Bulletinof
The Japanese Classification Society

Volume 1
December 1987
JAPAN

Contents
Statistical Analysis of Heavy Metals in Human Body Organs
Masaya Miyai
Changes in the Distribution of Accumulated Savings
Toji Makino and Takashi Hatada
Disease Classification in Traditional Chinese Medicine
Hideo Miyahara and Han Jing Xian
Automatic Thresholding Methods from Histograms
Nobuyuki Otsu
Writer Identification Experiment Using Reduced Variance Estimators
Isao Yoshimura and Mitsu Yoshimura
Hierarchical Classification of Large Datasets and its Applications
Miyuki Takahashi and Noboru Ohsumi
Characterization on the Parameter Sub-family of Agglomerative Hierarchical Clustering
Algorithms
and the Space Distorting Properties
Nagatomo Nakamura and Noboru Ohsumi
ANew Criterion for Classification Using AID
Toshiro Haga and Chikuma Hamada
AConversational AID System
Toshiro Haga and Motohiro Setoya

1 of 14JCS Bulletin Vol. 1

Statistical Analysis of Heavy Metalsin
Human Body Organs
Masaya Miyai
Himeji-Dokkyo University
7-2-1, Kamiohno, Himeji, Hyogo, JAPANAccording to the industrial developments of Japan, substances which exist formerly few inenvironment increase gradually over the natural and human area and finally harmed humanbodies.
In this report behaviours of mercury in human body organs are examined. Mercury,
especially methyl, is one of the dangerous substances for all life over the world. It attacks the
motor nerve, sensory nerve, optic nerve and so on, and causes complicate symptoms, called
typically by Hunter-Russel's syndrome.
Data are measured on human dead body organs who thought to be suffered by mercury
through environment originally exhausted from factory.
In this case organs means four parts, cerebrum, cerebelli, liber and kidney. Mercury means
two kinds, total and methyl. Number of case is ninety eight constituted by seventy six males
and twenty two females and have the other various kinds of characteristics each other, also they
are divided into two groups, admitted to be harmed by mercury or not by the committee.
Each of eight variates distributes nearly log-normally and transformed to nearly normal
distribution by logarithmic transformation.
Ratios methyl/total of each four organs distributes nearly log-normally and distributes wider
range compared with former reports.
Almost all of ratios cerebrum/liver and cerebelli/liver of methyl mercury distribute withinone. Half number of ratios of cerebrum/kidney and cerebelli/kidney of methyl mercury
distribute over one.
Principal component analysis is applied to the data to see the structure of mercuries inhuman body organs.
The results show that the first principal component is thought to be the total amounts of eachmercury, the second is the difference among organs, liver and kidney to cerebrum andcerebelli, the third is the difference between mercuries, methyl to total.
Each mark of admitted or not for ninety eight case is plotted on the sample score diagrams,
but it is difficult to see the relation among the components and the groups of admitted or not.
Discriminant analysis is applied to the data to see whether the amounts of mercuries of eachorgan contributes to the decision of admittance, and if so then what kinds of mercuries do so.
The results shown that the most effective one is total merculy of kidney and next one is total
merculy of cerebelli, but their contributions are not so great and each methyl mercury inquestion has few contributions.

2 of 14JCS Bulletin Vol. 1

Changes in DistributionofAccumulated Savings
Toji Makino
Takashi Hatada
Science University of Tokyo
Noda City, Chiba 278, Japan

Every autumn, the "Public Opinion Survey on Saving" conducted in the previous year by
the Saving Encouragement Central Committee is published in most newspapers. We discuss
here the distribution of accumulated savings in the light of this survey's results. It has alreadybeen shown that the distribution of savings, like income distribution, approximates to the lognormal distribution. In this presentation, we study the pattern of the distribution of
accumulated savings over the last five years.
In the first place, we calculate the parameters and mode of the corresponding log-normal
distribution from the mean and median of accumulated savings for the last five years. Further,
we examine the change in the mean, median and mode of these saving holdings year by year.
Secondly, we draw Pareto diagrams for the distribution of accumulated savings in the last
five years, and calculate the area of the bow shape that shows the degree of inequality. We
examine the year by year variation in this.
Finally, we do ABC analysis on the Pareto diagram. We also examine the yearly variationin the points of division derived from ABC analysis.
From these calculated results, no great changes are evident in the savings differentials overthe period of the study.

3 of 14JCS Bulletin Vol. 1

Disease Classification in Traditional
Chinese Medicine
Hideo Miyahara and Han Jing Xian
Dept. of Internal Medicine, School of Medicine, Kitasato University.
1-15-1, Kitasato, Sagamihara, Kanagawa, 228 Japan

By using 28 signs and symptoms (SS) commonly used in traditional Chinese medicine,
specialists in Chinese medicine classified 377 cases of chronic epigastralgia (wei won tong)
into 3 disease categories, that is D1, D2 and D3. The data collected in this study were analysedby principal component analysis and Hayashi's quantification method type III. Both 377 casesand 28 SS were arranged according to the similarity and were assigned on multidimensional
coordinates. On the 2 dimensional plane spanned by the first 2 components, 377 cases wereplotted and were examined whether the members of the same category could make an isolated
cluster and did not mix with those of the different categories. The examination revealed that themembers of D1 and those of D2 occupied the left half and the right half of one unseparable
group, respectively. On the other hand, those of D3 were scattered below the combined groupof D1 and D2. As a result, it was concluded that the separation of D1 and D2 was considerably
difficult from the standpoint of the numerical taxonomy alone, whereas that of D3 and the rest
members was achieved by using the coordinates of the 2nd component. Spatial allocation ofeach of the 28 SS was represented multidimensionally, and could not be reduced a few
dimensional space.

4 of 14JCS Bulletin Vol. 1

Automatic Thresholding Methods from Histograms
Nobuyuki Otsu
Electrotechnical Laboratory
1-1-4 Umezono, Tukuba-shi, Ibaraki, 305 JAPANThreshold selection is a basic problem in image processing and many other fields. It can beviewed as one-dimensional unsupervised classification, given a histogram as a mixture of
implicit class distributions. Many methods have been proposed including adhocs andheuristics. In this paper, we briefly survey those methods and reconsider the problem from thestandpoint of unsupervised statistical decision and also of information theory. Without loss ofgenerality, two class cases are considered.
As a direct and desirable criterion function J(T) to evaluate the goodness of a threshold T
and derive an automatic thresholding by optimization, we adopted the error rate ofclassification. The error rate would be minimized by the Bayesian thresholding if we knew
exactly the two class distributions (supervised cases). However, in our case of unsupervised
decision, we have to estimate the parameters and approximate the optimum decision in terms ofthreshold T.
Three methods are derived. One is based on the information theoretic equivocation as an
upper bound on the Bayes error rate. The others make use of linear approximations of Bayes
posteriors, for an upper bound and for the Bayesian thresholding in an iterative process,respectively. Some experimental results are shown to indicate the validity of the methods.

5 of 14JCS Bulletin Vol. 1

Writer Identification Experiment Using
Reduced Variance Estimators
Isao Yoshimura
Faculty of Engineering, Nagoya University
Chikusa-ku, Nagoya, 464, JAPANMitsu Yoshimura
Shotoku-Gakuen Women's Junior College
Nakauzura, Gifu, 500, Japan

The principal problem of writer identification is to infer the writer of a set of handwritten
characters whose writer is not known from a reference set of characters whose writers are
known. In this paper the following situation is considered:'The character Y in question is ob served as a p-variate vector and the writer of Y is known
to be among a persons Πi,i=1, ... , a. The reference set is composed of the sample characters{X ij;i=1, ... ,a, j=1,...,m} in which the same letter as Y is written m-times repeatedly by eachof the a persons. The writer of Y is inferred as the person the distance from which to Y is theminimum. The distance is assumed to have the form that
-1

2

D (Y :Π i.) = (Y -X i.)'Σ i (Y - X i.) + nΣ i ,(1)where X i
is the sample mean of {Xij ; i=1, ... ,a} and Σ i
is a suitably chosen estimator of the variance matrix ∑ i .
Concerning with such situations Inaba [1] suggested the use of reduced variance estimator
Wi(w) defined by (2).
W (w) = (1-w)V + w V.
i

i

(2)

where V is the sample variance matrix based oni{X ; i=1, ... ,a} and V. = (V1+ ... +Va)/a .
ij

He showed the effectiveness of this estimator in cases of small m through a Monte Carlostudy under normality assumptions. However in real situations the normality assumption is notalways assured. The effectiveness must be examined through experiments with real data,
which is performed in this paper.
The material used in the experiment is composed of repeated writings of a sentence with 25
letters by a=24 persons. The writings were done at 6 different occasions at least three weeks
6 of 14JCS Bulletin Vol. 1

apart and at each occasion the sentence was written 8 times repeatedly. Two characters of eachletter by each person at the first through the fourth occasions are contained in the reference set,that is, m=8. Four characters of each letter at each occasion by each person not included in thereference set are contained in the test set which is used for obtaining correct identification rates
in the experiment.
As the features 'the weighted direction index histogram' proposed by Kurita et al. [2] areused which determine p=64 dimensional vectors. The parameter w is set on the five values
0.00, 0.25, 0.50, 0.75, and 1.00
In the calculation of inverse matrix of Wi(w), eigenvalues less than the (m-1)-th aresubstituted with the (m-1)-th as is proposed in Yoshimura et al. [3].
The average correct recognition rates obtained in the experiment are as follows:This result implies the superiority of the pooled variance estimator which corresponds to the
reduced variance estimator with w=1.0. The result is out of the anticipation induced from
Inaba's Monte Carlo study. The reason must exist in the following:(a) The normality is violated.
(b) The population variances are equal.
(c) When m is less than p Inaba's result is not valid.
(d) When p and m are large Inaba's result is not valid.
(e) The substitution of eigenvalues yielded the result.
It is left for further study which is the true cause.
References[1] Inaba. (1987), "Reduced variance estimators for discriminant analysis, "(in Japanese) 9-th
symp. on applied statistics, 1987.10., Tokyo.
[2] Kurita et al. (1983), "Handwritten character recognition based on weighted direction index
histogram and pseudo Maharanobis' distance," (in Japanese) IECE Jpn., PRL. 82-79.
[3] Yoshimura et al. (1986), "Discriminant analysis applied to the writer identification," 2ndJapan-China Symp. Statistics, 1986.11, Fukuoka, Japan.

7 of 14JCS Bulletin Vol. 1

Hierarchical Classification of Large Data Sets
and its Applications
Miyuki Takahashi and Noboru Ohsumi
Department of Statistical Methodology
The Institute of Statistical Mathematics
Tokyo Japan

Various ideas will be required in order to classify large data sets under a computer
environment, such as a microcomputer or workstation, and then watch the clustering processby graphical and colored representation.
The objective of this paper is to propose some procedures for presenting the skeleton of thedata structure, and not to directly display the distribution of large-scale data. The clustering
process for classifying large-scale data sets (exceeding 10,000 observations) and representing
the classification results smoothly with computer graphics is provided. The process isessentially summarized as follows:
Step 1: A partial data set is sampled from a given data set at random in a certain
proportion.
Step 2: Sampled data is subjected to the initial classification. At this time, the numberof clusters is specified to be as large as possible to generate a large number of groups.
The distance table is not used because it requires the use of a large matrix. The kmeans method is used in a hierarchical manner to obtain the tree structure.
Step 3: The classification thereby completed is displayed by tree representation as the"preliminary classification."
Step 4: All individuals in the data set subjected to the initial classification areclassified
consecutively using the tree obtained in Step 3 as a decision tree (i.e., each individual isassigned consecutively to some cluster in the tree by a binary decision rule).
Step 5: After all individuals are assigned, "reclassification" or "refinement" isconducted using the centroid vector of each cluster at the terminal end of the tree. The
reclassification obtained in this step is the fine adjustment of the relationship between
each cluster and the individuals belonging to each cluster. At this time, the tree
structure is reconstructed using the RNN-rule (reciprocal nearest neighbor rule) and the
NN-chains (nearest neighbor chains).
Step 6: The tree structure obtained by reclassification is displayed and observed again,
and is compared with that obtained by the preliminary classification.
Next the experimental results obtained using artificial data sets and practical LANDSAT data
are presented. The following figures show the tree structures presenting the results ofpreliminary classification and the reclassification of these data sets. In Figures 1 and 2, it canbe seen that the figures of the original artificial data are fairly well exposed by the trees. On theother hand, in Figures 3 and 4, we can observe several tendencies indicating the characteristics
of LANDSAT data, that is, the heterogeneous classes (the difference between land and sea, thesates of ocean currents and clouds, and so on) within the data sets can be clearly observed fromthe shape of branches in the tree structures. These features will be more really emphasizedusing the color display monitor and color photo prints.

8 of 14JCS Bulletin Vol. 1

Figure 1
Result of preliminary classificationof artificial datasetof artificial dataset(61,440 observations)Figure 2
Result of reclassification(61,440 observations)Figure 3
Result of preliminary classificationof LANDSAT data
(61,440 observations)Figure 4
Result of reclassificationof LANDSAT data
(61,440 observations)

9 of 14JCS Bulletin Vol. 1

Characterization on the Parameter Sub-Familyof Hierarchical Clustering Algorithms
andthe Space-distorting Properties
Nagatomo Nakamura
The Postgraduate Course of Science and Technology
Nihon University
Tokyo Japan
Noboru Ohsumi
Department of Statistical Methodology
The Institute of Statistical Mathematics
Tokyo JapanThe objective of this paper is to present algebraically the agglomerative hierarchicalclustering algorithms (the AHC algorithms) and some properties of the clustering strategies.
Generally, the AHC algorithms, in which the dissimilarity or distance between a newly-mergedcluster Ci ∪Cj and the remainingd(C i∪C j,C k) = α id(C i,C k) + α jd(C j,C k) + βd(C i,C j) + γ d(C i,C k) - d(C j,C k)

(1)

In this formula, h(Ci ) is the height (or level) of cluster Ci a tree structure, such as adendrogram, and θ≡{α i . α i , β, γ} is a set of parameters that characterize the clustering
strategies. Some clustering strategies included in the AHC algorithms have the property called"reversals in the tree structure," i.e., it can contain clusters C1 and C2 for which C2 ⊃ C1 but
produce h(C1 ) < h(C2 ). With respect to this matter, Lance and Williams (1967) introduced theconcept of space distortion, explaining the clustering strategies as space-contracting, spaceconserving or space-dilating, but they discussed these concepts intuitively.
On the other hand, Milligan (1979) and Batagelj (1981) present necessary and sufficientconditions for suppressing the presence of reversals produced by the AHC algorithms.
Moreover, Dubien and Warde (1979) derive under some assumptions the more sophisticatedresults of space distortion among distances that are obtained at different levels in merging
clusters. However, these efforts are restricted to characterization of a sub-family of AHC
algorithms, the (β-γ) family in the algorithm described as expression (1).
This paper, therefore, discusses extension and sophistication of the concept of the spacedistortion. First, we define the concept as follows:
Letting∆={d(C i∪C j,C k) at (α i,α j,β,γ )d(C i,C j) < d(C i,C k) < d(C j,C k)(2)

be a set of all distances obtained from the result of an agglomeration or a clustering process, wedefine the conditions of space distortion as:
10 of 14JCS Bulletin Vol. 1

1) space-conserving:2) space-contracting:3) space-dialating"d(C j,C k) < d(C i∪C j,C k) < d(C j,C k) ; d(C i∪C j,C k) ∈ ∆d(C i,C k) ≥ d(C i∪C j,C k) ; d(C i∪C j,C k) ∈ ∆d(C j,C k) ≤ d(C i∪C j,C k) ; d(C i∪C j,C k) ∈ ∆The relationship between the parameter set θ and the space distortion is clearly examinedaccording to these definitions. Especially, the {α i , α j , β, γ} family of AHC algorithms and theregions of the Cartesian plane investigating a strategy for the {α i , α j , β, γ} algorithms arestrictly derived. Hence, this investigation clarifies the classification of most clustering
strategies that are characterized by the general agglomerative algorithms (1) belonging to the{α i , α j , β, γ} plane (for example, single-linkage, complete-linkage, centroid method, groupaverage method, median method, Ward's method, minimum variance method by Jambu, and soon). Experimental examples obtained using artificial data sets are discussed. In addition, therelationships between each parameter and the agglomerative clustering strategies are graphicallypresented within a domain specified by the {α i , α j , β, γ} plane.
References(1)V. Batagelj (1981):Note on ultrametric hierarchical clustering algorithms, Psychometrika,
46, 351-352.
(2)J. L. DuBien and W. D. Warde (1979): A mathematical comparison of the members of an
infinite family of agglomerative clustering algorithms, The Canadian Journal of Statistics, 7,
1, 29-38.
(3)G. N. Lance and W. T. Williams (1967): A general theory of classificatory sorting
strategies, I.Hierarchical systems, The Computer Journal, 9, 373-380.
(4)G. W. Milligan (1979):Ultrametric hierarchical clustering algorithms, Psychometrika, 44,343-346.

11 of 14JCS Bulletin Vol. 1

A New Criterion for Classification Using AID
Toshiro Haga and Chikuma Hamada
Department of Management Science
Science University of Tokyo

When predictor variables which are two-way classification types are used with the AID(Automatic Interaction Detector) program, assigning a score of 0 and 1, or 1 and 2, to thepredictor variables makes it possible to use maximization of the sum of squares between thegroups or of the variance ratio as the partitioning rule, as when the predictor variables arequantitative variables. Use of χ 2 in a contingency table and of entropy have both beensuggested, but the characteristics of these two methods differ little from those of the sum ofsquares between groups method. This report introduces a new classification criterion which
differs significantly from conventionally used criteria.
The validity of a classification result can be judged from the following two conditions:1) The difference between the average scores (ratios) for groups 1 and 2 is large.
2) The groups are partitioned so that the number of cases in both groups is approximately
the same.
Table 1 Cross totals (number of cases)
y
0
1
Total
Mean

Group 1adc1p1Group 2
b
e
c2
p2

Total
r1
r2
n
p

Table 2 Example
Previous
No
86
122
208
p
|p1 - p2|

Illness
Yes
1
25
26
0.587
0.375
F0
F1
F2

White Cell
<7000
19
70
89
0.962
0.256
13.9
23.2
54.3

Count
≥7000
68
77
145
0.787

Total
87
147
234
0.531

0.650

15.4
16.6
18.1

When the distribution of cases following classification is as shown in Table 1, the varianceratio between groups is expressed by the expression below. In order to distinguish this
12 of 14JCS Bulletin Vol. 1

variance ratio from the one we will introduce below, we shall call it F0.
F0 = (p1 - p2)2/{p(1 - p)(1/r1 + 1/r2)}
Under the null hypothesis that the expected values of p1 and p2 are equal, the variance ratio
is obtained using the variance within the group as that for the entire sample.
Since in partitioning using AID, the classification is conducted under the assumption that the
expected values of p1 and p2 will be different, a new variance ratio F2 is defined using theinternal variance within each group.
F2 = (p1 - p2)2/{p 1(1 - p1)/r1 + p2(1 - p2)/r2}
Finally, the variance ratio obtained from variance equalization through arc–sinetransformation performed on p1 and p2 is defined to be F1.
F1 = (sin–1√p1 - sin–1√p2)2/{(1/4)(1/r1 + 1/r2)}
Comparing the characteristics of the three variance ratios described here, we see that F0emphasizes condition 2 above and F2 emphasizes condition 1. F1, on the other hand, shows
characteristics intermediate between the two. Table 2 shows a concrete example of this.
Using the classification criteria presented here, we constructed diagnosis charts for cerebral
hemorrhage patients and cerebral infarction patients from clinical data and obtained specificpartitioning patterns for each partitioning criteria. For each partitioning, F2 tended to producegroups with highly skewed ratios (meaning that the diagnosis can be confirmed) despite thesmall number of samples contained in a group. F0 was able to produce groups with skewed
ratios only after repeated partitioning. These features were obtained using mathematicalsimulations.

13 of 14JCS Bulletin Vol. 1

A Conversational AID System
Toshiro Haga and Motohiro Setoya
Department of Management Science
Science University of Tokyo

It is extremely important that full technical study proper to the field of application be
conducted on any set of analysis results. In fact, it might be better said that such a study
should be conducted not on the "results" of an analysis, but rather on the "stages" of thatanalysis. One of the authors of this report has developed or been involved in the developmentof two programs: CDA (Conversational Data Analysis) and QCAS/MA1 (Quality Control
Assisting System/Multivariate Analysis Part 1). Both of these programs are used in analyzing
data conversationally on a personal computer. As its name indicates, AID (AutomaticInteraction Detector) is a program which conducts an analysis automatically, but by adding his
ideas during an analysis, an analyst can obtain binary trees with high applicability to a particularsite. In this report, the authors will introduce CID (Conversational Interaction Detector), which
operates as a CDAsubsystem.
The stages of conversational processing are as follows:
1 . The variables used in classifying and the partition locations are displayed on the screen inorder of F ratio. A graph is displayed on the upper part of the screen and the number of cases,
mean, classification F ratio, etc., for each group are displayed below that.
2 . With the information displayed on the screen and the analyst's technical knowledge, theanalyst selects a classification and instructs the computer as follows. To obtain theclassification with the largest F ratio, simply enter D (for divide). Thereafter specify D
followed by the variable number and classification location.
3 . When classification has progressed somewhat, to see the classification up to that point,
simply enter T (for tree) and the binary tree will be displayed. Except for branches which havebeen completely classified, it is possible to experiment with a different classifications. Thisfeature can be used in the following situation: when different (interactive) variables are used forclassifying two branches and the analyst wishes to cancel one of the classifications and use thesame variable for both.
4 . As Haga and Hamada have reported separately, changing the classification criterion willproduce different results. Not only is it possible to specify the criterion to be used at the
beginning of the analysis, but the criterion can be changed during the analysis in order to obtain
the optimum classification.
5 . As with the CART (Classification And Regression Tree) program, it is possible to specify
whether the category order will maintained for each variable during classification and tospecify
multiple classification stopping rules.
An example illustrating the merits of the conversational method follows. In diagnosing
patients with cerebral hemorrhage, the variable "Does your head hurt? often includes missing
value and cannot be used in classification. This is because many patients are unconscious, andthus, it is not possible to obtain an answer. However, the variable "Does your head hurt?"canbe used after a classification using the variable "Is the patient conscious?". In conversational
processing, it possible to give full consideration to this type of problem while progressing
through the classification.
14 of 14JCS Bulletin Vol. 1