Quantile regression forest

Overview

A quantile regression forest predicts conditional quantiles of the target instead of only its mean, so each input gets a predicted distribution. The problem it addresses: a mean hides the tails, and questions such as "what share of homes exceed a reference value" need the distribution, not a conditional mean 2310.11143 p. 2. The method predicts several quantiles per row and turns them into a distribution that can be sampled and aggregated 2310.11143 p. 7.

Only one card in the corpus names this method. Its evidence therefore comes from one application: indoor radon in Germany, trained on a survey of 14,053 measurements 2310.11143 p. 7. Treat the variant below as one worked design, not a benchmarked family.

Variants

Quantile readout with a per-row lognormal and Monte Carlo aggregation

The forest is a conditional random forest with 500 trees and mtry 4 2310.11143 p. 7. For each row, nine quantiles are predicted, at the 10th, 25th, 50th, 75th, 80th, 85th, 90th, 95th and 98th percentile 2310.11143 p. 7. A local offset (outdoor radon) is subtracted from the quantiles, and a three-parameter lognormal is fitted to the remainder 2310.11143 p. 7. Samples are drawn from each row's lognormal in proportion to the expected population of that row, the offset is added back, and the draws are grouped for summaries 2310.11143 p. 7. Card: High-resolution indoor radon map.

The paper's stated loss is RMSE on the untransformed target [arxiv:2310.11143 p. 6]. Two details that matter for reimplementation are in the authors' code, not the paper: a clamp that keeps the offset below the 10th percentile, and fit weights of 1 to 6 that rise with percentile [arxiv:2310.11143 code: IRC prediction individual buildings.R#L191-L195] [arxiv:2310.11143 code: IRC prediction individual buildings.R#L202].

When to use which

Implementation sketch

The sketch follows the paper's steps. Where the code adds a step that the paper does not describe, the comment says so.

# Fit a quantile regression forest on survey rows  [arxiv:2310.11143 p. 7]
# Input:  rows (target, predictors), spatial folds
# Output: forest that returns conditional quantiles

def fit_forest(rows, predictors, spatial_folds):
    # Forward selection by spatial-CV RMSE  [arxiv:2310.11143 p. 7]
    selected = forward_select(rows, predictors, spatial_folds)
    # Final fit: 500 trees, mtry 4  [arxiv:2310.11143 p. 7]
    return grow_forest(rows[selected], ntree=500, mtry=4)

# Sample one row's distribution  [arxiv:2310.11143 p. 7]
# Input:  row predictors, weight (expected people), offset
# Output: integer samples, or none when the weight is tiny

def sample_row(forest, row, weight, offset):
    # Nine conditional quantiles for this row  [arxiv:2310.11143 p. 7]
    q = forest.predict_quantiles(row, [0.10, 0.25, 0.5, 0.75, 0.8,
                                       0.85, 0.9, 0.95, 0.98])
    # Code-only clamp: keep the offset below the 10th percentile
    # [arxiv:2310.11143 code: IRC prediction individual buildings.R#L191-L195]
    offset = min(offset, q[0] - 1)
    # Fit a lognormal to the shifted quantiles; code uses weights 1..6
    # [arxiv:2310.11143 p. 7]
    # [arxiv:2310.11143 code: IRC prediction individual buildings.R#L202]
    shifted = [x - offset for x in q]
    mu, sigma = fit_lognormal(shifted, weights=[1, 2, 3, 4, 5, 6, 6, 6, 6])
    # Samples scale with weight; zero after rounding means no samples
    # [arxiv:2310.11143 p. 7]
    n = round(10 * weight)
    return [round(x + offset) for x in lognormal(mu, sigma, n)]

Contradictions

None across cards, since there is one card. Within the paper and its code, two things do not line up. The paper writes the exceedance threshold once as "exceedance of 300" and once as "> 300 Bq/m³", and it does not say whether equal values count; the code counts them 2310.11143 p. 9 2310.11143 p. 13 2310.11143 code: aggregate predictions.R#L119-L122. Pick one rule and state it. The paper's final predictor set cannot be checked against the code, because the selected set is not written to disk 2310.11143 code: model building.R#L66-L75.

Papers

• High-resolution indoor radon map: a quantile regression forest, per-floor lognormal fits and Monte Carlo sampling for German homes. Author-reported, with the survey data not public.