Similarity theory: Testing whether dimensions are separable or integral

Question

Note: I'm not referring to linear separability.

I've found the interesting comment in Edelman, Shahbazi: "Renewing the respect for similarity" that for integral dimensions, Euclidean distance is more appropriate, while for separable dimensions, Manhattan distance is more appropriate. Now I'm trying to dig up relevant literature, in particular for statistical tests for these two cases. (Most of the literature seems to be in psychology, some keywords are "human similarity perception").

Roughly said, integral vs. separable appears to be intuitively as follows:

• separable dimensions can be given independently. E.g. I can tell you my favorite book, and this fragment of information is somewhat complete and independent of, say, my shoe size.
• integral dimensions only make sense when given together. For example coordinates on a map: if I give you just a latitude, the information is not too useful. If you also have the longitude, it makes just much more sense. Similarly, giving the hue of a color lacks the information about the saturation; if the saturation is 0, the hue is meaningless. Or the pitch of a tone without information on the loudness (which could be 0).

As you can see, I gave some examples, but no mathematical definition or statistical test. In fact, I'm interested in literature discussing such tests, if there are any. It probably is closely related to correlation; but it's not that easy. If I take positions in a major city, they will probably not be highly correlated but still have a lot of clusters and "integral". Unless we are in a city grid, Euclidean distance is then more appropriate to measure similarity. Pitch and Volume - there probably is some correlation, but not of major interest again.

So do you know any good statistical test when to combine attributes in an "Euclidean" fashion (i.e. geometric mean joint similarity), and when to better combine attributes in a "Manhattan" fashion (i.e. arithmetic mean joint similarity)?

I know this question is a bit vague. Please bear with me; this is a research question, and probably not something that is discussed in textbooks.

Some related literature:

• Attneave: "Dimensions of Similarity", 1950
• Shepard: "Toward a universal law of generalization for psychological science", 1987
• Santini, Jain: "Similarity Measures", 1999
• Edelman, Shahbazi: "Renewing the respect for similarity", 2012

Answer 1

Following the exposition in Lee (2008), consider the case of defining a distance function for metric multidimensional scaling of similarity judgments. The distance between K-dimensional points $\mathbf{p}_i = [p_{1i} \ldots p_{Ki}]$ and $\mathbf{p}_j = [p_{1j} \ldots p_{Kj}]$ can be written as $$ D(\mathbf{p}_i, \mathbf{p}_j) = \Big[\sum_k^K |p_{ki}-p_{kj}|^r\Big]^{1/r} $$ When $r=2$ dimensions are integral (euclidean) and when $r=1$ they are separable (cityblock). So you have to estimate $r$ or compare a bunch of possibilities. Lee sets up a Bayes Net for the MDS problem, puts a prior on $r$, and gets a posterior for it in the light of some subject (dis)similarity judgments. See, e.g. Figure 3 in the paper.

This approach is not, strictly speaking, the 'test' that you are looking for, but it would perhaps still answer your question statistically by giving you an estimate of the dimension type.

You'll also notice that a lot of other psychologically relevant model choices are here implicitly fixed and you might reasonably suspect that some of them would affect inference about $r$. But it's a start.

Answer 2

Conceptual Spaces: The Geometry of Thought, by Peter Gärdenfors

This book contains a section that discusses what has been published to test empirically which representation holds for a given set of similarities or distances. Google books provides a preview of this section starting on page 24.