  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.
