Volume 5 β’ Issue 2 β’ PP: 15β21 β’ 2025
A Mathematical Framework for Adaptive Rolling Conformal Quantile Boosting under Temporal Distribution Shift: Application to Hour-Ahead PM2.5 Forecast Intervals
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Β© 2025 The Author(s). Published by ASPG. This article is licensed under the Creative Commons Attribution 4.0 International License (CC BY 4.0).
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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 πΌ, the calibration state follows a projected stochastic recurrence, πΌπ‘+1 = Ξ A{πΌπ‘ + πΎ(πΌ β ππ‘ )}, where ππ‘ 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 π(πβ1). The interval width admits the exact decomposition π€π‘ = π€(0) π‘ + 2ππ‘ , 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 πgmβ3 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.
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References
[1] Aggarwal, A., & Toshniwal, D. (2021). A hybrid deep learning framework for urban air quality forecasting. Journal of Cleaner Production, 329, 129660.
[2] Barber, R. F., CandΓ¨s, E. J., Ramdas, A., & Tibshirani, R. J. (2021). Predictive inference with the jackknife+. The Annals of Statistics, 49(1), 486β507.
[3] Barber, R. F., CandΓ¨s, E. J., Ramdas, A., & Tibshirani, R. J. (2023). Conformal prediction beyond exchangeability. The Annals of Statistics, 51(2), 816β845.
[4] Bian, M., & Barber, R. F. (2023). Training-conditional coverage for distribution-free predictive inference. Electronic Journal of Statistics, 17(2), 2044β2066.
[5] Chang, Y.-S., Chiao, H.-T., Abimannan, S., Huang, Y.-P., Tsai, Y.-T., & Lin, K.-M. (2020). An LSTM-based aggregated model for air pollution forecasting. Atmospheric Pollution Research, 11(8), 1451β1463.
[6] Gibbs, I., & CandΓ¨s, E. J. (2021). Adaptive conformal inference under distribution shift. Advances in Neural Information Processing Systems, 34, 1660β1672.
[7] Jin, N., Zeng, Y., Yan, K., & Ji, Z. (2021). Multivariate air quality forecasting with nested long short term memory neural network. IEEE Transactions on Industrial Informatics, 17(12), 8514β8522.
[8] Liao, Q., Zhu, M., Wu, L., Pan, X., Tang, X., & Wang, Z.
(2020). Deep learning for air quality forecasts: A review. Current
Pollution Reports, 6(4), 399β409.
[9] Lu, X., Ye, C., Shan, M., Qin, B., Wang, Y., Xing, H., Xie, X., & Liu, Z. (2023). The prediction of PM2.5 concentration using transfer learning based on ADGRU. Water, Air, & Soil Pollution, 234, 258.
[10] Ma, J., Ding, Y., Cheng, J. C. P., Jiang, F., Gan, V. J. L., & Xu, Z. (2020). A Lag-FLSTM deep learning network based on Bayesian optimization for multi-sequential-variant PM2.5 prediction. Sustainable Cities and Society, 60, 102237.
[11] Mao, W., Wang, W., Jiao, L., Zhao, S., & Liu, A. (2021). Modeling air quality prediction using a deep learning approach: Method optimization and evaluation. Sustainable Cities and Society, 65, 102567.
[12] Masood, A., & Ahmad, K. (2021). A review on emerging artificial intelligence (AI) techniques for air pollution forecasting: Fundamentals, application and performance. Journal of Cleaner Production, 322, 129072.
[13] Steinberger, L., & Leeb, H. (2023). Conditional predictive inference for stable algorithms. The Annals of Statistics, 51(1), 290β311.
[14] Torres, J. F., Hadjout, D., Sebaa, A., MartΓnez-Γlvarez, F., & Troncoso, A. (2021). Deep learning for time series forecasting: A survey. Big Data, 9(1), 3β21.
[15] Wang, Y., Wang, H., & Zhang, S. (2020). Prediction of daily PM2.5 concentration in China using data-driven ordinary differential equations. Applied Mathematics and Computation, 375, 125088.
[16] Xiao, F., Yang, M., Fan, H., Fan, G., & Al-Qaness, M. A. A. (2020). An improved deep learning model for predicting daily PM2.5 concentration. Scientific Reports, 10, 20988.
[17] Xu, C., & Xie, Y. (2023). Sequential predictive conformal inference for time series. Proceedings of the 40th International Conference on Machine Learning, PMLR 202, 38707β38727.
[18] Yang, J., Yan, R., Nong, M., Liao, J., Li, F., & Sun, W. (2021). PM2.5 concentrations forecasting in Beijing through deep learning with different inputs, model structures and forecast time. Atmospheric Pollution Research, 12(9), 101168.
[19] Zaini, N., Ean, L. W., Ahmed, A. N., Abdul Malek, M., & Chow, M. F. (2022). PM2.5 forecasting for an urban area based on deep learning and decomposition method. Scientific Reports, 12, 17565.
[20] Zeng, Y., Chen, J., Jin, N., Jin, X., & Du, Y. (2022). Air quality forecasting with hybrid LSTM and extended stationary wavelet transform. Building and Environment, 213, 108822.
[21] Zhang, Z., Zeng, Y., & Yan, K. (2021). A hybrid deep learning technology for PM2.5 air quality forecasting. Environmental Science and Pollution Research, 28, 39409β39422.
[22] Zhang, K., Yang, X., Cao, H., ThΓ©, J., Tan, Z., & Yu, H. (2023). Multi-step forecast of PM2.5 and PM10 concentrations using convolutional neural network integrated with spatialβtemporal attention and residual learning. Environment International, 171, 107691.
[23] Zheng, G., Liu, H., Yu, C., Li, Y., & Cao, Z. (2022). A new PM2.5 forecasting model based on data preprocessing, reinforcement learning and gated recurrent unit network. Atmospheric Pollution Research, 13(7), 101475.
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