(abridged version)
Maurice Clerc 1995
It is then possible to mathematically define some similarities, which are symmetrical, and have a kind of partial transitivity. Since memoware built with these principles has to be accepted by users as an extension of their ownmemory, it is important to verify that these properties are psychologically valid.
This paper presents the approach and some mathematical tools. A study was conducted with human subjects classifying pairs of objects (outlines of birds)on the basis of their similarity.
.
- on the one hand each intermediate outline is calculated according to the two extremes, by linear variation of only one numerical parameter.
- on the other hand, this simplicity is not obvious, especially when all
the sequence in order is not visible (see figure 1).
Figure 1. Stimuli. Each point of the first outline has a counterpart on the last one. For intermediate outlines, the equivalent point is calculated by linear interpolation.
Preliminary tests showed that identical pairs were always found unambiguously.So they are finally not included into the material available for the subjects,which is then made of 30 pairs on 3x3 cm "cards" , and a A4 sheet of paper for arranging them
Without speaking here in detail of tested population, and of protocol, let us simply say that at each session instructions are given to put each card more or less "high" on the sheet, depending of the similarity of the two outlines on the card. Figure 2 shows an example of test sheet which has been judged satisfactory by a subject.
Each subject has to do five sessions, with at least some days between each one and the next, and with five different sets of 15 cards, in order to eliminate some artifacts (e.g. some pairs are rotated of 180 degrees).
Figure 2.-Filled test-sheet (scale 1/3). Subject were told to arrange each card accordingly to the similarity of the two outlines, the more similar they are the more the card should be close to the top, and vice versa.
? The figure 3
shows a quite good correlation. In fact the probability fort he hypothesis
of symmetry to be true is 0.905, which is acceptable, but not perfect, and
it could be interesting to do more tests to examine this point.
Figure 3 Symmetry check. Each pair of outlines has been placed several times, sometimes with the card (i,j), and sometimes with the card(j,i). So it is possible to compute two separated averages, one for each order. The most important differences are for "intermediary" pairs, for which subjects have difficulty deciding the best position. On the whole, the correlation is however good.
,whose
estimated values are coming from measures, for one hand, and from the
constraint that the identity is valued to 1, for the other hand. This values
can be retrieved by a similarity function, in the mathematical meaning,
applied to fuzzy representations of the studied objects. Figure 4 gives
such are presentation for two descriptors.
Figure 4 Two descriptors Fuzzy Representation . Here, the sum ofthe values for each object is exactly 1, but it is not the general case.
Several kinds of similarity have been tested ("ensemblistic", "distance","angular"). The angular model is the only one which takes into account the typicality, in the sense that, for example, outlines 5 and 6 are indeed more similar than 3 and 4.
More precisely, a formula like 
gives
(with lambda=1.29 and mu=1.04) a excellent rank correlation (0.96),and a
small maximum relative difference (0.17).
"If the similarity between i and j is high, and also between jand k, then there must be some similarity between i and k"
and this can be defined by a strict archimedian t-norm T 
.where
ri,j is the similarity between the object i and the object
j.
More precisely, one must have
with, for example, for the automorphism j
It seems complicated, but this is only a generalization of the very
simple case 
,obtained for nu=1.
The exhaustive check is a bit boring, but it can be shown that the data
have indeed an underlying structure, which constrains the similarities
between the objects of a triplet. This can be visualized by a surface 
,drawn for nu=1.374, so that, for each
triplet 
,the point 
is"above"
this surface (figure 5). In passing, the subjects were completely unaware
of this kind of transitivity. And some of them contested it, even when it
could be found in their own test sheets.
Figure 5 Constraint surface for quasi-transitivity. For each triplet of objects, (bird outlines), the three similarities two by two,can be seen as the coordinates of a point. With the collected data all points are "above" the surface.
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