Rule Fit.en

Rule Fit.en

Last updated 2026-02-16 Read time 3 min
Summary
  • RuleFit extracts decision rules from tree ensembles and combines them with original features in a sparse linear model.
  • L1 regularization selects informative rules and features, improving interpretability without giving up nonlinear structure.
  • Rule depth, number of generated rules, and regularization strength strongly affect generalization and sparsity.

Intuition #

RuleFit turns tree paths into human-readable if-then indicators, then learns linear weights on those indicators. You get nonlinear interactions from trees and coefficient-level interpretability from sparse linear modeling.

Detailed Explanation #

1. Idea (with formulas) #

  1. Extract rules: each path to a leaf becomes a binary feature (r_j(x) \in {0,1}).
  2. Add scaled linear terms (z_k(x)) for continuous features.
  3. L1-regularized linear fit:
$$ \hat{y}(x) = \beta_0 + \sum_j \beta_j r_j(x) + \sum_k \gamma_k z_k(x) $$

L1 promotes sparsity so only influential rules/terms remain.

2. Dataset (OpenML: house_sales) #

King County housing prices (OpenML data_id=42092). Numeric columns only for clarity.

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import japanize_matplotlib
import matplotlib.pyplot as plt
import numpy as np
import pandas as pd
from sklearn.datasets import fetch_openml
from sklearn.model_selection import train_test_split
from sklearn.metrics import mean_squared_error, mean_absolute_error, r2_score

dataset = fetch_openml(data_id=42092, as_frame=True)
X = dataset.data.select_dtypes("number").copy()
y = dataset.target.astype(float)

X_train, X_test, y_train, y_test = train_test_split(
    X, y, test_size=0.2, random_state=42
)

3. Fit RuleFit #

Python implementation: christophM/rulefit

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from rulefit import RuleFit
import warnings
warnings.simplefilter("ignore")

rf = RuleFit(max_rules=200, tree_random_state=27)
rf.fit(X_train.values, y_train.values, feature_names=list(X_train.columns))

pred_tr = rf.predict(X_train.values)
pred_te = rf.predict(X_test.values)

def report(y_true, y_pred, name):
    rmse = mean_squared_error(y_true, y_pred, squared=False)
    mae  = mean_absolute_error(y_true, y_pred)
    r2   = r2_score(y_true, y_pred)
    print(f"{name:8s}  RMSE={rmse:,.0f}  MAE={mae:,.0f}  R2={r2:.3f}")

report(y_train, pred_tr, "Train")
report(y_test,  pred_te, "Test")

4. Inspect top rules #

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rules = rf.get_rules()
rules = rules[rules.coef != 0].sort_values(by="importance", ascending=False)
rules.head(10)
  • rule: if-then condition (type=linear denotes a linear term)
  • coef: regression coefficient (target units)
  • support: fraction of samples that satisfy the rule
  • importance: scaled score combining coefficient magnitude and support

5. Validate via visualization #

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plt.figure(figsize=(6, 5))
plt.scatter(X_train["sqft_above"], y_train, s=10, alpha=0.5)
plt.xlabel("sqft_above")
plt.ylabel("price")
plt.title("Relationship between sqft_above and price")
plt.grid(alpha=0.3)
plt.show()

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rule_mask = X["sqft_living"].le(3935.0) & X["lat"].le(47.5315)

applicable_data = np.log(y[rule_mask])
not_applicable  = np.log(y[~rule_mask])

plt.figure(figsize=(8, 5))
plt.boxplot([applicable_data, not_applicable],
            labels=["Rule satisfied", "Rule not satisfied"])
plt.ylabel("log(price)")
plt.title("Price distribution by rule satisfaction")
plt.grid(alpha=0.3)
plt.show()

6. Practical tips #

  • Handle outliers (Winsorization, clipping) for stable rules.
  • Clean categorical levels and encode only after grouping rare categories.
  • Transform skewed targets (log(y) or Box-Cox) if necessary.
  • Select rule counts/depths that stakeholders can read; cross-validate to pick limits.
  • Summarize the top rules in plain language for business reports.

7. References #

  • Friedman, J. H., & Popescu, B. E. (2008). Predictive Learning via Rule Ensembles. The Annals of Applied Statistics, 2(3), 916–954.
  • Christoph Molnar. (2020). Interpretable Machine Learning. https://christophm.github.io/interpretable-ml-book/rulefit.html