Non-crossing nonlinear regression quantiles by monotone composite quantile regression neural network, with application to rainfall extremes

Abstract

The goal of quantile regression is to estimate conditional quantiles for specified values of quantile probability using linear or nonlinear regression equations. These estimates are prone to ''quantile crossing'', where regression predictions for different quantile probabilities do not increase as probability increases. In the context of the environmental sciences, this could, for example, lead to estimates of the magnitude of a 10-year return period rainstorm that exceed the 20-year storm, or similar nonphysical results. This problem, as well as the potential for overfitting, is exacerbated for small to moderate sample sizes and for nonlinear quantile regression models. As a remedy, this study introduces a novel nonlinear quantile regression model, the monotone composite quantile regression neural network (MCQRNN), that (1) simultaneously estimates multiple non-crossing, nonlinear conditional quantile functions; (2) allows for optional monotonicity, positivity/ non-negativity, and generalized additive model constraints; and (3) can be adapted to estimate standard least-squares regression and non-crossing expectile regression functions. First, the MCQRNN model is evaluated on synthetic data from multiple functions and error distributions using Monte Carlo simulations. MCQRNN outperforms the benchmark models, especially for non-normal error distributions. Next, the MCQRNN model is applied to real-world climate data by estimating rainfall Intensity-Duration-Frequency (IDF) curves at locations in Canada. IDF curves summarize the relationship between the intensity and occurrence frequency of extreme rainfall over storm durations ranging from minutes to a day. Because annual maximum rainfall intensity is a non-negative quantity that should increase monotonically as the occurrence frequency and storm duration decrease, monotonicity and non-negativity constraints are key constraints in IDF curve estimation. In comparison to standard QRNN models, the ability of the MCQRNN model to incorporate these constraints, in addition to non-crossing, leads to more robust and realistic estimates of extreme rainfall.

Projected intensification of sub-daily and daily rainfall extremes in convection-permitting climate model simulations over North America: Implications for future Intensity–Duration–Frequency curves

“…However, it is possible that less restrictive forms of statistical model might reveal different future changes. For example, regional quantile regression models for IDF curves (Ouali and Cannon,20 2018), including those that explicitly incorporate temporal scaling relationships (Cannon, 2018), are free from the parametric assumptions of the GEVSS model and may provide additional insight into dependence of changes on event rarity and duration.…”

Snow Drought Risk and Susceptibility in the Western United States and Southwestern Canada

“…This physical basis provides confidence in the chosen methodology and outcomes and guidance for future studies. However, additional studies, using artificial neural networks or information theoretic polynomial selection (Cannon, 2018;Fleming, 2007;Fleming & Dahlke, 2014), for example, may provide additional insights into the relationship between T w and snow drought risk.…”

Projected intensification of sub-daily and daily rainfall extremes in convection-permitting climate model simulations over North America: implications for future intensity–duration–frequency curves

“…A relatively broad uniform prior distribution, with limits constrained to be positive multiples of i 0 CTRL , is used for both the location µ 0 and scale σ 0 parameters of the CTRL and PGW simulations. This choice is informed by past work showing (1) that end-of-century projected changes in annual rainfall extremes by CMIP5 models, which scale similarly to µ 0 and σ 0 , are expected to be less than half the upper limit of the prior (Kharin et al, 2013;Toreti et al, 2013;Cannon et al, 2015) -d) show the associated percentage changes in return levels for each duration: (b) with constant H , increasing µ 0 and σ 0 without changing the dispersion leads to relative increases in return levels for all durations that match the relative changes in the underlying parameters; (c) increasing dispersion leads to return period dependence of changes, with larger relative increases evident at longer return periods; (d) increasing H steepens the IDF curves, which leads to duration dependence of changes; and (e) increases in both H and dispersion result in greater intensification at longer return periods and shorter durations. Note that values in (e) are based on domain mean values from Sect.…”

“…To this end, hourly 4 km rainfall outputs from historical and end-of-century pseudo-global-warming convectionpermitting simulations by the Weather Research and Forecasting (WRF) model Liu et al, 2017) are used in conjunction with a parsimonious generalized extreme value simple scaling (GEVSS) model (Nguyen et al, 1998;Van de Vyver, 2015;Blanchet et al, 2016;Mélèse et al, 2018) to estimate historical and future IDF curves (1 to 24 h durations) over a domain covering northern Mexico, the conterminous US, and southern Canada. The pseudo-global-warming simulation perturbs historical boundary conditions with the climate change signal obtained from an ensemble of global climate models.…”