# Feature Selection

> More features is not better. The right features is better.

**Type:** Build
**Language:** Python
**Prerequisites:** Phase 2, Lessons 01-09, 08 (feature engineering)
**Time:** ~75 minutes

## Learning Objectives

- Implement filter methods (variance threshold, mutual information, chi-squared) and wrapper methods (RFE, forward selection) from scratch
- Explain why mutual information captures nonlinear feature-target relationships that correlation misses
- Compare L1 regularization (embedded selection) with RFE (wrapper selection) and evaluate their computational tradeoffs
- Build a feature selection pipeline that combines multiple methods and demonstrate improved generalization on held-out data

## The Problem

You have 500 features. Your model trains slowly, overfits constantly, and nobody can explain what it learned. You add more features hoping to improve performance. It gets worse.

This is the curse of dimensionality in action. As the number of features grows, the volume of the feature space explodes. Data points become sparse. Distances between points converge. The model needs exponentially more data to find real patterns. Noise features drown out signal features. Overfitting becomes the default.

Feature selection is the antidote. Strip away the noise. Remove the redundancy. Keep the features that carry actual information about the target. The result: faster training, better generalization, and models you can actually explain.

The goal is not to use all available information. It is to use the right information.

## The Concept

### Three Categories of Feature Selection

Every feature selection method falls into one of three categories:

```mermaid
flowchart TD
    A[Feature Selection Methods] --> B[Filter Methods]
    A --> C[Wrapper Methods]
    A --> D[Embedded Methods]

    B --> B1["Variance Threshold"]
    B --> B2["Mutual Information"]
    B --> B3["Chi-squared Test"]
    B --> B4["Correlation Filtering"]

    C --> C1["Recursive Feature Elimination"]
    C --> C2["Forward Selection"]
    C --> C3["Backward Elimination"]

    D --> D1["L1 / Lasso Regularization"]
    D --> D2["Tree-based Importance"]
    D --> D3["Elastic Net"]
```

**Filter methods** score each feature independently using a statistical measure. They do not use a model. Fast, but they miss feature interactions.

**Wrapper methods** train a model to evaluate feature subsets. They use model performance as the score. Better results, but expensive because they retrain the model many times.

**Embedded methods** select features as part of model training. L1 regularization drives weights to zero. Decision trees split on the most useful features. Selection happens during fitting, not as a separate step.

### Variance Threshold

The simplest filter. If a feature barely varies across samples, it carries almost no information.

Consider a feature that is 0.0 for 999 out of 1000 samples. Its variance is near zero. No model can use it to distinguish between classes. Remove it.

```
variance(x) = mean((x - mean(x))^2)
```

Set a threshold (e.g., 0.01). Drop every feature with variance below it. This removes constant or near-constant features without looking at the target variable at all.

When to use it: as a preprocessing step before other methods. It catches obviously useless features at near-zero cost.

Limitation: a feature can have high variance and still be pure noise. Variance threshold is necessary but not sufficient.

### Mutual Information

Mutual information measures how much knowing the value of feature X reduces uncertainty about target Y.

```
I(X; Y) = sum_x sum_y p(x, y) * log(p(x, y) / (p(x) * p(y)))
```

If X and Y are independent, p(x, y) = p(x) * p(y), so the log term is zero and I(X; Y) = 0. The more X tells you about Y, the higher the mutual information.

Key advantage over correlation: mutual information captures nonlinear relationships. A feature might have zero correlation with the target but high mutual information because the relationship is quadratic or periodic.

For continuous features, discretize into bins first (histogram-based estimation). The number of bins affects the estimate -- too few bins lose information, too many bins add noise. A common choice: sqrt(n) bins or Sturges' rule (1 + log2(n)).

```mermaid
flowchart LR
    A[Feature X] --> B[Discretize into Bins]
    B --> C["Compute Joint Distribution p(x,y)"]
    C --> D["Compute MI = sum p(x,y) * log(p(x,y) / p(x)p(y))"]
    D --> E["Rank Features by MI Score"]
    E --> F[Select Top K]
```

### Recursive Feature Elimination (RFE)

RFE is a wrapper method. It uses a model's own feature importance to iteratively prune:

1. Train the model with all features
2. Rank features by importance (coefficients for linear models, impurity reduction for trees)
3. Remove the least important feature(s)
4. Repeat until the desired number of features remains

```mermaid
flowchart TD
    A["Start: All N Features"] --> B["Train Model"]
    B --> C["Rank Feature Importances"]
    C --> D["Remove Least Important"]
    D --> E{"Features == Target Count?"}
    E -->|No| B
    E -->|Yes| F["Return Selected Features"]
```

RFE considers feature interactions because the model sees all remaining features together. Removing one feature changes the importance of others. This makes it more thorough than filter methods.

The cost: you train the model N - target times. With 500 features and a target of 10, that is 490 training runs. For expensive models, this is slow. You can speed it up by removing multiple features per step (e.g., remove the bottom 10% each round).

### L1 (Lasso) Regularization

L1 regularization adds the absolute value of weights to the loss function:

```
loss = prediction_error + alpha * sum(|w_i|)
```

The alpha parameter controls how aggressively features are pruned. Higher alpha means more weights go to exactly zero.

Why exactly zero? The L1 penalty creates a diamond-shaped constraint region in weight space. The optimal solution tends to land at a corner of this diamond, where one or more weights are zero. L2 regularization (ridge) creates a circular constraint where weights shrink but rarely hit zero.

This is embedded feature selection: the model learns during training which features to ignore. Features with zero weight are effectively removed.

Advantages: single training run, handles correlated features (picks one and zeros the others), built into most linear model implementations.

Limitation: only works for linear models. Cannot capture nonlinear feature importance.

### Tree-Based Feature Importance

Decision trees and their ensembles (random forests, gradient boosting) naturally rank features. Every split reduces impurity (Gini or entropy for classification, variance for regression). Features that produce larger impurity reductions are more important.

For a random forest with T trees:

```
importance(feature_j) = (1/T) * sum over all trees of
    sum over all nodes splitting on feature_j of
        (n_samples * impurity_decrease)
```

This gives a normalized importance score for each feature. It handles nonlinear relationships and feature interactions automatically.

Caution: tree-based importance is biased toward features with many unique values (high cardinality). A random ID column will appear important because it perfectly splits every sample. Use permutation importance as a sanity check.

### Permutation Importance

A model-agnostic method:

1. Train the model and record baseline performance on validation data
2. For each feature: shuffle its values randomly, measure the drop in performance
3. The bigger the drop, the more important the feature

If shuffling a feature does not hurt performance, the model does not depend on it. If performance collapses, that feature is critical.

Permutation importance avoids the cardinality bias of tree-based importance. But it is slow: one full evaluation per feature, repeated multiple times for stability.

### Comparison Table

| Method | Type | Speed | Nonlinear | Feature Interactions |
|--------|------|-------|-----------|---------------------|
| Variance threshold | Filter | Very fast | No | No |
| Mutual information | Filter | Fast | Yes | No |
| Correlation filter | Filter | Fast | No | No |
| RFE | Wrapper | Slow | Depends on model | Yes |
| L1 / Lasso | Embedded | Fast | No (linear) | No |
| Tree importance | Embedded | Medium | Yes | Yes |
| Permutation importance | Model-agnostic | Slow | Yes | Yes |

### Decision Flowchart

```mermaid
flowchart TD
    A[Start: Feature Selection] --> B{How many features?}
    B -->|"< 50"| C["Start with variance threshold + mutual information"]
    B -->|"50-500"| D["Variance threshold, then L1 or tree importance"]
    B -->|"> 500"| E["Variance threshold, then mutual info filter, then RFE on survivors"]

    C --> F{Using linear model?}
    D --> F
    E --> F

    F -->|Yes| G["L1 regularization for final selection"]
    F -->|No - trees| H["Tree importance + permutation importance"]
    F -->|No - other| I["RFE with your model"]

    G --> J[Validate: compare selected vs all features]
    H --> J
    I --> J

    J --> K{Performance improved?}
    K -->|Yes| L["Ship with selected features"]
    K -->|No| M["Try different method or keep all features"]
```

```figure
f3-feature-prune
```

## Build It

### Step 1: Generate synthetic data with known feature structure

```python
import numpy as np


def make_feature_selection_data(n_samples=500, seed=42):
    rng = np.random.RandomState(seed)

    x1 = rng.randn(n_samples)
    x2 = rng.randn(n_samples)
    x3 = rng.randn(n_samples)
    x4 = x1 + 0.1 * rng.randn(n_samples)
    x5 = x2 + 0.1 * rng.randn(n_samples)

    informative = np.column_stack([x1, x2, x3, x4, x5])

    correlated = np.column_stack([
        x1 * 0.9 + 0.1 * rng.randn(n_samples),
        x2 * 0.8 + 0.2 * rng.randn(n_samples),
        x3 * 0.7 + 0.3 * rng.randn(n_samples),
        x1 * 0.5 + x2 * 0.5 + 0.1 * rng.randn(n_samples),
        x2 * 0.6 + x3 * 0.4 + 0.1 * rng.randn(n_samples),
    ])

    noise = rng.randn(n_samples, 10) * 0.5

    X = np.hstack([informative, correlated, noise])
    y = (2 * x1 - 1.5 * x2 + x3 + 0.5 * rng.randn(n_samples) > 0).astype(int)

    feature_names = (
        [f"info_{i}" for i in range(5)]
        + [f"corr_{i}" for i in range(5)]
        + [f"noise_{i}" for i in range(10)]
    )

    return X, y, feature_names
```

We know the ground truth: features 0-4 are informative (plus 3 and 4 are correlated copies of 0 and 1), features 5-9 are correlated with informative features, features 10-19 are pure noise. A good selection method should rank 0-4 highest and 10-19 lowest.

### Step 2: Variance threshold

```python
def variance_threshold(X, threshold=0.01):
    variances = np.var(X, axis=0)
    mask = variances > threshold
    return mask, variances
```

### Step 3: Mutual information (discrete)

```python
def discretize(x, n_bins=10):
    min_val, max_val = x.min(), x.max()
    if max_val == min_val:
        return np.zeros_like(x, dtype=int)
    bin_edges = np.linspace(min_val, max_val, n_bins + 1)
    binned = np.digitize(x, bin_edges[1:-1])
    return binned


def mutual_information(X, y, n_bins=10):
    n_samples, n_features = X.shape
    mi_scores = np.zeros(n_features)

    y_vals, y_counts = np.unique(y, return_counts=True)
    p_y = y_counts / n_samples

    for f in range(n_features):
        x_binned = discretize(X[:, f], n_bins)
        x_vals, x_counts = np.unique(x_binned, return_counts=True)
        p_x = dict(zip(x_vals, x_counts / n_samples))

        mi = 0.0
        for xv in x_vals:
            for yi, yv in enumerate(y_vals):
                joint_mask = (x_binned == xv) & (y == yv)
                p_xy = np.sum(joint_mask) / n_samples
                if p_xy > 0:
                    mi += p_xy * np.log(p_xy / (p_x[xv] * p_y[yi]))
        mi_scores[f] = mi

    return mi_scores
```

### Step 4: Recursive Feature Elimination

```python
def simple_logistic_importance(X, y, lr=0.1, epochs=100):
    n_samples, n_features = X.shape
    w = np.zeros(n_features)
    b = 0.0

    for _ in range(epochs):
        z = X @ w + b
        pred = 1.0 / (1.0 + np.exp(-np.clip(z, -500, 500)))
        error = pred - y
        w -= lr * (X.T @ error) / n_samples
        b -= lr * np.mean(error)

    return w, b


def rfe(X, y, n_features_to_select=5, lr=0.1, epochs=100):
    n_total = X.shape[1]
    remaining = list(range(n_total))
    rankings = np.ones(n_total, dtype=int)
    rank = n_total

    while len(remaining) > n_features_to_select:
        X_subset = X[:, remaining]
        w, _ = simple_logistic_importance(X_subset, y, lr, epochs)
        importances = np.abs(w)

        least_idx = np.argmin(importances)
        original_idx = remaining[least_idx]
        rankings[original_idx] = rank
        rank -= 1
        remaining.pop(least_idx)

    for idx in remaining:
        rankings[idx] = 1

    selected_mask = rankings == 1
    return selected_mask, rankings
```

### Step 5: L1 feature selection

```python
def soft_threshold(w, alpha):
    return np.sign(w) * np.maximum(np.abs(w) - alpha, 0)


def l1_feature_selection(X, y, alpha=0.1, lr=0.01, epochs=500):
    n_samples, n_features = X.shape
    w = np.zeros(n_features)
    b = 0.0

    for _ in range(epochs):
        z = X @ w + b
        pred = 1.0 / (1.0 + np.exp(-np.clip(z, -500, 500)))
        error = pred - y

        gradient_w = (X.T @ error) / n_samples
        gradient_b = np.mean(error)

        w -= lr * gradient_w
        w = soft_threshold(w, lr * alpha)
        b -= lr * gradient_b

    selected_mask = np.abs(w) > 1e-6
    return selected_mask, w
```

### Step 6: Tree-based importance (simple decision tree)

```python
def gini_impurity(y):
    if len(y) == 0:
        return 0.0
    classes, counts = np.unique(y, return_counts=True)
    probs = counts / len(y)
    return 1.0 - np.sum(probs ** 2)


def best_split(X, y, feature_idx):
    values = np.unique(X[:, feature_idx])
    if len(values) <= 1:
        return None, -1.0

    best_threshold = None
    best_gain = -1.0
    parent_gini = gini_impurity(y)
    n = len(y)

    for i in range(len(values) - 1):
        threshold = (values[i] + values[i + 1]) / 2.0
        left_mask = X[:, feature_idx] <= threshold
        right_mask = ~left_mask

        n_left = np.sum(left_mask)
        n_right = np.sum(right_mask)

        if n_left == 0 or n_right == 0:
            continue

        gain = parent_gini - (n_left / n) * gini_impurity(y[left_mask]) - (n_right / n) * gini_impurity(y[right_mask])

        if gain > best_gain:
            best_gain = gain
            best_threshold = threshold

    return best_threshold, best_gain


def tree_importance(X, y, n_trees=50, max_depth=5, seed=42):
    rng = np.random.RandomState(seed)
    n_samples, n_features = X.shape
    importances = np.zeros(n_features)

    for _ in range(n_trees):
        sample_idx = rng.choice(n_samples, size=n_samples, replace=True)
        feature_subset = rng.choice(n_features, size=max(1, int(np.sqrt(n_features))), replace=False)

        X_boot = X[sample_idx]
        y_boot = y[sample_idx]

        tree_imp = _build_tree_importance(X_boot, y_boot, feature_subset, max_depth)
        importances += tree_imp

    total = importances.sum()
    if total > 0:
        importances /= total

    return importances


def _build_tree_importance(X, y, feature_subset, max_depth, depth=0):
    n_features = X.shape[1]
    importances = np.zeros(n_features)

    if depth >= max_depth or len(np.unique(y)) <= 1 or len(y) < 4:
        return importances

    best_feature = None
    best_threshold = None
    best_gain = -1.0

    for f in feature_subset:
        threshold, gain = best_split(X, y, f)
        if gain > best_gain:
            best_gain = gain
            best_feature = f
            best_threshold = threshold

    if best_feature is None or best_gain <= 0:
        return importances

    importances[best_feature] += best_gain * len(y)

    left_mask = X[:, best_feature] <= best_threshold
    right_mask = ~left_mask

    importances += _build_tree_importance(X[left_mask], y[left_mask], feature_subset, max_depth, depth + 1)
    importances += _build_tree_importance(X[right_mask], y[right_mask], feature_subset, max_depth, depth + 1)

    return importances
```

### Step 7: Run all methods and compare

The code file runs all five methods on the same synthetic dataset and prints a comparison table showing which features each method selects.

## Use It

With scikit-learn, feature selection is built into the pipeline:

```python
from sklearn.feature_selection import (
    VarianceThreshold,
    mutual_info_classif,
    RFE,
    SelectFromModel,
)
from sklearn.linear_model import Lasso, LogisticRegression
from sklearn.ensemble import RandomForestClassifier

vt = VarianceThreshold(threshold=0.01)
X_filtered = vt.fit_transform(X)

mi_scores = mutual_info_classif(X, y)
top_k = np.argsort(mi_scores)[-10:]

rfe_selector = RFE(LogisticRegression(), n_features_to_select=10)
rfe_selector.fit(X, y)
X_rfe = rfe_selector.transform(X)

lasso_selector = SelectFromModel(Lasso(alpha=0.01))
lasso_selector.fit(X, y)
X_lasso = lasso_selector.transform(X)

rf = RandomForestClassifier(n_estimators=100)
rf.fit(X, y)
importances = rf.feature_importances_
```

The from-scratch implementations show exactly what happens inside each method. Variance threshold is just computing `var(X, axis=0)` and applying a mask. Mutual information is counting joint and marginal frequencies in a contingency table. RFE is a loop that trains, ranks, and prunes. L1 is gradient descent with a soft-thresholding step. Tree importance accumulates impurity reductions across splits. No magic -- just statistics and loops.

The sklearn versions add robustness (e.g., mutual_info_classif uses k-NN density estimation instead of binning), speed (C implementations), and pipeline integration.

## Ship It

This lesson produces:
- `outputs/skill-feature-selector.md` -- a quick reference decision tree for choosing the right feature selection method

## Exercises

1. **Forward selection**: implement the opposite of RFE. Start with zero features. At each step, add the feature that improves model performance the most. Stop when adding features no longer helps. Compare the selected features against RFE results. Which is faster? Which gives better results?

2. **Stability selection**: run L1 feature selection 50 times, each time on a random 80% subsample of the data, with slightly different alpha values. Count how often each feature is selected. Features selected in > 80% of runs are "stable." Compare stable features against single-run L1 selection. Which is more reliable?

3. **Multicollinearity detection**: compute the correlation matrix for all features. Implement a function that, given a correlation threshold (e.g., 0.9), removes one feature from each highly-correlated pair (keeping the one with higher mutual information with the target). Test on the synthetic dataset and verify it removes the redundant correlated features.

4. **Feature selection pipeline**: chain variance threshold, mutual information filter, and RFE into a single pipeline. First remove near-zero-variance features, then keep the top 50% by mutual information, then run RFE on the survivors. Compare this pipeline against running RFE alone on all features. Is the pipeline faster? Is it equally accurate?

5. **Permutation importance from scratch**: implement permutation importance. For each feature, shuffle its values 10 times, measure the average drop in F1 score. Compare the ranking against tree-based importance. Find cases where they disagree and explain why (hint: correlated features).

## Key Terms

| Term | What people say | What it actually means |
|------|----------------|----------------------|
| Filter method | "Score features independently" | A feature selection approach that ranks features using a statistical measure without training a model, evaluating each feature in isolation |
| Wrapper method | "Use the model to pick features" | A feature selection approach that evaluates feature subsets by training a model and using its performance as the selection criterion |
| Embedded method | "The model selects features during training" | Feature selection that happens as part of model fitting, such as L1 regularization driving weights to zero |
| Mutual information | "How much one variable tells you about another" | A measure of the reduction in uncertainty about Y given knowledge of X, capturing both linear and nonlinear dependencies |
| Recursive Feature Elimination | "Train, rank, prune, repeat" | An iterative wrapper method that trains a model, removes the least important feature(s), and repeats until a target count is reached |
| L1 / Lasso regularization | "Penalty that kills features" | Adding the sum of absolute weight values to the loss function, which drives unimportant feature weights to exactly zero |
| Variance threshold | "Remove constant features" | Dropping features whose variance across samples falls below a specified threshold, filtering out features that carry no information |
| Feature importance | "Which features matter most" | A score indicating how much each feature contributes to model predictions, computed from split gains (trees) or coefficient magnitudes (linear) |
| Permutation importance | "Shuffle and measure the damage" | Evaluating feature importance by randomly shuffling each feature's values and measuring the resulting drop in model performance |
| Curse of dimensionality | "Too many features, not enough data" | The phenomenon where adding features increases the volume of the feature space exponentially, making data sparse and distances meaningless |

## Further Reading

- [An Introduction to Variable and Feature Selection (Guyon & Elisseeff, 2003)](https://jmlr.org/papers/v3/guyon03a.html) -- the foundational survey on feature selection methods, still widely referenced
- [scikit-learn Feature Selection Guide](https://scikit-learn.org/stable/modules/feature_selection.html) -- practical reference for filter, wrapper, and embedded methods with code examples
- [Stability Selection (Meinshausen & Buhlmann, 2010)](https://arxiv.org/abs/0809.2932) -- combines subsampling with feature selection for robust, reproducible results
- [Beware Default Random Forest Importances (Strobl et al., 2007)](https://bmcbioinformatics.biomedcentral.com/articles/10.1186/1471-2105-8-25) -- demonstrates the cardinality bias in tree-based importance and proposes conditional importance as an alternative
