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Figure 5.5
Euclidean space for samples a , b and c .
To embed our data matrix, X , into a Euclidean space, we need to find a coordinate
representation that will reproduce the matrix D when calculating the inter-sample
Pythagorean distances. A simple way to accomplish this is as follows: we construct
a two-dimensional Euclidean space and place a at the origin. Sample b now has to
beadistanceof0.39awayfrom a . This is accomplished by placing b at coordinates
(0.39, 0) as illustrated in Figure 5.4.
Next, c has to be embedded in the Euclidean space such that it is at a distance
0.39 from a and0.60from b . Finding the intersection (strictly one of the two possible
intersections) of the circles with radii 0.39 and 0.60 respectively, c is placed in the
two-dimensional Euclidean space as shown in Figure 5.5.
To embed d into a Euclidean space such that it has distances 0.60, 0.69 and 0.33 from
a , b and c respectively, it is necessary to turn to a three-dimensional Euclidean space.
The embedding of d is illustrated in Figure 5.6. In general, n samples are embedded
in an ( n
1)-dimensional Euclidean space. The process described here can be contin-
ued further, but cannot be visually represented in dimensions higher than 3. A more
straightforward algebraic way of finding a representation Y embedded in the Euclidean
space is to solve the eigenvector problem B = YY imposing the scaling Y Y = .This
is known as principal coordinate analysis (PCO) or classical scaling .Ofcourse,if B
is not positive semi-definite, it will not be possible to find such a configuration Y .As
an example, if the distances between a , b and c were such that the circles centred
at a and b did not intersect, there would be no Euclidean representation where c can
be embedded.
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