Wednesday, February 17, 2010

Prinicpal Components for Modeling

Problem Statement

Analysts constructing predictive models frequently encounter the need to reduce the size of the available data, both in terms of variables and observations. One reason is that data sets are now available which are far too large to be modeled directly in their entirety using contemporary hardware and software. Another reason is that some data elements (variables) have an associated cost. For instance, medical tests bring an economic and sometimes human cost, so it would be ideal to minimize their use if possible. Another problem is overfitting: Many modeling algorithms will eagerly consume however much data they are fed, but increasing the size of this data will eventually produce models of increased complexity without a corresponding increase in quality. Model deployment and maintenance, too, may be encumbered by extra model inputs, in terms of both execution time and required data preparation and storage.

Naturally, the goal in data reduction is to decrease the size of needed data, while maintaining (as much as is possible) model performance, this process must be performed carefully.

A Solution: Principal Components

Selection of candidate predictor variables to retain (or to eliminate) is the most obvious way to reduce the size of the data. If model performance is not to suffer, though, then some effective measure of each variable's usefulness in the final model must be employed- which is complicated by the correlations among predictors. Several important procedures have been developed along these lines, such as forward selection, backward selection and stepwise selection.

Another possibility is principal components analysis ("PCA" to his friends), which is a procedure from multivariate statistics which yields a new set of variables (the same number as before), called the principal components. Conveniently, all of the principal components are simply linear functions of the original variables. As a side benefit, all of the principal components are completely uncorrelated. The technical details will not be presented here (see the reference, below), but suffice it to say that if 100 variables enter PCA, then 100 new variables (called the principal components come out. You are now wondering, perhaps, where the "data reduction" is? Simple: PCA constructs the new variables so that the first principal component exhibits the largest variance, the second principal component exhibits the second largest variance, and so on.

How well this works in practice depends completely on the data. In some cases, though, a large fraction of the total variance in the data can be compressed into a very small number of principal components. The data reduction comes when the analyst decides to retain only the first n principal components.

Note that PCA does not eliminate the need for the original variables: they are all still used in the calculation of the principal components, no matter how few of the principal components are retained. Also, statistical variance (which is what is concentrated by PCA) may not correspond perfectly to "predictive information", although it is often a reasonable approximation.

Last Words

Many statistical and data mining software packages will perform PCA, and it is not difficult to write one's own code. If you haven't tried this technique before, I recommend it: It is truly impressive to see PCA squeeze 90% of the variance in a large data set into a handful of variables.

Note: Related terms from the engineering world: eigenanalysis, eigenvector and eigenfunction.

Reference

For the down-and-dirty technical details of PCA (with enough information to allow you to program PCA), see:

Multivariate Statistical Methods: A Primer, by Manly (ISBN: 0-412-28620-3)

Note: The first edition is adequate for coding PCA, and is at present much cheaper than the second or third editions.

Matthew said...

Very cool. Could you comment on principal components vs SVD and where you've seen the two applied? I'd also be interested to hear your take on the a priori assumption that the most important "directions of variance" should be orthogonal. In some bayesian work in genetics it's been found that using MCMC to approximate a non-orthogonal decomposition gives better results than PCA. Does this consideration ever come up in business analytics? Thanks! Really enjoy the blog.

Sandro Saitta said...

Nice post! I definitely think that PCA is a really nice tool for variable selection. However, its main drawback is to be a filter approach. This means PCA "selects" features independently of the learning algorithms used afterward. This is why I usually prefer the wrapper approaches (which of course are much more time consuming)

Jožo Kováč said...

PCA reduces dimensions, but also creates blackbox. And that's unacceptable for many business applications.