Introduction to spconform

Overview

spconform provides distribution-free, finite-sample prediction intervals for spatial and spatio-temporal data by relaxing the exchangeability assumption of standard conformal prediction. It offers two core procedures:

• scp_geostatistical() for point-referenced (geostatistical) data, using locally weighted split conformal prediction with spatial (and, optionally, temporal) Gaussian kernels.
• scp_areal() for areal (lattice) data, using a neighbourhood-weighted leave-one-out conformal procedure based on a graph adjacency structure.

Both procedures are model-agnostic: you supply your own point predictor (a linear model, a GAM, a random forest, kriging, or anything else), and spconform handles only the conformal calibration layer, producing prediction intervals with a guaranteed marginal coverage level regardless of whether your predictor is well specified.

This vignette illustrates both procedures on the meuse river dataset (Pebesma and Bivand 2005), a standard geostatistics benchmark distributed with the sp package.

library(sp)
data(meuse)

s <- as.matrix(meuse[, c("x", "y")])
y <- log(meuse$zinc)

We use log-transformed zinc concentration (ppm) as the response, and the projected coordinates as the spatial index. The 155 sampling locations trace a diagonal band following the natural course of the river.

plot(meuse$x, meuse$y, col = rgb(0.2, 0.4, 0.8, 0.5), pch = 19,
     xlab = "X coordinate", ylab = "Y coordinate",
     main = "Meuse Sampling Locations")

Geostatistical (point-referenced) prediction

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))
  cbind(1, s_new[, 1], s_new[, 2], s_new[, 1]^2, s_new[, 2]^2) %*% coef(fit)
}

set.seed(1)
n <- nrow(s)
idx <- sample(n, floor(0.7 * n))

s_train <- s[idx, ]; y_train <- y[idx]
s_test  <- s[-idx, ]; y_test  <- y[-idx]

out <- scp_geostatistical(s_train, y_train, s_test, pred_fun,
                          alpha = 0.1, seed = 1)
print(out)
#> <spconform> geostatistical conformal prediction
#> Target coverage: 90.0%
#> Number of prediction points: 47
#>    pred lower upper
#> 1 6.573 5.661 7.484
#> 2 6.255 5.344 7.167
#> 3 5.954 5.043 6.866
#> 4 6.455 5.544 7.367
#> 5 6.171 5.260 7.083
#> 6 5.773 4.862 6.685
#> ... (41 more)

coverage_report(out, y_test)
#> $coverage
#> [1] 0.9574468
#> 
#> $mean_width
#> [1] 2.210471

plot(out, y_true = y_test)

diag <- diagnose(out, y_true = y_test, s_test = s_test, plot = TRUE)
#> Note: Empirical coverage (0.957) exceeds nominal (0.9) by >5%. Consider reducing 'bandwidth' for tighter intervals.

print(diag)
#> === spconform Diagnostic Report ===
#> 
#> Marginal coverage:
#>   Empirical: 0.9574  (nominal: 0.9 )
#>   Mean width: 2.2105 
#>   n = 47 , covered = 45 
#> 
#> Conditional coverage by spatial bin:
#>   Q1-1: 1 (n=8, width=2.242)
#>   Q1-2: 1 (n=3, width=2.311)
#>   Q2-1: 1 (n=3, width=2.242)
#>   Q2-2: 1 (n=2, width=2.346)
#>   Q2-3: 0.8889 (n=9, width=2.346)
#>   Q3-1: 0 (n=1, width=2.242)
#>   Q3-2: 1 (n=4, width=2.294)
#>   Q3-3: 1 (n=6, width=2.346)
#>   Q4-3: 1 (n=1, width=2.346)
#>   Q4-4: 1 (n=10, width=1.865)
#> 
#> Boundary effect:
#>   Near boundary:   0.9167 (n=24)
#>   Far from boundary:1 (n=23)
#> 
#> Nonconformity scores:
#>   Mean: 1.1052 
#>   Median: 1.1212 
#>   SD: 0.0981 
#>   90% quantile: 1.1728

set.seed(123)
coverages <- numeric(50)
widths    <- numeric(50)

for (i in 1:50) {
  idx_i <- sample(n, floor(0.7 * n))
  s_tr <- s[idx_i, ]; y_tr <- y[idx_i]
  s_te <- s[-idx_i, ]; y_te <- y[-idx_i]

  out_i <- scp_geostatistical(s_tr, y_tr, s_te, pred_fun,
                              alpha = 0.1, seed = i)

  rep_i <- coverage_report(out_i, y_te)
  coverages[i] <- rep_i$coverage
  widths[i]    <- rep_i$mean_width
}

mean(coverages)
#> [1] 0.9204255
sd(coverages)
#> [1] 0.04358497
mean(widths)
#> [1] 1.973462

hist(coverages, breaks = 15, col = "lightblue", border = "white",
     main = "Empirical Coverage Across 50 Random Splits",
     xlab = "Empirical Coverage", xlim = c(0.7, 1))
abline(v = 0.90, col = "red", lwd = 2, lty = 2)
legend("topleft", legend = "Nominal target (0.90)",
       col = "red", lty = 2, bty = "n")

plot_df <- data.frame(
  x = s_test[, 1],
  y = s_test[, 2],
  width = out$upper - out$lower
)

plot(plot_df$x, plot_df$y,
     cex = plot_df$width, pch = 19,
     col = rgb(0.2, 0.4, 0.8, 0.5),
     xlab = "X coordinate", ylab = "Y coordinate",
     main = "Spatial Distribution of Interval Width")

Areal (lattice) prediction

scp_areal() targets data observed on a fixed set of areal units (e.g., counties, grid cells) linked by an adjacency structure, rather than continuous coordinates. To illustrate this on the same phenomenon, we aggregate the point-referenced Meuse data onto a regular \(6\times6\) grid, retaining occupied cells and taking the mean log-zinc concentration within each as the areal response.

xbreaks <- seq(min(meuse$x), max(meuse$x), length.out = 7)
ybreaks <- seq(min(meuse$y), max(meuse$y), length.out = 7)

meuse$cell_x  <- cut(meuse$x, xbreaks, include.lowest = TRUE, labels = FALSE)
meuse$cell_y  <- cut(meuse$y, ybreaks, include.lowest = TRUE, labels = FALSE)
meuse$cell_id <- (meuse$cell_y - 1) * 6 + meuse$cell_x

agg <- aggregate(log(zinc) ~ cell_id, data = meuse, FUN = mean)
names(agg) <- c("cell_id", "y")

cell_coords <- unique(meuse[, c("cell_id", "cell_x", "cell_y")])
agg <- merge(agg, cell_coords, by = "cell_id")
agg <- agg[order(agg$cell_id), ]

n_cells <- nrow(agg)
adj <- matrix(0, n_cells, n_cells)
for (i in 1:n_cells) {
  for (j in 1:n_cells) {
    if (i != j) {
      dx <- abs(agg$cell_x[i] - agg$cell_x[j])
      dy <- abs(agg$cell_y[i] - agg$cell_y[j])
      if (dx <= 1 && dy <= 1) adj[i, j] <- 1
    }
  }
}

adj is a binary adjacency matrix linking grid-adjacent cells. We now apply scp_areal() at a nominal 80% coverage level, using the default neighbourhood-mean predictor:

out2 <- scp_areal(agg$y, adjacency = adj, alpha = 0.2)
print(out2)
#> <spconform> areal conformal prediction
#> Target coverage: 80.0%
#> Number of prediction points: 21
#>    pred lower upper
#> 1 5.966 5.148 6.785
#> 2 5.749 4.854 6.643
#> 3 5.981 5.145 6.817
#> 4 5.502 4.608 6.396
#> 5 6.495 4.576 8.415
#> 6 5.934 5.116 6.752
#> ... (15 more)
summary(out2)
#> spconform summary
#> ------------------
#> Type:                  areal 
#> Target coverage:       80.0% 
#> Mean interval width:   1.8666 
#> Median interval width: 1.7888
coverage_report(out2, agg$y)
#> $coverage
#> [1] 0.7619048
#> 
#> $mean_width
#> [1] 1.866613

plot(out2, y_true = agg$y)

Most areal units show narrow intervals, with occasional exceptions at units with a sparse neighbourhood (e.g., boundary cells of the grid). This is a desirable property of scp_areal(): units with fewer graph neighbours have a smaller, less informative local calibration set, and their wider interval correctly reflects the higher predictive uncertainty at the periphery of the spatial domain.

boxplot(list(Geostatistical = out$upper - out$lower,
             Areal          = out2$upper - out2$lower),
        main = "Interval Width Comparison",
        ylab = "Interval Width",
        col  = c("lightblue", "lightgreen"))

Summary

Both procedures achieve empirical coverage close to their nominal targets on this real environmental dataset, using deliberately simple underlying predictors (a misspecified trend surface, and a neighbourhood mean), illustrating that the coverage guarantee comes from the localized conformal calibration layer itself.

Using your own predictor

Both scp_geostatistical() and scp_areal() accept an arbitrary prediction function: * For geostatistical data: any function function(s_train, y_train, s_new) returning numeric predictions (e.g., via gstat, mgcv, or ranger). * For areal data: any custom function(y_train, X_train, idx_train, idx_target, adjacency) in place of the default neighbourhood-mean predictor.

Platform portability

spconform is implemented in pure base R, importing only stats, graphics, and grDevices. It has been verified to pass R CMD check --as-cran with 0 errors, 0 warnings, and 0 notes across Linux, macOS, and Windows.

References

• Mao, H., Martin, R., and Reich, B. J. (2024). Valid Model-Free Spatial Prediction. Journal of the American Statistical Association, 119(546), 904–914. doi:10.1080/01621459.2022.2147531[cite: 1]
• Pebesma, E. J., and Bivand, R. S. (2005). Classes and Methods for Spatial Data in R. R News, 5(2), 9–13.[cite: 1]
• Vovk, V., Gammerman, A., and Shafer, G. (2005). Algorithmic Learning in a Random World. Springer.[cite: 1]