In the previous article (Benchmark Example in MLR Part I), SVM and logistic regression are benchmarked on German credit data - this data is from the credit scoring example in Chapter 4 of Applied Predictive Modeling. For SVM, the cost parameter (C) is tuned by repeated cross-validation before the test measure is compared to that of logistic regression.

A potential issue on that approach is the CV error can be optimistically biased to estimate the expected test error as discussed in Varma & Simon (2006) and Tibshirani and Tibshirani (2009). In this article, the issue is briefly summarized in its nature, remedies and cases where it can be outstanding. Among the two remedies, nested cross-validation is performed as (1) mlr provides this resampling strategy and (2) this resampling strategy is useful as it can be applied to other topics such as feasure selection.

## Summary of optimistic bias of CV error

CV error can be optimistically biased to estimate the expected test error (Tibshirani and Tibshirani (2009))

CV estimate of expected test error or CV error curve

CV error in the kth fold or the error curve computed from the predictions in the kth fold

Therefore

• first $e_{k}(\hat{\theta})\approx CV(\hat{\theta})$
• Yes, since both are error curves evaluated at their minima.
• and, for fixed $\theta$, $e_{k}(\hat{\theta})\approx E\Big[ L\left(y,\hat{f}\left(x,\hat{\theta}\right)\right)\Big]$
• Not perfect.
• RHS: $\left(x,y\right)$ is stochastically independent of the training data and hence of $\hat{\theta}$.
• LHS: $\left(x_{i},y_{i}\right)$ has some dependence on $\hat{\theta}$ as $\hat{\theta}$ is chosen to minimize the validation error across all folds, including the kth fold.

Tibshirani and Tibshirani (2009) show that the bias itself is only an issue when $p\gg N$ and its magnitude varies considerably depending on the classifier. Therefore it can be misleading to compare the CV error rates when choosing between models.

In order th tackle down this issue, Varma & Simon (2006) suggest nested cross-validation to eliminate the dependence in LHS. However this strategy is computationally intensive and can be impractical.

Tibshirani and Tibshirani (2009) propose a method for the estimation of this bias that uses information from the cross-validation process. Specifically

and if the fold sizes are equal

then

Let’s get started.

The bold-cased topics below are mainly covered.

1. Imputation, Processing …
3. Learner
4. Train
5. Predict
6. Performance
7. Resampling
8. Benchmark

The following packages are used.

## Preprocessing

mlr has different methods of preprocessing and splitting data to caret. For comparison that may be necessary in the future, these steps are performed in the same way.

80% of data is taken as the training set.

Task is set up using the training data and normalized as the original example.

## Control grid set up for tuning

As the original example, sigma (inverse kernel width) is estimated first using sigest() in the kernlab package. Then a control grid is made by varying values of C only.

In makeParamSet(), sigma and kernel are fixed as discrete parameters while C is varied from lower to upper in the scale that is determined by the argument of trafo. For numeric and integer parameters, it is possible to adjust increment by resolution. Note that the above set up can be relaxed, for example, by varying both C and sigma and, in this case, it would be more flexible to set sigma as a numeric parameter.

The resulting grid can be checked using generateGridDesign()

## Resampling

Repeated cross-validation is chosen for the outer resampling while cross-validation is set up for the inner resampling. The outer resampling is for estimating the test errors of the three learners and the inner resampling is for tuning the hyper parameter for SVM (lrn.nest) with the nested resampling strategy - the estimated hyper parameter in each fold is expected to be independent from the training data.

## Learner

Learners of support vector machine with and without nested resampling and logistic regression are set up - a wrapped learner is set up to be used with nested resampling. Note that the development version (v2.3) is necessary to fit logistic regression - see this article for installation information.

## Tuning

The parameter can be tuned using tuneParams() as shown below - the hyper parameter (C) of SVM without nested resampling is estimated here.

Fitting details can check as following.

## Benchmark

Once the hyper- or tuning parameter is determined, the learner can be updated using setHyperPars().

The tuned SVM learner can be bechmarked with the logistic regression learner. It is shown that SVM with nested resampling performs slightly worse than logistic regression.

I consider machine/statistical learning tasks are a combination of art (model/feature selection, feature engineering …) and science (model/algorithm). While the latter part can be relatively straightforward, the former would require good practice, caution, experience, domain knowledge… (generally things that can easily mislead the researcher/practitioner). In other words, fitting models alone is hardly effective. In this regard, I tried benchmarking with some standard resampling strategies first although I’m yet to be aware of a variety of useful models. As having a whole project in mind when learning a new model would be a lot faster to be practical with it, I’ll try to cover as much steps as I can in subsequent posts.