Statistical modelling of gaze behaviour as categorical time series: what you should watch to save soccer penalties

Abstract

Previous research on gaze behaviour in sport has typically reported summary fixation statistics thereby largely ignoring the temporal sequencing of gaze. In the present study on penalty kicking in soccer, our aim was to apply a Markov chain modelling method to eye movement data obtained from goalkeepers. Building on the discrete analysis of gaze employed by Dicks et al. (Atten Percept Psychophys 72(3):706–720, 2010b), we wanted to statistically model the relative probabilities of the goalkeeper’s gaze being directed to different locations throughout the penalty taker’s approach (Dicks et al. in Atten Percept Psychophys 72(3):706–720, 2010b). Examination of gaze behaviours under in situ and video-simulation task constraints reveals differences in information pickup for perception and action (Attention, Perception and Psychophysics 72(3), 706–720). The probabilities of fixating anatomical locations of the penalty taker were high under simulated movement response conditions. In contrast, when actually required to intercept kicks, the goalkeepers initially favoured watching the penalty taker’s head but then rapidly shifted focus directly to the ball for approximately the final second prior to foot-ball contact. The increased spatio-temporal demands of in situ interceptive actions over laboratory-based simulated actions lead to different visual search strategies being used. When eye movement data are modelled as time series, it is possible to discern subtle but important behavioural characteristics that are less apparent with discrete summary statistics alone.

Appendix: General model

Models were fitted separately to the data for each experimental condition. For a given condition, we let y _tk denote the observed gaze location at time t = 1,2,…,T (T = 54), in trial k = 1,2,…,120. This gaze location was categorised as discussed in the Data Analysis section, but because there were no observations were recorded in the ‘other’ category, the possible values for y _tk are 1,2,…,9.

We let π _i represent the probability that the gaze was directed at state i at t = 1, for i = 1,2,…,9. The probability of transition of gaze location from state i (at time t) to state j (at time t + 1) is denoted by $\psi_{ij}^{(t)}$, for t = 1,2,…,T − 1 and i, j = 1,2,…,9. Thus, the vector $\psi_{i}^{(t)}$ gives the probabilities of transition from state i at time t.

For each condition, we model

$$ y_{1k} \sim {\text{Categorical(}}\pi ) $$

$$ y_{tk} \sim {\text{Categorical}}\left(\psi_{y(t - 1)k}^{(t - 1)} \right) $$

The initial probability vector was given a Dirichlet prior distribution (with α a vector of ones of equal length to π):

$$ \pi \sim {\text{Dirichlet(}}\alpha ) $$

Similarly, each row of the first transition matrix was also given a Dirichlet (α) prior distribution.

Autoregressive smoothing

From the first transition matrix, we calculate (using the ninth column as the reference):

$$ \eta_{i}^{(1)} = m\log it(\psi_{i}^{(1)} ) $$

A first-order autoregressive process is then applied (for t = 2,…,T − 1):

$$ \eta_{i}^{(1)} \sim N(\eta_{i}^{(t - 1)} ,\tau ) $$

A Gamma prior distribution on the precision parameter was used:

$$ \tau \sim {\text{Gamma}}(0.5,0.5) $$

A range of values for the parameters in this prior distribution were tried, and it was found that the results were not sensitive to this choice.

With back-transformation, we get the rest of the transition probabilities:

$$ \psi_{ij}^{(t)} = {\frac{{\exp \left( {\eta_{ij}^{(t)} } \right)}}{{\sum\nolimits_{j = 1}^{9} {\exp \left( {\eta_{ij}^{(t)} } \right)} }}} $$

Summaries

We denote the marginal probabilities by γ _it , and these give the probability that the gaze is directed to state i at time t. These were calculated by:

$$ \gamma_{i1} = \pi_{i} $$

$$ \gamma_{it} = \sum\limits_{j = 1}^{9} {\psi_{ji}^{(t - 1)} \gamma_{j(t - 1)} } $$

Fixation probabilities, which were the probability of gaze remaining directed at state i from time t − 1 to time t, were denoted by ω _it . They were calculated using:

$$ \omega_{it} = \psi_{ii}^{(t - 1)} \gamma_{i(t - 1)} $$