Computational Efficiency and Precision for Replicated-Count and Batch-Marked Hidden Population Models.

We address two computational issues common to open-population N-mixture models, hidden integer-valued autoregressive models, and some hidden Markov models. The first issue is computation time, which can be dramatically improved through the use of a fast Fourier transform. The second issue is tractability of the model likelihood function for large numbers of hidden states, which can be solved by improving numerical stability of calculations. As an illustrative example, we detail the application of these methods to the open-population N-mixture models. We compare computational efficiency and precision between these methods and standard methods employed by state-of-the-art ecological software. We show faster computing times (a $\sim 6$ to $\sim 30$ times speed improvement for population size upper bounds of 500 and 1000, respectively) over state-of-the-art ecological software for N-mixture models. We also apply our methods to compute the size of a large elk population using an N-mixture model and show that while our methods converge, previous software cannot produce estimates due to numerical issues. These solutions can be applied to many ecological models to improve precision when logs of sums exist in the likelihood function and to improve computational efficiency when convolutions are present in the likelihood function. Supplementary materials accompanying this paper appear online. Supplementary materials for this article are available at 10.1007/s13253-022-00509-y.