A Semi-Markov framework for modeling football possessions and temporal expected threat.

This study presents a spatio-temporal semi-Markov framework to model football possessions and evaluate scoring probability within specific time windows. Current possession-value models, such as Expected Threat (xT), typically treat football as a memoryless Markov process where event timing is ignored. Our framework addresses this by jointly representing action type, spatial location, and non-exponential sojourn times. We estimate transition probabilities over a [Formula: see text] pitch grid and model event durations using Accelerated Failure Time models conditioned on distance and angle to the middle of the goal posts and next event. By using a quantile based criterion to select distributional families, we find that different transitions follow fundamentally different patterns. This contradicts the single-family assumptions used in prior research. We introduce a temporal xT metric that computes the probability of scoring within a fixed time horizon via backward induction. Evaluated on Ligue 1 and EURO event data, our model consistently outperforms a memoryless baseline on AUC and Brier score. We conclude that while infinite-horizon Markov models remain suitable for general play, temporal xT is the essential tool for time-critical decision support where the remaining clock is a binding constraint. This model provides an interpretable and data-driven foundation for analytics that accounts for how action duration directly affects scoring opportunities.