Tree-Based Proximity Measure

• Tree-Based Proximity Measure is a non-parametric method that uses decision trees to determine similarity by automatically discovering thresholds and interactions among predictors.
• It employs ensemble techniques like random forests and BART to aggregate partitions, capturing nonlinear, threshold effects with improved stability and accuracy.
• The approach outperforms traditional regression by revealing discontinuous and locally restricted effects, as demonstrated in studies on border proximity and gun control opinions.

A tree-based proximity measure quantifies the similarity or nearness of data points, features, or structures according to relationships revealed by a tree or a set of trees, rather than assuming a specific mathematical distance such as Euclidean or Manhattan distance. Tree-based methods leverage recursive partitioning, hierarchical relationships, and non-parametric estimation to capture both linear and highly complex, nonlinear relationships between proximity variables (e.g., spatial or temporal distance) and outcomes.

1. Non-Parametric Characterization of Proximity

Tree-based methods define proximity not by prespecified functional forms but by the capacity of decision trees or ensembles to partition the covariate space based on observed associations. At each node in a decision tree, the algorithm selects a variable (which may be spatial distance to a landmark, temporal gap since an event, or other contextual covariate) and a threshold that best separates the data according to the outcome of interest. The partitioning process naturally captures discontinuous effects, threshold phenomena, and interactions. For instance, the effect of distance to a border crossing on political attitudes may only manifest for individuals within a specific radius or may interact with demographic features. The tree learns these cutoffs automatically:

• At each split, a predictor is chosen to optimize reduction in dispersion (e.g., variance for regression trees).
• Splits form discrete strata or “bins” of proximity, so that proximity becomes encoded by tree path membership rather than continuous distance.

A single tree model continues partitioning until further divisions fail to appreciably reduce prediction error or terminal nodes become too small. Each direct tree path determines a set of “proximal” cases according to the learned criteria, which may combine distance features with others (e.g., party, year).

2. Ensemble Methods and Proximity Aggregation

While a single decision tree can reflect complex, conditional proximity measures, its stability is limited; small changes in data can yield different partitions. Ensemble methods, particularly @@@@1@@@@ and Bayesian Additive Regression Trees (BART), address this through aggregation:

• Random Forest: Constructs multiple trees on bootstrapped samples. Each tree may use a different subset of covariates, increasing robustness to noise and overfitting. Final predictions and variable importance are typically taken as the average (for regression) or majority vote (for classification) across all trees.
• BART: Employs a sum-of-trees framework, estimated using MCMC, that averages across many small trees, favoring simpler and mostly additive structures but allowing for flexible capture of nonlinear proximity effects.

In both cases, the tree-based proximity measure becomes a function of ensemble partitions: cases that frequently appear in similar leaf nodes or under similar splits are considered more proximate in the sense relevant to the outcome.

3. Methodological Tools: Variable Importance and Model Evaluation

To identify which proximity constructs (e.g., spatial or temporal distance, event attributes) have the greatest effect, permutation importance techniques are used. For a fitted ensemble model:

• Compute the baseline test mean squared error (MSE₀).
• For each predictor $p$, randomly permute its values across the test set, recompute the MSE ($\textrm{MSE}_{(p)}$), and calculate

$$\textrm{importance}_p = \left( \frac{\textrm{MSE}_{(p)}}{\textrm{MSE}_0} - 1 \right) \times 100$$

averaging across $K$ independent permutations for stability.

• Variables with large importance scores are understood to be essential for separating cases via the tree—indicating a strong “tree-based proximity” to the outcome under consideration.

This approach is explicitly used to compare the relative effect of spatial (or temporal) distance versus demographic or contextual predictors.

4. Comparison with Conventional Regression Techniques

Conventional regression models typically require explicit parametric assumptions about the proximity-outcome relationship: distances may enter linearly, quadratically, or as dichotomized values (e.g., “near” vs “far”, with arbitrary thresholds). These models may miss or misrepresent:

• Threshold effects, where an outcome jumps only beyond/within particular distances.
• Interactions between proximity and subordinate features (e.g., effect of distance may depend on party or age).
• Complex, non-monotonic forms.

Tree-based proximity measures avoid these limitations by selecting splits and interactions in a purely data-driven way, with the capacity to uncover highly variable or locally restricted effects not accessible to fixed-form regression. This results in models with improved mean squared error and a more nuanced mapping between proximity and outcome.

5. Empirical Applications

a. Proximity to Border Crossings and Immigration Attitudes

In the context of US-Mexico border proximity and support for immigration reform, conventional models used either continuous distance or binary indicators and often included modifiers such as crossing size. Tree-based methods were applied to the same dataset, allowing the algorithm to determine whether, how, and where spatial distance affected attitudes, potentially in interaction with party, age, or ethnicity. While the raw distance was not always selected as the primary predictor in shallow trees, permutation-based importance analyses showed the conditions under which spatial proximity became relevant, as well as identification of nonlinear or threshold effects.

b. Proximity to Mass Shootings and Gun Control Opinions

In examining support for gun control, tree-based models were fit to data with spatial distance to shootings as covariates. Both single trees and ensembles provided a mapping of how spatial and temporal proximity interacted with sociodemographics to modulate support for policy changes. For some subgroups (e.g., Independents), small increases in distance led to pronounced shifts in opinion—a detail that could be masked by a model imposing global linearity.

6. Interpretation and Model Diagnostics

Tree-based proximity measures inherently yield interpretable tree diagrams, where the meaning of proximity is explicitly given as a data-driven threshold (“within 200 km”) or a combination of conditions (“within 200 km and age < 40”). Such interpretability is a marked advantage over black-box distance metrics.

Permutation-based importance, in conjunction with mean squared error, allows not only more accurate estimation of outcomes but also a principled quantification of the explanatory role of each proximity variable, revealing both direct and conditional effects.

7. Implications for Social Science and Beyond

Tree-based proximity measures provide a rigorous, flexible alternative to conventional modeling when exploring the impact of spatial, temporal, or more abstract notions of “proximity” on outcomes. By relaxing functional-form assumptions and allowing for data-driven discovery of critical thresholds and interactions, tree-based approaches can reveal nuanced, heterogeneous effects, potentially leading to better domain understanding and more targeted interventions.

These advantages establish tree-based proximity as a powerful and generalizable framework for modeling complex proximity-outcome relationships in fields ranging from political science to epidemiology to spatial economics (Levin, 2024).