39 minute read

1. Introduction: Trimming the Fat from Your Data for Better Models

Imagine you’re preparing for a big trip and you need to pack a suitcase. You start by putting everything you might need into the suitcase. Then, you realize it’s too heavy and bulky! So, you start taking things out, one by one, starting with the least essential items, until your suitcase is lighter and contains only the really important stuff.

Backward Feature Elimination in machine learning is like that process of decluttering your suitcase, but for your data! It’s a feature selection technique that starts with all your features (columns in your data) and then iteratively removes the least important ones to simplify your model and potentially improve its performance.

Why would you want to remove features?

  • Simpler Models: Models with fewer features are often easier to understand and explain. Simplicity is valuable, especially when you need to communicate your model to non-technical audiences or when model interpretability is crucial.
  • Improved Performance: Counterintuitively, removing features can sometimes improve model performance. This happens when you remove irrelevant or noisy features that are actually confusing the model or leading to overfitting (fitting the training data too well but performing poorly on new data).
  • Faster Training and Prediction: Models with fewer features train faster and make predictions faster, which can be important for large datasets or real-time applications.
  • Reduced Data Collection Cost: If you identify that some features are not very important, you might be able to reduce the cost of data collection in the future by focusing only on the essential features.

Real-world examples where Backward Feature Elimination is useful:

  • Predicting Customer Churn: Imagine a telecommunications company wants to predict which customers are likely to stop using their service (churn). They might have a huge dataset with hundreds of features about customer demographics, usage patterns, billing information, customer service interactions, etc. Backward Feature Elimination can help identify the most important features that predict churn. Features like “minutes of calls per month,” “data usage,” “customer service call frequency,” might be found to be crucial, while less important features (that were initially included) can be removed to simplify the churn prediction model.

  • Medical Diagnosis: When predicting a disease based on various medical tests and patient characteristics, doctors might start with a broad panel of tests. Backward Feature Elimination can help identify which tests are the most diagnostically relevant. This could lead to reducing the number of tests needed for routine screening, making diagnosis more efficient and less burdensome for patients, without sacrificing accuracy. For instance, in predicting diabetes, initial data might include dozens of blood markers, lifestyle factors, family history elements. Backward elimination could pinpoint the most predictive markers (like blood glucose level, BMI, age, etc.) and eliminate less informative ones.

  • Financial Risk Assessment: In credit scoring or loan default prediction, banks might collect a wide range of financial and personal information about loan applicants. Backward Feature Elimination can help select the key financial indicators that best predict loan default risk. Focusing on these core indicators can streamline the credit assessment process and potentially improve the accuracy and robustness of risk models. Examples of initially collected features might be: income, credit history, employment duration, home ownership status, number of credit cards, and many more. Backward elimination can help choose the most vital few, such as credit score, debt-to-income ratio, and employment stability, and remove less impactful or redundant features.

In essence, Backward Feature Elimination is about “pruning” your dataset, removing the less useful branches (features), to reveal the strong, essential trunk (the most important features) that supports a robust and efficient model.

2. The Mathematics of Elimination: Step-by-Step Feature Pruning

Let’s look at how Backward Feature Elimination works mathematically. Don’t worry, we’ll keep it as simple as possible!

The core idea is to iteratively remove features based on their “importance” to the model. But how do we define “importance”? And how do we remove features step-by-step?

Step 1: Start with All Features

Backward Feature Elimination begins with a model trained using all available features in your dataset. Let’s say you have a dataset with features (X = [x_1, x_2, …, x_p]) and a target variable (y). You train a model (e.g., linear regression, logistic regression, tree-based model – the type of model doesn’t change the core Backward Elimination process) using all (p) features.

Step 2: Evaluate Feature Importance

Next, you need a way to assess the “importance” of each feature in the current model. There are different ways to do this:

  • Coefficient Magnitude (for Linear Models): If you’re using a linear model like linear regression or logistic regression, you can use the magnitude (absolute value) of the coefficients associated with each feature. Larger coefficient magnitude suggests a larger influence of that feature on the model’s predictions. You might consider features with smaller coefficient magnitudes as less important.

  • Feature Importance from Tree-Based Models: For tree-based models (like Decision Trees, Random Forests, Gradient Boosting Machines), many implementations provide feature importance scores. These scores estimate how much each feature contributes to reducing impurity (e.g., Gini impurity, entropy) in decision trees or improving prediction accuracy in tree ensembles. Features with lower importance scores are considered less crucial. Scikit-learn’s tree-based models have a .feature_importances_ attribute.

  • Performance Drop after Feature Removal (More General Approach): A more general and often more robust approach is to directly measure the drop in model performance when you remove a feature.

    • Train a model using all features. Evaluate its performance (e.g., using cross-validation R-squared for regression, cross-validation accuracy for classification). Let’s call this baseline performance (P_{all}).
    • For each feature (x_i), temporarily remove it from the dataset. Train a new model using the remaining features (all features except (x_i)). Evaluate the performance of this reduced-feature model. Let’s call this performance (P_{-i}) (performance without feature (x_i)).
    • Calculate the performance drop when removing feature (x_i): (\Delta P_i = P_{all} - P_{-i}). A smaller performance drop (\Delta P_i) suggests that feature (x_i) is less important, as removing it doesn’t hurt performance much. A larger drop means the feature is more important.

Step 3: Identify the Least Important Feature

Based on your chosen “importance” criterion (coefficient magnitude, feature importance score, performance drop), determine which feature is currently considered the least important. If using performance drop, the feature with the smallest performance drop (\Delta P_i) is the least important.

Step 4: Remove the Least Important Feature

Remove the least important feature (identified in Step 3) from your dataset. You now have a dataset with one fewer feature.

Step 5: Repeat Steps 2-4

Repeat steps 2-4. Train a new model using the reduced set of features (after removing one feature in the previous step). Again, evaluate the importance of the remaining features and remove the currently least important feature.

Step 6: Stopping Criterion

Continue iterating steps 2-5 until you reach a stopping criterion. Common stopping criteria are:

  • Reach a Desired Number of Features: Stop when you have reduced the number of features to a pre-defined number (e.g., reduce to the top 10 most important features).
  • Performance Degradation Threshold: Monitor model performance (e.g., cross-validation R-squared or accuracy) as you remove features. Stop when removing another feature leads to a significant drop in performance (e.g., performance drops below a certain acceptable level).
  • No Significant Improvement in Simplicity (Diminishing Returns): Stop when removing more features doesn’t significantly simplify the model further (in terms of number of features) or when the performance gain from further reduction is negligible.

Example: Backward Elimination Illustrated (using performance drop and linear regression)

Let’s say we have a dataset with a target variable (y) and three features (x_1, x_2, x_3). We’ll use Backward Elimination with linear regression and performance drop based on R-squared.

  1. Start with all features ({x_1, x_2, x_3}). Train a linear regression model with all features. Baseline R-squared (from cross-validation): 0.85.

  2. Step 1: Evaluate feature importance (performance drop):
    • Remove (x_1). Model with ({x_2, x_3}). R-squared: 0.84. Performance drop (\Delta P_1 = 0.85 - 0.84 = 0.01).
    • Remove (x_2). Model with ({x_1, x_3}). R-squared: 0.78. Performance drop (\Delta P_2 = 0.85 - 0.78 = 0.07).
    • Remove (x_3). Model with ({x_1, x_2}). R-squared: 0.83. Performance drop (\Delta P_3 = 0.85 - 0.83 = 0.02).
    • Feature (x_1) has the smallest performance drop (0.01). So, (x_1) is currently considered the least important.

  3. Step 2: Remove (x_1). Remaining features: ({x_2, x_3}).

  4. Step 3: Repeat evaluation with remaining features ({x_2, x_3}). Start with model using ({x_2, x_3}) (baseline R-squared 0.84 from Step 1).
    • Remove (x_2). Model with ({x_3}). R-squared: 0.65. Performance drop (\Delta P_{2}’ = 0.84 - 0.65 = 0.19).
    • Remove (x_3). Model with ({x_2}). R-squared: 0.70. Performance drop (\Delta P_{3}’ = 0.84 - 0.70 = 0.14).
    • Feature (x_3) now has the smaller performance drop (0.14) relative to the current baseline (model with ({x_2, x_3})). So, (x_3) is now considered less important between (x_2) and (x_3).

  5. Step 4: Remove (x_3). Remaining feature: ({x_2}).

  6. Stopping Criterion: Let’s say we decide to stop when we have reduced to just one feature. We are now left with ({x_2}) as the selected feature subset.

Result: Backward Elimination has selected feature (x_2) as the most important single feature in this example. The feature removal order was: (x_1), then (x_3), leaving (x_2).

Key Idea: Backward Feature Elimination is a systematic, iterative process of removing features one at a time, based on their perceived importance in a model, aiming to simplify the model while maintaining or even improving its performance.

3. Prerequisites and Preprocessing: Getting Ready for Backward Elimination

Before applying Backward Feature Elimination, it’s important to understand the prerequisites and any necessary preprocessing steps.

Prerequisites and Assumptions:

  • Dataset with Features and Target Variable: You need a dataset with clearly defined features (input variables) and a target variable (the variable you want to predict). Backward Elimination is about selecting a subset of these features.
  • Model Choice: You need to choose a specific machine learning model that will be used within the Backward Elimination process to evaluate feature importance. The choice of model influences how feature importance is assessed (e.g., coefficients for linear models, feature importance scores for trees, performance drop for any model type).
  • Evaluation Metric: You need to select an appropriate evaluation metric to assess model performance. For regression, it could be R-squared, RMSE, MSE, etc. For classification, it could be accuracy, precision, recall, F1-score, AUC-ROC, etc. The choice of metric guides the feature selection process.
  • Computational Resources: Backward Elimination involves repeatedly training and evaluating models as features are removed. For datasets with many features and/or slow-to-train models, it can be computationally intensive, especially if you use cross-validation for robust performance estimation in each step.

Testing Assumptions (and Considerations):

  • Model Suitability: Ensure that the chosen model type is reasonably suitable for your data and task. For example, if you choose linear regression, you should have some reason to believe that linear relationships might be present in your data (or be willing to work with linear approximations). If using tree-based models, they are generally more flexible in handling non-linearities.
  • Feature Importance Metric Relevance: Check if the chosen feature importance metric makes sense for your chosen model type and task. For linear models, coefficient magnitude might be reasonable, but for non-linear models, performance drop or tree-based feature importance scores might be more appropriate. If using performance drop, make sure your performance evaluation (e.g., cross-validation) is robust and reliable.
  • Dataset Size and Computational Cost: Consider the size of your dataset and the computational cost of training your chosen model repeatedly. If your dataset is very large or models are slow to train, Backward Elimination (especially with cross-validation in each step) might become very time-consuming. In such cases, consider more computationally efficient feature selection methods or approximation techniques.

Python Libraries for Backward Elimination:

  • sklearn (scikit-learn): Scikit-learn provides RFE (Recursive Feature Elimination) in sklearn.feature_selection. RFE can be used to implement Backward Elimination style feature selection with a given estimator (model).

  • mlxtend (Machine Learning Extensions): The mlxtend library offers SequentialFeatureSelector in mlxtend.feature_selection, which can be used to perform both Forward Selection and Backward Selection (and other sequential feature selection methods). It’s often more flexible for implementing Backward Elimination compared to sklearn.RFE because it directly allows you to specify forward or backward direction and use different scoring metrics.

# Python Libraries for Backward Elimination
import sklearn.feature_selection
from sklearn.feature_selection import RFE
import sklearn.linear_model
from mlxtend.feature_selection import SequentialFeatureSelector as SFS

print("scikit-learn version:", sklearn.__version__)
import mlxtend
print("mlxtend version:", mlxtend.__version__)

Ensure these libraries are installed. Install using pip if needed:

pip install scikit-learn mlxtend

4. Data Preprocessing: Scaling and Encoding Considerations

Data preprocessing can be relevant for Backward Feature Elimination, but the specific preprocessing needs can depend on the chosen model and the nature of your data.

Scaling of Features:

  • Importance for Distance-Based Models and Regularization: If you are using distance-based models (though less common in direct Backward Elimination, but might be used in evaluating performance) or regularization-based models (like linear models with L1 or L2 regularization) within your Backward Elimination process, feature scaling is often recommended.
    • Why: As explained in previous blog posts, scaling ensures that features with larger ranges do not disproportionately influence distance calculations or regularization penalties.
    • Methods: Standardization (Z-score scaling) or Min-Max scaling are common choices.
  • Less Critical for Tree-Based Models: For tree-based models (Decision Trees, Random Forests, Gradient Boosting), feature scaling is generally less critical, as discussed in earlier blogs. Tree-based models are less sensitive to feature scales in terms of split decisions.

Preprocessing Example (Scaling - same as in previous blogs):

from sklearn.preprocessing import StandardScaler, MinMaxScaler
import numpy as np

# Dummy data (example - replace with your actual data)
data = np.array([[10, 1000],
                 [20, 20000],
                 [15, 15000],
                 [5, 5000]])

# StandardScaler
scaler_standard = StandardScaler()
scaled_data_standard = scaler_standard.fit_transform(data)
print("Standardized data:\n", scaled_data_standard)

# MinMaxScaler
scaler_minmax = MinMaxScaler()
scaled_data_minmax = scaler_minmax.fit_transform(data)
print("\nMin-Max scaled data:\n", scaled_data_minmax)

Handling Categorical Features:

  • Encoding: Backward Feature Elimination itself doesn’t directly handle categorical features. You need to encode categorical features into numerical representations before applying Backward Elimination, as the models you use within Backward Elimination (e.g., linear models, tree-based models) typically require numerical inputs.
    • One-Hot Encoding: Convert categorical features into binary indicator variables (dummy variables). pandas (pd.get_dummies) or scikit-learn (OneHotEncoder) can be used.

Handling Missing Values:

  • Imputation or Removal: You need to address missing values before applying Backward Feature Elimination because most machine learning models (used for performance evaluation within Backward Elimination) require complete data.
    • Imputation: Fill in missing values with estimates (mean, median, mode, or more advanced imputation methods). sklearn.impute.SimpleImputer can be used.
    • Removal: Remove rows or columns with too many missing values (but be cautious about data loss).

When can preprocessing be ignored (or less strictly enforced)?

  • Tree-Based Models (Scaling): If you are using tree-based models as your estimator within Backward Elimination, scaling is generally less critical (though it might still sometimes help with convergence in gradient boosting). You could potentially skip scaling for tree-based models in some cases if you are primarily concerned about model simplicity and not as much about feature scale effects, but it’s usually safer to scale even for tree-based models.
  • Initial Exploration (Less Rigorous Preprocessing): During initial exploratory feature selection, if you’re just quickly trying out Backward Elimination to get a general sense of feature importance, you might skip some preprocessing steps (like scaling or imputation) for speed. However, for more rigorous evaluation and production use, proper preprocessing is essential.
  • If Features are Already on Comparable Scales (Less Common): If you are absolutely sure that your numerical features are already on very comparable scales, and you’re using a model that is not very sensitive to feature scale, you might consider skipping scaling. However, this is less common and generally not recommended best practice.

Best Practice: In most cases, especially when using models that are sensitive to feature scale or when aiming for robust and reliable feature selection, it’s generally recommended to scale your numerical features (e.g., using StandardScaler) and properly handle categorical features (e.g., using one-hot encoding) and missing values before applying Backward Feature Elimination. Consistent preprocessing is crucial for fair and meaningful feature evaluation during the elimination process.

5. Implementation Example: Backward Elimination with SequentialFeatureSelector

Let’s implement Backward Feature Elimination for regression using mlxtend.SequentialFeatureSelector and linear regression as the model. We’ll use a dummy dataset.

import numpy as np
import pandas as pd
from sklearn.linear_model import LinearRegression
from sklearn.metrics import r2_score
from sklearn.model_selection import train_test_split
from sklearn.preprocessing import StandardScaler
from mlxtend.feature_selection import SequentialFeatureSelector as SFS

# 1. Generate dummy regression data (same dummy data as in Forward Selection blog post)
np.random.seed(42)
X = np.random.rand(100, 5) # 100 samples, 5 features (some might be irrelevant)
y = 2*X[:, 0] + 0.5*X[:, 1] - 1.5*X[:, 3] + np.random.randn(100) # y depends mainly on features 0, 1, 3
feature_names = ['feature_1', 'feature_2', 'feature_3', 'feature_4', 'feature_5']
X_df = pd.DataFrame(X, columns=feature_names)
y_df = pd.Series(y, name='target')

# 2. Split data into training and testing sets
X_train, X_test, y_train, y_test = train_test_split(X_df, y_df, test_size=0.3, random_state=42)

# 3. Scale features (important for linear regression and feature selection stability)
scaler = StandardScaler()
X_train_scaled = scaler.fit_transform(X_train)
X_test_scaled = scaler.transform(X_test)
X_train_scaled_df = pd.DataFrame(X_train_scaled, columns=feature_names) # Keep column names
X_test_scaled_df = pd.DataFrame(X_test_scaled, columns=feature_names)   # Keep column names

# 4. Backward Elimination using SequentialFeatureSelector (from mlxtend)
linear_reg = LinearRegression()
backward_selector = SFS(linear_reg,
                            k_features=(1, 5), # Select 1 to 5 features (going backward from 5)
                            forward=False, # Set forward=False for Backward Selection
                            floating=False,
                            scoring='r2',      # Use R-squared for scoring
                            cv=5)             # 5-fold cross-validation

backward_selector = backward_selector.fit(X_train_scaled_df, y_train)

# 5. Print results of Backward Selection
print("Backward Selection Results:")
print("Selected feature indices:", backward_selector.k_feature_idx_)
print("Selected feature names:", backward_selector.k_feature_names_)
print("CV score (R-squared) for selected subset:", backward_selector.k_score_)
print("Best feature subsets and scores at each step (showing elimination process):\n", backward_selector.subsets_)

# 6. Evaluate model performance on test set using selected features
selected_feature_names = list(backward_selector.k_feature_names_)
X_train_selected = X_train_scaled_df[selected_feature_names]
X_test_selected = X_test_scaled_df[selected_feature_names]

final_model = LinearRegression()
final_model.fit(X_train_selected, y_train)
y_pred_test = final_model.predict(X_test_selected)
test_r2 = r2_score(y_test, y_pred_test)
print("\nR-squared on test set with selected features:", test_r2)

# 7. Save and load selected feature names (for later use)
import joblib # or pickle

# Save selected feature names
joblib.dump(selected_feature_names, 'selected_features_backward_selection.joblib')
print("Selected feature names saved to selected_features_backward_selection.joblib")

# Load selected feature names
loaded_feature_names = joblib.load('selected_features_backward_selection.joblib')
print("\nLoaded selected feature names:", loaded_feature_names)

Explanation of the Code and Output:

This code is very similar to the Forward Selection example code, with key differences in the SequentialFeatureSelector setup and interpretation of results related to Backward Elimination:

  1. Data Generation, Splitting, Scaling: Steps 1-3 are identical to the Forward Selection example – we use the same dummy regression data, train-test split, and feature scaling with StandardScaler.
  2. Backward Selection with SequentialFeatureSelector:
    • SFS(linear_reg, k_features=(1, 5), forward=False, ...): We initialize SequentialFeatureSelector similar to Forward Selection, but with:
      • forward=False: This is set to False to specify Backward Selection (instead of Forward Selection).
      • k_features=(1, 5): We still specify k_features=(1, 5), but now it means we want to explore subsets ranging from 5 features down to 1 feature (going backward from the initial set of 5 features).
      • Other parameters (estimator=linear_reg, scoring='r2', cv=5, floating=False) are the same as in Forward Selection, using linear regression as the model, R-squared as the scoring metric, and 5-fold cross-validation for performance estimation.
    • backward_selector.fit(...): We fit the backward selector on the scaled training data and target variable.
  3. Print Results:
    • backward_selector.k_feature_idx_, backward_selector.k_feature_names_, backward_selector.k_score_: These outputs are interpreted similarly to Forward Selection – they show the indices, names, and CV score (R-squared) for the best feature subset found (in this case, the best subset identified during the backward elimination process).
    • backward_selector.subsets_: This is particularly important for Backward Elimination. It shows the feature subsets and scores at each step of the elimination process. In Backward Elimination, it shows how performance changes as features are removed. It allows you to see the sequence of feature removals and how each removal affects the model performance.

  4. Evaluate on Test Set: Step 6 (evaluation on the test set with selected features) is the same as in Forward Selection.

  5. Save and Load Selected Feature Names: Step 7 (saving and loading selected feature names) is also the same as in Forward Selection.

Interpreting the Output (especially subsets_ for Backward Elimination):

When you run this code, you will see output similar to this (output might vary slightly due to randomness):

Backward Selection Results:
Selected feature indices: (0, 1, 3)
Selected feature names: ('feature_1', 'feature_2', 'feature_4')
CV score (R-squared) for selected subset: 0.925...
Best feature subsets and scores at each step (showing elimination process):
 {5: {'feature_idx': (0, 1, 2, 3, 4), 'cv_scores': array([0.92..., 0.90..., 0.89..., 0.93..., 0.94...]), 'avg_score': 0.92..., 'std_dev': 0.01...}, ...
 {4: {'feature_idx': (0, 1, 2, 3), 'cv_scores': array([0.92..., 0.91..., 0.89..., 0.93..., 0.94...]), 'avg_score': 0.92..., 'std_dev': 0.01...}, ...
 {3: {'feature_idx': (0, 1, 3), 'cv_scores': array([0.92..., 0.91..., 0.89..., 0.93..., 0.94...]), 'avg_score': 0.92..., 'std_dev': 0.01...}, ...
 {2: {'feature_idx': (0, 1), 'cv_scores': array([0.84..., 0.83..., 0.78..., 0.90..., 0.89...]), 'avg_score': 0.85..., 'std_dev': 0.04...}, ...
 {1: {'feature_idx': (0,), 'cv_scores': array([0.79..., 0.82..., 0.72..., 0.87..., 0.87...]), 'avg_score': 0.81..., 'std_dev': 0.05...}}

R-squared on test set with selected features: 0.93...
Selected feature names saved to selected_features_backward_selection.joblib

Loaded selected feature names: ['feature_1', 'feature_2', 'feature_4']

  • Selected Features and Performance: Backward Selection, like Forward Selection in the previous blog post, has identified features ‘feature_1’, ‘feature_2’, and ‘feature_4’ (indices 0, 1, 3) as the best subset (in this specific run - results may vary slightly due to randomness). The CV score (R-squared) and test set R-squared are also similar to the Forward Selection results.

  • subsets_ Output for Backward Elimination: This is the key to understanding Backward Elimination. It shows the process of feature removal:
    • {5: {'feature_idx': (0, 1, 2, 3, 4), ... 'avg_score': 0.92...}, ...}: Starting with 5 features (all features), the CV R-squared is around 0.92.
    • {4: {'feature_idx': (0, 1, 2, 3), ... 'avg_score': 0.92...}, ...}: After removing the least important feature in the first step (which is feature ‘feature_5’ in this particular run, though the specific feature removed in the first step might vary), with 4 features, the CV R-squared is still around 0.92 (no performance drop, or even slight improvement in some runs).
    • {3: {'feature_idx': (0, 1, 3), ... 'avg_score': 0.92...}, ...}: After removing another feature (in the second step), with 3 features, the CV R-squared is still around 0.92 (performance maintained with fewer features).
    • {2: {'feature_idx': (0, 1), ... 'avg_score': 0.85...}, ...}, {1: {'feature_idx': (0,), ... 'avg_score': 0.81...}: As you continue removing features (down to 2, then 1 feature), the CV R-squared starts to decrease more noticeably. This indicates that you are now starting to remove features that are actually important for model performance.
  • Feature Removal Order Implied by subsets_: By examining the feature_idx sets in subsets_, you can infer the order in which features were removed by Backward Elimination. The feature removed at each step is the one that is present in the feature set of the previous step (e.g., 5 features, then 4 features) but missing in the feature set of the current step (e.g., from 5 features set to 4 features set). In this example, you can infer that the removal sequence (approximate, may vary run to run) was something like: feature_5, then feature_3, then feature_2, then feature_4, leaving feature_1 as the last remaining (and considered most important single feature in this sequence).

R-squared and Output Interpretation (same as in Forward Selection Blog): See the “Interpreting the Output” section in the Forward Selection blog post for explanation of R-squared, test set R-squared, saving and loading features. The R-squared value and its interpretation are the same for both Forward and Backward Selection examples in this blog series.

6. Post-Processing: Analyzing Feature Elimination Order and Model Insights

Post-processing for Backward Feature Elimination primarily focuses on analyzing the feature elimination order, understanding the impact of feature removal on model performance, and gaining insights into feature importance.

Analyzing Feature Elimination Order:

  • Examine backward_selector.subsets_ Output: The subsets_ attribute (from mlxtend.SequentialFeatureSelector) is key to understanding the Backward Elimination process. It shows you:
    • The sequence of feature subsets explored (starting with all features and progressively removing features).
    • The cross-validated performance (e.g., CV R-squared or accuracy) for each feature subset.
    • The average cross-validation score (avg_score) and standard deviation (std_dev) for each subset size.
  • Feature Removal Sequence: By inspecting the feature_idx (or feature_names) in the subsets_ output at each step, you can infer the order in which features were removed by Backward Elimination. The features removed earlier in the process are generally considered less important than those removed later or those that are retained in the final selected subset.
  • Performance Trend with Feature Removal: Observe the trend of model performance (e.g., avg_score) as features are removed (as subset size decreases in subsets_).
    • Initial Performance Stability: In the early steps of Backward Elimination, you might see that removing a few features has little impact on model performance (performance might remain roughly constant or even slightly improve in some cases). This indicates that those initially removed features were indeed less important or redundant.
    • Performance Drop-off Point: As you continue to remove more features, you’ll eventually reach a point where removing a feature starts to cause a more significant drop in performance (the avg_score decreases more noticeably). This drop-off point suggests that you are now removing features that are important for model performance.

Identifying Important Variables:

  • Selected Feature Subset: The final feature subset chosen by Backward Elimination (e.g., given by backward_selector.k_feature_names_) is considered to be the set of most important features identified by this process, for the specified number of features to select (k_features parameter).
  • Features Retained Longest: Features that are removed later in the Backward Elimination sequence (i.e., features that are present in larger feature subsets in subsets_ before being removed in later steps) are generally considered more important than those removed earlier. Features that are never removed (if you stop before removing all but one feature) are considered the most important overall.
  • Feature Ranking (Implicit): Backward Elimination implicitly provides a ranking of features in terms of their “importance” (or rather, lack of importance). Features removed earlier are ranked lower in importance, and features removed later are ranked higher. You can create a ranked list of features based on their removal order from Backward Elimination.

Further Analysis and Testing:

  • Model Re-evaluation (with Selected Features): After Backward Elimination selects a feature subset, it’s crucial to re-evaluate the performance of a model trained only on the selected features on a held-out test set (as we did in the implementation example). This gives a more robust estimate of how well the model with selected features generalizes to unseen data. Compare the test set performance with the baseline model (using all features) and with models built using different feature subsets.
  • Comparison with Forward Selection or Other Methods: Compare the feature subsets selected by Backward Elimination to those selected by other feature selection methods (like Forward Selection, Regularization-based methods, tree-based feature importance). Do different methods select similar features? Are there overlaps or differences? Understanding the consistency or variability in feature selection across different methods can provide a more comprehensive view of feature importance.
  • Domain Expertise Interpretation: Relate the selected features and the feature elimination order back to your domain knowledge. Do the selected features make sense in the context of the problem you are trying to solve? Are they consistent with your expectations or prior understanding of the data? Domain expertise is essential for validating and interpreting the results of any feature selection method.
  • No AB Testing or Hypothesis Testing Directly on Feature Selection Output (like for visualization methods): Backward Feature Elimination is a feature selection technique, not an experimental design or statistical inference method in itself. You don’t directly perform AB testing or hypothesis testing on the output of Backward Elimination (the selected features) in the same way you might for experimental results or model predictions. However, you would use statistical hypothesis testing in the evaluation stage of your machine learning pipeline (e.g., to compare the performance of models with different feature subsets, or to statistically assess if feature selection leads to a significant improvement in performance).

In summary: Post-processing for Backward Feature Elimination is about analyzing the feature elimination sequence, understanding the performance trade-offs at each step, identifying the most important features (based on retention and model impact), and using domain knowledge to validate and interpret the feature selection results. It’s about turning the algorithmic feature pruning process into meaningful insights about feature importance and model simplification.

7. Hyperparameters of Backward Elimination: Tweaking the Selection Process

Backward Feature Elimination, when implemented using mlxtend.SequentialFeatureSelector or similar tools, has hyperparameters that you can tune to control the feature selection process. The most important ones relate to:

Key “Hyperparameters” and Choices in Backward Elimination:

  • k_features (Number of Features to Select):

    • Effect: The k_features parameter in SequentialFeatureSelector (and similar parameters in other implementations) determines the stopping point for Backward Elimination. It specifies the minimum number of features that the algorithm will reduce down to.
      • k_features=(1, p) (Range from 1 to initial number of features): If you set k_features to a range like (1, p) where (p) is the initial number of features, Backward Elimination will try to find the best feature subset for each possible number of features, from (p) down to 1. It explores all possible subset sizes within this range. This is common when you want to explore performance for different levels of feature reduction and choose a subset size based on a performance vs. simplicity trade-off.
      • k_features=n (Fixed Number, e.g., k_features=10): You can set k_features to a fixed integer value (e.g., 10). In this case, Backward Elimination will continue removing features until it reaches a subset size of exactly (n) features. It will select the best subset of size (n) according to the chosen evaluation metric. Use this when you have a specific target number of features in mind (e.g., for model complexity constraints or interpretability goals).
    • Tuning:
      • Explore Performance vs. Number of Features: When using k_features as a range (e.g., (1, p)), you can plot model performance (e.g., cross-validated R-squared or accuracy) against the number of features in the subset. This plot helps you visualize how performance changes as you remove features.
      • Elbow Method (in Performance Plot): Look for an “elbow” point in the performance vs. number of features plot. The “elbow” (where performance starts to plateau or degrade significantly as you remove more features) can suggest a good trade-off point between model simplicity (fewer features) and performance. Choose the number of features around the elbow.
      • Cross-Validation Scores for Different Subset Sizes: Examine the cross-validated performance scores (e.g., avg_score in subsets_ output) for different feature subset sizes in mlxtend.SequentialFeatureSelector. Compare the scores and standard deviations. Choose a subset size that provides a good balance of high average performance and acceptable variability (standard deviation) across cross-validation folds.
      • Domain Knowledge and Interpretability: Consider domain knowledge and interpretability goals. You might choose a smaller number of features for simplicity and better interpretability, even if performance is slightly lower than with a larger feature set.

  • scoring (Evaluation Metric):

    • Effect: The scoring parameter in SequentialFeatureSelector (and RFE and similar tools) defines the metric used to evaluate model performance during feature selection.
    • Tuning: Choose a scoring metric that is appropriate for your machine learning task (regression or classification) and your objectives.
      • Regression: 'r2', 'neg_mean_squared_error', 'neg_mean_absolute_error', etc. (Note: For metrics like MSE or MAE where lower is better, you might need to use negative versions like 'neg_mean_squared_error' if the feature selection algorithm is designed to maximize the score).
      • Classification: 'accuracy', 'f1', 'roc_auc', 'precision', 'recall', etc.

  • cv (Cross-Validation Strategy):

    • Effect: The cv parameter specifies the cross-validation strategy used to estimate model performance in each step of Backward Elimination. Cross-validation provides a more robust and reliable estimate of performance than using a single train-validation split.
    • Tuning: Choose an appropriate cross-validation strategy (k-fold CV, stratified k-fold for imbalanced classification, Leave-One-Out CV if dataset is very small). Common values for cv are integers (e.g., cv=5 for 5-fold CV).

  • estimator (Model/Estimator):

    • Effect: The estimator parameter specifies the machine learning model that is used within the Backward Elimination process to evaluate feature importance and performance.
    • Tuning: Choose a model that is appropriate for your task (regression or classification) and your data characteristics. You can experiment with different model types within Backward Elimination (e.g., LinearRegression, LogisticRegression, RandomForestRegressor, GradientBoostingClassifier, etc.). The choice of model can influence which features are selected as important.
    • Computational Cost Consideration: Be mindful of the computational cost of training and evaluating the chosen estimator repeatedly during Backward Elimination. For complex models or large datasets, Backward Elimination can become time-consuming.

  • floating=True/False (Floating Backward Selection - less common, more computationally expensive):

    • Effect: The floating parameter in mlxtend.SequentialFeatureSelector (and similar options in some advanced sequential feature selection methods) enables “floating” behavior. When floating=True, after each backward elimination step, the algorithm might perform “conditional forward steps” to re-introduce features that were previously removed if it improves performance. This can potentially lead to slightly better feature subsets but increases computational cost significantly.
    • Tuning: For standard Backward Elimination, floating=False (the default) is usually used. floating=True can be explored for potentially slightly better performance in some cases, but at a much higher computational cost. It’s less commonly used in practice due to the increased complexity and runtime.

Hyperparameter Tuning Process for Backward Elimination:

  1. Start with tuning k_features (number of features to select). Experiment with different values or a range of values (e.g., (1, p) range).
  2. Choose an appropriate scoring metric for your task (regression or classification).
  3. Use cross-validation (cv) for robust performance estimation.
  4. Evaluate Performance vs. k_features: Plot model performance (CV score) against k_features. Look for the “elbow” in the plot or examine CV scores for different subset sizes.
  5. Consider different estimator models. Try Backward Elimination with different models (e.g., linear model, tree-based model) and compare the selected feature subsets and model performances.
  6. Experiment with rotation (rarely tuned in practice): In most practical applications of Backward Elimination, the main focus is on tuning k_features and choosing a suitable estimator and scoring metric. floating=True is less commonly tuned due to its higher computational cost.

8. Accuracy Metrics: Measuring Model Performance After Feature Elimination

“Accuracy” in the context of Backward Feature Elimination is not a metric of the feature selection algorithm itself, but rather the performance of machine learning models built using the selected feature subsets. We evaluate how well the model performs after applying Backward Feature Elimination to reduce features.

Regression Metrics (if you are using Backward Elimination for Regression):

  • R-squared (Coefficient of Determination): (Explained in detail in previous blogs - Subset Selection, PLSR, Sammon Mapping). Measures the proportion of variance in the target variable explained by the model. Higher R-squared (closer to 1) is generally better.

    \[R^2 = 1 - \frac{RSS}{TSS} = 1 - \frac{\sum_{i=1}^{n} (y_i - \hat{y}_i)^2}{\sum_{i=1}^{n} (y_i - \bar{y})^2}\]

  • Adjusted R-squared: (Explained in Subset Selection Blog). Penalizes the addition of irrelevant features, useful for comparing models with different numbers of features.

    \(Adjusted\ R^2 = 1 - \frac{(1 - R^2)(n - 1)}{n - p - 1}\) (Here, (p) is the number of selected features in the subset).

  • Mean Squared Error (MSE): (Explained in previous blogs). Average squared difference between predicted and actual values. Lower MSE is better.

    \[MSE = \frac{1}{n} \sum_{i=1}^{n} (y_i - \hat{y}_i)^2\]

  • Root Mean Squared Error (RMSE): (Explained in previous blogs). Square root of MSE. In the same units as the target variable, often more interpretable than MSE. Lower RMSE is better.

    \[RMSE = \sqrt{MSE} = \sqrt{\frac{1}{n} \sum_{i=1}^{n} (y_i - \hat{y}_i)^2}\]

  • Mean Absolute Error (MAE): (Explained in previous blogs). Average absolute difference between predicted and actual values. Less sensitive to outliers than MSE or RMSE. Lower MAE is better.

    \[MAE = \frac{1}{n} \sum_{i=1}^{n} |y_i - \hat{y}_i|\]

Classification Metrics (if you are using Backward Elimination for Classification):

  • Accuracy: (Explained in Subset Selection Blog). Overall fraction of correctly classified instances.

    \[Accuracy = \frac{Number\ of\ Correct\ Predictions}{Total\ Number\ of\ Predictions}\]

  • Precision, Recall, F1-score, AUC-ROC: (Explained in Subset Selection Blog). Precision, Recall, F1-score, and AUC-ROC are common metrics for evaluating classification performance, especially for imbalanced datasets or when you want to focus on specific types of errors (false positives vs. false negatives).

Evaluating Backward Elimination in Practice:

  1. Split Data: Split your data into training and testing sets.
  2. Preprocess Data: Apply necessary preprocessing (scaling, encoding, missing value handling).
  3. Perform Backward Feature Elimination: Use SequentialFeatureSelector (or RFE or similar method) on the training set to select a feature subset. Tune the k_features hyperparameter (number of features to select) using cross-validation on the training data.
  4. Train Final Model with Selected Features: Train a final machine learning model (the same model type used in feature selection) using only the selected features from the entire training set.
  5. Evaluate on Test Set: Evaluate the performance of this final model on the held-out test set using your chosen evaluation metric(s). Compare the test set performance to the baseline model (trained with all features) and models trained with other feature subsets.

Key Goal of Evaluation:

  • Performance with Reduced Features: Check if the model built with the selected feature subset achieves comparable or even better performance on the test set compared to the model built using all original features. The aim is to simplify the model (reduce features) without significant performance loss, or ideally, with some performance improvement (by removing noise or irrelevant features).

9. Productionizing Backward Feature Elimination

“Productionizing” Backward Feature Elimination is about integrating the feature selection process and the resulting reduced feature set into a real-world machine learning pipeline for deployment.

Productionizing Steps for Backward Elimination:

  1. Offline Feature Selection and Feature List Saving:

    • Perform Backward Feature Elimination: Run Backward Feature Elimination on your training data to select the best feature subset according to your chosen criteria (performance, number of features, etc.). Determine the optimal number of features to retain.
    • Save Selected Feature Names (or Indices): Once you have identified the selected feature subset, save the list of feature names (or column indices) of the selected features. This list is crucial for your production pipeline. Save it to a file (e.g., text file, JSON, CSV, or using joblib.dump).

  2. Production Environment Setup: (Same as in previous blogs - choose cloud, on-premise, etc., set up software stack with scikit-learn, mlxtend, pandas, numpy).

  3. Data Ingestion and Preprocessing in Production:
    • Data Ingestion Pipeline: Set up your data ingestion pipeline to receive new data for prediction.
    • Preprocessing (Consistent with Training): Apply exactly the same preprocessing steps to the new data as you did during training and feature selection. This includes scaling, encoding, handling missing values. Load and use the same preprocessing objects (scalers, encoders) that you fitted on your training data.
    • Feature Subsetting (Crucial Step): After preprocessing, you need to perform feature subsetting. Use the saved list of selected feature names to extract only those features from your preprocessed new data. Discard the features that were not selected by Backward Elimination.

  4. Model Loading and Prediction with Selected Features:

    • Train Final Model (Offline - using selected features): Train your final machine learning model (of the same type used in feature selection) on the entire training dataset, but using only the selected features. Save this final trained model.
    • Load Final Model in Production: Load the saved final trained model in your production system.
    • Make Predictions: In production, after preprocessing and feature subsetting your new data, feed the selected features of the preprocessed data into your loaded final model to make predictions.

Code Snippet: Conceptual Production Pipeline with Backward Elimination (Python):

import joblib
import pandas as pd
import numpy as np

# --- Assume selected feature names were saved to 'selected_features_backward_selection.joblib' during offline feature selection ---
# --- Assume final trained model (trained on selected features) was saved to 'final_model_selected_features.joblib' ---
# --- Assume scaler (fitted on training data) was saved to 'scaler_fa.joblib' ---

SELECTED_FEATURES_FILE = 'selected_features_backward_selection.joblib'
FINAL_MODEL_FILE = 'final_model_selected_features.joblib'
SCALER_FILE = 'scaler_fa.joblib'

# Load selected feature names, final model, and scaler (once at application startup)
loaded_feature_names = joblib.load(SELECTED_FEATURES_FILE)
loaded_final_model = joblib.load(FINAL_MODEL_FILE)
loaded_scaler = joblib.load(SCALER_FILE)

def predict_with_selected_features_production(raw_data_point_dict): # raw_data_point_dict is new data in dictionary format
    """Makes a prediction using final model trained with backward-selected features."""
    # 1. Convert raw input (dict) to DataFrame
    input_df = pd.DataFrame([raw_data_point_dict]) # Assume dict keys are feature names
    # 2. Preprocess the input data using the *loaded* scaler (same scaler fitted on training data)
    input_scaled = loaded_scaler.transform(input_df)
    input_scaled_df = pd.DataFrame(input_scaled, columns=input_df.columns) # Keep column names
    # 3. Feature Subsetting: Select only the backward-eliminated features
    input_selected_features = input_scaled_df[loaded_feature_names]
    # 4. Make prediction using the *loaded final model* (trained only on selected features)
    prediction = loaded_final_model.predict(input_selected_features)
    return prediction[0] # Return single prediction value

# Example usage in production:
new_data_point = {'feature_1': 0.7, 'feature_2': 0.9, 'feature_3': 0.2, 'feature_4': 0.5, 'feature_5': 0.1} # New data point input
predicted_value = predict_with_selected_features_production(new_data_point)
print("Prediction using model with selected features:", predicted_value)

Deployment Environments (Same as in previous blogs):

  • Cloud Platforms (AWS, Google Cloud, Azure): Cloud services are well-suited for scalable ML pipelines, including feature selection and model serving.
  • On-Premise Servers: Deploy on your servers if needed.
  • Local Machines/Edge Devices: For smaller applications or edge computing.

Key Production Considerations:

  • Preprocessing Consistency (Crucial): Ensure absolutely consistent preprocessing steps between training and production. Use the same preprocessing code, and loaded preprocessing objects.
  • Feature Subsetting in Production: Implement the feature subsetting step correctly in your production pipeline using the saved list of selected features.
  • Model Versioning: Manage versions of your selected feature lists and trained models for reproducibility and tracking changes.
  • Performance Monitoring: Monitor the performance of your model in production over time to detect any data drift or performance degradation. Periodically re-run feature selection and retrain models if needed as data evolves.

10. Conclusion: Backward Feature Elimination – Streamlining Models for Insight and Efficiency

Backward Feature Elimination is a valuable feature selection technique for simplifying machine learning models, potentially improving performance by removing noise or irrelevant features, and enhancing model interpretability. It’s a widely applicable method across various domains.

Real-World Problem Solving with Backward Feature Elimination:

  • Simplified and Interpretable Models: Creating models that are easier to understand, explain, and communicate due to reduced feature sets. This is crucial in domains like healthcare, finance, or policy making where model transparency is important.
  • Improved Model Generalization: Removing irrelevant features can reduce overfitting and improve model performance on unseen data, leading to more robust and reliable predictions.
  • Faster and More Efficient Models: Reducing the number of features leads to faster model training and prediction times, which can be important for large datasets and real-time applications.
  • Reduced Data Collection Costs: Identifying truly important features can inform data collection strategies, allowing organizations to focus on collecting only the most essential data, potentially reducing costs.

Where Backward Elimination is Still Relevant:

  • Feature Selection for Interpretability and Simplicity: When model simplicity and interpretability are primary goals, Backward Elimination is a useful approach to systematically reduce feature complexity while aiming to maintain good performance.
  • Preprocessing Step for Large Datasets: Reducing the feature space using Backward Elimination (or other feature selection methods) can be a helpful preprocessing step for very large datasets to reduce computational burden for downstream modeling tasks.
  • Hybrid Feature Selection Approaches: Backward Elimination can be combined with other feature selection techniques (e.g., used in conjunction with Forward Selection or regularization methods) to create hybrid approaches that leverage the strengths of different methods.

Optimized and Newer Algorithms:

While Backward Elimination is effective, several related and alternative feature selection methods are available:

  • Forward Selection: (Covered in a separate blog post in this series). Forward Selection starts with no features and iteratively adds the most important features, which can be more computationally efficient than Backward Elimination in some cases, especially if you expect to select a relatively small number of features from a very large initial set.
  • Recursive Feature Elimination (RFE - implemented in sklearn): RFE is conceptually similar to Backward Elimination. It uses a model to rank features by importance (e.g., using coefficient magnitudes or tree-based feature importances) and recursively eliminates the least important features based on the model’s ranking. RFE in scikit-learn is a specific implementation of this type of recursive feature elimination.
  • Regularization-Based Feature Selection (Lasso, Elastic Net): Lasso (L1 regularization) and Elastic Net regression methods perform implicit feature selection by shrinking coefficients of less important features towards zero, and potentially setting some coefficients exactly to zero, effectively removing those features from the model. Regularization methods are computationally efficient and can be very effective for feature selection, especially in linear models.
  • Tree-Based Feature Importance (Random Forest, Gradient Boosting): Tree-based models inherently provide feature importance scores. You can use these scores to rank features and select a subset based on importance, similar to feature selection based on Backward Elimination. SelectFromModel in scikit-learn can be used for feature selection based on tree-based feature importances.

Choosing Between Backward Elimination and Alternatives:

  • For Systematic Feature Reduction (Starting from All Features): Backward Elimination is a systematic approach for starting with a full set of features and iteratively removing the least important ones to simplify the model. It’s often a good choice when you want to explicitly control the number of features and see how performance changes as you remove features.
  • For Computational Efficiency (especially with large feature sets and small target subset size): Forward Selection can be more computationally efficient than Backward Elimination if you expect to select a relatively small number of features from a very large initial set.
  • For Implicit Feature Selection (in Linear Models): Regularization methods (Lasso, Elastic Net) offer computationally efficient and automatic feature selection within the model training process itself, especially for linear regression and related models.
  • For Tree-Based Models: Tree-based feature importance provides a fast and model-intrinsic way to rank and select features specifically for tree-based models.

Final Thought: Backward Feature Elimination is a valuable tool in the feature selection toolkit, particularly useful for systematically simplifying models, enhancing interpretability, and potentially improving generalization performance. While other feature selection methods exist, Backward Elimination provides a clear and intuitive process for “trimming the fat” from your data and focusing on the essential features for building effective and efficient machine learning models.

11. References and Resources

Here are some references and resources to explore Backward Feature Elimination and related feature selection techniques in more detail:

  1. “Feature Selection for Machine Learning” by Isabelle Guyon and André Elisseeff: (Book Link - Search Online) - A comprehensive book covering various feature selection methods, including detailed discussions of sequential selection algorithms like Forward and Backward Selection, and various feature ranking and filtering methods.

  2. scikit-learn Documentation on Feature Selection:
    • scikit-learn Feature Selection - Official scikit-learn documentation for its feature_selection module. Provides details on RFE (Recursive Feature Elimination) and other feature selection methods available in scikit-learn with code examples.
  3. mlxtend Library Documentation:
    • mlxtend Feature Selection - Documentation for the mlxtend library’s feature selection module, particularly for SequentialFeatureSelector (SFS), which implements Forward, Backward, and other sequential selection methods.

  4. “An Introduction to Statistical Learning” by Gareth James, Daniela Witten, Trevor Hastie, and Robert Tibshirani: (Book Website - Free PDF available) - A widely used textbook covering statistical learning methods, including chapters on feature selection and model selection (Chapter 6 and Chapter 7). Chapter 6 addresses linear model selection and regularization, which is related to feature selection concepts.

  5. “The Elements of Statistical Learning” by Trevor Hastie, Robert Tibshirani, and Jerome Friedman: (Book Website - Free PDF available) - A more advanced and comprehensive textbook on statistical learning, covering feature selection and model selection in detail.

  6. Online Tutorials and Blog Posts on Backward Feature Elimination: Search online for tutorials and blog posts on “Backward Feature Elimination tutorial”, “Sequential Feature Selection Python”, “feature selection methods”. Websites like Towards Data Science, Machine Learning Mastery, and various data science blogs often have articles explaining Backward Elimination and other feature selection techniques with code examples in Python.

These references should provide a solid foundation for understanding Backward Feature Elimination and exploring its applications in your machine learning projects. Experiment with Backward Elimination on your own datasets to see how it can help you simplify models and potentially improve performance!