A Mathematical Framework for Adaptive Rolling Conformal

Quantile Boosting under Temporal Distribution Shift: Application

to Hour-Ahead PM2.5 Forecast Intervals

Aiyared Iampan1,* Said Broumi2

1 School of Science, University of Phayao, 19, Moo 2, Tambon Mae Ka, Amphur Mueang, Phayao 56000, Thailand

2 Laboratory of Information Processing, Faculty of Science Ben MSik, University of Hassan II, Casablanca, Morocco

Emails: aiyared.ia@up.ac.th; broumisaid78@gmail.com

ABSTRACT

Prediction intervals for temporally dependent data require both conditional quantile estimation and a calibration mechanism capable

of responding to distribution shift. An adaptive rolling conformal quantile boosting (ARCQB) formulation is developed in which

boosted quantile functions provide a nonlinear base interval and a sequential state variable controls the empirical conformal quantile.

For target miscoverage 𝛼, the calibration state follows a projected stochastic recurrence, 𝛼𝑡+1 = ΠA{𝛼𝑡 + 𝛾(𝛼 − 𝑒𝑡 )}, where 𝑒𝑡 is

the realized miss indicator. A telescoping identity links the time-averaged miss frequency to the state displacement and projection

residuals; in the unprojected bounded case, the calibration error is 𝑂(𝑇−1). The interval width admits the exact decomposition

𝑤𝑡 = 𝑤(0)

𝑡 + 2𝑞𝑡 , separating predictive sharpness from conformal inflation. Numerical evaluation uses a strictly chronological

one-hour-ahead design on hourly Beijing air-quality measurements. For nominal 90% coverage, raw boosted quantiles attain

83.18%, static conformal calibration 87.40%, and rolling conformal calibration 89.87%. ARCQB attains 90.05% with mean width

47.14 𝜇gm−3 and the lowest interval score, 72.86. Its maximum seasonal coverage deviation is 0.38 percentage points, compared

with 7.99 points for the uncalibrated interval. The numerical behavior is therefore consistent with the feedback relation predicted by

the calibration dynamics, while high-pollution regimes remain the principal source of conditional under-coverage.

Keywords: Conformal prediction ■ Adaptive calibration ■ Quantile regression ■ Stochastic recurrence ■ Distribution

shift ■ Time-series uncertainty

1. INTRODUCTION

For a temporally ordered process, a point predictor and an

uncertainty set solve different mathematical problems. The

first estimates a conditional location functional; the second

must control the frequency with which future observations fall

outside a data-dependent set. In air-quality forecasting these

objectives diverge sharply because pollutant concentrations

are persistent, heteroscedastic, seasonal, and subject to abrupt

regime changes. Recent forecasting models have improved

deterministic accuracy using recurrent, decomposition-based,

spatial–temporal, and transfer-learning architectures [5, 7, 9–

11, 20, 22]. Their predictive intervals, when available, do

not automatically retain nominal coverage after the error

distribution changes.

Conformal prediction supplies a model-agnostic calibration

layer based on empirical score ranks. Exact finite-sample

arguments are strongest under exchangeability [2], while recent

results extend conformal reasoning to non-exchangeable and

dependent observations [3, 17]. Online formulations replace a

fixed calibration level by a state that reacts to realized coverage

errors [6]. This feedback interpretation is especially natural

for environmental series: forecast errors arrive sequentially,

and the uncertainty layer can adapt even if the underlying

nonlinear predictor remains fixed.

Recent reviews describe the rapid expansion of machinelearning

air-quality forecasting [8, 12], including spatially

informed deep predictors [16], alternative input and horizon

structures [18], and recurrent hybrid schemes [23]. Applied

mathematical formulations based on data-driven differential

equations provide a complementary perspective [15]. For