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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
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  • $\begingroup$ What would it mean to have a "statistical test" of "when to combine attributes in an 'Euclidean' fashion..."? What are you testing? What would the data be? What would be the hypothesis? $\endgroup$ – gung - Reinstate Monica Oct 2 '13 at 19:29
  • $\begingroup$ E.g. the hypothesis "Euclidean distance is better than Manhattan distance", or: "The dimensions are integral, not separable". $\endgroup$ – Erich Schubert Oct 4 '13 at 13:30
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    $\begingroup$ Better how? What does that mean? Eg, the prototypical statistical test would be a t-test on the response of a patient to a candidate medication vs a placebo to see if the mean of the treatment group is higher than that of the control group. It is not at all clear how to create an analogy b/t that & Euclidean vs Manhattan distance. It is not clear that this is even a meaningful statistical issue; it seems more like a theoretical issue that might better fit on cognitive sciences.SE. $\endgroup$ – gung - Reinstate Monica Oct 4 '13 at 15:04
  • $\begingroup$ For whatever definition of "better" that someone already managed to formulate a test for. I'm open-minded. But the point is that I'm looking for a test based on the observations; what you find in cognition is usually "supervised", as in the literature I gave. An expert can decide that pitch and volume are integral, whereas city blocks are separable; but I'm looking for the situation that I don't have such an expert available. $\endgroup$ – Erich Schubert Oct 4 '13 at 16:29
  • $\begingroup$ If someone has already managed to formulate a test, you can use what they did. Otherwise, you need to define "better" and have a way to measure it. Imagine if a drug company wants to develop a new medication, & they ask you to test if its "better" than a placebo. This can't be done until they specify what they mean by "better", eg, lower blood pressure, greater lung capacity, fewer proteins per mL urine, greater bone density, etc. There is no "better" in the abstract. $\endgroup$ – gung - Reinstate Monica Oct 4 '13 at 16:52
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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.

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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.

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