0

I would like to use a bayesian model to determine which position (X, Y) is most likely the best position to score a goal in soccer. For this I have a dataset of a soccer club with all its goals and shots with X and Y coordinates and the type of shot for some seasons. To make it more clear, imagine we have the following visualization of goals scored at certain positions:

In this plot I didn't show the no goals, but I hope this makes it more clear. As I said I also have the `(1|shotType) of the shot/goal. A goal/shot can be taken in the following ways:

  • LeftFoot
  • RightFoot
  • Head

I was thinking of using a Bayesian Binary Logistic Regression (If you think something is better feel free to let me know) since we have a binary Target (Goal or No goal) and two numeric variable and one grouping variable. So I fitted the following model (result2 -> 0 is no goal and 1 is goal):

library(brms)  
fit <- brm(result2 ~ X + Y + (1|shotType), 
           data = df, 
           family = bernoulli(link = "logit"))
fit
#> Warning: There were 55 divergent transitions after warmup. Increasing
#> adapt_delta above 0.8 may help. See
#> http://mc-stan.org/misc/warnings.html#divergent-transitions-after-warmup
#>  Family: bernoulli 
#>   Links: mu = logit 
#> Formula: result2 ~ X + Y + (1 | shotType) 
#>    Data: df (Number of observations: 2728) 
#>   Draws: 4 chains, each with iter = 2000; warmup = 1000; thin = 1;
#>          total post-warmup draws = 4000
#> 
#> Group-Level Effects: 
#> ~shotType (Number of levels: 4) 
#>               Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
#> sd(Intercept)     1.02      0.69     0.24     2.76 1.01      525      771
#> 
#> Population-Level Effects: 
#>           Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
#> Intercept   -12.18      1.16   -14.51    -9.96 1.00      988      606
#> X            11.34      1.04     9.27    13.39 1.00     2463     2335
#> Y             0.23      0.47    -0.66     1.13 1.01     2025     2420
#> 
#> Draws were sampled using sampling(NUTS). For each parameter, Bulk_ESS
#> and Tail_ESS are effective sample size measures, and Rhat is the potential
#> scale reduction factor on split chains (at convergence, Rhat = 1).

Created on 2023-11-01 with reprex v2.0.2

But I am now at the point, that I don't understand how to interpet these results. I was assuming you could see a CI of what the best range is of to score a goal, for example X (0.8-0.9) and Y (0.25-0.3) which is like a square (good to know, the X and Y are in range of 0 to 1 and not in meters). So I was wondering if anyone could help me how to interpret these results and if I'm using the right model for determining the most likely positions to score a goal?

Here is some reproducible data of df:

structure(list(result2 = c(0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 
1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 
1, 0, 1, 0, 0, 1, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 
1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 
0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 
0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 
0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 
0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 
0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 
0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 
0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 1, 0, 1, 1, 0, 0, 0, 1, 0, 0, 
0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 
0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 
0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 1, 0, 
0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 
0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 
0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 
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0, 1, 0, 0, 1, 1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 
0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 
0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 
1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 0, 0, 
0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 1, 0, 
0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 
0, 0, 0, 1, 0), X = c(0.736, 0.895, 0.706, 0.901, 0.852, 0.952, 
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), shotType = c("LeftFoot", "RightFoot", "RightFoot", "Head", 
"LeftFoot", "RightFoot", "LeftFoot", "Head", "Head", "RightFoot", 
"RightFoot", "LeftFoot", "RightFoot", "RightFoot", "RightFoot", 
"RightFoot", "OtherBodyPart", "Head", "RightFoot", "RightFoot", 
"RightFoot", "RightFoot", "RightFoot", "RightFoot", "RightFoot", 
"RightFoot", "LeftFoot", "RightFoot", "LeftFoot", "Head", "Head", 
"RightFoot", "Head", "RightFoot", "RightFoot", "RightFoot", "LeftFoot", 
"Head", "LeftFoot", "RightFoot", "RightFoot", "RightFoot", "Head", 
"RightFoot", "RightFoot", "Head", "RightFoot", "RightFoot", "RightFoot", 
"LeftFoot", "LeftFoot", "Head", "OtherBodyPart", "RightFoot", 
"LeftFoot", "LeftFoot", "LeftFoot", "RightFoot", "LeftFoot", 
"RightFoot", "LeftFoot", "RightFoot", "RightFoot", "RightFoot", 
"Head", "LeftFoot", "LeftFoot", "Head", "RightFoot", "LeftFoot", 
"LeftFoot", "Head", "LeftFoot", "LeftFoot", "RightFoot", "RightFoot", 
"LeftFoot", "RightFoot", "LeftFoot", "Head", "RightFoot", "RightFoot", 
"LeftFoot", "Head", "LeftFoot", "LeftFoot", "RightFoot", "RightFoot", 
"LeftFoot", "RightFoot", "LeftFoot", "RightFoot", "RightFoot", 
"RightFoot", "LeftFoot", "LeftFoot", "LeftFoot", "RightFoot", 
"RightFoot", "RightFoot", "RightFoot", "LeftFoot", "Head", "LeftFoot", 
"Head", "Head", "Head", "LeftFoot", "LeftFoot", "LeftFoot", "Head", 
"LeftFoot", "RightFoot", "RightFoot", "RightFoot", "LeftFoot", 
"LeftFoot", "RightFoot", "LeftFoot", "Head", "RightFoot", "LeftFoot", 
"Head", "RightFoot", "RightFoot", "LeftFoot", "RightFoot", "RightFoot", 
"RightFoot", "Head", "LeftFoot", "RightFoot", "RightFoot", "RightFoot", 
"RightFoot", "RightFoot", "LeftFoot", "Head", "LeftFoot", "RightFoot", 
"LeftFoot", "LeftFoot", "RightFoot", "RightFoot", "RightFoot", 
"RightFoot", "LeftFoot", "LeftFoot", "RightFoot", "RightFoot", 
"Head", "LeftFoot", "RightFoot", "RightFoot", "RightFoot", "RightFoot", 
"LeftFoot", "RightFoot", "Head", "RightFoot", "LeftFoot", "RightFoot", 
"RightFoot", "RightFoot", "RightFoot", "RightFoot", "RightFoot", 
"RightFoot", "LeftFoot", "LeftFoot", "LeftFoot", "RightFoot", 
"Head", "RightFoot", "RightFoot", "LeftFoot", "RightFoot", "RightFoot", 
"RightFoot", "RightFoot", "LeftFoot", "RightFoot", "Head", "RightFoot", 
"RightFoot", "LeftFoot", "LeftFoot", "RightFoot", "LeftFoot", 
"RightFoot", "RightFoot", "RightFoot", "RightFoot", "LeftFoot", 
"LeftFoot", "LeftFoot", "Head", "RightFoot", "RightFoot", "RightFoot", 
"RightFoot", "Head", "LeftFoot", "RightFoot", "Head", "Head", 
"LeftFoot", "LeftFoot", "Head", "RightFoot", "LeftFoot", "LeftFoot", 
"LeftFoot", "RightFoot", "RightFoot", "Head", "RightFoot", "RightFoot", 
"LeftFoot", "LeftFoot", "RightFoot", "Head", "RightFoot", "LeftFoot", 
"LeftFoot", "RightFoot", "LeftFoot", "RightFoot", "RightFoot", 
"RightFoot", "LeftFoot", "LeftFoot", "LeftFoot", "RightFoot", 
"LeftFoot", "LeftFoot", "LeftFoot", "LeftFoot", "LeftFoot", "Head", 
"LeftFoot", "LeftFoot", "RightFoot", "RightFoot", "LeftFoot", 
"RightFoot", "LeftFoot", "RightFoot", "RightFoot", "LeftFoot", 
"RightFoot", "RightFoot", "Head", "Head", "RightFoot", "Head", 
"LeftFoot", "LeftFoot", "LeftFoot", "RightFoot", "RightFoot", 
"RightFoot", "RightFoot", "Head", "RightFoot", "RightFoot", "RightFoot", 
"Head", "RightFoot", "RightFoot", "LeftFoot", "LeftFoot", "LeftFoot", 
"LeftFoot", "RightFoot", "LeftFoot", "RightFoot", "RightFoot", 
"Head", "LeftFoot", "RightFoot", "RightFoot", "RightFoot", "LeftFoot", 
"RightFoot", "LeftFoot", "RightFoot", "RightFoot", "RightFoot", 
"Head", "Head", "LeftFoot", "LeftFoot", "Head", "LeftFoot", "RightFoot", 
"Head", "LeftFoot", "RightFoot", "RightFoot", "LeftFoot", "LeftFoot", 
"RightFoot", "RightFoot", "Head", "LeftFoot", "RightFoot", "LeftFoot", 
"RightFoot", "RightFoot", "RightFoot", "RightFoot", "RightFoot", 
"RightFoot", "RightFoot", "LeftFoot", "RightFoot", "RightFoot", 
"RightFoot", "Head", "LeftFoot", "LeftFoot", "LeftFoot", "RightFoot", 
"Head", "RightFoot", "Head", "RightFoot", "Head", "RightFoot", 
"LeftFoot", "RightFoot", "RightFoot", "RightFoot", "Head", "Head", 
"RightFoot", "RightFoot", "LeftFoot", "RightFoot", "LeftFoot", 
"LeftFoot", "RightFoot", "RightFoot", "RightFoot", "LeftFoot", 
"LeftFoot", "RightFoot", "LeftFoot", "Head", "RightFoot", "LeftFoot", 
"RightFoot", "Head", "Head", "RightFoot", "RightFoot", "RightFoot", 
"RightFoot", "RightFoot", "LeftFoot", "RightFoot", "RightFoot", 
"RightFoot", "Head", "LeftFoot", "RightFoot", "RightFoot", "Head", 
"Head", "LeftFoot", "RightFoot", "LeftFoot", "RightFoot", "RightFoot", 
"RightFoot", "LeftFoot", "RightFoot", "LeftFoot", "Head", "RightFoot", 
"RightFoot", "LeftFoot", "RightFoot", "LeftFoot", "RightFoot", 
"LeftFoot", "RightFoot", "LeftFoot", "RightFoot", "RightFoot", 
"RightFoot", "RightFoot", "Head", "RightFoot", "Head", "RightFoot", 
"RightFoot", "LeftFoot", "RightFoot", "RightFoot", "RightFoot", 
"LeftFoot", "LeftFoot", "RightFoot", "LeftFoot", "LeftFoot", 
"RightFoot", "RightFoot", "LeftFoot", "Head", "RightFoot", "RightFoot", 
"RightFoot", "RightFoot", "RightFoot", "LeftFoot", "LeftFoot", 
"LeftFoot", "LeftFoot", "LeftFoot", "LeftFoot", "LeftFoot", "LeftFoot", 
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Quinten
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    Just a quick comment: logistic regression assumes that the odds of scoring a goal vary linearly with changes in x-y coordinates on the field. I don't think this is plausible a-priori, and furthermore it won't have a well-defined maximum (the predicted log-odds of goal will be described by a tilted plane). I think the model would need at least also quadratic and interaction terms. Alternatively, something more sophisticated like a generalized addiditve model (GAM). – matteo Nov 06 '23 at 15:52
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    Why not say that the best position is the one whose 1-meter circle contains the greatest number of goal shots? That may not be Bayesian but it’s as simple as you could want and reasonable enough. – Matt F. Nov 07 '23 at 01:21
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    The best place may be roughly X = 1, Y = 0.5, in front of the middle of the net — and if so, it’s probably not worth the time to do elaborate modeling. – Matt F. Nov 07 '23 at 07:29
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    A classic Bayesian example of coordinate inference is "Gull's Lighthouse Problem". PDFs (including Gull's) and videos discussing a Bayesian coordinate inference problem are available on the internet. – krkeane Nov 09 '23 at 13:26
  • Bayesian methods are about expressing and manipulating uncertainty of knowledge. For continuous parameters, such as those in your problem, its not Bayesian in spirit to focus on the most likely ** point **. See discussion https://stats.stackexchange.com/a/148475/43149 – krkeane Nov 09 '23 at 13:33
  • Have you thought of re-parameterizing it in terms of angle and distance? – Björn Nov 12 '23 at 14:06

1 Answers1

1

the best position

You could say that the best position is the position where we have the largest probability of a goal. That is

$$p(goal|x,y)$$

With a Bayesian approach you would compute

$$p(goal|x,y,data)$$

and you could have some parametric model that describes $p(goal|x,y)$ based on a few parameters $\beta_i$ for which we have prior distributions. Then based on the data we can update the distributions for those variables and compute the posterior.

$$f(\beta|data) \propto f(data|\beta) f(\beta)$$

with the given $f(\beta|data)$ you can describe also $p(goal|x,y,data)$ and figure out which place is the best.

The computations are not easy and the choice of model with coëfficiënts $\beta$ and their prior is subjective. I would go instead with MattF's comment and just compute at every point x,y the frequency of goals for all attempts within a certain range.

Sextus Empiricus
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  • Thank you for your clear answer! I was really looking for a way to describe this in an bayesian approach. I will have a further look in the coefficients and MattF's comment, but this brings me in the right direction. – Quinten Nov 13 '23 at 12:14