{"id":2314,"date":"2022-04-07T05:34:25","date_gmt":"2022-04-07T05:34:25","guid":{"rendered":"http:\/\/optimumsportsperformance.com\/blog\/?p=2314"},"modified":"2022-11-08T03:37:36","modified_gmt":"2022-11-08T03:37:36","slug":"bayes-adjusted-3-point","status":"publish","type":"post","link":"https:\/\/optimumsportsperformance.com\/blog\/bayes-adjusted-3-point\/","title":{"rendered":"Tyrese Maxey’s 3pt%, Bayes, Shrinkage"},"content":{"rendered":"

Some friends were discussing Philadelphia 76er’s point guard, Tyrese Maxey’s, three point% today. They were discussing how well he has performed over 72 games with a success rate of 43% behind the arc (at the time this data was scraped, 4\/6\/2022). While his percentage from 3pt range is very impressive I did notice that he has 294 attempts, which is less than 3 out of the 4 player’s that are ahead of him (Kyrie only has 214 attempts and he is ranked 3rd at the time of this writing) and Steph Curry is just behind Maxey in the ranking (42.4% success) with nearly 70 more attempts.<\/p>\n

The question becomes, how can we be of Maxey’s three point percentage relative to those with more attempts? We will take a Bayesian approach, using a beta conjugate, to consider the success rate of these players relative to what we believe the average three point success rate is for an NBA shooter (our prior), which we will determine from observing 3 point shooting over previous 3 seasons.<\/p>\n

NOTE:<\/strong> On basketball-reference.com<\/a><\/strong><\/span>, they have a nice check box that automatically will filter out players that are non-qualifiers for rate stats. After playing around with this, it appears that 200 attempts is their cut off. So, I will keep that and filter the data down to only those with 200 or more 3pt attempts.<\/p>\n

All of the code, web scrapping, and csv files of the data (if you are looking to run it prior to when I scrapped it) are available on my GITHUB PAGE<\/a><\/strong><\/span>.<\/p>\n

Exploratory Data Analysis<\/strong><\/p>\n

First, let’s view the top 10 three point shooters this season (size of the dot represents the number of three point attempts taken).<\/p>\n

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Visualize the distribution of three point attempts and three point% for the 2022 season, so far.<\/p>\n

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Establishing Our Prior<\/strong><\/p>\n

Since we are dealing with a binary outcome of successes (made the shot) and failures (missed the shot) we will use the beta distribution, which is the conjugate prior for the binomial distribution.<\/p>\n

The beta distribution has two parameters, alpha and beta. To determine what these parameters should be, we will use the method of moments with the data from the previous three seasons.<\/p>\n

To do this, we need to first find the mean and variance for the previous three seasons.<\/p>\n

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Next, we create a function that calculates alpha and beta based on the mean and variance from our observed data.<\/p>\n

\r\n# function for calculating alpha and beta\r\nbeta_parameters <- function(dist_avg, dist_var){\r\n  alpha <- dist_avg * (dist_avg * (1 - dist_avg)\/dist_var - 1)\r\n  beta <- alpha * (1 - dist_avg)\/dist_avg\r\n  list(alpha = alpha,\r\n       beta = beta)\r\n}\r\n<\/pre>\n
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\nThe function works to produce the two parameters we need. The data is returned in list format, so we will call each element of the list and store the respective values in their own variable.<\/p>\n
The function works to produce the two parameters we need. The data is returned in list format, so we will call each element of the list and store the respective values in their own variable.<\/p>\n
\"\"<\/a>
\nThe alpha and beta parameters derived from our method of moments function appear to produce the mean and standard deviation correctly. We can plot this distribution to see what it looks like.\"\"<\/a><\/p>\n
Update the 3pt% of the players in the 2022 season with our meta prior<\/strong><\/p>\n
We calculate our Bayes adjusted three point percentage for the players by adding their successes to `alpha` and their failures to `beta` and then calculating the new posterior percentage as<\/p>\n
alpha \/ (alpha + beta)<\/strong><\/em><\/p>\n
and the posterior standard deviation as<\/p>\n
sqrt((alpha * beta) \/ ((alpha + beta)^2 * (alpha + beta + 1)))<\/strong><\/em><\/p>\n
\r\ntbl2022 <- tbl2022 %>%\r\n  mutate(three_pt_missed = three_pt_att - three_pt_made,\r\n         posterior_alpha = three_pt_made + alpha,\r\n         posterior_beta = three_pt_missed + beta,\r\n         posterior_three_pt_pct = posterior_alpha \/ (posterior_alpha + posterior_beta),\r\n         posterior_three_pt_sd = sqrt((posterior_alpha * posterior_beta) \/ ((posterior_alpha + posterior_beta)^2 * (posterior_alpha + posterior_beta + 1))))\r\n<\/pre>\n
Have any of the players in the top 10 changes following in the adjustment?\"\"<\/a><\/p>\n