Using Markov Chains to Analyze a Volleyball Rally

Transcription

1 1 Introduction Using Markov Chains to Analyze a Volleyball Rally Spencer Best Carthage College sbest@carthage.edu November 3, 212 Abstract We examine a volleyball rally between two volleyball teams. Using the archived statistics of Carthage matches, we calculate the transition probabilities of the match. Results indicate the validity of our model. Volleyball is more than a sport where two teams play for a victory. Rather, it consists of carefully comprised teams, approximately six states in which the ball can be played, and a significant number of possible plays. Taking all of this into consideration we use a Markov Chain to analyze a rally to improve scoring efficiency. We transform statistics of previous matches into scoring probabilities. With this information, we are able to input those probabilities into a transition matrix, which yields the probabilities of scoring from each designated state. 2 Definitions and Development We will outline definitions and formulas that were utilized in our study. Our analysis illustrates that our data was derived by calculating probabilities from a compilation of team statistics and entering those probabilities into our matrix. We determined a rally to be our area of interest. We will discuss how we are going to divide a rally into areas of analysis. To do this we have to determine the states of a rally. Definition 1 A state is the row and column values analyzed within the matrix. In this analysis a state is described as the position the ball can be on the volleyball court. For our purpose, the states are serve, pass, set, hit, free ball, block and point. A serve is the first contact with the ball, beginning on one side of the court and transferring to the other, thus starting the rally. When a player utilizes their forearms together, we describe that as a pass. An overhead pass with two hands is characterized as a set. A hit, also labeled as an attack, is a strong hit, from one team to the other, usually following a set. In a time of desperation, a simple pass from one team to the other is called a free ball. When the other team defends a hit at the net, we call that a block. Point represents a team scoring a point and the end of the rally. The states are labeled and referenced in this order throughout the analysis. Definition 2 A state is considered absorbing if it is impossible to leave that state. In a rally point states are absorbing. In order to determine the input values, we calculate the transition probabilities and inserted those probabilities into our matrix. Definition 3 A transition matrix is collection of transition probabilities arranged in a square matrix.

2 Figure 2 illustrates our transition matrix for our first analysis. This matrix includes the states labeled with the number one or two. The purpose for this is to distinguish the states between Carthage, labeled by one, and the other team, labeled by two. Therefore, for our first analysis Serve 1 represents a serve from Carthage, and continues through the states respectively. Also, a Free Ball 2 distinguishes a free ball pass from Illinois Wesleyan. Definition 4 The transition probability is the calculated probability of moving from one state to the next. In order to calculate the transition probabilities, we gathered archived statistics of Carthage College, Illinois Wesleyan as well as UW Oshkosh and manipulated those statistics to yield probabilities. For example, to find the pass percentage we took the total defensive passes divided by the total passes. Similarly, we calculate the transition from pass to point by dividing the total passes from the total service errors. After calculating all of the transition probabilities, the values are then inserted into a transition matrix. 3 Results We are now able to begin our analysis. The first part of the analysis begins with a volleyball game between Carthage College and Illinois Wesleyan. To begin calculating the transition probabilities we reference the original statistics that can be found in Table 1. Kills Hitting Total Attacks Serve Aces Set Assists Serve Service Defensive Digs Block Aces Block Carthage College Illinois Wesleyan Table 1: Gathered statistics of the match between Carthage and Illinois Wesleyan. Utilizing the statistics, we begin by computing the probability a serve from Carthage will result in a point for Carthage. This is achieved by taking the Serve Aces from Carthage and dividing it by the Total Passes of the other team. Because the total number of passes was not given in the original statistics, we assume that the Total Passes is the sum of Defensive Digs and Service. Conversely, to compute the probability that Carthage loses a point from a serve, we divide the Total Passes of Illinois Wesleyan from the Serve from Carthage. We acknowledge that there is a small probability a serve will result in a free ball, thus we assign a probability. Lastly, to calculate the probability a serve will result in a pass for the other team, we sum the calculated probabilities for that state and subtract from one. Calculating the Pass Percentage for Carthage is done by dividing the Total Passes from Carthage by the recorded Defensive Digs. Next, using the statistics, the Service from Carthage divided by Carthage s Total Passes reveals the probability a pass from Carthage will

3 score a point for Illinois Wesleyan. Summing the calculated passing probabilities and subtracting from one, we divide the remaining number evenly between the states of Hit 1 and Free Ball 1. We do this because there is a small probability a pass will transition to a hit or free ball, although unable to be calculated from the given statistics, thus assigning those probabilities from the difference. Similarly, to find the probability a set from Carthage will transition to a hit for Carthage, we take the total Set Assists for Carthage and divide by the Total Passes for Carthage. Then, since the total number of sets is not given to us, to find the probability that a set from Carthage results in a point for Illinois Wesleyan, we calculate the difference between the total Kills and Set Assists for Carthage and divide by the total Set Assists of Carthage. Analogous to the remaining calculations we did for passing, we will add the calculated probabilities for setting and subtract that sum from one. We conclude that there is an insignificant probability to transition from a set from Carthage to another set for Carthage as well as a set to a free ball, thus we divide the remaining probability equally between those two transition states. Computing the transition probabilities for Carthage s hit is similar to the same process discussed earlier. The probability a hit from Carthage will transition to a pass for Illinois Wesleyan is derived by starting with the Total Passes of Illinois Wesleyan, subtract Block Aces of Illinois Wesleyan and then subtract Hitting from Carthage. Once we have found that difference, we then divide it by the Total Attacks from Carthage. Adding Kills from Carthage to the Service of Illinois Wesleyan and then dividing by the Total Attacks from Carthage will yield the probability a hit from Carthage will result in a point for Carthage. Conversely to find the probability a hit from Carthage will result in a point for Illinois Wesleyan we take the Hitting from Carthage, plus the Block Assists of Illinois Wesleyan and divide that sum by the Total Attacks from Carthage. The probability a block will occur from a hit is found by dividing the Block Assists of Illinois Wesleyan by the Total Attacks from Carthage. Again, we believe that the probability a hit from Carthage will transition into a hit, free ball is so small that we sum the calculated probabilities of a hit and dividing the difference from one equally between those two states. Unfortunately, the archived statistics for volleyball matches do not include any records for a free ball. Therefore, for this analysis, we assumed that the probabilities for the free ball transitions were the same between each team. Also, when watching a volleyball match, there is a high probability that a free ball from one team leads to a pass from the other team, thus, we assigned the transition from a free ball to a pass with a probability of.95. The next area of assignment resides with the probability a free ball will transition to a set. Although it is less likely to occur we assign the transition with a probability of.3. Then, we divide the remaining probabilities to a transition from a free ball from Carthage to a set, hit, free ball, and point for Illinois Wesleyan as well as a point for Carthage. Lastly, we do a similar process to the block calculations as we did for the free ball probabilities. Because we don t have a recorded path the ball took following a block, we took our knowledge of a volleyball game and inferred the transition probabilities. A block is the only state in the game that the ball has a probability of transitioning to all states except serve and block for both teams. Note that the equations in which we used to calculate probabilities for Carthage are the same equations we used for Illinois Wesleyan. Once we have calculated the transition probabilities for Carthage and Illinois Wesleyan, we input those probabilities in the matrix respectively. Those values can be seen in Figure 2. Notice that the values for the second and eighth column

4 are zero. These columns represent the serving state for both teams. Because the serve is the start of the rally, no state has a non-zero probability of entering that state. Serve 1 Pass 1 Set 1 Hit 1 Free Ball 1 Block 1 Serve 2 Pass 2 Set 2 Hit 2 Free Ball 2 Block2 Point 1 Point 2 Serve Pass Set Hit Free Ball Block Serve Pass Set Hit Free Ball Block Point 1 1 [ Point 2 1 ] Figure 2: Transition Matrix of Carthage vs. Illinois Wesleyan. We raise the transition matrix to the power of 1 to find the probability of being in a specific state after 1 touches. By doing this we are able to conclude the probability the absorbing state is reached from a given state. In this case, the Figure 3 illustrates the probability a point will be scored from a given state. When analyzing this matrix we see that when Carthage makes an attack there is a.66 probability that it will result in a point for Carthage. At the same time there is a.33 probability that a hit from Carthage will result in a point for Illinois Wesleyan. This matrix also shows that there is.51 probability Carthage will gain a point from a block along with a.48 probability that Illinois Wesleyan will gain a point from Carthage s block. Therefore we can assume that Illinois Wesleyan gained many of their points from Carthage s block errors. However, it can be seen that Illinois Wesleyan has a higher probability of scoring a point off of their hits with.53 than Carthage has scoring off of Illinois Wesleyan s hitting errors [ 1 ] Figure 3: Transition Matrix raised to a power of 1. From here, our next step is to analyze the probability a team will score or lose a point on a serve. Selecting the matrix displayed in Figure 3 and multiplying it by a vector of (1,,,,,,,,,,,,,) which represents Carthage serving, reveals the scoring probabilities. This can be seen in Figure 4. The last two values in Figure 4 represent the probability Carthage will score on its own serve or lose a point on its own serve. Notice that Carthage has a.59 probability that they will score on their serve. We do the same thing to calculate the serving probabilities for Illinois Wesleyan, except we multiply by a vector of (,,,,,,1,,,,,,,). Figure 5 conveys the serving probabilities for Illinois Wesleyan. We see that Illinois Wesleyan only had a.53 probability of scoring a point off of their serve. Carthage won this match.

5 [. 4 ] Figure 4: Serving matrix for Carthage [. 46 ] Figure 5: Serving matrix for Illinois Wesleyan. The final calculations of this analysis deliver the expected number of touches until the absorption as well as the estimated number of steps to the end of the rally from each state. Thus, we can see that when Carthage is serving, there is an estimated 5.47 touches until absorption from that state, which is expected. Also, the estimated number of steps to the end of the rally when Carthage sends a free ball is relatively close to the estimated number of steps when they are serving. This is due to the fact that a free ball, although received in the middle of rally, simulates the same response when a team is to serve. A more thorough display of these results can be found in Table 6.

6 Expected touches until absorption Number of steps to the end of the rally from each state Carthage Serve Carthage Pass Carthage Set Carthage Hit Carthage Free Ball Carthage Block Illinois Wesleyan Serve Illinois Wesleyan Pass Illinois Wesleyan Set Illinois Wesleyan Hit Illinois Wesleyan Free Ball Illinois Wesleyan Block Table 6: Expected and estimated touches until the end of the rally. The second part of our analysis looks at a match between Carthage and UW Oshkosh. Let s first look at the original statistics. Kills Hitting Total Attacks Serve Aces Set Assists Serve Service Defensive Digs Block Aces Block Carthage College UW Oshkosh Table 7: Gathered statistics of the match between Carthage and UW Oshkosh. A preliminary observation of this match from Table 7 is that UW Oshkosh is the better team. We ll see throughout our second analysis that UW Oshkosh seems to prevail with all skill areas. Again, we use the original statistics to calculate our transition probabilities for our matrix. We used the same equations as we did previously to calculate probabilities. These probabilities can be seen in Figure 8.

7 Serve 1 Pass 1 Set 1 Hit 1 Free Ball 1 Block 1 Serve 2 Pass 2 Set 2 Hit 2 Free Ball 2 Block2 Point 1 Point 2 Serve Pass Set Hit Free Ball Block Serve Pass Set Hit Free Ball Block Point 1 1 [ Point 2 1 ] Figure 8: Transition Matrix of Carthage vs. Illinois Wesleyan. Our next step is to raise the transition matrix to a power of 1. Recall that by doing this, it ensures the rally has ended and illustrates the probability a point will be scored from a given state. However, this particular matrix provides some interesting insight. We notice that when Carthage sends a free ball there is.58 probability that UW Oshkosh will capitalize off of that free ball, resulting in a point. In the game of volleyball, we understand that a free ball is a shot of desperation, resulting in a simple pass from one team to another. Therefore, when a team receives a free ball, they should be able to gain a point from that free ball. Yet, we see in Figure 9 that when Illinois Wesleyan sends a free ball, there is a.55 probability that they will win a point off of that free ball. That means that there is only a.44 probability that Carthage will gain a point from receiving a free ball. This alludes to the assumption that Carthage did not utilize the free ball opportunity to the fullest potential. However, we are able to see that hitting was an area of strength for Carthage, giving them a.5 probability of scoring a point from a hit [ 1 ] Figure 9: Transition Matrix raised to a power of 1. Another important factor of this match is the serving. Carthage s serving is outlined in Figure 1 and UW Oshkosh s serving is outlined in Figure 11. Serving appears to have been another area of struggle for Carthage due to a.54 probability that Carthage will lose a point from their own serve, compared to a.45 probability of gaining a point from their own serve. Similarly, UW Oshkosh has a significantly higher probability of gaining a point from their own serve with a probability of.57, as seen in Figure 11.

8 [. 54 ] Figure 1: Serving matrix for Carthage [. 57 ] Figure 11: Serving matrix for UW Oshkosh The final calculations of this match illustrate the expected number of touches until the absorption as well as the estimated number of steps to the end of the rally from each state. We observe that when Carthage is serving there are an estimated 5.4 touches until absorption from that state. Also, the estimated number of steps to the end of the rally when Carthage sends a free ball is actually greater than the estimated number of steps when they are serving. This would be attributed to the lack of skill level regarding their free balls. A more detailed display of these results can be found in Table 12.

9 Expected touches until absorption Number of steps to the end of the rally from each state Carthage Serve Carthage Pass Carthage Set Carthage Hit Carthage Free Ball Carthage Block UW Oshkosh Serve UW Oshkosh Pass UW Oshkosh Set UW Oshkosh Hit UW Oshkosh Free Ball UW Oshkosh Block Table 12: Expected and estimated touches until the end of the rally. 4 Conclusion and Direction for Further Research We have concluded that using Markov Chains creates an accurate representation of the archived statistics and calculated results, thus supporting the validity of our model. Following raising our matrix to a higher power, we are able to see the probability a team will score a point from a given state. This grants the observer an inside look on the team skill levels as well as an approximate outline of the match. It can also be seen that the probabilities of scoring a point from a serve is identical to the probabilities found in our matrix. Furthermore, the expected number of touches to absorption as well as the estimated number of until the rally ends projects an approximate rally length. A coach has the potential to use this model for further research of his/her teams skill level as well as the skill level of the opposing team. Also, to achieve genuine statistics, it is suggested to video tape a volleyball match and calculate the transitions probabilities from the exact observed statistics of the match.

10 References [1] Grinstead. C, Snell, J., Grinstead and Snell s Introduction to Probability, American Mathematical Society, Providence, 26. [2] Kemeny, J., & Snell, J., Finite Markov Chains, Van Nostrand, Princeton, 196.

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