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How To Perform Feature Selection for Regression Problems

In this article I explain what feature selection is and how to perform it before training a regression model in Python. Serafeim Loukas, PhD August 8, 20206 min read

1. Introduction

What is feature selection ?

Feature selection is the procedure of selecting a subset (some out of all available) of the input variables that are most relevant to the target variable (that we wish to predict). Target variable here **** refers to the** variable that we wish to predic**t. For this article we will assume that we only have numerical input variables and a numerical target for regression predictive modeling. Assuming that, we can easily estimate the relationship between each input variable and the target variable. This relationship can be established by calculating a metric such as the correlation value for example.

Learn this step by step with the interactive AI and Data Scientist and Machine Learning roadmaps.

2. The main numerical feature selection methods

The 2 most famous feature selection techniques that can be used for numerical input data and a numerical target variable are the following:

Correlation is a measure of how two variables change together. The most widely used correlation measure is the Pearson's correlation that assumes a Gaussian distribution of each variable and detects linear relationship between numerical variables. This is done in 2 steps:

  1. The correlation between each regressor and the target is computed, that is, ((X[:, i] - mean(X[:, i])) (y - mean_y)) / (std(X[:, i]) std(y)).
  2. It is converted to an F score then to a p-value.

Mutual information originates from the field of information theory. The idea is that the information gain (typically used in the construction of decision trees) is applied in order to perform the feature selection. Mutual information is calculated between two variables and measures as the reduction in uncertainty for one variable given a known value of the other variable.

3. The dataset

We will use the boston house -prices dataset. This dataset contains information collected by the U.S Census Service concerning housing in the area of Boston Mass. The dataset consists of the following variables:

  1. CRIM - per capita crime rate by town
  2. ZN - proportion of residential land zoned for lots over 25,000 sq.ft.
  3. INDUS - proportion of non-retail business acres per town.
  4. CHAS - Charles River dummy variable (1 if tract bounds river; 0 otherwise)
  5. NOX - nitric oxides concentration (parts per 10 million)
  6. RM - average number of rooms per dwelling
  7. AGE - proportion of owner-occupied units built prior to 1940
  8. DIS - weighted distances to five Boston employment centres
  9. RAD - index of accessibility to radial highways
  10. TAX - full-value property-tax rate per $10,000
  11. PTRATIO - pupil-teacher ratio by town
  12. B - 1000(Bk - 0.63)² where Bk is the proportion of blacks by town
  13. LSTAT - % lower status of the population
  14. MEDV - Median value of owner-occupied homes in $1000's

4. Python Code & Working Example

Let's load and split the dataset into training (70%) and test (30%) sets. text

from sklearn.datasets import load_boston
from sklearn.model_selection import train_test_split
from sklearn.feature_selection import SelectKBest
from sklearn.feature_selection import f_regression
import matplotlib.pyplot as plt
from sklearn.feature_selection import mutual_info_regression

text

# load the data
X, y = load_boston(return_X_y=True)

text

# split into train and test sets
X_train, X_test, y_train, y_test = train_test_split(X, y, test_size=0.30, random_state=0)

We will use the well known scikit-learn machine library.

Case 1: Feature selection using the Correlation metric

For the correlation statistic we will use the f_regression() function. This function can be used in a feature selection strategy, such as selecting the top k most relevant features (largest values) via the SelectKBest class. text

# feature selection
f_selector = SelectKBest(score_func=f_regression, k='all')

text

# learn relationship from training data
f_selector.fit(X_train, y_train)

text

# transform train input data
X_train_fs = f_selector.transform(X_train)

text

# transform test input data
X_test_fs = f_selector.transform(X_test)

text

# Plot the scores for the features
plt.bar([i for i in range(len(f_selector.scores_))], f_selector.scores_)

text

plt.xlabel("feature index")
plt.ylabel("F-value (transformed from the correlation values)")
plt.show()

Reminder : For the correlation statistic case:

  1. The correlation between each regressor and the target is computed, that is, ((X[:, i] - mean(X[:, i])) (y - mean_y)) / (std(X[:, i]) std(y)).
  2. It is converted to an F score then to a p-value.

Feature Importance plot Feature Importance plot The plot above shows that feature 6 and 13 are more important than the other features. The y-axis represents the F-values that were estimated from the correlation values.

Case 2: Feature selection using the Mutual Information metric

The scikit-learn machine learning library provides an implementation of mutual information for feature selection with numeric input and output variables via the mutual_info_regression() function. text

# feature selection
f_selector = SelectKBest(score_func=mutual_info_regression, k='all')

text

# learn relationship from training data
f_selector.fit(X_train, y_train)

text

# transform train input data
X_train_fs = f_selector.transform(X_train)

text

# transform test input data
X_test_fs = f_selector.transform(X_test)

text

# Plot the scores for the features
plt.bar([i for i in range(len(f_selector.scores_))], f_selector.scores_)

text

plt.xlabel("feature index")
plt.ylabel("Estimated MI value")
plt.show()

Feature Importance plot Feature Importance plot The y-axis represents the estimated mutual information between each feature and the target variable. Compared to the correlation feature selection method we can clearly see many more features scored as being relevant. This may be because of the statistical noise that might exists in the dataset.

5. Conclusion

In this article I have provided two ways in order to perform feature selection. Feature selection is the procedure of selecting a subset (some out of all available) of the input variables that are most relevant to the target variable. Target variable here **** refers to the** variable that we wish to predic**t. Using either the Correlation metric or the Mutual Information metric , we can easily estimate the relationship between each input variable and the target variable. Correlation vs Mutual Information: Compared to the correlation feature selection method we can clearly see many more features scored as being relevant. This may be because of the statistical noise that might exists in the dataset.

You might also like:

LSTM Time-Series Forecasting: Predicting Stock Prices Using An LSTM Model Time-Series Forecasting: Predicting Stock Prices Using An ARIMA Model

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