Fig. 2: Identifying dairy facilities likely to exceed regulatory ammonia thresholds.
a Bayesian non-linear regression fit to model relationship between all NH3 observations from dairy facilities in the NAEMS and animal inventory. Shaded green region shows 50% predictive interval. Dashed vertical green line indicates minimum inventory (1173) with 50% likelihood of exceeding 100 lbs NH3/day (dashed gray horizontal line). Shaded blue region shows 95% predictive interval. Dashed vertical blue line indicates minimum inventory (2067) with 95% likelihood of exceeding 100 lbs NH3/day. b Histogram showing the number of cows (thousand heads) in each facility size class for all dairy facilities in the U.S. ranging from facilities having 1–9 head to facilities with 2500 or more heads. c Histogram showing the number of dairy facilities in the U.S. in each facility class size. Predictive interval modeling details: We fit a non-linear model of the form a + b*Inventory + d*Inventory^2 to the relationship between NH3 emissions and Inventory using Markov Chain Monte Carlo (MCMC) with normally-distributed priors. To account for heteroskedasticity, we modeled the precision of the expectation using the relationship 1/(s*Inventory)^3, where a, b, d, and s are all fitted parameters. The prior for s was log-normally distributed. We ran the MCMC using the R package (rjags) and the software Jags with 3 chains, a burn-in period of 1,000 iterations, and 100,000 iterations following the burn-in period. We used the Gelman and Rubin’s diagnostic to assess convergence of chains and achieved an effective sample size of over 4700 for each parameter. We fit 50% and 95% predictive intervals for the model by sampling posterior predictions and identified the Inventory value where the lower bound of each predictive interval crosses the 100 lb/day (45.4 kg/day) threshold.
