Continuous-time state-space modelling of the hot hand in basketball

AStA Advances in Statistical Analysis ◽

10.1007/s10182-021-00410-y ◽

2021 ◽

Author(s):

Sina Mews ◽

Marius Ötting

Keyword(s):

State Space ◽

Continuous Time ◽

State Space Model ◽

Serial Dependence ◽

Uhlenbeck Process ◽

Space Model ◽

State Process ◽

Hot Hand ◽

Percentage Points ◽

Using Data

AbstractWe investigate the hot hand phenomenon using data on 110,513 free throws taken in the National Basketball Association. As free throws occur at unevenly spaced time points within a game, we consider a state-space model formulated in continuous time to investigate serial dependence in players’ success probabilities. In particular, the underlying state process can be interpreted as a player’s (latent) varying form and is modelled using the Ornstein-Uhlenbeck process. Our results support the existence of the hot hand, but the magnitude of the estimated effect is rather small as the underlying success probabilities are elevated by only a few percentage points.

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Using approximate Bayesian inference for a “steps and turns” continuous-time random walk observed at regular time intervals

PeerJ ◽

10.7717/peerj.8452 ◽

2020 ◽

Vol 8 ◽

pp. e8452

Author(s):

Sofia Ruiz-Suarez ◽

Vianey Leos-Barajas ◽

Ignacio Alvarez-Castro ◽

Juan Manuel Morales

Keyword(s):

State Space ◽

Continuous Time ◽

Time Scale ◽

Animal Movement ◽

State Space Model ◽

Movement Direction ◽

Time Intervals ◽

Space Model ◽

Regular Time ◽

Approximate Bayesian

The study of animal movement is challenging because movement is a process modulated by many factors acting at different spatial and temporal scales. In order to describe and analyse animal movement, several models have been proposed which differ primarily in the temporal conceptualization, namely continuous and discrete time formulations. Naturally, animal movement occurs in continuous time but we tend to observe it at fixed time intervals. To account for the temporal mismatch between observations and movement decisions, we used a state-space model where movement decisions (steps and turns) are made in continuous time. That is, at any time there is a non-zero probability of making a change in movement direction. The movement process is then observed at regular time intervals. As the likelihood function of this state-space model turned out to be intractable yet simulating data is straightforward, we conduct inference using different variations of Approximate Bayesian Computation (ABC). We explore the applicability of this approach as a function of the discrepancy between the temporal scale of the observations and that of the movement process in a simulation study. Simulation results suggest that the model parameters can be recovered if the observation time scale is moderately close to the average time between changes in movement direction. Good estimates were obtained when the scale of observation was up to five times that of the scale of changes in direction. We demonstrate the application of this model to a trajectory of a sheep that was reconstructed in high resolution using information from magnetometer and GPS devices. The state-space model used here allowed us to connect the scales of the observations and movement decisions in an intuitive and easy to interpret way. Our findings underscore the idea that the time scale at which animal movement decisions are made needs to be considered when designing data collection protocols. In principle, ABC methods allow to make inferences about movement processes defined in continuous time but in terms of easily interpreted steps and turns.

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