Data from: Penalized likelihood methods improve parameter estimates in occupancy models

1. Occupancy models are employed in species distribution modelling to account for imperfect detection during field surveys. While this approach is popular in the literature, problems can occur when estimating the model parameters. In particular, the maximum likelihood estimates can exhibit bias and large variance for data sets with small sample sizes, which can result in estimated occupancy probabilities near 0 and 1 (‘boundary estimates’). 2. In this paper, we explore strategies for estimating parameters based on maximizing a penalized likelihood. Penalized likelihood methods augment the usual likelihood with a penalty function that encodes information about what parameter values are undesirable. We introduce penalties for occupancy models that have analogues in ridge regression and Bayesian approaches, and we compare them to a penalty developed for occupancy models in prior work. 3. We examine the bias, variance and mean squared error of parameter estimates obtained from each method on synthetic data. Across all of the synthetic data sets, the penalized estimation methods had lower mean squared error than the maximum likelihood estimates. We also provide an example of the application of these methods to point counts of avian species. Penalized likelihood methods show similar improvements when tested using empirical bird point count data. 4. We discuss considerations for choosing among these methods when modelling occupancy. We conclude that penalized methods may be of practical utility for fitting occupancy models with small sample sizes, and we are releasing R code that implements these methods.1. 占有模型（Occupancy models）在物种分布建模中被用于解决野外调查中的不完全检测问题。尽管该方法在学术文献中应用广泛，但在估计模型参数时仍可能出现问题。具体而言，对于小样本量的数据集，极大似然估计（maximum likelihood estimates）可能存在偏差与较大方差，进而导致估计的占有概率趋近于0或1（即"边界估计"）。2. 本文探讨了基于最大化惩罚似然（penalized likelihood）的参数估计策略。惩罚似然方法通过惩罚函数对常规似然进行扩充，该函数可编码关于"哪些参数值不可取"的信息。我们针对占有模型引入了与岭回归（ridge regression）及贝叶斯方法类似的惩罚项，并将其与先前研究中为占有模型开发的惩罚项进行对比。3. 我们基于合成数据集（synthetic data）检验了各方法所得参数估计的偏差、方差与均方误差（mean squared error）。在所有合成数据集上，惩罚似然估计方法的均方误差均低于极大似然估计。我们还提供了将这些方法应用于鸟类物种样点计数（avian species point counts）的实例。在使用实测鸟类样点计数数据进行测试时，惩罚似然方法同样展现出类似的性能提升。4. 我们讨论了在构建占有模型时选择这些方法所需考虑的因素。最终得出结论：对于小样本量的占有模型拟合任务，惩罚似然方法具备实际应用价值；同时我们将发布实现了这些方法的R语言代码（R code）。