In celebration of the baseball Hall of Fame elections today, we decided to investigate the distribution of hits necessary for a player to achieve the offensive triple crown and the minimal number of hits needed to accomplish this feat in 2016. The offensive triple crown goes to the player who leads the league in home runs (HR), batting average (BA), and runs batted in (RBI). However, many baseball analysts, including myself, ignore RBIs as a measure of skill because they depend too strongly on factors outside a player's control. Instead, we will use slugging percentage (SLG). The last player to receive the triple crown was Miguel Cabrera in 2012, who was the first player since 1967 to achieve this accolade. This is in part due to the specialization of hitters in today’s game. For instance, last year three different players led the league in all three categories, DJ LeMahieu of the Rockies had a .348 BA, Mark Trumbo of the Orioles had 47 HRs, and David Ortiz of the Red Sox had a .620 SLG. Therefore, using an average of full season of atbats, our Triple Crown player must have the following stats:
Quick refresher of the equations defining BA, SLG, and H:
We can now set up our equation to minimize and its required constraints: Equation to Minimize:
Constraints:
Because we have four variables, graphing the solution in 4 dimensions would be difficult. Thus, we are using the simplex optimization method from Scipy in Python. Feel free to follow along by plugging the equations into WolframAlpha! Since the minimized BA of .350 is unbounded, there are many results that return the minimized hit count of 210. However, not all of the provided distributions match well with real players. For instance, one of the first results was the following: 1B = 155, 2B = 0, 3B = 0, HR = 55 BA = .350, SLG = .625, H = 210 There are a couple of issues with this distribution. First is the lack of diversity within the player’s hits. Second, the last time someone hit 55 home runs in a season was in 2002 by Alex Rodriguez, right in the middle of the steroid age. Taking this into account, we decided to remove the inequality for home run and instead make it an assignment. HR = 48. The results are below: 1B = 152, 2B= 0, 3B = 10, HR = 48 BA = .350, SLG = .623, H = 210 Now instead of just mashing home runs and what we assume are long singles, our player additionally hits triples. However, we were still not happy with our lack of doubles. This fact, combined with the knowledge that triples rely on speed and our player fits the persona of a muscly, but slow, slugger, we decided to limit him to 1 triple. We also decided to implement another constraint, maxing out the slugging percentage at a reasonable level for our superstar. Since the end of the steroid era around 2008, the two highest slugging percentages have been Albert Pujols with .659 in 2009 and Bryce Harper with .649 in 2015. Thus we settled on .650 and the equations now become: Equation to Minimize:
Constraints:
The results are as follows: 1B = 143, 2B = 18, 3B = 1, HR = 48 BA = .350, SLG = .623, H = 210 These stats now look feasible, for a great player at least! Interestingly, Adrian Beltre, the player we examined last week, had similar statistics in his 2004 campaign where he finished second in MVP voting (behind Barry Bonds). Here are his numbers: 1B = 120, 2B = 32, 3B = 0, HR = 48 BA = .334, SLG = .629, H = 200, AB = 598 The conclusion? Our algorithm favors singles to minimize SLG more so than reallife players hit them. Also, the best players during the steroid era could handily win multiple offensive stats categories in today’s game. Maybe it is a good thing the game has evolved, but watching Mark McGwire and Sammy Sosa mash home runs on their way to the record books was still magic. The SaberSmart Team
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