Locally Weighted Split Conformal Prediction for Geostatistical (Point-Referenced) Data

Constructs distribution-free prediction intervals at new spatial (or spatio-temporal) locations by combining split conformal prediction with spatial-distance kernel weights, relaxing the exchangeability assumption that standard conformal prediction relies on.

Usage

scp_geostatistical(
  s_train,
  y_train,
  s0,
  pred_fun,
  alpha = 0.1,
  split = 0.5,
  bandwidth = NULL,
  t_train = NULL,
  t0 = NULL,
  temporal_bandwidth = NULL,
  seed = NULL
)

Arguments

s_train: Numeric matrix of training coordinates, one row per observation (e.g. longitude/latitude, or projected easting/northing).
y_train: Numeric vector of training responses, same length as nrow(s_train).
s0: Numeric matrix of coordinates at which prediction intervals are desired, one row per target location.
pred_fun: A function with signature function(s_train, y_train, s_new) that returns a numeric vector of point predictions for the locations in s_new, fitted using s_train/y_train. This lets the user plug in any spatial predictor (kriging, random forest, GAM, etc.); spconform handles only the conformal calibration layer.
alpha: Miscoverage level; prediction intervals target \(1 - \alpha\) coverage. Default 0.1 (90 percent intervals).
split: Proportion of the training data used for model fitting; the remainder is used as the calibration set for conformal scores. Default 0.5.
bandwidth: Optional spatial kernel bandwidth passed to spatial_kernel_weights; if NULL, chosen automatically per target location.
t_train, t0, temporal_bandwidth: Optional time indices (and temporal bandwidth) for spatio-temporal weighting; see spatial_kernel_weights.
seed: Optional integer seed for the train/calibration split, for reproducibility.

Value

An object of class "spconform", a list with components:

pred: Point predictions at s0.
lower, upper: Lower/upper prediction interval bounds at s0.
alpha: The requested miscoverage level.
s0: The target coordinates.
call: The matched call.

Details

This implements a localized split-conformal procedure in the spirit of Mao, Martin and Reich (2020, "Valid model-free spatial prediction") and subsequent spatio-temporal extensions: nonconformity scores from a held-out calibration set are combined into a weighted empirical quantile where weights decay with distance (in space, and optionally time) from the target location, so nearby, more relevant observations dominate the calibration of the interval while validity is retained under a local exchangeability argument.

References

Mao, H., Martin, R., and Reich, B. J. (2020). Valid model-free spatial prediction. arXiv:2006.15640.

Examples

set.seed(42)
n <- 200
s <- matrix(runif(2 * n), ncol = 2)
f <- function(s) sin(4 * s[, 1]) + cos(3 * s[, 2])
y <- f(s) + rnorm(n, sd = 0.3)

idx <- sample(n, 150)
s_train <- s[idx, ]; y_train <- y[idx]
s0 <- s[-idx, ][1:10, ]

pred_fun <- function(s_train, y_train, s_new) {
  fit <- lm(y_train ~ s_train[, 1] + s_train[, 2] +
              I(s_train[, 1]^2) + I(s_train[, 2]^2))
  nd <- data.frame(s_train = I(s_new))
  as.numeric(cbind(1, s_new[, 1], s_new[, 2], s_new[, 1]^2, s_new[, 2]^2) %*%
               coef(fit))
}

out <- scp_geostatistical(s_train, y_train, s0, pred_fun, alpha = 0.1, seed = 1)
print(out)
#> <spconform> geostatistical conformal prediction
#> Target coverage: 90.0%
#> Number of prediction points: 10
#>     pred  lower  upper
#> 1 -0.543 -1.075 -0.010
#> 2  1.060  0.495  1.625
#> 3 -0.816 -1.355 -0.277
#> 4  0.619  0.054  1.184
#> 5  1.091  0.526  1.656
#> 6 -1.373 -1.906 -0.841
#> ... (4 more)