Stochastic differential equations (SDEs) are popular tools to analyse time series data in many areas, such as mathematical finance, physics, and biology. They provide a mechanistic description of the phenomenon of interest, and their parameters often have a clear interpretation. These advantages come at the cost of requiring a relatively simple model specification. We propose a flexible model for SDEs with time-varying dynamics where the parameters of the process are nonparametric functions of covariates, similar to generalized additive models. Combining the SDE and nonparametric approaches allows for the SDE to capture more detailed, non-stationary, features of the data-generating process. We present a computationally efficient method of approximate inference, where the SDE parameters can vary according to fixed covariate effects, random effects, or basis-penalty smoothing splines. We demonstrate the versatility and utility of this approach with three applications in ecology, where there is often a modelling trade-off between interpretability and flexibility.

Supplementary materials accompanying this paper appear online.

随机微分方程（SDE）是在许多领域分析时间序列数据的流行工具，例如数学金融，物理学和生物学。它们提供了对感兴趣现象的机械描述，并且它们的参数通常具有清晰的解释。这些优点是以需要相对简单的模型规格为代价的。我们为时变动力学的SDE提出了一个灵活的模型，该过程的参数是协变量的非参数函数，类似于广义加性模型。将SDE与非参数方法相结合，可以使SDE捕获数据生成过程的更详细，非平稳的特征。我们提供了一种计算有效的近似推理方法，其中SDE参数可以根据固定的协变量效应，随机效应，或基本惩罚平滑样条线。我们在生态学中的三个应用中证明了这种方法的多功能性和实用性，在生态学中，可解释性和灵活性之间通常需要进行模型取舍。

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