10.1 Goals of Feature Selection
In practice, we often have found that collaborators desire to have a model that has the best predictive ability and is interpretable. But this scenario infrequently occurs due to the trade-off between predictive performance and model interpretability. A misunderstanding of this trade off leads to the belief that simply filtering out uninformative predictors will help elucidate which factors are influencing the outcome. Then an explanation can be constructed as to why the remaining predictors are related to the outcome. This rationale is problematic for several reasons. First, consider the case when the number of predictors is much greater than the number of samples. In this scenario, there are likely to be many mutually exclusive subsets of predictors that result in models of nearly equivalent predictive performance (e.g. local optima). To find the best global solution, i.e. the subset of predictors that has best performance, would require evaluating all possible predictor subsets and may be computationally infeasible. But even if it is possible to find the global optimum, the identified subset may not be the true global optimum (if one exists) due to the inherent noise in the available predictors and outcome for the available data.
Many models are complex in the ways that they relate the predictors to the outcome. For such models it is nearly impossible to decipher the relationship between any individual predictor and the outcome. One approach that attempts to gain insight for an individual predictor in a complex model is to fix all other selected predictors to a single value, then observe the effect on the outcome by varying the predictor of interest. This approach is overly simplistic and only provides a small sliver of insight on the predictors true impact.
Here we would like to refocus the motivations for removing predictors from a model. The primary motivations should be to either mitigate a specific problem in the interplay between predictors and a model, or to reduce model complexity. For example:
Some models, notably support vector machines and neural networks, are sensitive to irrelevant predictors. As will be shown below, superfluous predictors can sink predictive performance in some situations.
Other models like linear or logistic regression are vulnerable to correlated predictors (see Chapter 6). Removing correlated predictors will reduce multicollinearity and thus enable these types of models to be fit.
Even when a predictive model is insensitive to extra predictors, it makes good scientific sense to include the minimum possible set that provides acceptable results. In some cases, removing predictors can reduce the cost of acquiring data or improve the throughput of the software used to make predictions.
The working premise here is that it is generally better to have fewer predictors in a model. For the remaining chapters, the goal of feature selection will be re-framed to
Reduce the number of predictors as far as possible without compromising predictive performance.
There are a variety of methods to reduce the predictor set. The next section provides an overview of the general classes of feature selection techniques.