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Data miners deal with growing data size and dimensionality more often than not these days. As such, feature selection plays an increasingly crucial role as it improves predictive model interpretability, and helps to avoid overfitting and multicollinarity. Also, the *speed* of feature selection is very important when dealing with big data.

Variance inflation factor (VIF) regression is a fast algorithm that does feature selection in large regression problems. VIF regression handles a large number of features in a streamwise fashion. Streamwise regression method has its advantages over traditional stepwise regression because it offers faster computational speed without loss of accuracy. You can implement the VIF algorithm in SAS language using the SAS platform or server -- which usually stores big data sets.

For a more detailed explanation of the algorithm, please read my paper, **VIF Regression: A SAS Application to Feature Selection in Large Data Sets**. The paper includes SAS code that is used to implement the algorithm in the Appendix.

The VIF algorithm is essentially characterized by the following two components:

- The evaluation step approximates the partial correlation of each candidate variable with the response variable by correcting the marginal correlation using variance inflation factor (VIF), calculated from a small pre-sampled set.
- The search step tests each variable sequentially using an α-investing rule (Foster and Stine 2008). Variables are added only when they can reduce a statistically sufficient variance in the predictive model. The α-investing rule guarantees no model-overfitting and provides highly accurate models.

Table 1 uses pseudo-code to illustrate the VIF regression algorithm.

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**Table 1**: VIF Regression Algorithm

This example uses the communities and crime data that are available in the UCI Machine Learning Repository. The data set consists of 2,215 observations on 125 predictors, and the regression model aims to predict the number of assaults in 1995. In the EM flow as shown in Figure 1, we make comparable analyses for stepwise regression, LARS and VIF regression. In the comparison, VIF regression and stepwise regression select similar numbers of variables, but the results from stepwise regression imply an overfitting problem. LARS yields the largest average squared error (ASE) in both training and validation data after 20 iteration steps.

**Figure 1**: Comparison Analysis Flow

Figure 2 demonstrates that VIF regression outperforms the other two regression methods in terms of achieving the minimum average square error from validation. It selects 20 predictors in the setting of m=200, w0=0.05, Δw=0.05. Using AIC as a selection criterion, stepwise regression can scale down the set of predictors to 21, a little larger size than VIF regression; however, it suffers from overfitting. It has the smallest error in training data among all three methods, the second lowest is 157860.5. On the other hand, stepwise regression obtains an average square error that is obviously higher than VIF regression in validation data. LARS, in this example, has the largest error in both training and validation data.

**Figure 2**: Model Comparison Results

Multicollinearity is a known danger for causing overfitting in regression analysis. It produces large standard errors in the related independent variables and might introduce a large prediction error. Removing such data redundancy (variables with high correlation) improves statistical robustness for regression models.

VIF regression, which is based on VIF and fast robust estimates, is a streamwise regression approach to select variables. It has been proven to be an efficient algorithm in finding good subsets of variables from a huge space of candidates, and it can apply to online problems when features are generated and added to the model dynamically. You can see the SAS code that is used to implement the algorithm in the Appendix in the paper mentioned earlier.

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