Characteristic function distance balancing

bw_cfd() specifies characteristic function distance balancing, also called kernel balancing, for balance(). The weights minimize a kernel measure of the distance between the reweighted exposure groups and a target sample, following Wong and Chan. Every kernel but "energy" is positive semidefinite, so the objective is a convex quadratic program; the energy kernel is conditionally positive semidefinite alone, so its quadratic form is indefinite and the solve routes to the ADMM backend. Either way the reweighting improves multivariate covariate balance without positing a propensity model. Characteristic function distance balancing supports binary and categorical exposures.

Usage

bw_cfd(
  ...,
  kernel = c("gaussian", "matern", "laplace", "t", "energy"),
  smoothness = 1.5,
  degrees_of_freedom = 5,
  simulation_draws = 5000,
  improved = TRUE,
  weight_penalty = 1e-04,
  min_weight = 1e-08,
  convergence_tolerance = NULL,
  max_iterations = NULL
)

Arguments

...: Reserved for future extensions; must be empty. Tuning parameters must be passed by name.
kernel: The covariate kernel the objective is built on, one of "gaussian", "matern", "laplace", "t", or "energy".
smoothness: The Matern smoothness order, one of 0.5, 1.5, or 2.5. Used only by the Matern kernel.
degrees_of_freedom: The degrees of freedom of the t kernel's frequency distribution, a finite number greater than two. Used only by the "t" kernel. The t kernel approaches the Gaussian kernel as the degrees of freedom grow, so that limit is requested as kernel = "gaussian" rather than as an infinite degrees of freedom, which names no frequency distribution to draw from.
simulation_draws: The number of Monte Carlo frequency projections for the "t" kernel.
improved: Whether to add the between-group term of the improved variant for the average treatment effect with a discrete exposure.
weight_penalty: The L2 penalty on the weights, which stabilizes the quadratic program.
min_weight: The smallest permitted weight.
convergence_tolerance: The quadratic-program solver tolerance, or NULL for the core default.
max_iterations: The maximum solver iterations, or NULL for the core default.

Value

A bw_cfd specification, a balance_method.

Details

The objective is the sum of each group's kernel distance to the target sample, built from a covariate kernel matrix rather than a pairwise distance matrix. kernel selects the kernel. The "gaussian", "laplace", and "matern" kernels are distance based: their bandwidth is the median of the pairwise covariate distances, so the fit is invariant to a uniform rescaling of the covariates. smoothness sets the Matern order, one of 0.5, 1.5, or 2.5. The "t" kernel approximates a heavy-tailed spectral kernel by Monte Carlo: simulation_draws frequency vectors are drawn from a multivariate t distribution with degrees_of_freedom degrees of freedom, on the R side under R's random number generator, so a fixed seed reproduces the weights. Raising the degrees of freedom takes the t kernel toward the Gaussian kernel, which is available exactly as kernel = "gaussian" and needs no Monte Carlo draws. The improved variant for the average treatment effect adds the between-group term of the kernel mean embedding, balancing the groups against one another as well as against the sample.

Setting kernel = "energy" uses the negative pairwise distance, which reproduces bw_energy() with its "scaled_euclidean" distance on the same data and constraints. That equivalence is the reference point for the method, since the energy kernel has an external implementation while the other kernels do not.

Characteristic function distance balancing belongs to the quadratic-program family, which has no estimating equations, so a fit produces no estimating-equations container and the guarantee is objective-level rather than exact moment balance: without moment constraints the kernel objective drives balance, and with them the constraint rows hold within tolerance. The tolerance in balance_terms() relaxes any added moment constraints rather than selecting an inexact solver. A tolerance supplied without moment constraints has nothing to relax, so it is warned and ignored.

What the optimization pins is the objective and the balance it achieves, not the individual weights. A kernel matrix built on a smooth kernel is ill-conditioned by construction, since nearby units contribute nearly proportional rows, and a quadratic program whose objective is flat along such directions has a set of solutions rather than one. Two quadratic-program backends given the same ill-conditioned problem agree on the objective to seven significant figures while individual unit weights differ materially; a Gaussian kernel at its median bandwidth is the case this was measured on. Treat the per-unit weights of a bw_cfd() fit as one member of an equivalence set rather than as a uniquely determined quantity: what the method determines is the reweighted distribution, so estimands and balance statistics computed from the weights are stable where a claim about a particular unit's weight is not.

References

Wong, R. K. W. and Chan, K. C. G. (2018). Kernel-based covariate functional balancing for observational studies. Biometrika, 105(1), 199-213.

Huling, J. D. and Mak, S. (2024). Energy balancing of covariate distributions. Journal of Causal Inference, 12(1), 20220029.

Examples

n <- 200
x1 <- rnorm(n)
x2 <- rnorm(n)
df <- data.frame(
  exposure = rbinom(n, 1, plogis(0.5 * x1 - 0.5 * x2)),
  x1 = x1,
  x2 = x2
)
fit <- balance(df, exposure, c(x1, x2), method = bw_cfd())
#> ℹ Treating `.exposure` as binary
fit
#> 
#> ── Characteristic function distance balancing ──────────────────────────────────
#> Exposure: "exposure" (binary)
#> Estimand: "ate"
#> Observations: 200
#> Solver: converged in 1825 iterations
#> Constraints: 2 terms (tolerance 0)
#> Largest imbalance: 0.0117 (standardized mean difference)