ARC Centre of Excellence for Mathematical and Statistical Frontiers, Queensland University of Technology, Brisbane, Australia.
PLoS One. 2020 May 20;15(5):e0233019. doi: 10.1371/journal.pone.0233019. eCollection 2020.
Many methods of spatial smoothing have been developed, for both point data as well as areal data. In Bayesian spatial models, this is achieved by purposefully designed prior(s) or smoothing functions which smooth estimates towards a local or global mean. Smoothing is important for several reasons, not least of all because it increases predictive robustness and reduces uncertainty of the estimates. Despite the benefits of smoothing, this attribute is all but ignored when it comes to model selection. Traditional goodness-of-fit measures focus on model fit and model parsimony, but neglect "goodness-of-smoothing", and are therefore not necessarily good indicators of model performance. Comparing spatial models while taking into account the degree of spatial smoothing is not straightforward because smoothing and model fit can be viewed as opposing goals. Over- and under-smoothing of spatial data are genuine concerns, but have received very little attention in the literature.
This paper demonstrates the problem with spatial model selection based solely on goodness-of-fit by proposing several methods for quantifying the degree of smoothing. Several commonly used spatial models are fit to real data, and subsequently compared using the goodness-of-fit and goodness-of-smoothing statistics.
The proposed goodness-of-smoothing statistics show substantial agreement in the task of model selection, and tend to avoid models that over- or under-smooth. Conversely, the traditional goodness-of-fit criteria often don't agree, and can lead to poor model choice. In particular, the well-known deviance information criterion tended to select under-smoothed models.
Some of the goodness-of-smoothing methods may be improved with modifications and better guidelines for their interpretation. However, these proposed goodness-of-smoothing methods offer researchers a solution to spatial model selection which is easy to implement. Moreover, they highlight the danger in relying on goodness-of-fit measures when comparing spatial models.
已经开发出许多空间平滑方法，既适用于点状数据也适用于面状数据。在贝叶斯空间模型中，这是通过有针对性设计的先验或平滑函数来实现的，这些函数将估计值平滑到局部或全局平均值。平滑具有多个重要原因，其中最重要的原因是它提高了预测的稳健性并降低了估计的不确定性。尽管平滑具有优势，但在模型选择方面，这一属性几乎被忽略。传统的拟合优度度量侧重于模型拟合和模型简约性，但忽略了“平滑优度”，因此不一定是模型性能的良好指标。在考虑空间平滑程度的情况下比较空间模型并不简单，因为平滑和模型拟合可以看作是相互矛盾的目标。过度平滑和欠平滑的空间数据是真实存在的问题，但在文献中很少受到关注。
本文通过提出几种量化平滑程度的方法，展示了仅基于拟合优度进行空间模型选择的问题。将几种常用的空间模型拟合到真实数据上，然后使用拟合优度和平滑优度统计量进行比较。
提出的平滑优度统计量在模型选择任务中具有高度一致性，并且倾向于避免过度或欠平滑的模型。相反，传统的拟合优度标准通常不一致，可能导致模型选择不佳。特别是，著名的偏差信息准则往往会选择过度平滑的模型。
一些平滑优度方法可以通过改进和更好的解释指南进行改进。然而，这些建议的平滑优度方法为研究人员提供了一种易于实施的空间模型选择解决方案。此外，它们强调了在比较空间模型时仅依赖拟合优度度量的危险。
