Generalized additive models are generalized linear models in which the linear predictor includes a sum of smooth functions of covariates, where the shape of the functions is to be estimated. They have also been generalized beyond the original generalized linear model setting to distributions outside the exponential family and to situations in which multiple parameters of the response distribution may depend on sums of smooth functions of covariates. The widely used computational and inferential framework in which the smooth terms are represented as latent Gaussian processes, splines, or Gaussian random effects is reviewed, paying particular attention to the case in which computational and theoretical tractability is obtained by prior rank reduction of the model terms. An empirical Bayes approach is taken, and its relatively good frequentist performance discussed, along with some more overtly frequentist approaches to model selection. Estimation of the degree of smoothness of component functions via cross validation or marginal likelihood is covered, alongside the computational strategies required in practice, including when data and models are reasonably large. It is briefly shown how the framework extends easily to location-scale modeling, and, with more effort, to techniques such as quantile regression. Also covered are the main classes of smooths of multiple covariates that may be included in models: isotropic splines and tensor product smooth interaction terms.

中文翻译：

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广义加法模型


广义加法模型是广义线性模型，其中线性预测器包括协变量的平滑函数之和，其中要估计函数的形状。它们也已从最初的广义线性模型设置推广到指数族之外的分布，以及响应分布的多个参数可能取决于协变量的平滑函数之和的情况。回顾了广泛使用的计算和推理框架，其中平滑项表示为潜在的高斯过程、样条或高斯随机效应，特别注意通过模型项的先验秩减少获得计算和理论可处理性的情况。采用实证贝叶斯方法，并讨论了其相对较好的频率主义表现，以及一些更明显的模型选择频率主义方法。涵盖了通过交叉验证或边际似然估计组件函数的平滑度，以及实践中所需的计算策略，包括当数据和模型相当大时。简要介绍了该框架如何轻松扩展到位置比例建模，以及通过更多努力扩展到分位数回归等技术。此外，还涵盖了模型中可能包含的多个协变量平滑的主要类别：各向同性样条和张量积平滑交互项。