TX_1~ABS:AT/ADD:TX_2~ABS:AT 8 http://journals.cihanuniversity.edu.iq/index.php/cuesj CUESJ 2025, 9 (1): 8-13 ReseaRch aRticle Analyzing the Football Sports Outcomes by the Two Types Branching Markov Process: Real Madrid Football Club as a Case Study Bestoon M. Abdullkarim¹, Azhin M. Khudhur¹, Dler H. Kadir ¹'² ¹Department of Statistics and Informatics, College of Administration and Economics, Salahaddin University-Erbil, Iraq ²Department of Business Administration, Cihan University-Erbil, Kurdistan Region, Iraq ABSTRACT In this research the football sport outcomes are analyzed by the two-type branching Markov process, assuming that the outcomes occur as a Poisson process and are described by the two-state Markov process. For this study, the number of losses, wins, and trophy results of the Real Madrid football club for 24 years (2000–2024) are obtained from the KOOORA Website. This study aimed to determine the probabilities of two state outcomes, the time the process stays in each state and the expected number of the two types for successive periods. In conclusion, the expected number of Real Madrid football sports outcomes for successive game times are continually decreasing until the number of games. Keywords: Two-state Markov process, two-type branching process, probability generating function, trophy, win and lose INTRODUCTION The stochastic branching process played a great role in various scientific applications and sports activities. In this work, the two types of branching Markov process have been applied to football sport outcomes. This study is in three sections, the first one includes the concept of the Markov process, the transition probabilities of two states, the expected time interval the process stays in each state, and the two types of branching process with their probability generating function. The second section specified for application, involves describing and analyzing the data about the game outcomes of Real Madrid football club during (2000–2024). The final section dedicated to conclusions, indicates the deduces which are found from the results of the application. METHODOLOGY In this study, the Markov process, the transition probabilities of two states, two types of branching process, and the probability generating function for two types. Markov Processes A stochastic process with continuous time parameter {X (t); t > 0} is called the Markov process if for any set of periods t1 < t2, <…, 0 𝑓1(𝑡) = 𝜇𝑒−𝜇𝑡 t > 0 Where the win and loss (λ and µ) rates are estimated by the maximum likelihood estimate (MLE), and they equal 1 / t . In addition, the average time ( t ) represents the average of spending in a specific state during each transition to that state in a two-state Markov process such as: ( ) ( )0 0 1 1 1 1 ,t E T t E T   = = = = They related the process parameters 𝜆 and 𝜇 to the directly observable characteristics of the process and the expected lengths of visits to each state {0,1}. ( ) ( )1 1 0 and 1E E   = = Thus, crude estimates of the process parameters are obtained. λ µ= = 1 0 1 1E E( ) , ( ) Two Type Branching Processes Considering a branching Galton–Watson process where two different types may be distinguished, either type will produce or possibly both types independently. Let U𝑛 and V𝑛 be a random variable representing type (1) and type (2), respectively, in the nth period,[6,7] which can be written as U X Xn j U j j V j n n + = = = +∑ ∑1 1 1 1 2 V Y Yn j U j j V j n n + = = = +∑ ∑1 1 1 1 2 When the branching process is one of the applications of Markov sequent then the transition probability law of the Markov process is: P (𝑋𝑗 𝑖 = K, 𝑌𝑗 𝑖 = L) = 𝑃𝑖 (k, l) k, l = 0,1,2, 3… for j = 1,2,3,… and i = 1,2 Where X Yj i j i, are independent identically distributed (iid) random vectors with probability mass functions 𝑃𝑖 (k, l) as above. The simplest situation assumes the process begins with a single type then we define for initial conditions. U0 = 1 𝑎𝑛𝑑 𝑉0 = 0 Or U0 = 0 𝑎𝑛𝑑 𝑉0 = 1 Moreover, the probability extinction of the two types (𝛱1, 𝛱2) is determined as follows: For type (1)  1 0 0 1 0= = = = =P U V U Vn n n n� ,� ,� ][ | And for type (2)  2 00 0 0 1= = = = =P U V U Vn n n�[ ,� | ,� ] Probability Generating Function Relations for Two Types of Branching Process In this section, we show some relation for the two-dimensional probability generating function of (U𝑛) and (𝑉𝑛) of the number of types of 𝑛𝑡h periods,[8,9] as Abdullkarim, et al.: Two types of branching Markov Process 10 http://journals.cihanuniversity.edu.iq/index.php/cuesj CUESJ 2025, 9 (1): 8-13 ( ) ( )1 2 1 1 1 2 , 0 , , ( , )i i k l k l n n i k l q q E q q P k l q q ∞ =  = =  ∑ (5) 0 ≤ 𝑞1, 𝑞2 ≤ 1 𝑎𝑛𝑑 𝑖 = 1,2 Where: 1. q1 is the auxiliary variable representing the weight assigned to the number of type (1) individuals (Un) 2. Moreover, q2 is the auxiliary variable representing the weight assigned to the number of type (2) individuals (Vn). Moreover, their multiple-step generalizations ( ) ( )1 1 2 0 0 1 2 , 0 , ( ,  | 1,  0) k l n n n k l q q p U k V l U V q q ∞ = = = = = =∑ (6) ( ) ( )2 1 2 0 0 1 2 , 0 , ( ,  | 0,  1) k l n n n k l q q p U k V l U V q q ∞ = = = = = =∑ With 𝑛 ≥ 0, it is clear that the generation function for type (1) is: ( ) ( )1 0 1 2 1,  q q q = ( ) ( )1 1 1 1 2 1 2,  ( ,( )q q q q = Moreover, for type (2) is ( ) ( )2 0 1 2 2,  q q q = ( ) ( )2 2 1 1 2 1 2,  ( , )q q q q = In general, the method can be used as ( ) ( ) ( ) ( )1 2 1 2 1 2 1 2, ( , , , )i i n m m n nq q q q q q   + = The Expected Numbers of the Two Types of Branching Process The method of generating function is the most important tool in the study of branching processes, it is useful to determine the mean matrix of the nth period. When (Un) and (Vn) the number of type 1 and 2 of the nth period,[9,10] then the expected numbers are given by ( ) 1 2 (1) 1 2 ( ) 0 0 111 1 , [ | 1, 0]n n nq q q q E U U V m q  = = ∂ = = = = ∂ ( ) 1 2 (2) 1 2 ( ) 0 0 12[ | 1, 0] q q q q E U U V m = = ∂ = = = = ∂ (7) ( ) 1 2 (1) 1 2 ( ) 0 0 211 1 , [ | 0, 1]n n nq q q q E V U V m q  = = ∂ = = = = ∂ ( ) 1 2 (2) 1 2 ( ) 0 0 221 2 , [ | 0, 1]n n nq q q q E V U V sssm q  = = ∂ = = = = ∂ Then the expected matrix for the nth period for two types of branching process is defined as, M m m m m =       11 12 21 22 (8) So M m m m m n n n n n ( ) ( ) ( ) ( ) ( )=         11 12 21 22 ( ) 1 1,1 1,2ij j m j q  = = ∂ = 1 j And ( ) 2 1,1 1,2ij j m j q  = = ∂ = 2 j Here M =       λ λ µ µ 11 12 21 22 According to the distribution of the two types (1) and (2) the probability generation function is given as ( )( ) ( ) ( )1 2 11 121 ,   . q q q q  = ( ) ( )11 1 12 21 1q qe − + −= ( )( ) ( )1 2 21 222 , . ( )q q q q  = ( ) ( )21 1 22 21 1q qe − + −= In addition, the expected number of the two types (1) and (2) for successive periods is G U Vn n n= ( )  (9) Then E G G G Mn s n n s( |+ =�� ) Application This section is specified for analyzing the football sports outcomes of Real Madrid football club by the two-type branching Markov process, where the football outcomes (loss and win) are denoted by the two states (0 and 1), respectively. Thus, the transition probabilities of the two- state Markov process and the expected length of time that the process spends in each state are defined, also the matrix of the expected numbers to four possible transitions of the two states (0,1) is prepared. Data Description For this study, the data about the football outcomes of the Real Madrid team, which has played annually with 19 other Spanish teams in periodic competitions for 24 years have been taken from the KOOORA website. Abdullkarim, et al.: Two types of branching Markov Process 11 http://journals.cihanuniversity.edu.iq/index.php/cuesj CUESJ 2025, 9 (1): 8-13 Table 1 shows the total outcome of 748 games played between 2000 and 2024, which included 142 losses with a probability of 0.2, and 606 wins with a probability of 0.8. Among these, 14 losses with a probability of 0.58 and 10 wins with a probability of 0.42 were associated with trophy outcomes. Estimation of the Loss and Win Rates The loss and win rates of the football outcomes are estimated concerning the trophy results, hence the loss rate ( ̂ )is estimated from the total number of losses, which is 101 from 448 games of 14 losses trophies in the average of 32 games annually 101 0.225 /   44 ˆ 8 gametime = = 0 225 32 7 2. . �� �/× = games annum Furthermore, the win rate ( ̂ ) is estimated from the total number of wins, which is 259 from 300 games of 10 wins trophies in an average of 30 games annually 259 0.863 /    0 ˆ 3 0 gametime = = 0 863 30 25 9. . � �/�× = games annum The Transition Probabilities of the Two- state Football Sport Outcomes The transition probabilities of the two-state football sport P i jij( , , )= 0 1 for successive game times ( )t0 are referred to in (1). Table 2 shows the transition probabilities between the two state football sports outcomes (loss and win) for successive game times (t = 1,2,…,10). Hence, the stationary distribution of the loose ( )0 and win 1( ) is defined in (2). 0 0.863 0.79 0.225 0.863  = = + 1 0.225 0.21 0.225 0.863  = = + The Expected Time the Football Sport (Process) Stays in Each Outcome (State) The expected length of time ( ) ( , 0,1)ij t i j = the football sport spent in any one of the states (0,1) for game time (t) is defined in (3). Table 3 shows the expected length of time the process spends in any football state during each transition for given game times (t = 5,10,15,20,25,30,35). The Expected Numbers of the Two-state Football Sport Outcomes The four expected numbers of the two-state (0,1) football sports m00, m01, m10, m11 are computed from the transition counts of the number of loss and win games according to the loss (state 0) and win (state 1) trophy of Real Madrid which has got during (2000–2024), as defined: Table 1: The number of loss and win outcomes with the trophy results Years Number of losses Number of wins Total Trophy 2000–2001 6 24 30 Win 2001–2002 10 19 29 Loss 2002–2003 4 22 26 Win 2003–2004 10 21 31 Loss 2004–2005 9 24 33 Loss 2005–2006 8 20 28 Loss 2006–2007 8 23 31 Win 2007–2008 7 27 34 Win 2008–2009 10 25 35 Loss 2009–2010 4 31 35 Loss 2010–2011 4 29 33 Loss 2011–2012 2 32 34 Win 2012–2013 5 26 31 Loss 2013–2014 5 27 32 Loss 2014–2015 6 30 36 Loss 2015–2016 4 28 32 Loss 2016–2017 3 29 32 Win 2017–2018 6 22 28 Loss 2018–2019 12 21 33 Loss 2019–2020 3 26 29 Win 2020–2021 3 21 24 Win 2021–2022 4 26 30 Win 2022–2023 8 24 32 Loss 2023–2024 1 29 30 Win Trophy : 101000110001000010011101 The transitions are as noted in Table 4. Table 4 shows the transition counts of the losses and wins of the Real Madrid football club. Then the four expected numbers mij (i,j = 0,1) of the football sports outcomes are represented in the following matrix, defined in (8). M m m m mij =     00 01 10 11 ij ij i ij m m i j= = ∑ ������, ,0 1 Mij� . . . . =     0 58 0 69 0 42 0 31 The Expected Number of Football Sport Outcomes for Successive Games The expected number of two state football outcomes (loss and win) with their probabilities for successive game times (t) is defined as in (9). Abdullkarim, et al.: Two types of branching Markov Process 12 http://journals.cihanuniversity.edu.iq/index.php/cuesj CUESJ 2025, 9 (1): 8-13 Table 4: The transition counts of the football sport outcomes State 0 1 Total 0 261 187 448 1 186 84 270 Total 447 271 718 Table 2: The transition probabilities of football sports outcomes with game times Time P00(t) P01(t) P10(t) P11(t) 1 0.860746816 0.139253184 0.523857216 0.476142784 2 0.813833866 0.186166134 0.700339264 0.299660736 3 0.798029382 0.201970618 0.759794231 0.240205769 4 0.792705015 0.207294985 0.77982399 0.22017601 5 0.790911291 0.209088709 0.786571808 0.213428192 6 0.790307005 0.209692995 0.788845078 0.211154922 7 0.790103427 0.209896573 0.789610919 0.210389081 8 0.790034843 0.209965157 0.789868923 0.210131077 9 0.790011738 0.209988262 0.789955841 0.210044159 10 0.790003955 0.209996045 0.789985123 0.210014877 Table 3: The expected time interval of the process spent in each state with game time Time µ00(t) µ01(t) µ10(t) µ11(t) 5 4.139175498 0.860824502 3.223167823 1.776832177 10 8.089996422 1.910003578 7.170013747 2.829986253 15 12.03999998 2.960000016 11.12000006 3.87999994 20 15.99 4.01 15.07 4.93 25 19.94 5.06 19.02 5.98 30 23.89 6.11 22.97 7.03 35 27.84 7.16 26.92 8.08 Table 5: The expected number of football sports outcomes for successive games Time (t) E (Gn+t | Gn) = Gn M (t) Pr. of loss game Pr. of win game Loss Win 1 337 286 0.54 0.46 2 154 126 0.55 0.45 3 73 65 0.53 0.47 4 35 37 0.49 0.51 5 17 24 0.41 0.59 6 8 16 0.33 0.67 7 4 11 0.27 0.73 8 2 7 0.22 0.78 9 1 5 0.2 0.8 E G G � �G �Mn t n n t( | ) ( ) + = = ( )      ( ) ( )142 606 0 58 0 69 0 42 0 31 ��� t t t t . . . . ( ) ( ) Table 5 shows the expected number of losses and wins with their probabilities of Real Madrid football outcomes for successive game times (t = 1, 2,…, 9). CONCLUSIONS From the results of the application, the following conclusions are found: 1. For given some successive game times, the transition probabilities P00 (t) and P11 (t) of the football sports outcomes are decreasing, but P01 (t) and P10 (t) are increasing 2. The expected time interval the process (football sport) spends in any one of the outcomes during each transition is continually increasing 3. The stationary distribution of the football sports outcomes shows that the Real Madrid team will win a game with the lowest probability and lose a game with the highest probability in a long time 4. The expected number of Real Madrid football sports outcomes for successive game times are continually decreasing till the number of games becomes five, then the mentioned team has won four games against one lost game. 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