Using Set Theory to analyse Recommendation relationships

Posted: February 7th, 2009 | Author: David | Filed under: Computer Science | Tags: , , , , , | Comments Off

This is a continuation of the post Good Recommendations.

Recommendations are found in the extension of the preferences of an Advisee to those of an Advisor, where the Advisee and Advisor share some known common interest. I’ll call someone’s existing preferences their Preference Set, and the shared preferences between Advisee and Advisor the Common Set.

The following diagram shows the Preference Sets of an Advisor (A) and an Advisee (B).

The total knowledge about Things they currently like is expressed within the union of these two Preference Sets (A∪B). Note that they are equally sized, indicating equivalent Domain knowledge, and for now we’ll assume that each preference within the set has equal significance, and that the two sets exhibit equal variation. It’s also important to state that A∪B does not represent total knowledge of all Things in the Domain. From this we can also state that no single Advisor has total knowledge of every Thing that could possibly be recommended, and no Advisee already has knowledge of every Thing they could possibly like.

The intersection of A and B (A∩B) indicates the Common Set of things that A and B both know and like. A minus B (A−B) is the subset of things where A’s recommendations to B would be found. I’ll call this the Recommendation Set. B minus A (B−A) is the subset of things A doesn’t know about, or wouldn’t recommend to B as he hasn’t already expressed a preference for.

This case represents a balanced proportion of knowledge about the Advisee’s Preference Set, and ability to recommend something the Advisor doesn’t already know.

The next diagram illustrates the case where there is a limited Common Set of preferences. It also shows the potential for a wider range of recommendations,.

The following diagram shows the opposite case where there is a large overlap of preferences, but a limited scope for recommendation.

So far, we’ve established that the scope for recommendation is tied to the size of the Common Set (A∩B) in proportion to the size of the Preference Sets (A∪B). In other words, the similarity of the Advisor and Advisee. This is more commonly represented by a statistic known as the Jaccard co-efficient. However, I’ve also only been concerned with Preference Sets of equal size; whereas in reality they will most likely be different according to an individual’s Domain knowledge.

The next diagram represents the relationship between an Advisor (A) who has more Domain knowledge than the Advisee (B).

Whilst the Advisor does have a far larger Preference Set  (n(A) > n(B)), it could be argued that this is also a good match given the relative size of the Common Set (A∩B) to the Advisee’s Preference Set (B) and the large Recommendation Set (A−B) from which to draw recommendations.

The opposite is illustrated by the following diagram.

Here the Advisor (A) has a smaller Preference Set than the Advisee (B), and a small Recommendation Set (A−B) from which to produce recommendations. This would appear to be a difficult scenario from which to produce anything other than obvious recommendations since the size of the Recommendation Set to the size of the Advisee’s Preference Set is so small. From these two scenarios we can deduce that the proportional sizes of the Advisor’s Recommendation Set and the Advisee’s Exclusive Set (B-A) is important.

In the next post I’ll be looking how the variation within a Preference Set affects the types of recommendations that can be drawn.

Comments are closed.