The linear classifier we presented in the previous section could look too simple. What if we use a higher degree polynomial? What if we also take as features not only the sepal length and width, but also the petal length and the petal width? This is perfectly possible, and depending on the sample distribution, it could lead to a better fit to the training data, resulting in higher accuracy. The problem with this approach is that now we must estimate not only the three original parameters (the coefficients for x1, x2, and the interception point), but also the parameters for the new features x3 and x4 (petal length and width) and also the product combinations of the four features.
Intuitively, we would need more training data to adequately estimate these parameters. The number of parameters (and consequently, the amount of training data needed to adequately estimate them) would rapidly grow if we add more features or higher order terms. This phenomenon, present in every machine learning method, is called the idem curse of dimensionality: when the number of parameters of a model grows, the data needed to learn them grows exponentially.
This notion is closely related to the problem of overfitting mentioned earlier. As our training data is not enough, we risk producing a model that could be very good at predicting the target class on the training dataset but fail miserably when faced with new data, that is, our model does not have the generalization power. That is why it is so important to evaluate our methods on previously unseen data.
The general rule is that, in order to avoid overfitting, we should prefer simple (that is, with less parameters) methods, something that could be seen as an instantiation of the philosophical principle of Occam's razor, which states that among competing hypotheses, the hypothesis with the fewest assumptions should be selected.
However, we should also take into account Einstein's words:
"Everything should be made as simple as possible, but not simpler."
The idem curse of dimensionality may suggest that we keep our models simple, but on the other hand, if our model is too simple we run the risk of suffering from underfitting. Underfitting problems arise when our model has such a low representation power that it cannot model the data even if we had all the training data we want. We clearly have underfitting when our algorithm cannot achieve good performance measures even when measuring on the training set.
As a result, we will have to achieve a balance between overfitting and underfitting. This is one of the most important problems that we will have to address when designing our machine learning models.
Other key concepts to take into account are the idem bias and variance of a machine learning method. Consider an extreme method that, in a binary classification setting, always predicts the positive class for any new instance. Its predictions are, trivially, always the same, or in statistical terms, it has null variance; but it will fail to predict negative examples: it is very biased towards positive results. On the other hand, consider a method that predicts, for a new instance, the class of the nearest instance in the training set (in fact, this method exists, and it is called the 1-nearest neighbor). The generalization assumptions that this method uses are very small: it has a very low bias; but, if we change the training data, results could dramatically change, that is, its variance is very high. These are extreme examples of the bias-variance tradeoff. It can be shown that, no matter which method we are using, if we reduce bias, variance will increase, and vice versa.
Linear classifiers have generally low-variance: no matter what subset we select for training, results will be similar. However, if the data distribution (as in the case of the versicolor and virginica species) makes target classes not separable by a hyperplane, these results will be consistently wrong, that is, the method is highly biased.
On the other hand, kNN (a memory-based method we will not address in this book) has very low bias but high variance: the results are generally very good at describing training data but tend to vary greatly when trained on different training instances.
There are other important concepts related to real-world applications where our data will not come naturally as a list of real-valued features. In these cases, we will need to have methods to transform non real-valued features to real-valued ones. Besides, there are other steps related to feature standardization and normalization, which as we saw in our Iris example, are needed to avoid undesired effects regarding the different value ranges. These transformations on the feature space are known as data preprocessing.
After having a defined feature set, we will see that not all of the features that come in our original dataset could be useful for resolving our task. So we must also have methods to do feature selection, that is, methods to select the most promising features.
In this book, we will present several problems and in each of them we will show different ways to transform and find the most relevant features to use for learning a task, called feature engineering, which is based on our knowledge of the domain of the problem and/or data analysis methods. These methods, often not valued enough, are a fundamental step toward obtaining good results.