# π Exercise M4.02#

In the previous notebook, we showed that we can add new features based on the original feature x to make the model more expressive, for instance x ** 2 or x ** 3. In that case we only used a single feature in data.

The aim of this notebook is to train a linear regression algorithm on a dataset with more than a single feature. In such a βmulti-dimensionalβ feature space we can derive new features of the form x1 * x2, x2 * x3, etc. Products of features are usually called βnon-linearβ or βmultiplicativeβ interactions between features.

Feature engineering can be an important step of a model pipeline as long as the new features are expected to be predictive. For instance, think of a classification model to decide if a patient has risk of developing a heart disease. This would depend on the patientβs Body Mass Index which is defined as weight / height ** 2.

We load the dataset penguins dataset. We first use a set of 3 numerical features to predict the target, i.e. the body mass of the penguin.

Note

If you want a deeper overview regarding this dataset, you can refer to the Appendix - Datasets description section at the end of this MOOC.

import pandas as pd

columns = ["Flipper Length (mm)", "Culmen Length (mm)", "Culmen Depth (mm)"]
target_name = "Body Mass (g)"

# Remove lines with missing values for the columns of interest
penguins_non_missing = penguins[columns + [target_name]].dropna()

data = penguins_non_missing[columns]
target = penguins_non_missing[target_name]
data

Flipper Length (mm) Culmen Length (mm) Culmen Depth (mm)
0 181.0 39.1 18.7
1 186.0 39.5 17.4
2 195.0 40.3 18.0
4 193.0 36.7 19.3
5 190.0 39.3 20.6
... ... ... ...
339 207.0 55.8 19.8
340 202.0 43.5 18.1
341 193.0 49.6 18.2
342 210.0 50.8 19.0
343 198.0 50.2 18.7

342 rows Γ 3 columns

Now it is your turn to train a linear regression model on this dataset. First, create a linear regression model.

# Write your code here.


Execute a cross-validation with 10 folds and use the mean absolute error (MAE) as metric.

# Write your code here.


Compute the mean and std of the MAE in grams (g). Remember you have to revert the sign introduced when metrics start with neg_, such as in "neg_mean_absolute_error".

# Write your code here.


Now create a pipeline using make_pipeline consisting of a PolynomialFeatures and a linear regression. Set degree=2 and interaction_only=True to the feature engineering step. Remember not to include a βbiasβ feature (that is a constant-valued feature) to avoid introducing a redundancy with the intercept of the subsequent linear regression model.

You may want to use the .set_output(transform="pandas") method of the pipeline to answer the next question.

# Write your code here.


Transform the first 5 rows of the dataset and look at the column names. How many features are generated at the output of the PolynomialFeatures step in the previous pipeline?

# Write your code here.


Check that the values for the new interaction features are correct for a few of them.

# Write your code here.


Use the same cross-validation strategy as done previously to estimate the mean and std of the MAE in grams (g) for such a pipeline. Compare with the results without feature engineering.

# Write your code here.


Now letβs try to build an alternative pipeline with an adjustable number of intermediate features while keeping a similar predictive power. To do so, try using the Nystroem transformer instead of PolynomialFeatures. Set the kernel parameter to "poly" and degree to 2. Adjust the number of components to be as small as possible while keeping a good cross-validation performance.

Hint: Use a ValidationCurveDisplay with param_range = np.array([5, 10, 50, 100]) to find the optimal n_components.

# Write your code here.


How do the mean and std of the MAE for the Nystroem pipeline with optimal n_components compare to the other previous models?

# Write your code here.