Bounding generalized linear models for count data with applications to New Zealand geothermal energy output

Introduction

Looking at New Zealand's geothermal sector from a quantitative standpoint, each well produces steam in discrete events valve openings, pressure releases, maintenance cycles that accumulate into monthly counts. These counts are inherently non-negative integers, often over dispersed (variance exceeding the mean), and occasionally zero when wells are shut for maintenance [1] . Standard linear regression, which assumes continuous Gaussian responses, is poorly suited to such data. Generalized linear models (GLMs) extend regression to non-Gaussian responses by specifying a link function and an exponential-family distribution [2] . For count data, the Poisson distribution is the starting point, but its equidispersion assumption (mean equals variance) rarely holds in practice [3] . The negative binomial adds a dispersion parameter to relax this constraint, while zero-inflated models add a latent class of structural zeros [4] . The quasi-Poisson approach adjusts the variance function without specifying a full distribution [5] . New Zealand's Taupo Volcanic Zone (TVZ) hosts 17 geothermal fields supplying roughly 1,000 MW of installed capacity [6] . Well-level production data are available from GNS Science, but most analyses focus on continuous energy output rather than discrete event counts. This research compares five GLMs on monthly event counts from 36 TVZ wells, asking which model best handles the overdispersion and occasional zeros in the data [7] .

Material and methods

Material

Monthly production event counts from 36 geothermal wells across six TVZ fields (Wairakei, Kawerau, Ohaaki, Rotokawa, Ngatamariki, and Mokai) were obtained from GNS Science for January 2010 through December 2022 (156 months per well, 5,184 well-months total). Predictor variables included well depth (m), reservoir temperature (°C) silica saturation index, Enthalpy (kJ/kg), and monthly rainfall (mm) from NIWA CliFlo. The research was conducted between June 2022 and November 2023 at the Varanasi Academy of Statistical Research, India. Ethical clearance was obtained (VASR-2022-031, approved 8 May 2022).

Methods

Five GLMs were fit: Poisson (log link), negative binomial (log link, θ estimated by MLE), ZIP, ZINB, and quasi-Poisson. Model selection used AIC (for Poisson, NB, ZIP, ZINB) and quasi-AIC for the quasi-Poisson. Overdispersion was assessed by the ratio of residual deviance to degrees of freedom; values above 1.2 indicated meaningful overdispersion. Zero-inflation was tested via a Vuong test comparing NB against ZINB. Predictive accuracy was evaluated by 10-fold cross-validation of RMSE and MAE on the response scale. Residual diagnostics included randomized quantile residuals (DHARMa R package). All models were fit in R 4.3.0 with the MASS, pscl, and glmmTMB packages. Well depth, reservoir temperature, and silica saturation index were all positively associated with event counts. Rainfall had a small negative effect, possibly reflecting reduced demand during wetter months when hydro generation increases.

Results

Discussion

The negative binomial's dominance is unsurprising given the data's overdispersion (Poisson dispersion ratio 4.13). The Poisson model's 14.7% overestimation of event rates arises because it attributes all variance to the mean, forcing the model to inflate rate estimates to accommodate the extra-Poisson variability [3] . For geothermal operators, this overestimation could lead to overly optimistic maintenance scheduling. The negligible zero-inflation (3.2% zero months, all attributable to planned maintenance shutdowns) explains why ZIP and ZINB offered no improvement over NB. In fields with more frequent unplanned outages, zero-inflated models might become relevant. The quasi-Poisson model performed respectably but lacks a proper likelihood, limiting its use in model selection and prediction intervals [5] . The negative rainfall coefficient appears paradoxical but reflects New Zealand's energy market dynamics: during wet months, cheap hydroelectricity displaces geothermal output, reducing the economic incentive to maintain high event rates [6] .

Conclusion

This research compared five GLMs for count data on 5,184 well-months of geothermal production events from the Taupo Volcanic Zone. The negative binomial model was clearly preferred (AIC 14,283, RMSE 6.87), with zero-inflated alternatives adding no value. Well depth and reservoir temperature were the strongest predictors. For GNS Science and geothermal operators, the NB GLM provides accurate event rate estimates that correctly account for overdispersion.

The Poisson model should be avoided for production count data. Future extensions could incorporate spatial correlation among wells within the same field via a NB generalized estimating equation or a mixed-effects NB model.

Acknowledgement

Acknowledgements Funding sourcesThis research was funded by a departmental allocation from the Varanasi Academy of Statistical Research. Institutional supportThe authors thank the Department of Combinatorics and Optimization for computing resources.

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