Biomechanics and Medicine in Swimming XI

tack and double-center attack) and one man-up offense (6-5) as well as
specific game situations (turn over, driving-in, and goal/penalty) (Table
1). The final state of “goal/penalty” is defined as the criterion of performance,
necessary for judging the performance relevance of all other tactical
behaviour. The typical tactical behaviour of the teams is expressed
in the transition between two subsequent states (“tactical patterns”).
Table 1. State model used for game analysis in water polo (for both
teams)
No. State Abbrev.
Turn over TO
Counter-attack CA
One-center-attack OCA
Back-position in OCA BPOCA
Flanker-position OCA FPOCA
Center-position in OCA CPOCA
Driving-in DI
Double-center-attack DCA
Back-position in double-center attack BPDCA
Flanker-position in double-center attack FPDCA
Center-position in double-center attack CPDCA
Man-up-Attack MUA
Back-position man-up (BPMUA) BPMUA
Flanker-position man-up (FPMUA) FPMUA
Center-position man-up (CPMUA) CPMUA
Goal/penalty G
Stochastic model: The transition probabilities between two states describe the
water polo match as a process that can be understood as a first order Markov
chain, when the following two properties are given: (1) the probability
for the next state depends only on the current state (Markov-property), and
(2) the transition probability from one state to another is independent of
their chronological position in the match process (chain-property).
The transition probabilities between the states can be transformed
into a two dimensional transition matrix. Each element of this matrix
has the property p ij (n) ≥ 0 and the line sum is equal to 1. In the theory
of a Markov chain, several kinds of states are distinguished. Absorbing
states are important, because the process ends in these states, and a new
process starts. For our purpose of performance diagnosis, the state “goal/
penalty” is defined as the absorbing state. The transition probability to
the absorbing state is called the goal probability (GP). The GP can be
calculated for both opponents by multiplication of a start vector (distribution
of the states) with the observed (empirical) transition matrix. Inso-doing,
the probabilities for the absorbing condition of “goal/penalty”
are attained for both teams.
Simulation to quantify the performance relevance of tactical behaviour
patterns: Based on the empirical transition matrix of a water polo
match, it is also possible to calculate the GP on the basis of a numerically
manipulated (simulated) transition matrix. In order to determine
the performance relevance of a tactical behaviour pattern of interest,
the empirical transition probability between these two states was manipulated
by a certain percentage in each observed match. After this the
goal probability was once more calculated and defined the performance
relevance (δGP) of a tactical behaviour pattern as the difference between
the goal probability (GP) as calculated by the original (observed) transition-matrix
and the goal probability as calculated by the manipulated
transition-matrix.
With reference to this simulation method there are two special
problems: 1. to find the adequate quantitative size of the manipulation
of the investigated transition probability, and 2. to establish an algorithm
that determines how the other (not manipulated) transition probabilities
have to be changed, so that the stochastic character of the matrix
chaPter4.trainingandPerformance
(sum of each row equals 1.00) is preserved. Regarding (1): according to
Lames (1991), the function to deflect the transition probabilities is: δTP
(TP) = C+B∙4∙TP (1-TP). TP is the transition probability and δTP the
change of the transition probability of the investigated tactical pattern.
The constant values applied in the present study were C = 1 and B = 5,
which were also determined by Lames (1991) and thoroughly tested
by Pfeiffer (2004, 2005). Regarding (2): each compensation procedure
was based on a proportional compensation of all other transition probabilities,
as no a priori information existed to justify any targeted compensation:
δTP yi = -(T yi /(1-TP)) ∙δTP x . Figure 1 demonstrates how the
performance relevance (δGP) of a tactical behaviour is calculated by a
simulation on the basis of a standardized manipulation of the transition
probability.
Figure 1. Part of a transition matrix of a water polo match with an example
of the simulative calculation of the performance relevance (δGP)
of the tactical pattern “back position in position attack” to “driving-in”
(BPPA 1 – DI 1) of team 1.
The increase of the transition probability between the states “back position
in position attack” to “driving-in” of team 1 by 2.48 % elevates the
GP by 0.6 % (Figure 1). This value documents the performance relevance
(δGP) of the manipulated tactical behaviour pattern. Thus, the
performance relevance (δGP) of any tactical pattern is quantified by the
scale in which the goal probability (GP) changes after the transition
matrix has been modified in terms of the tactical behaviour in question.
results
In the present study, eleven matches of the World League Final 2007
in Berlin were analysed in regard to eleven tactical patterns (transition
probabilities) (Table 2).
Table 2. Analysed tactical patterns (in each case for both teams)
No. Tactical pattern (transition) Abbrev.
Turn over – Counter-attack TO-CA
Turn over – One-center-attack TO-OCA
Back-position – Center-position (both “One-center-attack”) BPOCA-CPOCA
Back-position (One-center-attack) – Driving-in BPOCA-DI
Back-position (One-center-attack) – Double-center attack BPOCA-DCA
Back-position (One-center-attack) – Man-up-attack BPOCA-MUA
Flanker-position – Center-position (both “One-centerattack”)
FPOCA-CPOCA
Flanker-position (One-center-attack) – Driving-in FPOCA-DI
Flanker-position (One-center-attack) – Double-centerattack
FPOCA-DCA
Flanker-position (One-center-attack) – Man-up-attack FPOCA-MUA
Center-position (One-center-attack) – Man-up-attack CPOCA-MUA
The differences between the German and Serbian teams were analyzed
by t-test (independent two-sample). In all tests, p < 0.05 was accepted
as significant.
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