 

Copper Copper 

n Manganese Manganese 

Zinc Zinc 

Aluminum Aluminum 

m Silicon Silicon 

Depth Depth 

REFERENCES CITED 

Botbol, J.M., 1987, Geochemical enrichment vectors applied to 

manganese oxide phases in the northern Pacific: U.S. Geological Survey Open-File Report 87-197, 17 p. 

Botbol, J .M., and Evenden, G .1., in press, Descriptive statistics 

and spatial distributions of geochemical variables associated 

with manganese oxide-rich phases in the northern Pacific: 

U.S. Geological Survey Bulletin 1863. 

Davis, J.C., 1973, Statistics and data analysis in geology: New 

York, John Wiley and Sons, 550 p. 

Frazer, J.Z., and Fisk, M.B., 1980, Availability of copper, 

nickel, cobalt, and manganese from ocean ferromanganese 

30 Multivariate Clustering Based on Entropy 

Uncertainty coefficient 

Enrichment Raw data 

Iron Cobalt 

Lead Lead 

Cobalt 

Nickel Nickel 

Copper Copper 

Manganese Iron 

Zinc Manganese 

Zinc 

Aluminum Aluminum 

Silicon Silicon 

Depth Depth 

nodules, pt. III: La Jolla, Calif., Scripps Institution of 

Oceanography, SIO reference 80-16 [prepared for U.S. 

Bureau of Mines], 117 p. 

Huntsberger, D. V., 1963, Elements of statistical inference: Boston, Allyn and Bacon, 291 p. 

McKelvey, V.E., Wright, N.A., and Bowen, R.W., 1983, 

Analysis of the world distribution of metal-rich subsea 

manganese nodules: U.S. Geological Survey Circular 886, 55 

p. 

Press, W.H., Flannery, B.P., Teukolsky, S.A., and Vetterling, 

W. T. , 1988, Numerical recipes in C-The art of scientific 

computing: Cambridge, England, Cambridge University 

Press, p. 498. 

Singh, A., 1966, Great ideas in information theory, language, and 

cybernetics: New York, Dover Publications, 338 p. 


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