Package 'BORT'

BORT: Multi-objective Regression Trees

Description

Constructs a multiobjective regression tree or a Pareto-consistent family of trees based on a top-down generalisation. The partitioning process selects the hyper-rectangle with the maximum Lebesgue measure. The split thresholds are chosen to minimise the sum of the Weighted Mean Squared Error across dimensions.

Usage

bort(X, Y, k = 1, type = c("PARETO", "BORT"), minSample = NULL)

Arguments

X	A numeric matrix of size $n \times p$ containing the input variables (features) for $n$ samples.
Y	A numeric matrix of size $n \times q$ containing the target variables (objectives) to be predicted.
k	An integer scalar specifying the number of trees to generate. Default is 1.
type	A character string indicating the modelling strategy. "PARETO" generates a Pareto-consistent family of trees weighting the error via a single Dirichlet-sampled vector per tree. "BORT" updates the weight vector dynamically at each split iteration to adaptively explore the Pareto front.
minSample	An integer indicating the minimum number of samples a node must contain to be eligible for further splitting. If NULL, it defaults to 5% of $n$.

Details

This implementation maps continuous multiobjective functions $f:\mathbb{R}^{p}\rightarrow\mathbb{R}^{q}$. It partitions the Cartesian space $D$ bounded by $[min(X[,i]) - \delta, max(X[,i]) + \delta]$, where $\delta= 0.1*(max(X[,i])-min(X[,i]))$. For type = "PARETO", scalarisation of errors uses a constant weight vector across the tree depth. For type = "BORT", a novel weighting approach selects random weights at every partition, achieving efficient single-model dominance over entirely consistent families. Both approaches are explained in de Paz (2025).

Value

A list of length k containing R functions. Each function accepts a numeric vector x of length $p$ and returns a predicted numeric vector y of length $q$.

References

de Paz, E.G.G., Hernández-Aguirre, A., Cruz-Aceves, I. (2026). Beyond Pareto: A High-Efficiency Approach to Bi-objective Regression Trees. In Pattern Recognition. Springer Nature Switzerland. doi:10.1007/978-3-032-28393-1_2

de Paz, E.G.G., Vaquera Huerta, H., Albores Velasco, F.J., Bauer Mengelberg, J.R., Romero Padilla, J.M. (2025). A Splitting Criterion for CART Models Based on Bayesian Optimisation. In Statistics, Society and Environment. Springer Nature Switzerland. doi:10.1007/978-3-031-78401-9_2

Examples

# Ensure the C shared library is loaded before running
  # dyn.load(paste0("multiobjective_tree", .Platform$dynlib.ext))

  # Prepare the iris dataset
  data(iris)
  X <- as.matrix(iris[, 1:2]) # Sepal.Length, Sepal.Width
  Y <- as.matrix(iris[, 3:4]) # Petal.Length, Petal.Width

  # Generate a single BORT model
  bort_models <- bort(X, Y, k = 1, type = "BORT")

  # Predict for the first instance
  test_point <- X[1, ]
  prediction <- bort_models[[1]](test_point)
  print(prediction)

  # Generate a Pareto-consistent family of 5 trees
  pareto_models <- bort(X, Y, k = 5, type = "PARETO")
  pred_pareto <- pareto_models[[1]](test_point)
  print(pred_pareto)

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