Approach is first to cluster and then to compute spatial boundaries for resulting clusters. Next step is to compute, from set of Monte Carlo samples obtained from scrambled data, estimates of probabilities of obtaining at least as many points within boundaries as were actually observed in original data

Borucki, W. J.

Card, D. H.

Lyle, G. C.

English

NASA Technical Reports Server

July 1975	 B75-10140 
NASA TECH BRIEF 
Ames Research Center
	
- 
NASA Tech Briefs announce new technology derived from the U.S. space program. They are issued to encourage 
commercial application. Tech Briefs are available on a subscription basis from the National Technical Information 
Service, Springfield, Virginia 22151. Requests for individual copies or questions relating to the Tech Brief program may 
be directed to the Technology Utilization Office, NASA, Code KT, Washington, D.C. 20546. 
Method of Identifying Clusters Representing 
Statistical Dependencies in Multivariate Data 
.
The analysis and interpretation of environmental 
and medical data often is difficult because there are 
many simultaneously interacting variables and be-
cause there seldom is an explicit theory that suggests 
a relationship between observables and quantities of 
interest. Classical multivariate statistical techniques, 
such as multiple regression and principal component 
analysis, may provide a satisfactory analysis and in-
terpretation when data can satisfy the rather stringent 
linearity requirements imposed by the techniques; 
however, the classical linear methods are inappro-
priate when the underlying relationships between 
variables are highly nonlinear. 
Cluster analysis is a multivariate technique for 
detection of nonlinear relationships in data; it seeks 
to find, from the structure of the data itself, natural 
groupings or categories in the multivariate observa-
tions made on a set of objects. Unfortunately, most 
algorithms for cluster analysis provide no information 
that can be used to distinguish predictive clusters 
from the clusters resulting from statistical fluctuations 
in the data. 
There has been developed a new method that 
combines cluster analysis with statistical tests of sig-
nificance by Monte Carlo sampling procedures to 
detect the presence of statistical relationships among 
variables in multidimensional data. The basic hypoth-
esis is that the population marginal distributions of 
the variables are statistically independent; assuming 
that there are sufficient data to be assured that the 
sample marginals are representative of population 
marginals, independent computer sampling of the 
marginals is used to determine how often, by chance, 
the generated samples show cluster structure similar 
to that of the data. 
The new method is comprised of two parts: (1) an
"extended group clustering algorithm" that finds 
compact clusters and then merges them to form 
extended clusters; and (2) a "cluster classification 
algorithm" that determines whether the clusters found 
by the extended. group clustering algorithm could 
have arisen by chance from independent marginal 
distributions of the variables. After clustering has 
been accomplished, the final testing of clusters utilizes 
a Monte Carlo method of scrambling the data by 
sampling independently from the marginal distribu-
tion of variables. 
The approach is first to cluster and then to com-
pute spatial boundaries for the resulting clusters; the 
next step is to compute, from the set of Monte Carlo 
samples obtained from scrambled data, estimates of 
the probabilities of obtaining at least as many points 
within the boundaries as were actually observed in 
the original data. Clusters with low probability are 
retained for further examination because these clus-
ters may represent statistical association between 
variables. 
In biological systems, it is quite common to find 
many simultaneously interacting variables. Analysis 
of data from biological systems is often begun by 
plotting one variable against another. By using video 
outputs on modern computer systems, the technique 
can be extended to pseudo-three- or pseudo-four-
dimensional displays of groups of variables; never-
theless, it cannot be expected that three- or four-
dimensional displays will always meet the require-
ments for finding multidimensional statistical rela-
tionships in complex multivariate data. To illustrate 
this difficulty and the use of the present technique to 
discover relationships in three dimensions that are 
not obvious in two-dimensional plots, an artificial 
data set was generated to form three scattergrams, 
(continued overleaf) 
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https://ntrs.nasa.gov/search.jsp?R=19750000140 2020-03-19T20:22:36+00:00Z
0 each of two dimensions. Visual examination of the plots did not reveal any strong relationship in the 
data, yet in a space of the correct number of dimen-
sions, the relationhip becomes obvious and the data 
represent a noisy spiral. Random (normally distrib-
uted) noise was added to the spiral-generating func-
tion, and the final plot resulting from use of this 
technique is shown in the diagram. 
The solid line in the diagram is the generating 
function, and the open symbols represent the modes 
found by the clustering technique. The dashed line 
connects modes to their two most similar neighbors, 
providing that the clusters of which they are members 
have at least one point in common. The break (just 
to the left of the Z-axis) in the derived curve is the 
result of a statistical fluctuation in the density of 
points, but the clustering algorithm resumed following 
the generating function after this break. The modes 
for the last 1800 of the spiral are not shown because 
it would make the relationship between the generat-
ing function and the modes very difficult to visualize 
in the two-dimensional representation. 
The Monte Carlo estimation of the probability of 
forming the first extended group at random from 
independent marginal distributions was less than 
10-27,
 far too small for this group reasonably to be a 
type II cluster. Hence, either this cluster is recognized 
as one representing a statistical dependence between 
variables (type I cluster), or it is an artifact (type III 
cluster). The latter possibility could be eliminated by 
gathering new data from the same process, and an-
alyzing it in the same manner. If the same spiral 
appears, evidence becomes overwhelming that it 
represents real structure in the data. Even after the 
spiral has been detected and has been proven to be 
real, the task of visualizing or interpreting the rela-
tionship remains; this difficulty is always present. 
The new technique only searches for statistical asso-
ciation between variables, and can be thought of as a 
higher dimensional analog of the process of visual 
inspection of two-dimensional and three-dimensional 
scattergrams. 
Note: 
Requests for further information may be directed 
to:
Technology Utilization Officer 
Ames Research Center 
Moffett Field, California 94035 
Reference: TSP 75-10140 
Source: William J. Borucki,

Don H. Card, and Gilbert C. Lyle

Ames Research Center

(ARC-10744)
. 
. 
B75-10140	 Category 09


Method of identifying clusters representing statistical dependencies in multivariate data

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