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SAILS: Surrogate-based Analysis of Interactions via Local Effect Smooths

The paper introduces SAILS, a model-agnostic framework that utilizes interpretable generalized additive model surrogates fitted to local effects to not only detect pairwise feature interactions but also categorize their functional forms and provide tailored visualizations, thereby addressing a key gap in existing explainable AI methods.

Original authors: Timo Heiß, Julia Herbinger, Bernd Bischl, Giuseppe Casalicchio

Published 2026-06-09

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Timo Heiß, Julia Herbinger, Bernd Bischl, Giuseppe Casalicchio

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine you have a super-smart, but mysterious "Black Box" machine that makes predictions. Maybe it predicts how much electricity a city will use, or how a patient might respond to a drug. You know the machine works well, but you have no idea how it thinks. It's like a chef who makes a delicious stew but refuses to tell you the recipe.

In the world of machine learning, the "recipe" often involves interactions. This means the machine doesn't just look at one ingredient (like "temperature") in isolation; it looks at how ingredients mix together (like "temperature" + "time of day"). If it's hot and it's evening, the machine might predict high energy use (for air conditioning). But if it's hot and it's morning, the prediction might be totally different.

The problem is that most tools used to peek inside these Black Boxes are like looking at a blurry photo. They can tell you that two ingredients are interacting, or they can show you a messy 3D graph that's hard to read. They can't clearly explain what that interaction actually looks like.

Enter Sails: The "Recipe Decoder"

The paper introduces a new tool called Sails (Surrogate-based Analysis of Interactions via Local effect Smooths). Think of Sails as a detective that doesn't just say "Hey, these two things are related!" but actually draws a clear map of how they relate.

Here is how Sails works, using a simple analogy:

1. The "Slice and Dice" Strategy

Imagine the Black Box machine is a giant, complex cake. Sails doesn't try to eat the whole cake at once. Instead, it takes a knife and slices the cake into thin, manageable layers based on one specific ingredient (let's call it the "Focus Ingredient," like "Time of Day").

For each slice (or time interval), Sails asks: "If I hold the Time of Day constant, how does the prediction change when I tweak the other ingredients?"

2. The "Surrogate Detective"

For each slice, Sails builds a tiny, simple, and transparent model (a "surrogate") to mimic the Black Box's behavior just for that specific slice. It's like hiring a local guide to explain the terrain of just one small hill, rather than trying to map the whole mountain range at once.

These guides are mathematical models called GAMs (Generalized Additive Models). They are special because they are easy for humans to read. They can tell you if the relationship is a straight line, a curve, or a complex twist.

3. Three Superpowers

Once Sails has gathered these local guides, it performs three main tasks:

Detecting the Interaction (The "Is it there?" Test):
Sails checks if the local guides are doing anything interesting. If the guides for different slices all look exactly the same, there's no interaction. But if the guides change shape depending on the slice, Sails flags it: "Aha! These two ingredients are definitely interacting!" It uses a statistical "hearing test" (like a significance test) to make sure it's not just hearing noise.
Categorizing the Shape (The "What does it look like?" Test):
Sails doesn't just stop at "they interact." It asks, "What kind of interaction is this?"
- Linear: Is it a simple, straight-line relationship? (e.g., "The hotter it gets, the more we use AC, always.")
- Product-Separable: Is it a relationship where one ingredient just scales the other up or down? (e.g., "The effect of temperature is the same shape, just multiplied by how windy it is.")
- Complex/Non-Separable: Is it a messy, unique twist where the two ingredients create something entirely new?
  Sails uses math to measure how well the data fits these shapes, giving a score like "90% likely to be linear."
Visualizing the Story (The "Drawing the Map"):
Based on the category, Sails draws the best possible picture:
- If it's Linear, it draws a simple line.
- If it's Product-Separable, it draws a shape that shows how one ingredient scales the other.
- If it's Complex, it draws a series of curves (one for each slice) that show exactly how the interaction changes across the whole range. This avoids the confusing 3D graphs of the past.

Real-World Example: The City's Power Grid

The authors tested Sails on a real dataset about electricity consumption in Tetouan, Morocco. They wanted to understand how "Solar Radiation" (sunlight) interacted with other factors like "Time of Day" and "Temperature."

What they found: Sails revealed that the interaction wasn't just a simple rule. For example, on cloudy days (low radiation), electricity use went up during the day because people turned on lights. But on sunny days, high radiation only mattered during the evening when people came home.
The Result: Sails drew clear, colorful curves showing these patterns. It showed that the Black Box model was making very logical, human-understandable decisions about when to use power, based on the complex mix of weather and time.

The Limits

The paper is honest about where Sails might stumble:

Correlated Ingredients: If two ingredients are almost always the same (like "Temperature" and "Time of Day" in some seasons), it gets harder for Sails to separate their effects perfectly.
Complex Mixes: If three or more ingredients are interacting at once (a "triple threat"), Sails is currently designed for pairs, so it might miss some of the nuance.

The Bottom Line

Sails is a new tool that fills a gap in the "Explainable AI" toolbox. Instead of just telling you that a machine learning model is using complex interactions, Sails pulls back the curtain to show you exactly what those interactions look like, turning a black box into a transparent, understandable recipe.

Technical Summary: Sails – Surrogate-based Analysis of Interactions via Local Effect Smooths

Problem Statement

Feature interactions are a primary driver of predictive power in complex machine learning (ML) models, yet existing Explainable AI (XAI) methods struggle to fully characterize them. Current approaches typically focus on either:

Detection and Quantification: Methods like the H-statistic or Shapley interaction indices identify that an interaction exists and measure its strength but do not reveal its functional form.
Restricted Visualization: Methods like 2D-ALE (Accumulated Local Effects) or REPID (Regional Effects) visualize interactions but often produce difficult-to-interpret 3D plots, fail to capture continuous heterogeneity, or are limited to specific interaction types (e.g., product-separable).

There is a notable gap in the XAI toolbox for an integrated, model-agnostic framework that can detect pairwise interactions, categorize their functional form (linear, product-separable, or non-product-separable), and provide tailored, interpretable visualizations for each type.

Methodology: The Sails Framework

Sails (Surrogate-based Analysis of Interactions via Local effect Smooths) is a model-agnostic framework designed to analyze pairwise interactions of a Feature of Interest (FOI) by fitting interpretable Generalized Additive Model (GAM) surrogates to the local effects of a black-box model. The framework operates in two sequential steps followed by three integrated capabilities.

1. Estimation of Local Effects

Sails exploits the local decomposability of ALE (Accumulated Local Effects) derivatives. For a black-box model $\hat{f}$ , the local effect with respect to a feature $x_j$ is defined as $h(x_j, x_{-j}) = \partial_j \hat{f}(x_j, x_{-j})$ .

Interval Partitioning: The FOI ( $X_j$ ) is partitioned into $K$ intervals.
Finite Differences: Local effects are estimated for observations within each interval using finite differences of the model predictions, normalized by interval width.
Theoretical Basis: According to the functional ANOVA (FANOVA) decomposition, if $X_j$ is fixed, the variation in local effects arises solely from interactions involving $X_j$ . Thus, heterogeneity in these local effects across other features indicates interaction.

2. GAM Surrogate Fitting

For each interval $k$ of the FOI, Sails fits a separate GAM surrogate model:

Inputs: The remaining features $X_{-j}$ .
Target: The estimated local effects $\hat{h}^{(i)}_k$ .
Structure: $\hat{s}^{(k)}(x_{-j}) = \hat{s}^{(k)}_0 + \sum_{l \neq j} \hat{s}^{(k)}_l(x_l)$ .
Interpretation: The smooth terms $\hat{s}^{(k)}_l$ are conjectured to approximate the partial derivatives of the interaction components ( $\partial_j g_{jl}$ ) up to a centering constant.

3. Integrated Capabilities

Based on the fitted surrogates, Sails performs three tasks:

(i) Interaction Detection

Sails tests the significance of the smooth terms for each feature $X_l$ within each interval using F-tests on the basis coefficients.

Aggregation: Since tests across intervals are dependent, p-values are aggregated using the harmonic mean p-value to provide a global significance measure.
Correction: A Benjamini-Hochberg false discovery rate (FDR) correction is applied for multiple comparisons.
Heuristic: Due to the lack of additive noise in local effect calculations and the use of penalized splines, the resulting p-values are treated as a heuristic for global interaction significance rather than exact statistical values.

(ii) Interaction Categorization

Sails categorizes detected interactions into three types using $R^2$ -based heuristic measures derived from the smooth terms:

Linear: If the smooth terms $\hat{s}^{(k)}_l$ are identical across intervals (up to a constant), the interaction is linear with respect to the FOI. Measured by $R^2_{lin}$ , which fits a single curve to pooled evaluations.
Product-Separable: If the smooth terms share the same shape but differ only by a scaling factor, the interaction is product-separable ( $g_{jl}(x_j, x_l) = \phi_j(x_j)\phi_l(x_l)$ ). Measured by $R^2_{prod}$ , which fits a single curve to the ratios of smooth terms at a reference point.
Non-Product-Separable: If neither condition holds, the interaction is categorized as general/non-separable.

(iii) Tailored Visualization

Depending on the categorization, Sails generates specific visualizations:

Linear: Visualizes the univariate smoother $\phi_l(x_l)$ .
Product-Separable: Visualizes the shape of $\phi_l(x_l)$ via the ratio-based smoother.
General: For non-separable interactions, Sails integrates the smooth terms $\hat{s}^{(k)}_l$ across intervals to reconstruct the interaction function $g_{jl}$ , applying double centering to enforce FANOVA constraints. This provides a 2D visualization of how the interaction effect varies with the interacting feature for different intervals of the FOI.

Empirical Evaluation and Results

The framework was validated through controlled simulations and a real-world application.

Simulations

Experiments were conducted on synthetic data with varying correlation structures and interaction orders (linear, product-separable, non-separable, and higher-order).

Detection: Sails correctly identified interacting features with low false positive rates under both independent and correlated features. It remained robust even with strong correlations, correctly identifying noise features as non-interacting.
Categorization: Under independent features, Sails perfectly discriminated between linear, product-separable, and non-separable interactions.
- Limitations: $R^2_{prod}$ degraded under strong correlations (due to centering shifts) and higher-order interactions (due to surrogate misspecification). $R^2_{lin}$ was more robust but also degraded slightly with non-linear higher-order terms.
Visualization:
- Linear and product-separable visualizations closely matched ground truths under independence.
- The general integration-based visualization successfully recovered complex interaction shapes (e.g., sine waves) under independence.
- Under correlations and higher-order interactions, visualizations showed artifacts or reduced support but remained informative approximations.

Real-World Application

Sails was applied to an XGBoost model predicting power consumption in Tetouan, Morocco, using general_diffuse_flows (solar radiation) as the FOI.

Results: The framework identified interactions with Hour, Temperature, UnixTime, and WindSpeed.
Interpretation:
- Hour: Revealed that intermediate radiation levels correlate with increased evening consumption (dusk lighting), while low radiation correlates with daytime consumption (overcast lighting).
- Temperature: Showed synergistic effects where low radiation + low temperature (heating + lighting) and high radiation + high temperature (cooling) both increased consumption.
- WindSpeed: Exhibited a near product-separable form, with curves sharing a consistent shape across intervals.
Surrogate Fit: The average in-sample $R^2$ of the GAM surrogates was 0.96, indicating they captured the majority of local effect variation.

Significance and Claims

The paper claims that Sails fills a critical gap in the XAI toolbox by moving beyond mere interaction detection to characterizing the functional form of pairwise interactions.

Model-Agnostic: It works with any black-box model.
Integrated Workflow: It combines detection, categorization, and visualization in a single pipeline.
Interpretability: By categorizing interactions, it allows for tailored visualizations that are easier to interpret than generic 3D contour plots.
Robustness: It demonstrates effectiveness in detecting interactions even in the presence of feature correlations, a common challenge in real-world data.

Limitations and Future Work

The authors acknowledge several limitations:

Task Scope: Currently limited to regression tasks; extension to classification is future work.
Theoretical Guarantees: The conjecture that smooth terms approximate FANOVA partial derivatives lacks rigorous theoretical proof.
Correlation Sensitivity: The categorization measures ( $R^2_{prod}$ ) are sensitive to strong feature correlations.
Interaction Order: The framework assumes pairwise interactions; higher-order interactions can cause surrogate misspecification and artifacts.
Future Directions: Potential extensions include using GA2M surrogates for higher-order interactions, applying the method to other local effect methods (e.g., ICE, Shapley values), and unifying the approach with GADGET for regional effects.