A univariate continuous function can always be decomposed as the sum of a non-increasing function and a non-decreasing one. Based on this property, we propose a non-parametric regression method that combines two spline-fitted monotone curves. We demonstrate by extensive simulations that, compared to standard spline-fitting methods, the proposed approach is particularly advantageous in high-noise scenarios. Several theoretical guarantees are established for the proposed approach. Additionally, we present statistics to test the monotonicity of a function based on monotone decomposition, which can better control Type I error and achieve comparable (if not always higher) power compared to existing methods. Finally, we apply the proposed fitting and testing approaches to analyze the single-cell pseudotime trajectory datasets, identifying significant biological insights for non-monotonically expressed genes through Gene Ontology enrichment analysis. The source code implementing the methodology and producing all results is accessible at https://github.com/szcf-weiya/MonotoneDecomposition.jl.

翻译：一元连续函数总能分解为一个非增函数和一个非减函数之和。基于这一性质，我们提出一种结合两条样条拟合单调曲线的非参数回归方法。通过大量模拟实验证明，与标准样条拟合方法相比，该方法在高噪声场景下具有显著优势。我们为所提方法建立了若干理论保证。此外，基于单调分解提出检验函数单调性的统计量，该方法能更好控制第一类错误，并在功效方面（若非始终）达到可比甚至更高的水平。最终，我们将所提拟合与检验方法应用于单细胞伪时间轨迹数据集分析，通过基因本体富集分析识别出非单调表达基因的重要生物学意义。实现该方法并生成所有结果的源代码见https://github.com/szcf-weiya/MonotoneDecomposition.jl。