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SR4-Fit: A Unified Interpretable Rule-Based Machine Learning Framework for Informative and Trustworthy Decision-Making

Shyam Sundar Murali Krishnan, Dean Frederick Hougen

arXiv:2609.34019v1cs.LGcs.AI

TL;DR

Black-box models can predict well but are difficult to interpret, while existing rule-based approaches can lack stability or predictive strength. SR4-Fit combines tree-derived rules with sparse optimization for classification and regression. Across four experimental settings, it Pareto-dominates RuleFit and Decision Tree, while trading modest predictivity for greater understandability than Random Forest.

  • Problem

    Black-box election models can be accurate but obscure which demographic patterns drive predictions, while rule-based methods can have limited stability or predictive strength.

    Source: §1 · pp. 2–3

  • Method

    SR4-Fit combines rules extracted from decision-tree ensembles with sparse linear modeling for classification and regression.

    Source: §1 · p. 3, §3 · p. 5

  • Results

    Across election classification, election regression, public classification, and public regression, SR4-Fit Pareto-dominates RuleFit and Decision Tree, while trading modest predictivity for greater rule understandability than Random Forest.

    Source: §7 · p. 25

  • Takeaways & Limitations

    The U.S. House election analysis illustrates that interpretable rules can reveal demographic connections to electoral outcomes, while benchmark results support applicability across multiple datasets.

    Source: §7 · p. 25

    Prior party vote percentages can change rapidly between elections and may reflect candidate availability as well as party loyalty.

    Source: §4.1.1 · p. 8

Abstract

from arXiv · show

In many high-stakes applications, machine learning is dominated by black-box models that require post hoc explanations to justify their predictions. These explanations are often unreliable because they do not reflect the model's actual computations, limiting accountability and trust. A natural alternative is to use models that are interpretable by design. However, existing rule-based approaches, such as RuleFit and decision trees, while transparent, often lack stability and predictive strength, reinforcing a perceived trade-off between traditional performance measures and model understandability. To address this, we propose Sparse Relaxed Regularized Regression Rule-Fit (SR4-Fit), an intrinsically interpretable algorithm for both classification and regression that produces compact and stable rule sets without sacrificing performance. Using demographic data from the U.S. Census Bureau's American Community Survey, SR4-Fit predicts U.S. House election outcomes with high accuracy and interpretability while uncovering demographic interactions missed by black-box models. We further validate SR4-Fit across fourteen benchmark datasets (six classification and eight regression), where it outperforms existing rule-based methods, including RuleFit and decision trees in terms of accuracy, stability, and compactness while remaining competitive with black-box models in predictivity. These results demonstrate that interpretability and predictive reliability need not be mutually exclusive, offering a practical and transparent alternative for high-stakes decision-making.

1. Introduction

Black-box models can predict congressional elections accurately, but their opaque computations and the instability or limited predictive strength of rule-based alternatives leave a continuing challenge: balancing predictivity with interpretability. SR4-Fit extends RuleFit with sparse optimization to produce compact, stable rules for classification and regression, and is evaluated on U.S. House elections and fourteen benchmark datasets.

1. Introduction

  • In electoral forecasting, demographic and structural factors relate to voting patterns, district security, and representation, making transparent explanations relevant to accountability.The passage identifies race, education, income, and age as characteristics correlated with voting patterns and party preference.Source: §1 · p. 2
  • Post-hoc explanations such as SHAP and LIME can diverge from a model’s actual computations, so they do not fully resolve opacity.They provide approximate insights such as feature importance or local decision boundaries.Source: §1 · p. 3
  • Rule-based methods can lose predictive depth, address only one task type, or produce overlapping and run-varying rules that complicate interpretation and trust.RuleFit can struggle with stability, consistency, and predictivity, particularly on high-dimensional or noisy data.Source: §1 · p. 3
  • SR4-Fit extends RuleFit with SR3 sparse optimization to produce compact, stable, human-readable rule sets for binary and multiclass classification and regression.The method is designed to maintain competitive predictive accuracy.Source: §1 · p. 3
  • Across fourteen benchmarks, SR4-Fit performs competitively with black-box and interpretable methods while election rules highlight demographic interactions, including education, racial composition, and age distribution.The benchmark suite includes six classification and eight regression datasets.Source: §1 · p. 4

2. Contributions

The paper presents SR4-Fit as a unified rule-based approach to classification and regression, alongside independent measures of rule understandability and combined interpretability-predictivity. It evaluates the approach on election data and fourteen public benchmarks.

2. Contributions

  • SR4-Fit is proposed as an intrinsically interpretable algorithm producing compact, stable, human-readable rules for classification and regression.The contribution contrasts its scope with limitations attributed to RuleFit and decision trees.Source: §2 · p. 4
  • Rule Understandability Score combines stability, compactness, and complexity without predictive performance, enabling Pareto analysis against predictivity.This allows model selection without fixing a weighting between understandability and predictivity.Source: §2 · p. 4
  • Enhanced Interpretability Score adds rule complexity to accuracy, stability, and compactness; SR4-Fit is reported to achieve consistently high scores across classification and regression.Source: §2 · p. 4
  • Election-data rules are reported to uncover demographic interactions, and fourteen public benchmarks—six classification and eight regression—are used to assess generalizability and robustness.Source: §2 · p. 4

3. Methodology

SR4-Fit combines tree-derived rules with sparse optimization in a shared framework for classification and regression. Its stages construct an expanded feature representation, optimize predictive weights with separate sparsity control, prune inactive predictors, and use the remaining features and rules for predictions.

3. Methodology

  • SR4-Fit combines RuleFit-style rule extraction with SR3 optimization and uses logistic loss for classification and squared-error loss for regression.The pipeline comprises forest generation, rule extraction, feature construction, sparse optimization, and pruning.Source: §3 · p. 5

3.1. Random Forest Generation and Rule Extraction

  • Classification uses a separate one-versus-rest forest for each class, while regression uses one forest trained on the continuous target; rule extraction is capped by r_max.Source: §3.1 · p. 5

3.2. Feature Construction and Optimization Objective

  • The model combines original input features with binary indicators for extracted rules in an extended feature matrix.Each extracted rule corresponds to a root-to-terminal path in a tree and evaluates to one when its conditions hold, otherwise zero.Source: §3.2 · p. 5, §3.1 · p. 5
  • The optimization uses task-specific loss and alternates updates to predictive weights and sparsity-enforcing weights; λ controls sparsity and κ controls how closely β follows w.Classification uses logistic loss, regression uses squared error, and the iterative process stops when relative objective change falls below a tolerance, typically 10^-4.Source: §3.2 · pp. 5–6

3.3. Model Pruning and Rule Selection

  • Soft-thresholding updates the auxiliary weights, and zero-weight predictors are pruned to retain a compact rule set in both tasks.Separating predictive coefficients from sparsity weights is described as more stable and interpretable than applying sparsity directly to predictive coefficients.Source: §3.3 · pp. 6–7

3.4. Classification and Regression Prediction

  • For classification, the model converts its linear score to a probability and uses one-versus-rest classifiers for multiclass prediction; regression uses the score directly as a continuous estimate.Nonzero regression coefficients quantify contributions of their corresponding rules or features.Source: §3.4 · pp. 7–8

4. Experimental Setup

The study evaluates SR4-Fit on U.S. House election data and 14 standard benchmarks, using comparisons designed to assess prediction, understandability, stability, and interpretability. Its analysis combines Pareto comparisons, composite scores, statistical tests, and visualizations.

4.1.1. Election Data

  • Election experiments compare demographic feature configurations with and without party percentages, and regression targets Democratic or Republican vote share with or without the opposing party’s percentage.Four demographic datasets vary in feature scope, from a minimum set to expanded attributes or prior-election party winner.Source: §4.1.1 · p. 8

4.1. Datasets

  • Experiments cover 14 standard benchmarks—six classification and eight regression datasets—alongside U.S. House election data evaluated across classification and regression settings.The election data spans 2006–2016, with 2006–2013 for training and 2014–2016 for testing.Source: §4.1.1 · p. 8, §4.1 · p. 8, §4.1.2 · p. 9

4.1.2. Standard Benchmark Datasets

  • Six classification benchmarks span binary, multiclass, and imbalanced tasks, while eight regression benchmarks cover varied prediction problems.The datasets include medical, biological, vehicle, document, environmental, housing, and other outcomes.Source: §4.1.2 · p. 9

4.2. Analysis Methods

  • Analysis combines Pareto comparisons, a composite predictivity-understandability score, statistical testing, and visual summaries of performance distributions.The evaluation uses 30 independent trials per model on the U.S. demographic datasets and assesses rule-set stability with the Dice-Sorensen index.Source: §4.2 · p. 9, §4.1.1 · p. 9, §4.2.3 · p. 10

4.2.1. Pareto Analysis and the Rule Understandability Score

  • RUS combines rule-set stability, rule count, and conditions per rule, weighting the three components equally while excluding predictive performance to keep Pareto axes independent.F1 score measures classification predictivity and R^2 measures regression predictivity in the Pareto analysis.Source: §4.2.1 · pp. 9–10

4.2.2. Extended Interpretability Score (E-IPS)

  • E-IPS combines predictive performance, stability, rule count, and rule complexity with equal component weights, and is compared only among rule-based models.Black-box models and logistic or LASSO regression are excluded because they do not produce rule sets.Source: §4.2.2 · p. 10

4.2.3. Statistical Analysis

  • Wilcoxon signed-rank tests compare SR4-Fit with RuleFit across performance metrics, using p < 0.05 as the significance threshold.The comparison focuses on these two rule-based models because they share a similar interpretability objective.Source: §4.2.3 · p. 10

4.2.4. Visualizations

  • Aggregate Pareto plots average results across dataset variants, while violin and line plots show performance variability, central tendency, and distributions.Figure 3 plots mean F1 for classification or mean R^2 for regression against RUS.Source: §4.2.4 · p. 10, Figure 3 · p. 11

5. Results

Across election and public benchmark tasks, SR4-Fit consistently balances predictivity and interpretability, outperforming RuleFit and Decision Tree on both aggregate axes while trading some predictivity for substantially higher RUS than Random Forest. Dataset-level results show how its interpretability and stability vary with task and feature availability.

5.1. Pareto Analysis

  • On public benchmarks, SR4-Fit matches RuleFit’s top classification RUS (0.7350) with higher mean F1 (0.8164 vs. 0.8119), and attains mean R² 0.6369 and RUS 0.9312 in regression.In public regression, it Pareto-dominates RuleFit and Decision Tree; Random Forest has the highest mean R² (0.6441) but lowest RUS (0.0800).Source: §5.1 · pp. 11–12, Figure 4 · p. 12
  • Across all four aggregate views, SR4-Fit Pareto-dominates RuleFit and Decision Tree, while Random Forest has marginally higher predictivity and SR4-Fit substantially higher RUS.This pattern spans election and public benchmark classification and regression tasks.Source: §5.1 · p. 12

5.2. Classification Results

  • With party percentages included, SR4-Fit leads E-IPS in three of four demographic classification datasets and has the highest predictive performance across all four.Without those features, its E-IPS lead is limited to the expanded dataset, but its rule structures remain more consistent than RuleFit’s and Decision Tree’s.Source: §5.2 · pp. 12–13

5.3. DEM Regression Results

  • When predicting Democratic vote percentage with Republican share included, SR4-Fit has tighter, more stable E-IPS distributions than baselines and comparable errors, while XGBoost has the lowest average error across most datasets.Without Republican share, error rises for all models; SR4-Fit leads E-IPS in the previous dataset (0.8427).Source: §5.3 · pp. 13–14, Figure 5 · p. 14

5.4. REP Regression Results

  • When predicting Republican vote percentage with Democratic share included, SR4-Fit leads E-IPS in three demographic datasets and maintains tighter, more stable distributions than baselines.Without Democratic share, E-IPS declines across models, but SR4-Fit leads in the previous dataset (0.6639); errors increase and XGBoost remains lowest.Source: §5.4 · pp. 14–15, Figure 6 · p. 15

5.5. Results on Standard Public Classification Datasets

  • Across six public classification benchmarks, SR4-Fit leads E-IPS in three datasets and achieves the highest stability across all six, while maintaining competitive predictive performance.Its predictive distributions are tightly clustered across 30 trials; Random Forest and Decision Tree have near-zero stability scores.Source: §5.5 · p. 16, Figure 8 · p. 17

5.6. Results on Standard Public Regression Datasets

  • Across eight public regression benchmarks, SR4-Fit leads E-IPS in two datasets and stability in six, with competitive predictive accuracy and fewer rules than Random Forest.Its rule sets are comparable in size to RuleFit but have lower rule complexity in most cases.Source: §5.6 · pp. 16–17, Figure 9 · p. 17

6. Discussion

The discussion interprets SR4-Fit’s election rules in terms of prior party support and demographic combinations, including patterns found when prior party percentages are unavailable. Its regression rules likewise connect prior Republican vote share and demographic conditions to predicted Democratic vote percentage.

6.1. Including Prior Party Voting Percentages (Classification)

  • With prior party percentages included, classification rules primarily use earlier vote shares, while other patterns associate age, race, education, income, and diversity with predicted party outcomes.For example, moderately divided support paired with higher socioeconomic indicators and diversity is associated with Democratic classifications in the expanded dataset.Source: §6.1 · pp. 18–19

6.2. Without Prior Party Voting Percentages (Classification)

  • Without prior party percentages, classification rules associate predicted outcomes with combinations of racial composition, age, education, income, marital status, and other demographic features.In the expanded dataset, marital-status composition recurs across many rules; diverse, more educated districts appear in Democratic predictions, while other demographic combinations appear in Republican predictions.Source: §6.2 · pp. 19–20

6.3. Including REP Percentage and DEM Percentage as Label (Regression)

  • For Democratic vote percentage regression, rules repeatedly associate low prior Republican support with higher predictions, sometimes alongside demographic conditions such as Black population share or age composition.The reported prediction increases range from 3.91 to 23.72 across the described rules.Source: §6.3 · p. 20

6.4. Without REP Percentage and DEM Percentage as Label (Regression)

Across three regression settings, SR4-Fit’s rules associate district demographics and prior electoral alignment with predicted House vote percentages. The identified contributions vary with the dataset and whether prior-party vote share is included.

6.4. Without REP Percentage and DEM Percentage as Label (Regression)

  • 17.82: In the minimum dataset without prior Republican vote share, low White non-Hispanic population share increases predicted Democratic percentage; education and racial composition also feature in the rules.The passages report increases of 18.73 for a low White-share condition with specified male-population ranges, and 14.04 for higher education with higher White share.Source: §6.4 · p. 20
  • 19.53: In the standard dataset without prior Republican vote share, low White and very high Black population shares increase predicted Democratic vote share; Asian population share and age conditions also contribute.The passages report increases of 18.08 for higher Asian share with fewer younger residents, and 15.06 to 15.21 for larger 25–44 age groups under specified conditions.Source: §6.4 · pp. 20–21
  • 27.95: In the expanded dataset without prior Republican vote share, high-school graduation combined with lower median income produces the strongest increase in predicted Democratic vote share.The rules also identify marital status, racial diversity, language composition, and age distribution as relevant predictors.Source: §6.4 · p. 21

6.5. Including DEM Percentage and REP Percentage as Label (Regression)

  • 15.11: In the standard dataset with prior Democratic vote share included, extremely low prior Democratic support gives the strongest increase in predicted Republican vote percentage.The minimum-dataset rule reports a 6.47 increase when prior Democratic vote share is relatively low; the previous-dataset rule reports up to 12.92 under a low-Black-share condition.Source: §6.5 · p. 21
  • 16.18: In the expanded dataset with prior Democratic vote share included, extremely low prior Democratic support increases predicted Republican vote percentage, particularly under a specified income condition.The passage also reports increases of 9.95 and 6.24 for less extreme prior Democratic vote-share ranges with marital-status conditions.Source: §6.5 · p. 21

6.6. Without DEM Percentage and REP Percentage as Label (Regression)

  • 20.05: In the minimum dataset without prior Democratic vote share, low bachelor’s-degree share combined with lower median age gives the strongest increase in predicted Republican vote share.The same passage reports a 12.90 contribution for higher bachelor’s-degree share.Source: §6.6 · p. 22
  • 9.69: In the expanded dataset without prior Democratic vote share, lower never-married share combined with higher less-than-high-school education share increases predicted Republican vote percentage, particularly in later election periods.The rules also identify age, education, ethnicity, income, and poverty conditions; the previous political dataset reports a 14.06 increase for prior Republican alignment with a graduate-degree condition.Source: §6.6 · p. 22
  • 14.06: In the previous political dataset without prior Democratic vote share as a feature, prior Republican alignment combined with graduate or professional degree share above the stated threshold gives the strongest increase in predicted Republican vote percentage.Other reported contributions include 6.40 for population aged 65–74 and 5.74 for White non-Hispanic population share.Source: §6.6 · p. 22

6.7. Interpretation SR4-Fit Rules for Public Classification Datasets

SR4-Fit’s public-dataset rules express predictions through combinations of domain-specific features. The examples span six classification datasets and eight regression datasets, with some combinations contributing more strongly than others.

6.7. Interpretation SR4-Fit Rules for Public Classification Datasets

  • In breast-cancer rules, malignant classifications combine elevated cellular irregularity and abnormal-nuclei features, while benign rules include lower shape irregularity and, in one pattern, low marginal adhesion despite larger cell size.The malignant examples include clump thickness above 0.7301 with bare nuclei above −0.4242; a benign pattern allows uniformity of cell size above 0.4405 when marginal adhesion is ≤−0.4647.Source: §6.7 · pp. 22–23
  • E. coli rules distinguish cytoplasmic and inner-membrane protein classes through differing combinations of alm1, mcg, alm2, and aac values, while page-block rules emphasize geometry and pixel features.Page-block class 1 rules commonly involve smaller block heights and low eccentricity; class 2 rules include larger or taller blocks with specific eccentricity values.Source: §6.7 · p. 23
  • Pima diabetes rules associate diabetic predictions with elevated glucose, often alongside BMI or other factors; lower glucose and BMI feature in non-diabetic rules.Some rules also use pregnancy count, insulin, blood pressure, age, or Diabetes Pedigree Function.Source: §6.7 · p. 23
  • Vehicle rules distinguish bus, car, and van classifications using silhouette descriptors including aspect ratios, elongatedness, circularity, variance, and rectangularity.Bus rules include elongated or compact rectangular shapes, car rules tend toward compact and circular shapes, and van rules toward box-like silhouettes.Source: §6.7 · p. 23
  • Yeast rules predict cytosolic proteins from varied combinations of weak compartment signals or particular high mcg and gvh values.Examples include very low vac or alm values and low mitochondrial indicators.Source: §6.7 · p. 23

6.8. Interpretation SR4-Fit Rules for Public Regression Datasets

  • In Abalone and Bone regression rules, shell weight and age feature in positive predictions, respectively, with the reported Bone contributions small.Abalone examples include contributions of 0.56 for high shell weight with moderate shucked weight and 0.48 for mid-range shell weight; Bone examples report increases of 0.01 for younger age intervals.Source: §6.8 · pp. 23–24
  • Diabetes regression rules show that predicted progression contributions vary with combinations of BMI and biomarkers, including S2, S5, S6, and blood pressure.The largest cited contribution is 68.7 for high BMI, low S2, and S5 above −0.0210; high BMI with larger S2 yields smaller contributions.Source: §6.8 · p. 24
  • Housing, Machine, MPG, Ozone, and prostate regression rules connect predictions to domain-specific feature combinations, including rooms, hardware capacity, vehicle characteristics, atmospheric conditions, and clinical markers.Examples include housing increases of 12.12 to 15.15 with high room counts and favorable neighborhood conditions, a 49.42 Machine increase with high MMAX and CHMAX, MPG increases with small engines and newer years, an Ozone increase of 8.23 under hot, stagnant conditions, and prostate increases associated with high tumor volume.Source: §6.8 · p. 24

7. Conclusions

The conclusion presents SR4-Fit as a compact, stable rule-based framework for classification and regression, applied to U.S. House election data and public benchmarks. Its results show a balance of predictive performance and interpretability relative to rule-based baselines.

7. Conclusions

  • SR4-Fit produces stable, compact, human-readable rules for analyzing U.S. House vote percentages from demographic and socioeconomic characteristics.The election analysis describes how district-level demographic patterns relate to electoral outcomes.Source: §7 · p. 25
  • Election rules combine demographic, socioeconomic, and historical voting variables to explain Democratic and Republican vote shares.The cited examples include race, age distribution, education, income, and marital status.Source: §7 · p. 25
  • Public regression and classification benchmarks extend the evaluation beyond elections and show consistent performance alongside interpretable rule structures.The conclusion describes these experiments as spanning diverse domains.Source: §7 · p. 25
  • Across election and public benchmark settings, SR4-Fit Pareto-dominates RuleFit and Decision Tree while trading some predictivity for greater rule understandability relative to Random Forest.The conclusion reports competitive F1 and R2 performance and stable, compact rules across the four settings.Source: §7 · p. 25

A. Statistical Significance Test (Wilcoxon Test)

Wilcoxon signed-rank tests compare SR4-Fit and RuleFit across election datasets and feature configurations. Significant differences favor SR4-Fit, most consistently in classification, while regression significance varies by target, metric, and configuration.

A. Statistical Significance Test (Wilcoxon Test)

  • In classification, SR4-Fit significantly exceeds RuleFit on all four metrics in every dataset without prior party percentages, and in three datasets when they are included.With prior percentages included, none of the four metrics is significant in the previous dataset; without them, most p-values are below 0.0001.Source: A. Statistical Significance Test… · p. 28, Table 6 · p. 29
  • For DEM regression, significant differences favor SR4-Fit on selected metrics and datasets, but none is significant in the previous dataset under either feature configuration.With prior Republican share included, significance occurs for RMSE and R2 in the Standard and Expanded datasets and for MAE in Minimum; without it, RMSE and R2 are significant in Expanded and Previous.Source: A. Statistical Significance Test… · p. 28, Table 7 · p. 29
  • For REP regression, prior Democratic share included yields significant MAE differences across all four datasets, while removing it yields significant RMSE and R2 differences in Standard, Expanded, and Previous.MAE is not significant in any dataset after the feature is removed.Source: A. Statistical Significance Test… · pp. 28–29, Table 8 · p. 30
  • Across the three comparisons, no statistically significant result favors RuleFit; significant differences consistently favor SR4-Fit.The reported significance pattern is strongest and most consistent for classification, and more selective across regression metrics and configurations.Source: A. Statistical Significance Test… · p. 29

B. Results of U.S House of Representative Elections Classification

In U.S. House classification, SR4-Fit combines competitive predictive performance with stable rules across datasets and feature configurations. Removing prior party percentages lowers predictive performance across models, while the leading model and compactness patterns vary by dataset.

B. Results of U.S House of Representative Elections Classification

  • Across election-classification configurations, SR4-Fit is reported as the best combination of stability, accuracy, and compactness among the compared baselines.The assessment draws on line plots, violin plots, stability tables, and rule-complexity analyses, with and without party-percentage features.Source: B. Results of U.S House of… · p. 30
  • In the Minimum dataset, SR4-Fit has the smoothest metric trajectories and tighter distributions than other models with prior party percentages, but lower accuracy and smoother trends after their removal.With prior percentages removed, XGBoost has the highest predictive scores; SR4-Fit has the highest Dice–Sørensen Index among rule-based models in both configurations.Source: B. Results of U.S House of… · p. 30, Figure 10 · p. 31, Figure 11 · p. 32
  • In the Standard and Expanded datasets, SR4-Fit has the smoothest trajectories and outperforms all models when prior party percentages are included; without them, SVM has the highest predictive scores.SR4-Fit has the highest Dice–Sørensen Index among rule-based models in both Expanded configurations, while RuleFit leads and SR4-Fit closely follows in Standard.Source: B. Results of U.S House of… · pp. 30–31, Figure 12 · p. 33, Figure 13 · p. 34
  • In the Previous dataset, SR4-Fit has the smoothest trajectories and highest Dice–Sørensen Index among rule-based models in both configurations, while Random Forest leads prediction scores without prior party percentages.Decision Tree is the most compact by rule count in both configurations; Random Forest has lower average rule complexity without prior percentages, while SR4-Fit does with them.Source: B. Results of U.S House of… · pp. 31–32, Table 11 · p. 35, Figure 16 · p. 37

C. Results of U.S House of Representative Elections Regression (DEM)

Across the four DEM-label election datasets, removing REP percentage worsens error for all models; XGBoost often has the lowest error, while SR4-Fit’s stability and complexity vary by dataset.

Minimum dataset

  • In the minimum dataset, removing REP raises error across models; SR4-Fit has the highest rule-based stability in both settings.Random Forest has slightly lower average error with REP included, while XGBoost does so without REP; the Decision Tree is most compact, and SR4-Fit has lower rule complexity with REP included.Source: C. Results of U.S House of… · p. 43, Table 12 · p. 44, Figure 22 · p. 44

Standard dataset

  • In the standard dataset, removing REP worsens performance; XGBoost has slightly lower average error in both settings.RuleFit leads rule-based stability with REP included, while SR4-Fit leads without it; the Decision Tree is most compact, and rule-complexity leaders differ by setting.Source: C. Results of U.S House of… · p. 43, Table 12 · p. 44, Figure 24 · p. 45

Expanded dataset

  • In the expanded dataset, removing REP worsens performance; XGBoost has slightly lower average error, while SR4-Fit leads rule-based stability with REP included.RuleFit leads stability without REP; the Decision Tree is most compact in both settings, while SR4-Fit has lower rule complexity with REP included and the Decision Tree without it.Source: C. Results of U.S House of… · p. 43, Table 12 · p. 44, Figure 26 · p. 46

Previous dataset

  • In the previous dataset, removing REP worsens performance; XGBoost has slightly lower average error, and SR4-Fit leads rule-based stability in both settings.The Decision Tree is most compact in both settings; RuleFit has lower rule complexity with REP included, while SR4-Fit does without REP.Source: C. Results of U.S House of… · p. 43, Table 12 · p. 44, Figure 28 · p. 47

D. Results of U.S House of Representative Elections Regression (REP)

Across the four REP-label election datasets, removing DEM percentage worsens performance in nearly all reported comparisons; XGBoost often has the lowest error, while stability and complexity leaders vary.

Minimum dataset

Standard dataset

  • In the standard dataset, removing DEM worsens performance; XGBoost has slightly lower average error in both settings.SR4-Fit leads rule-based stability with DEM included, while RuleFit leads without it; the Decision Tree is most compact, and it also has lower complexity with DEM included, while RuleFit does without it.Source: D. Results of U.S House of… · p. 52, Table 15 · p. 53, Figure 36 · p. 54

Expanded dataset

Previous dataset

  • In the previous dataset, removing DEM worsens performance; XGBoost has lower average error with DEM included, while Random Forest does slightly better without it.RuleFit leads rule-based stability in both settings, and the Decision Tree is most compact; complexity leaders differ by setting.Source: D. Results of U.S House of… · p. 52, Table 15 · p. 53, Figure 40 · p. 56

E. Results on Standard Public Datasets Classification

Across six classification datasets and 30 trials, SR4-Fit balances predictive performance with stable, low-variance behavior, although other models sometimes lead on accuracy. Its stability often matches RuleFit’s, while compactness and rule complexity vary by dataset.

Overall results

Dataset-level results

  • In breast cancer, page blocks, Pima Indians, and vehicle, SR4-Fit trails the prediction-metric leader closely; it shares top stability with RuleFit in the reported datasets.XGBoost leads prediction metrics in breast cancer, Random Forest in page blocks and Pima Indians, and SVM in vehicle; SR4-Fit and RuleFit achieve the highest Dice–Sørensen Index in those four datasets.Source: E. Results on Standard Public Datasets… · p. 61
  • SVM leads prediction metrics on E. coli, while SR4-Fit and RuleFit share the highest Dice–Sørensen Index.The Decision Tree has a compact structure, while Random Forest has lower average rule complexity than the rule-based models.Source: E. Results on Standard Public Datasets… · p. 61, Table 18 · p. 62, Table 19 · p. 63
  • On yeast, Random Forest leads prediction metrics, while SR4-Fit and RuleFit share the highest Dice–Sørensen Index.The Decision Tree has a compact structure, and Random Forest has lower average rule complexity than the rule-based models.Source: E. Results on Standard Public Datasets… · p. 61, Table 18 · p. 62, Table 19 · p. 63

F. Results on Standard Public Datasets Regression

Across eight regression datasets and 30 trials, SR4-Fit balances predictive performance with stable, low-variance behavior, although other models sometimes achieve lower errors. Stability, compactness, and rule-complexity leaders vary by dataset.

Overall results

Dataset-level results

  • SR4-Fit trails the lowest-error model closely on abalone, bone, diabetes, machine, ozone, and prostate; it leads stability in all six.SVM has the lowest errors on abalone, ozone, and prostate; Random Forest on bone; Linear Regression on diabetes; and XGBoost on machine.Source: F. Results on Standard Public Datasets… · p. 70

Model structure

  • Rule-complexity leaders vary: the Decision Tree is lowest on abalone, bone, and diabetes, while RuleFit leads on housing.The Decision Tree is most compact on abalone, bone, and housing; SR4-Fit has lower average rule complexity than other rule-based models on machine and ozone.Source: F. Results on Standard Public Datasets… · p. 70

Dataset-level results

  • On housing, XGBoost has the lowest errors and RuleFit leads stability; on MPG, SVM has the lowest errors and the Decision Tree leads stability.SR4-Fit trails the lowest-error model closely on housing; SR4-Fit and RuleFit do so on MPG.Source: F. Results on Standard Public Datasets… · p. 70

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