Inertia around the diagonal (What is seriation ?)
The inertia of dissimilarities around the diagonal of D is measured
by the following criterion:
.
where |i-j| represents the distance from the cell (i,j)
to the diagonal of D and d(i,j) the weight of
the cell (i,j) of D.
The seriation based on this criterion searches for the permutation
which maximizes this criterion.
This is equivalent to moving high dissimilarities away from the diagonal of
D, and, as a corollary, the low values are moved near the diagonal.
Similar to the criterion used by the "Unidimensional Scaling" method,
this criterion is global, for it takes alldissimilarities of D into
account. Then it takes general trends into account.
The difference with the UDS approach is the weighting attributed to the cells
in D.
The inertia criterion gives more weight to the cells situated far from the
diagonal, then it tends to oppose the extremes rather than to bring together
contiguous items.
However, experience has shown that both approaches give similar and possibly
complementary results.
The optimization procedures for this criterion are analogous to those used
in the UDS method:
In the general case, the numerical heuristic searches for specific directions,
whereas the combinatory approach proceeds by successive transpositions in
accordance with the maximum gradient method.
Under a tree constraint, an ascendant greedy heuristic is usually used to
determine, at each stage, the position of each subtree which maximizes the
inertia criterion.