Modeling Likelihood differences in Men’s professional tennis rally Lengths

Jared Grooms

Dr. Gilbert Fellingham

Dr. Garritt L. Page

Dr. Nate Sandholtz

We seek to understand the distribution of rally lengths

Double Faults are Length 0

Aces are Length 1

1

Each Subsequent Ball in Play Adds 1 to Rally Length

1

2

3

Data gathered from Tennis Abstract: Match Charting Project

• Volunteer based
• 3,348 men’s singles matches between 2010 and 2020
• 333,354 rallies
• Rally length was constructed as described before
• 6f37b3* (down the t serve, shallow forehand return to backhand side, backhand winner to backhand side)

Need a likelihood that fits rally length well across all three surfaces

Poisson and Geometric alone struggle to capture densities at early rally lengths

We construct a Modified Geometric Distribution for both first and second serve rallies

Point masses at shorter rally lengths account for early server advantage

• In tennis, serving lends a large advantage due to the pace of the shot and the uncertainty around the spin and direction
• We define the loss of server advantage as the point in the rally, or the rally length, where the MG(r) and TMG(r) begin to resemble a geometric distribution
• We will determine this point by comparing WAIC scores across the MG(1) – MG(5) and TMG(1) – TMG(5)
• When model fit no longer improves by adding additional point masses, we will say that we have reached a server independent rally

Models are nested so that WAIC score improvement is driven only by the likelihood

Our primary objectives are as follows

• Determine the best fitting likelihood for each serve type across all three surfaces
• Assess the consistency of the best fitting likelihood across surfaces within each serve type
• Evaluate the difference in the number of touches required for rally length to approximate a Geometric Distribution between first and second serves
• Make inference on parameter differences when applicable

TMG(5) and MG(3) fit best across all three surfaces

• With two less additional point masses needed on second serves than first serves we conclude that server advantage takes two less balls in play to fall off when going from a first serve rally to a second serve rally

Significant differences across P0, P1, and P on second serve rallies for all three surfaces

Differences in P1, with less distinguishable differences in P for first serves on all three surfaces

Significant differences in P1-P3 justify choice to model first and second serve rallies separately

Modified Geometric Probabilities Against Data (Grass)

Modified Geometric Probabilities Against Data (Clay)

Modified Geometric Probabilities Against Data (Grass)

Final Posterior Probabilities for TMG(5) and MG(3)

Conclusion

• First serve rallies require two additional balls in play for server advantage to dissipate than second serve rallies
• Grass courts had the highest probability of aces and the shortest rallies
• First and second serve rallies appeared to have significantly different distributions, especially at short rally lengths

Further Work

• Explore other modified distributions
• Apply the same research to women’s data
• Estimate player effects using hierarchical models
• Investigate parameter differences in tiebreaker rallies

Acknowledgments

• Tyler Ward
• Anna Fellars

Thank You