Model shooting as a generative process

[huffyhenry]

Shooting is best modelled as a two-step process, where shot quality is first sampled from an appropriate distribution, and conversion is then a Bernoulli experiment. The (unnormalized) likelihood of a datapoint then becomes the definite integral over (0,1) of xi*pdf(xi | theta)dxi for a goal and (1-xi)*pdf(xi | theta)dxi for a miss, where pdf is the probability density function of the distribution of shot qualities, and theta are the parameters of that distribution, which can depend on any factors of interest, especially team identities and game state.

Cf. #6.

[huffyhenry]

The exponential distribution is a better fit on the (shooting) team level too:

library(dplyr)
library(MASS)

expoL <- function(vec){
  return(logLik(fitdistr(vec, "exponential")))
}

betaL <- function(vec){
  return(logLik(fitdistr(vec, "beta", start=list(shape1=0.5, shape2=0.5))))
}

read.csv("../sbs-xg-review/data/sb.csv") %>%
  filter(competition_name == "Premier League") %>%
  filter(shot_set_play != "penalty") %>%
  group_by(team_name) %>%
  summarize(
    expo=expoL(shot_xg),
    beta=betaL(shot_xg)
  ) %>% 
  mutate(choice=ifelse(beta > expo, "beta", "expo"))

yields

That said, it is entirely possible that a beta (or another) model, where both teams modify the distribution parameters, is a better fit.