## Visualising Shrinkage

A useful property of mixed effects and Bayesian hierarchical models is that lower level estimates are shrunk towards the more stable estimates further up the hierarchy.

To use a time honoured example you might be modelling the effect of a new teaching method on performance at the classroom level. Classes of 30 or so students are probably too small a sample to get useful results. In a hierarchical model the data are pooled so that all classes in a school are modelled together as a hierarchy and even all schools in a district.

At each level in this hierarchy an estimate for the efficiency of the teaching method is obtained. You will get an estimate for the school as a whole and for the district. You will even get estimates for the individual classes. These estimates will be weighted averages of the estimates for the class and the estimate for the school (which in turn is a weighted average of the estimate for the school and the district.) The clever part is that this weighting is itself determined by the data. Where a class is an outlier, and therefore the overall school average is less relevant, the estimate will be weighted towards the class. Where it is typical it will be weighted towards the school. This property is known as shrinkage.

I’m often interested in how much shrinkage is affecting my estimates and I want to see it. I’ve created this plot which I find useful. It’s done in R using ggplot and is very simple to code.

The idea is that the non shrunk estimates bi (i.e. the estimates that would be obtained by modelling classes individually) are plotted on along the line x=y at the points (bi, bi). The estimates they are being shrunk towards ai are plotted at the points (bi, ai). Finally we plot the shrunk estimates si at (bi, si) and connect the points with an arrow to illustrate the direction of the shrinkage.

Here is an example. You can see the extent of the shrinkage by the the distance covered by the arrow towards the higher level estimate.

Note the arrows do sometimes point away from the higher level estimate. This is because this data is for a single coefficient in a hierarchical regression model with multiple coefficients. Where other coefficients have been stabilized by shrinkage this causes this particular coefficient to be revised.

The R code is as follows:

```# *--------------------------------------------------------------------
# | FUNCTION: shrinkPlot
# | Function for visualising shrinkage in hierarchical models
# *--------------------------------------------------------------------
# | Version |Date      |Programmer  |Details of Change
# |     01  |31/08/2013|Simon Raper |first version.
# *--------------------------------------------------------------------
# | INPUTS:  orig      Estimates obtained from individual level
# |                    modelling
# |          shrink    Estimates obtained from hierarchical modelling
# |          prior     Priors in Bayesian model or fixed effects in
# |                    mixed effects model (i.e. what it is shrinking
# |                    towards.
# |          window    Limits for the plot (as a vector)
# |
# *--------------------------------------------------------------------
# | OUTPUTS: A ggplot object
# *--------------------------------------------------------------------
# | DEPENDS: grid, ggplot2
# |
# *--------------------------------------------------------------------
library(ggplot)
library(grid)
shrinkPlot&lt;-function(orig, shrink, prior, window=NULL){
group&lt;-factor(signif(prior,3))
data&lt;-data.frame(orig, shrink, prior, group)
g&lt;-ggplot(data=data, aes(x=orig, xend=orig, y=orig, yend=shrink, col=group))
g2&lt;-g+geom_segment(arrow = arrow(length = unit(0.3, &quot;cm&quot;))) +geom_point(data=comp.in, aes(x=coef, y=mean))
g3&lt;-g2+xlab(&quot;Estimate&quot;)+ylab(&quot;Shrinkage&quot;)+ ggtitle(&quot;Shrinkage Plot&quot;)
if (is.null(window)==FALSE){
g3&lt;-g3+ylim(window)+xlim(window)
}
print(g3)
}
```

## Book Recommendations from Beyond the Grave: A Mahout Example

In H P Lovecraft’s The Case of Charles Dexter Ward the villainous Curwen, having taken possession of the body of Charles Dexter Ward, uses a combination of chemistry and black magic to bring back from the dead the wisest people who have ever lived. He then tortures them for their secrets. Resurrection of the dead would be a bit of an over claim for machine learning but we can get some of the way: we can bring them back for book recommendations!

It’s a grandiose claim and you can judge for yourself how successful it is. Really I just wanted an opportunity to experiment with Apache Mahout and to brush up on my Java. I thought I’d share the results.

Apache Mahout is a scalable machine learning library written in Java (see previous post). The recommender part of the library takes lists of users and items (for example it could be all users on a website, the books they purchased and perhaps even their ratings of those books). Then for any given user, say Smith, it works out the users that are most similar to that user and compiles a list of recommendations based on their purchases (ignoring of course anything Smith already has). It can get more complicated but that’s the gist of it.

So all we need to supply the algorithm with is a list of users and items.

Casting about for a test data set I thought it might be interesting to take the influencers data set that I extracted from Wikipedia for a previous post. This data set is a list of who influenced who across all the famous people covered on Wikipedia. For example the list of influences for T. S. Elliot is: Dante Alighieri, Matthew Arnold, William Shakespeare, Jules Laforgue, John Donne, T. E. Hulme, Ezra Pound, Charles Maurras and Charles Baudelaire.

Why not use x is influencing y as a proxy for y has read x? Probably true in a lot of cases. We can imagine that the famous dead have been shopping on Amazon for their favourite authors and we now have the database. Even better we can tap into their likes and dislikes to give us some personalised recommendations.

It works (sort of) and was very interesting to put together. Here’s a demo:

Say you liked English Romantic poets. Then you might submit the names:

http://simpleweb-deadrecommender.rhcloud.com/?names=john keats, william wordsworth, samuel taylor coleridge

Click on the link to see what the web app returns. You’ll find that you need to hit refresh on your browser whenever you submit something. Not sure why at the moment.

Or you might try some painters you like:

http://simpleweb-deadrecommender.rhcloud.com/?names=Pablo Picasso, Henri Matisse, Paul Gauguin

Or even just …

Or you could upset things a bit by adding in someone incongruous!

http://simpleweb-deadrecommender.rhcloud.com/?names=john keats, william wordsworth, samuel taylor coleridge, arthur c clarke

Have a go. All you need to do is add some names separated by commas after the names= part in the http request. Note although my code does some basic things to facilitate matching, the names will have to roughly match what is in wikipedia so if you’re getting an error just look the person up there.

What are we getting back?

The first list shows, in Mahout terminology, your user neighborhood, that is individuals in wikipedia who are judged to have similar tastes to you. I have limited the output to 20 though in fact the recommender is set to use 80. I will explain this in more detail in later posts. They are not in rank order (I haven’t figured out how to do this yet.)

The second list is the recommendations derived from the preferences of your ‘neighbours’. Again I’ll explain this in more detail later (or you could look at Mahout in Action where it is explained nicely)

The code is available on github

Sometimes the app goes dead for some reason and I have to restart it. If that happens just drop me a line and I’ll restart it.

I plan to do several more posts on how this was done. At the moment I’m thinking

• A walk though of the code
• An explanation of how the recommender was tuned
• How it was deployed as a web app on Open Shift

But if anyone out there is interested in other things or has questions just let me know.

Simon