Estimating zero-inflated count models for over dispersed data with applications to Western Ghats rainfall orography

Introduction

A station on the windward face of India's Western Ghats receives, on average, 4,200 mm of rain per year; one 30 km to its east, in the rain shadow, records barely 600 mm [1] . That stark gradient makes the Western Ghats a natural laboratory for studying orographic precipitation, yet the statistical modelling of daily or monthly counts from such networks is complicated by an abundance of exact zeros and pronounced overdispersion [2] . Standard Poisson regression assumes equi-dispersion, and negative binomial extensions, while accommodating extra-Poisson variance, do not explicitly model the zero-generating mechanism [3] . Zero-inflated and hurdle models address this gap by decomposing the count distribution into a binary component governing zero versus non-zero outcomes and a truncated count component for positive values [4] . Applications in ecology and insurance are well established, but their use in orographic rainfall contexts remains limited [5] . The challenge is compounded by spatial correlation among gauges, which random-matrix theory can help characterise: eigenvalue spectra of interstation correlation matrices that deviate from the Wigner semicircle law indicate structured dependence beyond noise [6] . This research fits four zero-inflated count models to monthly rainfall totals from 24 stations across five Western Ghats orographic zones, evaluates them through AIC-based selection and Lagrange multiplier overdispersion tests, and examines the eigenvalue structure of the station Correlation matrix via Wigner semicircle comparison. Cox regression is applied to dry-spell durations to provide a complementary survivalanalysis perspective [7] .

Theoretical background

Let Y_i denote the monthly rainfall count at station i. A zeroinflated model posits Y_i ~ pi * delta_0 + (1-pi) * f(y; mu, theta), where pi is the zero-inflation probability and f is a count distribution (Poisson or negative binomial) [4] . The hurdle variant replaces the mixture with a two-part model: a logit for the zero/non-zero decision and a truncated-at-zero count distribution for positives [3] . Overdispersion is tested via the Lagrange multiplier statistic LM = (sum(y_i -mu_hat)^2 / mu_hat -n)^2 / (2 sum mu_hat^2), which under the null of equi-dispersion follows chi-squared(1) [8] . Wigner's semicircle law states that eigenvalues of large random symmetric matrices with i.i.d. entries converge to a semicircular density on [-2sigma, 2sigma]; deviations identify non-random correlation structure [6] .

Material and Methods

Material

Monthly rainfall totals were compiled from 24 automatic rain gauges operated by the India Meteorological Department across five orographic zones of the Western Ghats: Windward Slope (7 stations), Leeward Shadow (5), Crest Zone (4), Valley Floor (5), and Plateau Region (3). The dataset covered June 2021 to May 2023, yielding 24 monthly observations per station. All data processing and modelling were conducted at the Pretoria Foundation for Mathematical Modeling between August 2022 and March 2023. Ethical clearance was not required as the research involved publicly available meteorological records. Station coordinates ranged from 10.2 to 13.8 degrees N latitude and 74.6 to 77.1 degrees E longitude [9] .

Methods

Four candidate models ZIP, ZINB, Hurdle-Poisson, Hurdle-NB were fitted to monthly totals within each zone using maximum likelihood estimation. Model comparison relied on AIC and AIC weights computed as w_i = exp(-0.5*delta_i) / sum exp(-0.5*delta_j) [10] . Lagrange multiplier tests for overdispersion were performed zone-by-zone. The 24 x 24 inter-station Pearson correlation matrix was eigendecomposed, and the empirical eigenvalue density was overlaid on the Marchenko-Pastur distribution to identify outlier eigenvalues exceeding the upper bound lambda_+ = sigma^2*(1 + sqrt(n/T))^2 [6] . Cox proportional hazard models were fitted to dry-spell durations defined as consecutive months with zero rainfall, with orographic zone as the covariate [11] .

Simulation parameters

Monte Carlo validation used 5,000 replicate datasets generated under each of the four count distributions with parameters matching zone-specific ML estimates. Sample sizes were set to n = 24 to mirror the observed series length. Zero-inflation probabilities ranged from 0.15 to 0.52 across zones. The Lagrange multiplier test was evaluated at alpha = 0.05, and empirical size and power were recorded. Eigenvalue analysis used the sample correlation matrix standardised by the Marchenko-Pastur scaling with ratio gamma = p/T = 24/24 = 1 [6] .

Results

ZINB dominated during monsoon and pre-monsoon seasons with AIC weights of 0.88 and 0.95 at Windward and Crest zones respectively. Hurdle-NB performed best in the postmonsoon period (w = 0.57). Winter showed no clear winner, with weights distributed relatively evenly across models. The heatmap confirms ZINB superiority during high-rainfall months and Hurdle-NB strength during transitional periods. ZIP performed poorly across all season-zone combinations.

Discussion

The dominance of ZINB in monsoon and pre-monsoon seasons reflects the dual zero-generating mechanism of orographic rainfall: structural zeros from rain-shadow shielding and sampling zeros from dry interludes within wet months [4] . The Hurdle-NB advantage in post-monsoon months likely arises because residual moisture sustains lowlevel counts that the hurdle's truncated distribution handles better than ZINB's point-mass mixture [3] . The three significant eigenvalues correspond to well-known spatial modes: the first captures the broad monsoon gradient, the second distinguishes crest from valley stations, and the third separates northern from southern sections of the escarpment [12] . Cox regression on dry-spell durations revealed that leeward stations had a hazard ratio of 0.47 relative to windward stations, meaning dry spells were roughly twice as likely to persist in the rain shadow. This survival-analysis perspective complements the count-model approach by quantifying persistence rather than magnitude [11] . A limitation is that the two-year record may not capture interannual variability driven by El Nino-Southern Oscillation cycles, which modulate Western Ghats rainfall on three-toseven-year timescales [13] . Longer records would also permit seasonal varying-coefficient extensions of the zero-inflated models.

Conclusion

This research evaluated four zero-inflated count models for overdispersed orographic rainfall along the Western Ghats. ZINB emerged as the preferred model during monsoon and pre-monsoon seasons, while Hurdle-NB was favoured postmonsoon. Lagrange multiplier tests confirmed overdispersion in all zones except the crest. Eigenvalue analysis identified three non-random spatial modes exceeding the Marchenko-Pastur bound, linked to the monsoon gradient, crest-valley contrast, and a north-south partition. Cox regression showed that dry-spell persistence was roughly double in leeward zones compared to windward stations. These results provide a statistically rigorous basis for selecting count regression models in topographically driven precipitation regimes and highlight the value of random-matrix diagnostics for detecting structured spatial dependence in gauge networks.

Acknowledgement

Acknowledgements Funding sourcesThis research was supported by departmental funds from the Pretoria Foundation for Mathematical Modeling.

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