Modeling in Sport

Along with the operationally defined concepts of dynamical systems theory comes a rich arsenal of mathematical  methods  and  modeling  tools  that may  be  usefully  employed  in  the  study  of  complex  sports  behaviors.  This  entry  highlights  just a handful of those, which are all centered on the “problem”  of  dimensionality  reduction.  Before going  into  modeling,  the  entry  first  outlines some useful approaches to data analysis aimed at system identification, starting with more heuristic approaches  that  capitalize  on  an  educated  guess about which state variable is (most) relevant, followed by more unbiased, statistical approaches.

System Identification

Parameter Extraction: Stability and Regularity

When seeking to identify the main characteristics  of  a  system’s  dynamics,  one  typically  applies conventional  statistics  in  combination  with  stability-related  measures  like  Lyapunov  or  Floquet exponents.  Lyapunov  exponents  are  the  method of  choice  in  estimating  the  stability  properties  of a  dynamical  system  involving  the  emergence  and disappearance  of  macroscopic  patterns,  such  as the transitions from walking to running, and from ankle to hip strategy in upright posture.

The Lyapunov exponent measures the exponential divergence of nearby trajectories and thus the robustness of a (movement) pattern against (small) perturbations. Only if the exponential divergence is negative, a perturbation will be compensated and the pattern is (locally) stable; otherwise, a transition  is  likely  to  occur  or  is  permanently  present (as in chaotic systems or self-organized criticality). In the study of movement coordination, Lyapunov exponents  and  other  stability-related  measures have proved rather useful in addressing movement flexibility, stability, and adaptability.

Of  similar  importance  for  the  identification of  dynamics  are  entropic  measures.  In  physics, entropy is used to quantify the degree of disorder in a statistical ensemble such as a gas. When looking at movement trajectories, in contrast, entropy may serve  to  quantify  the  degree  of  regularity.  A  frequently used measure in the analysis of noisy time series, such as movement data, is sample entropy. Entropic measures have been used, among others, to assess balance in gymnastics and dance, and the effects  of  sports-related  anxiety  and  brain  injury on  postural  control.  Entropy  can  be  readily  estimated, irrespective of the type and dimensionality of the data in question. For instance, predictability of  competitive  balance  in  sporting  contests  has been assessed through entropy using the example of the 2006–2007 English Premier League results.

Dimensionality Reduction: Principal Component Analysis and More

Advances  in  data  acquisition  have  allowed  for simultaneous  recordings  of  multiple  signals  for considerable  time  spans,  resulting  in  huge  data sets. This development has increased the need of a priori data reduction. Principal component analysis (PCA) suits this purpose.

When   analyzing   movement,   an   estimate of  the  covariance  matrix  between  trajectories (or  time  series)  is  central  to  PCA.  Eigenvalues of  this  covariance  matrix  rank  the  degree  to which  a  so-called  principal  component  contributes  to  the  entire  data  variance  (which  provides an  index  of  its  strength).  The  corresponding eigenvectors  are  the  principal  components,  also referred to as factor loadings. When projected onto these components, the movement trajectories yield factor  scores  (here  the  time  series  of  the  components). PCA has found wide application in sports, and many examples may be found in the literature.

Alternative but related approaches include, for example, clustering techniques and artificial neural networks, which all pursue the same goal: reducing dimensionality. However, an even more principled and  rigorous  approach  to  pursue  that  goal  is  to focus on phase transitions because theory dictates that in the immediate vicinity of the critical point the  dynamics  of  the  complex  system  under  study is  reduced  to  a  (small)  set  of  order  parameters. Whenever  that  scenario  occurs,  mathematical modeling in terms of dynamical systems becomes feasible.

Identifying Dynamics: Capitalizing on Noise

This  section  highlights  a  recently  established method   for   identifying   stochastic   dynamics. Originally,  this  analysis  had  been  developed  to study  stochastic  processes  in  the  context  of  turbulent flows, but of late, it has also been applied successfully  in  the  analysis  of  a  broad  variety  of experimental  data  including  human  movement. The method produces numerical values of dynamical equations as a function of phase space coordinates. From these numerical values, specific model terms can be estimated, but they can also be analyzed  directly  by  reconstructing  vector  fields  and comparing  those  across  experimental  conditions; see Figure 1.


modeling-sports-psychologyFigure 1  Identifying Dynamics

To  model  a  complex  dynamical  system,  consider xkas state variable dependent on time t. Further, let the system be high-dimensional (e.g., of dimension n) such that the entire system can be cast inthe vector-form X = (x1, x2, . . . , xn). Its stochastic dynamics may then read

= F(X, t) + noise,        (1)

where  the  dot-notation  refers  to  the  derivative with respect to time. In the presence of phase transitions one can split the state variables into those that  become  unstable  (u,  representing  the  switch in  macroscopic  patterns)  and  those  that  stay  stable  (s).  One  may  formally  write  =  (us).  The previously  explained  separation  of  time  scales  in the vicinity of the phase transition implies that the stable parts can be eliminated adiabatically,1which somewhat  counterintuitively  has  the  mathematical  form     =  0.  This  adiabatic  elimination  allows for expressing the stable components as mere functions  of  the  unstable  ones,  i.e.,  M(u),  which known as the center manifold theorem according to  which  the  stable  components  are  “enslaved” by the unstable ones. The latter are hence the so-called order parameters whose dynamics read

= N(u, t) + noise.        (2)

Importantly,  the  dimension  of  is  always  much  smaller  than  that  of  by  which  one  can  the rigorously  analyze  the  system  through  well-established mathematical techniques.

As an example, consider the case in which the order parameter dynamics can be written using the gradient of a potential V—that is,    = –V’(u). If V is depicted as a landscape, then the systems stability can be readily determined by means of the hills and valleys of the landscape: the former represent the unstable states and the latter the stable states of the dynamics.

Movement Coordination: Low-Dimensional Dynamics

The  previously  given  approach  had  a  major impact on the study of the coordination of rhythmic  movement  through  the  seminal  work  of J.  A.  Scott  Kelso  and  colleagues  on  frequency-induced phase transitions in rhythmic finger movements.  This  transition  was  modeled  by  means  of potential for the relative phase between the finger movements, which changed shape as a function of movement frequency, as shown in Figure 2, and an underlying system of two nonlinearly coupled nonlinear oscillators. The model motivated a flurry of research on rhythmic bimanual coordination, with ramifications  to  learning  and  development,  the accompanying brain activity, and to interpersonal coordination  in  dyads  and  teams.  Through  the intimate  connection  between  the  study  of  motor systems  has  found  many  applications  in  recent years.  Complex  networks  link  to  dynamical  systems by virtue of the phase transitions that occurwhen   a   network   grows   and   connections   areformed;  in  random  networks,  transitions  occur control  and  learning  and  sports  behavior,  it  also penetrated the domain of sport science.


Figure 2  Potential Underlying the Dynamics of the Relative Phase

modeling-sports-psychology-f3Figure 3   Effective Value of the Lung Volume as a Function of Locomotion and Respiratory Frequency Ratio

Entrainment of Locomotion and Respiration— Also Low-Dimensional

A  closely  related  approach  revealed  how  the optimization  of  oxygen  extraction  in  animal  respiration  can  explain  the  multi-frequency  synchronization of breathing and locomotion observed in competitive rowing. In this case, the to-be-analyzed potential is formed by the negative effective value of  lung  volume  (modulated  by  the  cyclic  muscle contraction due to locomotion). The corresponding variational principles (searches for hills and valleys) do  not  only  predict  steady—that  is,  optimal— states  but  also  (points  of)  transitions  between those   states   whenever   control   parameters— again, movement speed but also fatigue—are being altered; see Figure 3.

Graph Theory

The  analysis  of  complex  networks  as  a  discretized   and   distributed   version   of   complex toward  (almost)  fully  connected  networks  dependent on growth rate and the probability to connect. It  is  duly  recognized  in  the  literature  that  match play  in  (ball)  sports  are  amenable  in  terms  of  an analysis  in  terms  of  network  theory.  And  indeed all  ingredients  for  this  are  available:  The  players represent  the  nodes,  and  they  are  connected  to each other via ball passes and via perception-based collective  motion.  These  connections  represent the  edges.  The  number  of  passes  between  players  provides  a  strength  of  their  connection  and so  does—although  more  difficult  to  quantify— coherent  motion.  An  analysis  of  a  sports  match in those terms is already arguably more revealing than  an  analysis  in  more  conventional  terms  like average  speed,  distance  covered,  or  areas  visited. Using a variety of so-called centrality measures, it may provide information about the importance of nodes  (hubs)  and  edges  in  the  network  and  thus reveal  strategic  behaviors.  Various  attempts  have been  made  in  this  regard,  but  a  crucial  element has so far been missing—namely the fact that the connections between players serve the purpose of making  and  preventing  goals,  implying  that  goal effectiveness and directionality should be incorporated  into  the  analysis.  In  other  words,  network theory needs to be appropriated to match analysis in  order  to  make  it  transcend  the  current  divide between  match  play  analyses  based  on  temporal correlations  (e.g.,  relative  phase)  and  spatial  correlations (e.g., Voronoi diagrams).


Concepts of dynamical systems build on a potent arsenal  of  mathematical  tools.  Exploiting  this arsenal  yields  a  highly  sophisticated  assessment of complex sports performance. Models in motor control  are  seminal  for  the  predictive  capacity of  this  approach.  In  particular,  the  macroscopic switches  in  behavior  can  be  analyzed  with  great precision. Like the general focus on dynamical systems  in  sports  science,  also  modeling  efforts  are still  in  their  infancies.  The  rigor  of  the  employed mathematical tools, however, renders the approach comprehensive and appealing.


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  8. Yue, Z. Y., Broich, H., Seifriz, F., & Mester, J. (2008).Mathematical analysis of a soccer game. Parts I & II.Studies in Applied Mathematics, 121(3), 223–243,245–261

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