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This paper presents a generative statistical model for analyzing time series of planar shapes. Using elastic shape analysis, we separate object kinematics (rigid motions and speed variability) from morphological evolution, representing the latter through transported velocity fields (TVFs). A principal component analysis (PCA) based dimensionality reduction of the TVF representation provides a finite-dimensional Euclidean framework, enabling traditional time-series analysis. We then fit a vector auto-regressive (VAR) model to the TVF-PCA time series, capturing the statistical dynamics of shape evolution. To characterize morphological changes,we use VAR model parameters for model comparison, synthesis, and sequence classification. Leveraging these parameters, along with machine learning classifiers, we achieve high classification accuracy. Extensive experiments on cell motility data validate our approach, demonstrating its effectiveness in modeling and classifying migrating cells based on morphological evolution—marking a novel contribution to the field.