This thesis presents an in-depth analysis of time alignment methods for biological data, specifically in the area of pseudotime alignment. The thesis begins by introducing the motivation and topics, followed by a detailed examination of three popular single-cell trajectory inference methods: Slingshot, PAGA, and MATCHER. A comprehensive comparison of these methods is presented, including assessments of their performance on linear, bifurcating, and converging datasets. This thesis then delves into numerical studies of pseudotime alignment based on synthetic multi-cell data, investigating the effects of different mathematical functions and noise levels on alignment accuracy. An exploration of the impact of varying the number of cells and time points per cell is also considered. Lastly, this thesis introduces stochastic processes and Brownian motion, offering insights into their properties, construction, and applications to the time alignment of biological data.
