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structures to be explained in Chap.
5
are essentially linear or non-linear. Another
issue is the scale-up question. Both algorithms handled the 20,000-point Swiss-roll
data well. It is a promising direction to investigate the potential of applying such
algorithms to concept mapping and science mapping data.
3.4
Concept Mapping
Concept maps provide a visual representation of knowledge structures and argument
forms. In many disciplines various forms of concept map are used as formal
knowledge representation systems, for example, semantic networks in artificial
intelligence, bond graphs in mechanical and electrical engineering, Petri nets in
communications, and category graphs in mathematics. Here we describe an example
from William Trochim of Cornell University (Trochim
1989
; Trochim et al.
1994
;
Trochim and Linton
1986
).
3.4.1
Card Sorting
Card sorting is one of the earliest methods used for concept mapping. Earlier works
on card sorting include George Miller's “A psychological method to investigate
verbal concepts” (Miller
1969
) and Anthony Biglan's “The characteristics of subject
matter in different academic areas” (Biglan
1973
).
We illustrate the process of concept mapping with the following example drawn
from William Trochim and his colleagues at Cornell University, see for example
(Trochim
1989
). They follow a similar process as what we see in Chap.
2
for creating
a thematic map - a base map is superimposed by a thematic overlay (See Fig.
3.39
).
In particular, the process utilizes MDS and clustering algorithms.
The process started with a brainstorm session, in which individual participants
were asked to sort a large set of
N
statements on a chosen topic into piles.
They should put statements into the same pile if they thought they were similar.
The results of each individual participant's sorting were represented as an N
N
similarity matrix. If a participant put statement
i
and statement
j
into the same pile,
the value of
e
ij
in the matrix was set to 1; if they were not in the same pile, the value
was set to 0. Then they aggregated the matrices of all the participants into a matrix
(
E
ij
). The value of
E
ij
therefore is the number of participants who had put statement
i
and statement
j
into the same pile. Because a statement is always sorted into the
same pile as itself, the diagonal of the aggregated matrix always equals N.
The structure of the similarity matrix was depicted through a two-dimensional
non-metric MDS configuration, which was followed by a hierarchical cluster
analysis of the MDS coordinates to divide the spatial configuration into district-
like groups. Finally, participants were led through a structured interpretation session
designed to help them understand the maps and label them in a meaningful way.
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