Asymmetry in MDS: Question on Unilateral Love & Hate

[noowutah]

I am perplexed by an example given in an article about asymmetry in MDS (multi-dimensional scaling). You don't need to know anything about this to answer my question. It's all very intuitive.

There are six people. They have various likes and dislikes for each other, which are asymmetric (as one would expect). Here is the table:

http://tinyurl.com/oshtv7p

MDS tries to give a graphical representation of the relationships. In the paper

Chino, N. (1978). A graphical technique for representing the asymmetric relationships between N objects. Behaviormetrika, 5, 23-40.

Chino presents a model and then uses this example to provide the following graph:

http://tinyurl.com/oefndcs

Chino explains these relationships as follows:

(1) For example, the skewness between persons 1 and 4 is the greatest of all, as the angle between the lines P0P1 and P0P4 is pi/2. Further, we find it easy to see that person 4 likes person 1 very much, though person 1 doesn't like person 4 at all. It should be noted that the co-ordinate system is assumed to be right-handed.

(2) Persons 1 and 6 like each other, as the angle between the lines P0P1 and P0P6 is 0.

(3) For example, persons 1 and 3 hate each other, as the angle between the lines POP1 and P0P3 is pi.

(4) For example, person 4 likes person 5 very much, but person 5 neither likes nor dislikes him, as the angle between the lines P0P4 and P0P5 is pi/4. Such a relationship might be called "unilateral love". On the other hand, person 5 neither likes nor dislikes person 2, but person 2 hates him very much, as the angle between the lines P0P5 and P0P2 is 3pi/4. Such a relation ship might be interesting in contrast to that of "unilateral love".

I have tried to think about this and interpret what the angles and the distances in the diagram mean, but I can't wrap my head around it. Why, just looking at the graph, does P1 not like P4 at all, but P5 is indifferent towards P4?

 

[mfb]

Without explicit rules how to construct such a graph, the description looks arbitrary. It is easy to introduce person 7 where 1 and 7 like each other (so their relative angle should be zero?) and 6 and 7 hate each other (so their relative angle should be pi?).
It is not possible to "properly" map all 30 degrees of freedom of the table to the 12 degrees of freedom of the graph.

 

[noowutah]

I don't even understand how "X likes Y" and to what degree is represented in the graph. Any help?

 

[mfb]

"4 likes 1 very much" because the entry in the fourth column, first row is positive and large.
And the representation in the graph looks odd.
Apparently "going from 4 to 1 goes in the positive direction around the center" is related to "4 likes 1", and a large distance (from 1 to 4 in positive direction is 3/4 of the circle) seems to indicate dislike. But then 5 should really hate 4 with the 7/8 circumference distance, but 5 is neutral towards 4...

 

[noowutah]

I agree, mfb. Chino's comments just don't seem to make sense looking at the diagram. The same article has another diagram on page 32, but it is just as uninformative about how to understand what the relative positions mean.

 

[noowutah]

I asked Chino about this, and he graciously explained it to me. The lack of symmetry is expressed by the size of the parallelogram between the two vectors. The feelings between P1 and P6 are symmetrical (they like each other). The feelings between P1 and P3 are also symmetrical (they hate each other). The liking and disliking is represented by the size of the angle. The angle between P1 and P6 is zero, thus like. The angle between P1 and P3 is pi (that's maximal, given symmetry), thus dislike. P4 and P1 have a very asymmetrical relationship, because the parallelogram between them couldn't be any larger (given the length of the vectors). P4 likes P1, because the angle from P4 to P1 goes in the positive direction. The asymmetry then requires that P1 doesn't like P4. With P4 and P5, it is more nuanced. The asymmetry is not as strong, so P4 likes P5, and P5 doesn't care about P4 (i.e. doesn't like or dislike P4).

It is interesting to think about distinguishing the following cases:

(1) P4 likes P5, but P5 doesn't care about P4.

(2) P5 hates P3, but P3 doesn't care about P5.

The asymmetry is the same in (1) and (2), so the sizes of the parallelograms are the same. The angle gives away the difference. An angle between zero and pi/2 expresses like as the dominant, an angle between pi/2 and pi expresses dislike as the dominant. The formulas in Chino make this clear, but they need careful attention.

 

[mfb]

Okay, so where do we draw the vector of person 7, where 1 and 7 like each other and 6 and 7 hate each other? The first condition requires the vector to be parallel to P1, the second one requires it to be antiparallel to P6, but P1 and P6 are parallel. That cannot all work at the same time.

 

[noowutah]

P1 and P6 as well as P1 and P3 already fulfill this scenario. P1 and P6 like each other, P1 and P3 hate each other. Maybe your question is, what about P7 who is in a mutual hate relationship with P3 AND in a mutual hate relationship with P1, given that P1 and P3 already hate each other mutually. That would be a problem, I suppose. Chino gives a formal method how to get these coordinates from a like/dislike matrix. It would be interesting to run these cases and see what they yield.