A comparison between forward feature selection with cross-validation, forward selection guided by AIC/BIC, and Lasso regularization with Python Code

Forward feature selection with cross-validation incorporates cross-validation at each step to get a reliable estimate of how well a model with a particular set of features is likely to perform on unseen data. Without cross-validation, you might just pick the feature that gives the best performance on the training data, which could lead to overfitting and poor generalization. Cross-validation helps you choose features that are consistently beneficial across different subsets of your data.

Here’s how the process typically unfolds:

1. Initialization:
   • Start with an empty set of selected features.
   • Choose a predictive model (like linear regression, logistic regression, a decision tree, etc.).
   • Decide on a cross-validation strategy (like k-fold cross-validation, often with k=5 or k=10).
   • Select a performance metric to evaluate the model (e.g., accuracy, precision, recall, F1-score for classification; mean squared error, R-squared for regression).
2. Iteration:
   • For each feature not currently in the selected set:
     • Temporarily add this feature to the current set of selected features.
     • Train and evaluate the chosen model using cross-validation on the data with the temporarily expanded feature set. Calculate the average performance metric across all the folds.
   • Once you’ve evaluated adding each of the remaining features, identify the feature that resulted in the best cross-validated performance metric.
   • If the best performance with the added feature is better than the performance without it (or meets a predefined improvement threshold), permanently add this feature to your set of selected features.
   • Repeat this iteration until adding any of the remaining features does not improve the cross-validated performance (or until you reach a desired number of features).
3. Final Model:
   • Once the selection process is complete, train your chosen model on the entire dataset using the final set of selected features. This is your final predictive model.

Potential Considerations:

• Computational Cost: Forward selection can be computationally expensive, especially with a large number of features, as you train and evaluate the model multiple times in each iteration.
• Suboptimal Solutions: Forward selection is a greedy algorithm. It makes locally optimal choices at each step, which doesn’t guarantee finding the globally optimal set of features. There might be a combination of features that performs better than the one found by forward selection, but this combination might not have been reached through the sequential addition process.
• Choice of Model and Metric: The effectiveness of forward feature selection can depend on the choice of the predictive model and the evaluation metric. You should select these based on the nature of your problem and the goals of your analysis.

A comparison between forward feature selection with cross-validation, forward selection guided by AIC/BIC, and Lasso regularization
1. Forward Feature Selection with Cross-Validation:

• Guidance Metric: Uses the cross-validated performance of the chosen model (e.g., accuracy, MSE) to decide which feature to add at each step.
• Focus: Directly optimizes for predictive performance on unseen data. The chosen features are those that demonstrably improve the model’s ability to generalize.
• Model Dependency: Heavily tied to the specific model you choose. The selected features might be optimal for a linear regression but different for a tree-based model.
• Computational Cost: Can be computationally expensive, especially with many features and folds in cross-validation, as you need to train and evaluate the model multiple times in each iteration.
• Overfitting Control: Cross-validation provides a robust mechanism to control overfitting during the feature selection process. You’re selecting features that consistently improve out-of-sample performance.
• Stopping Criterion: Typically based on whether adding a feature leads to a statistically significant or practically meaningful improvement in the cross-validated performance.

2. Forward Feature Selection using AIC (Akaike Information Criterion) and BIC (Bayesian Information Criterion):

• Guidance Metric: Uses information criteria (AIC or BIC) to evaluate the quality of a model with a given set of features. These criteria balance the goodness of fit with the complexity (number of parameters) of the model.
  • AIC: Tends to favor more complex models (more features) as it penalizes complexity less heavily than BIC.
  • BIC: Tends to favor simpler models (fewer features) with a stronger penalty for the number of parameters.
• Focus: Aims to find a model that balances fit and parsimony. The idea is to select a model that explains the data well without being overly complex and prone to overfitting.
• Model Dependency: Usually applied within the context of likelihood-based models, where the likelihood function can be calculated (e.g., linear regression, logistic regression).
• Computational Cost: Generally less computationally expensive than cross-validation, as you primarily need to fit the model and calculate the AIC or BIC at each step.
• Overfitting Control: AIC and BIC provide an indirect way to control overfitting by penalizing model complexity. BIC is generally considered more effective at preventing overfitting in large datasets.
• Stopping Criterion: You stop when adding a feature no longer decreases the AIC or BIC (you want to minimize these criteria).

3. Lasso (Least Absolute Shrinkage and Selection Operator):

• Mechanism: Lasso is a regularization technique that adds an penalty to the cost function during model training:
  
  where is the regularization strength.
• Feature Selection Property: The penalty has the crucial property of shrinking the coefficients of less important features to exactly zero, effectively performing feature selection as part of the model training process.
• Focus: Simultaneously performs model training and feature selection. It aims to build a sparse model with good predictive accuracy.
• Model Dependency: Can be applied to various linear models (e.g., linear regression, logistic regression).
• Computational Cost: The computational cost depends on the size of the dataset and the complexity of the model. Efficient algorithms exist for fitting Lasso models.
• Overfitting Control: The penalty directly controls overfitting by shrinking coefficients and simplifying the model. The regularization strength is a key hyperparameter that needs to be tuned (often using cross-validation!).

Here’s a table summarizing the key differences:

Feature	Forward Selection with CV	Forward Selection with AIC/BIC	Lasso
Guidance Metric	Cross-validated performance	AIC/BIC	penalty during training
Focus	Predictive performance	Balance of fit and parsimony	Sparse model with good prediction
Model Dependency	Strong (tied to chosen model)	Likelihood-based models	Linear models
Computational Cost	Potentially high	Lower	Moderate to high (depending on tuning)
Overfitting Control	Direct via cross-validation	Indirect via complexity penalty	Direct via regularization
Sequential?	Yes	Yes	No (simultaneous)
Hyperparameter Tuning	Cross-validation strategy	None (AIC/BIC are calculated)	Regularization strength (), often via CV

When to Choose Which Approach:

• Forward Selection with Cross-Validation: A good choice when your primary goal is to maximize predictive performance on unseen data and you’re willing to pay the computational cost. It’s flexible and can be used with any model.
• Forward Selection with AIC/BIC: Useful when you want a more interpretable model with a good balance of fit and parsimony, especially with likelihood-based models. BIC might be preferred for larger datasets to avoid overfitting. It’s computationally more efficient than cross-validation.
• Lasso: An excellent option when you suspect that many of your features are irrelevant and you want an automatic way to perform feature selection and model training simultaneously, resulting in a sparse and potentially more interpretable model. It often requires tuning the regularization strength using cross-validation.

In practice, the best approach often depends on the specific dataset, the goals of the analysis, and the computational resources available. It’s not uncommon to experiment with different methods and compare their performance using appropriate evaluation metrics and cross-validation.

Python code

Here’s a Python script illustrating a comparison between forward feature selection with cross-validation, forward selection guided by AIC/BIC, and Lasso regularization. This example assumes a regression setting with synthetic data.

import numpy as np
import pandas as pd
from sklearn.datasets import make_regression
from sklearn.model_selection import train_test_split, cross_val_score
from sklearn.linear_model import LinearRegression, LassoCV
from sklearn.metrics import mean_squared_error
import statsmodels.api as sm
from itertools import combinations

# Generate synthetic dataset
X, y = make_regression(n_samples=500, n_features=15, noise=0.1, random_state=42)
X_train, X_test, y_train, y_test = train_test_split(X, y, test_size=0.2, random_state=42)

# Forward selection using cross-validation
def forward_selection_cv(X, y):
    remaining_features = list(range(X.shape[1]))
    selected_features = []
    best_score = float('inf')
    
    while remaining_features:
        scores = []
        for feature in remaining_features:
            features_to_test = selected_features + [feature]
            model = LinearRegression()
            score = -np.mean(cross_val_score(model, X[:, features_to_test], y, cv=5, scoring='neg_mean_squared_error'))
            scores.append((score, feature))
        
        scores.sort()
        if scores[0][0] < best_score:
            best_score, best_feature = scores[0]
            selected_features.append(best_feature)
            remaining_features.remove(best_feature)
        else:
            break
    
    return selected_features

selected_features_cv = forward_selection_cv(X_train, y_train)
model_cv = LinearRegression().fit(X_train[:, selected_features_cv], y_train)
y_pred_cv = model_cv.predict(X_test[:, selected_features_cv])
print("Forward selection with CV MSE:", mean_squared_error(y_test, y_pred_cv))

# Forward selection using AIC/BIC
def forward_selection_aic_bic(X, y, criterion='AIC'):
    remaining_features = list(range(X.shape[1]))
    selected_features = []
    best_criterion_value = float('inf')
    
    while remaining_features:
        criterion_values = []
        for feature in remaining_features:
            features_to_test = selected_features + [feature]
            X_test = sm.add_constant(X[:, features_to_test])
            model = sm.OLS(y, X_test).fit()
            criterion_value = model.aic if criterion == 'AIC' else model.bic
            criterion_values.append((criterion_value, feature))
        
        criterion_values.sort()
        if criterion_values[0][0] < best_criterion_value:
            best_criterion_value, best_feature = criterion_values[0]
            selected_features.append(best_feature)
            remaining_features.remove(best_feature)
        else:
            break
    
    return selected_features

selected_features_aic = forward_selection_aic_bic(X_train, y_train, criterion='AIC')
X_train_aic = sm.add_constant(X_train[:, selected_features_aic])
X_test_aic = sm.add_constant(X_test[:, selected_features_aic])
model_aic = sm.OLS(y_train, X_train_aic).fit()
y_pred_aic = model_aic.predict(X_test_aic)
print("Forward selection with AIC MSE:", mean_squared_error(y_test, y_pred_aic))

# Lasso regularization
lasso = LassoCV(cv=5).fit(X_train, y_train)
y_pred_lasso = lasso.predict(X_test)
print("Lasso regularization MSE:", mean_squared_error(y_test, y_pred_lasso))

Run on Colab

Output of result on test set:

Forward selection with CV MSE: 0.010051744290126037
Forward selection with AIC MSE: 0.010042954153151939
Lasso regularization MSE: 0.17016100246753524

More detailed code explanation:

import numpy as np
import pandas as pd
from sklearn.datasets import make_regression
from sklearn.model_selection import train_test_split, cross_val_score
from sklearn.linear_model import LinearRegression, LassoCV
from sklearn.metrics import mean_squared_error
import statsmodels.api as sm
from itertools import combinations

Explanation: These are the necessary imports. The code uses libraries like NumPy and Pandas for data handling, scikit-learn for regression and cross-validation, statsmodels for statistical modeling, and itertools for combinations during forward selection.

# Generate synthetic dataset
X, y = make_regression(n_samples=500, n_features=15, noise=0.1, random_state=42)
X_train, X_test, y_train, y_test = train_test_split(X, y, test_size=0.2, random_state=42)

Explanation: A synthetic regression dataset is created using make_regression with 500 samples and 15 features. The dataset is split into training (80%) and test (20%) sets using train_test_split.

# Forward selection using cross-validation
def forward_selection_cv(X, y):
    remaining_features = list(range(X.shape[1]))
    selected_features = []
    best_score = float('inf')
    
    while remaining_features:
        scores = []
        for feature in remaining_features:
            features_to_test = selected_features + [feature]
            model = LinearRegression()
            score = -np.mean(cross_val_score(model, X[:, features_to_test], y, cv=5, scoring='neg_mean_squared_error'))
            scores.append((score, feature))
        
        scores.sort()
        if scores[0][0] < best_score:
            best_score, best_feature = scores[0]
            selected_features.append(best_feature)
            remaining_features.remove(best_feature)
        else:
            break
    
    return selected_features

Explanation: This function implements forward selection with cross-validation. Features are added one by one to improve cross-validated performance (measured by negative mean squared error). If no further improvement occurs, the process stops.

selected_features_cv = forward_selection_cv(X_train, y_train)
model_cv = LinearRegression().fit(X_train[:, selected_features_cv], y_train)
y_pred_cv = model_cv.predict(X_test[:, selected_features_cv])
print("Forward selection with CV MSE:", mean_squared_error(y_test, y_pred_cv))

Explanation: The forward_selection_cv function is applied to the training set to determine the selected features. A linear regression model is fitted using these features, and the mean squared error (MSE) is calculated on the test set.

# Forward selection using AIC/BIC
def forward_selection_aic_bic(X, y, criterion='AIC'):
    remaining_features = list(range(X.shape[1]))
    selected_features = []
    best_criterion_value = float('inf')
    
    while remaining_features:
        criterion_values = []
        for feature in remaining_features:
            features_to_test = selected_features + [feature]
            X_test = sm.add_constant(X[:, features_to_test])
            model = sm.OLS(y, X_test).fit()
            criterion_value = model.aic if criterion == 'AIC' else model.bic
            criterion_values.append((criterion_value, feature))
        
        criterion_values.sort()
        if criterion_values[0][0] < best_criterion_value:
            best_criterion_value, best_feature = criterion_values[0]
            selected_features.append(best_feature)
            remaining_features.remove(best_feature)
        else:
            break
    
    return selected_features

Explanation: This function performs forward feature selection using AIC (or optionally BIC) as the selection criterion. Features are added one by one, optimizing the criterion value (lower AIC/BIC indicates a better model). The process stops when no further improvement is observed.

selected_features_aic = forward_selection_aic_bic(X_train, y_train, criterion='AIC')
X_train_aic = sm.add_constant(X_train[:, selected_features_aic])
X_test_aic = sm.add_constant(X_test[:, selected_features_aic])
model_aic = sm.OLS(y_train, X_train_aic).fit()
y_pred_aic = model_aic.predict(X_test_aic)
print("Forward selection with AIC MSE:", mean_squared_error(y_test, y_pred_aic))

Explanation: The forward_selection_aic_bic function is used to select features based on AIC. A linear regression model is fitted using these features, and the test set MSE is calculated.

# Lasso regularization
lasso = LassoCV(cv=5).fit(X_train, y_train)
y_pred_lasso = lasso.predict(X_test)
print("Lasso regularization MSE:", mean_squared_error(y_test, y_pred_lasso))

Explanation: Lasso regression with cross-validation is performed using LassoCV, which selects features by shrinking some coefficients to zero. The MSE for Lasso regression is then calculated on the test set.