Predicting Preferences

This post is a continuation of Good Recommendations, Using Set Theory to analyse Recommendation relationships and Variation within Preferences.

Does all of this just mean that the underlying rule is to pair the Advisee to the Advisor with biggest Preference Set? And how does that relate to what was concluded earlier about the similarity of the two sets; the Jaccard co-effient. Taking an extreme scenario illustrated here:

Advisor (A) has a Preference Set of a large order of magnitude greater than the Advisee (B). Assuming that there’s a similar variance within the two sets, and that the previous assertions were correct, then this would clearly be approaching a Utopian case.

In reality we would find it difficult to assume so much about such a small sample relative to the larger one. It would be much better if we could transform the smaller set in to a larger sample on the basis of predicted future recommendations. Such as in the following diagram, where the dashed line shows the expansion of the Advisee’s original Preference Set to the Advisee’s Predicted Preference Set (B).

However, this makes the assumption that growth would be uniformly distributed out from the current Preference Set, and that the coupling of Advisor (A) with Advisee (B) was a good one to begin with and one that holds throughout the introduction of future recommendations.

My assumption would be that the variation within the Preference Sets holds the solution for this. Taking the difference in variation in the Common Set (A∩B) and the Recommendation Set (A—B) as ‘pull’ factors towards the Advisor and the difference in variation between the Common Set (A∩B) and B—A as ‘push’ factors away from the Advisor, we could infer growth of the Common Set with a the centre weighted towards or away from the Advisee. A greater pull could be shown as in the following diagram:

Whilst a greater push could be illustrated as:

The Jaccard co-effient could then be applied to work out the predicted similarity between the two sets. And from this, predict whether the original relationship is likely to provide more interesting, average, or more obvious recommendations.

Formally this can be represented as:

A∩PPS(B) / A∪PPS(B)

where: PPS(X)  = weighted Predicted Preference Set of X

Summary: Between Obvious and Interesting

Variation within Preferences

This post is a continuation of Good Recommendations and Using Set Theory to analyse Recommendation Relationships.

Until now, I’ve kept the assumption that both the Advisor’s and Advisee’s Preference Sets exhibit a similar level of variation within the Domain. That is to say that any member of the Recommendation Set would be as likely to be as good a recommendation as any other member, which is clearly unrealistic. The following diagram shows quadrants on a cartesian plane that show the relationships between increasing variety of a Preference Set within a Domain (from specialist to generalist), and the size of a preference set.

A more specialist Preference Set is one that contains more elements from within a particular sub division (e.g. Genre or sub-Domain) of the Domain than a more generalist Preference Set.

I am interested in the influence that each of these groupings is likely to have on the other, assuming a constant proportion of commonality between them. I would offer that a specialist Preference Set is less likely to receive good recommendations from a generalist Preference Set, but a smaller specialist Preference Set is more likely to receive good recommendations from a larger Preference Set of the same specialism.

I would also suggest that a larger generalist Preference Set would be more likely to receive a good recommendation from a specialist Preference Set of any size, than a smaller generalist Preference Set.

The following matrix summarises my assumptions about the suitability of each group to produce a good recommendation to another. (• indicates a good match between groups).

From this, a good recommendation is more likely in the case where the following holds true:

v(B) ≥ v(A) and n(A) ≥ n(B)

where:

v(X) = variation of set X

n(X) = number of members of set X

That is to say the recommendation will be better received when the Advisee is more flexible about what he likes within the Domain, and the Advisor knows more about the Domain than the Advisee. Which also seems a bit of a no brainer when put like that.

Calculating Variation

Variation needs to indicate the range of member groupings within a set. This is problematic as we don’t necessarily know what all the groups are, how their boundaries lie, and where to allocate members as there may be many levels of sub-Domains or Genres within a Domain. I will make an assumption that there is some method of calculating variation within a set through some other means of set member classification (e.g. Naive Bayes, k-Nearest Neighbour, or otherwise).

In the next post, I’ll examine the possibility of predicting Preference Sets as a method to normalize preferences for comparative analysis.

Using Set Theory to analyse Recommendation relationships

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.

Good Recommendations

A recommendation can said to be advice about some Thing based up on an Advisor’s prior experience of the Thing, knowledge of the wider Domain, and knowledge about an Advisee. I’ve home-brewed that definition from a variety of dictionary sources, but I’m hoping it doesn’t push the levels of acceptability too far. It suits what is to follow quite well, and I’ve even drawn a diagram:

From personal experience, people generally recommend things that they know a bit about and won’t often recommend things they don’t personally like. If the assumption holds that an Advisor will have better knowledge about the Things they like, recommendations should be best made about things the advisor personally rates.

From this we can approximate ‘things that people like’ to equal ‘things they are likely to recommend’.

Knowledge about the preferences of the Advisee matters too. I’m more likely to provide a well received recommendation to someone I know than someone I don’t. In this regard, I might also be able to provide a recommendation about some Thing I don’t necessarily like; although my knowledge of the Thing is likely to be more limited.

It follows that the more knowledge the Advisor has about the Thing, and about the wider Domain, and about the Advisee, the better the recommendation the Advisor will probably make.

Are recommendations best made from an Advisor who knows more about the Thing or the Domain than the Advisee? Most people I know don’t like being told things they already know, but self affirmation is nice sometimes.

Is an Advisee with only a small amount in common with the Advisor more likely to receive a recommendation less in line with their current preferences, but one that may be more interesting as a consequence? Conversely, is the Advisee with a large amount in common with the Advisor more likely to receive a recommendation in line with their current preferences, but is likely to be more obvious as a consequence? How much does the variety / specialism of the Advisors and Advisee’s current preferences matter?

It seems there is potential for a sliding scale between interesting and obvious recommendations, both of which may be good for different reasons.

A ‘good’ recommendation depends entirely on the Advisees expectation of the type of recommendation anticipated from the Advisor. How much can this be inferred by size of and the variation within the Domain shown in their initial preferences? It could well be that an Advisee that already shows more variety in their current preferences will be more ‘willing’ to accept off-kilter recommendations than one which is already more specialist. But it could also be that an Advisor who exhibits similar variation in their preferences to their Advisees will also make more acceptable recommendations.

We can infer that recommendations containing a good balance of interestingness while in keeping with existing preferences, are best made when an Advisee has a good proportion of current preferences in common with an Advisor. But should this be relative to their own preferences, or relative to their Advisor’s preferences? And to what extent does the variation of Domain preferences matter?

In the next post I’ll be introducing Set Theory as a mechanism for analysing these relationships.

Evaluating Feeds

A not so uncommon situation I’m finding is that a website will have more than one feed associated with it. This is sometimes just to point to alternative markup (e.g. different versions of RSS spec, or a site offering both RSS and Atom feeds, or combinations thereof), or to hook up with feed aggregation services (Feedburner easily being the most prevalent), but the content of the feed can also sometimes be quite different.

Initially, I had made the crude assumption that for me, RSS is more useful than Atom (as I had written a very lightweight RSS parser). Now that I’m incorporating the ROME Java API for feed processing, I’m not so bothered about the choice of tech, or the spec of that tech, but I am quite interested in hooking up with the best feed for my purposes. I also don’t want to have to approve a few hundred feeds manually.

So what’s the best feed for my purposes? Assuming that these feeds are concerning the same subjects (i.e. new posts to the blog), then the best purpose feed is most likely going to be the one with the most content.

A really simple algorithm for deriving the feed with the most content

The first task is to pre-process the content of each feed to determine a value for the content of each post of each feed, measured by the number of words in the description and the largest number of words in each representation of the post content, once all markup has been removed.

We’re then left with a representation of feeds to lists of word counts for relative posts, such as:

feed_1..n → { words_post1, words_post2, .. words_postx }

Since the number of posts in each feed could vary (and the number of posts a feed covers shouldn’t be a discriminating factor), we take the minimum length of all the word count lists, and sum the word counts within that range for each feed. We can then select the feed which has the highest word count as the preferred feed to use.

This method assumes that the feed entries are in the same order and about the same posts in each feed, on the basis that each feed is most likely to originate from the same blog management system and therefore either dynamically produced, or published at a similar time.