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<p>We study shot angles in ice hockey as circular time series and propose a hidden Markov model with an explicit <i>uniform transient</i> state to capture brief, unstructured play. The structured states use wrapped-Cauchy emissions to model concentrated aim toward the goal; the uniform state accounts for diffuse angles that arise during scrambles or broken plays. Parameters are estimated by an iterative forward–backward moment-update routine using responsibility-weighted trigonometric moments, and model size is chosen by the Bayesian information criterion (BIC). Simulations varying the proportion of diffuse shots, the separation between directional modes, and state persistence show that BIC prefers the simpler two-state wrapped-Cauchy model when no diffuse layer is present and selects the uniform-augmented model otherwise. When diffuse shots exist, separating them improves the stability of the estimated centers and concentrations of the structured states. Applied to NHL 2021 shot data, the angular distribution is effectively unimodal with a gradual decline away from the primary direction and only a small, context-dependent diffuse share. In focused subsets (even-strength by team; power-play one-timer candidates), two fitted centers lie on either side of the <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(0/2\pi \)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mn>0</mn> <mo stretchy="false">/</mo> <mn>2</mn> <mi>π</mi> </mrow> </math></EquationSource> </InlineEquation> cut, indicating a single physical lane wrapped on the circle. Adding state responsibilities to a baseline expected-goals model using distance and angle yields no material out-of-sample gains. The approach is interpretable, computationally light, and readily extendable to circular–linear features and covariate-dependent transitions.</p>

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Hidden Markov modeling of hockey shot angles with a uniform transient state

  • 期刊论文

  • 出版模式：

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  • 发表日期：

    2026年7月9日

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  • Abdolnasser Sadeghkhani

摘要

We study shot angles in ice hockey as circular time series and propose a hidden Markov model with an explicit uniform transient state to capture brief, unstructured play. The structured states use wrapped-Cauchy emissions to model concentrated aim toward the goal; the uniform state accounts for diffuse angles that arise during scrambles or broken plays. Parameters are estimated by an iterative forward–backward moment-update routine using responsibility-weighted trigonometric moments, and model size is chosen by the Bayesian information criterion (BIC). Simulations varying the proportion of diffuse shots, the separation between directional modes, and state persistence show that BIC prefers the simpler two-state wrapped-Cauchy model when no diffuse layer is present and selects the uniform-augmented model otherwise. When diffuse shots exist, separating them improves the stability of the estimated centers and concentrations of the structured states. Applied to NHL 2021 shot data, the angular distribution is effectively unimodal with a gradual decline away from the primary direction and only a small, context-dependent diffuse share. In focused subsets (even-strength by team; power-play one-timer candidates), two fitted centers lie on either side of the \(0/2\pi \) 0 / 2 π cut, indicating a single physical lane wrapped on the circle. Adding state responsibilities to a baseline expected-goals model using distance and angle yields no material out-of-sample gains. The approach is interpretable, computationally light, and readily extendable to circular–linear features and covariate-dependent transitions.