Skip to main content
Browse Subject Areas

Click through the PLOS taxonomy to find articles in your field.

For more information about PLOS Subject Areas, click here.

  • Loading metrics

The relationship between movement speed and duration during soccer matches

  • Kai Roecker ,

    Affiliations Furtwangen University, Applied Public Health (AGW), Furtwangen, Germany, Albert-Ludwigs-University Freiburg, Institute of Sports and Sports Science (IfSS), Freiburg im Breisgau, Germany

  • Hubert Mahler,

    Affiliations Furtwangen University, Applied Public Health (AGW), Furtwangen, Germany, Albert-Ludwigs-University Freiburg, Institute of Sports and Sports Science (IfSS), Freiburg im Breisgau, Germany

  • Christian Heyde,

    Affiliations Furtwangen University, Applied Public Health (AGW), Furtwangen, Germany, Albert-Ludwigs-University Freiburg, Institute of Sports and Sports Science (IfSS), Freiburg im Breisgau, Germany, adidas AG, Future Team, Herzogenaurach, Germany

  • Mareike Röll,

    Affiliation Furtwangen University, Applied Public Health (AGW), Furtwangen, Germany

  • Albert Gollhofer

    Affiliation Albert-Ludwigs-University Freiburg, Institute of Sports and Sports Science (IfSS), Freiburg im Breisgau, Germany


The relationship between the time duration of movement (t(dur)) and related maximum possible power output has been studied and modeled under many conditions. Inspired by the so-called power profiles known for discontinuous endurance sports like cycling, and the critical power concept of Monod and Scherrer, the aim of this study was to evaluate the numerical characteristics of the function between maximum horizontal movement velocity (HSpeed) and t(dur) in soccer. To evaluate this relationship, GPS data from 38 healthy soccer players and 82 game participations (≥30 min active playtime) were used to select maximum HSpeed for 21 distinct t(dur) values (between 0.3 s and 2,700 s) based on moving medians with an incremental t(dur) window-size. As a result, the relationship between HSpeed and Log(t(dur)) appeared reproducibly as a sigmoidal decay function, and could be fitted to a five-parameter equation with upper and lower asymptotes, and an inflection point, power and decrease rate. Thus, the first three parameters described individual characteristics if evaluated using mixed-model analysis. This study shows for the first time the general numerical relationship between t(dur) and HSpeed in soccer games. In contrast to former descriptions that have evaluated speed against power, HSpeed against t(dur) always yields a sigmoidal shape with a new upper asymptote. The evaluated curve fit potentially describes the maximum moving speed of individual players during the game, and allows for concise interpretations of the functional state of team sports athletes.


Metabolic, locomotor, and physical performance factors such as maximum sprint speed, endurance, or repeated sprint ability [1] contribute to the success of players in team sports such as soccer (association football). These factors can be observed as locomotor events using a relatively new type of wearable measurement device comprising high-resolution GPS combined with accelerometers and/or gyroscopes [2,3]. These micro-sensors not only continually measure GPS locations and thus running speeds but also, through the use of accelerometers and gyroscopes, register complex body movement dynamics [4,5].

However, uncategorized time-ordered motion data from players of soccer and other team sports are generally chaotic and not obviously interpretable, nor even functionally describable, between repeated samples [68]. Thus, the locomotor activities performed during soccer matches remain difficult to correctly classify into categories and are not clearly interpretable by either coaches or athletes.

Most past efforts to classify sports-related movements have used automated or semi-automated visual video analyses [9]; thus, fixed velocity limits or classifications by comparative analysis of distinct locomotion types (e.g., standing, walking, jogging, and sprinting) have been proposed for the interpretation of sports-specific metabolic or locomotor demands [1014]. However, recently researchers have increasingly used so-called wearable tracking devices, and have mainly focused on factors such as cumulative running distances within classified speed categories [1417], tactical maneuvers, or specific metabolic load indices for locomotor activities, such as "Player Load" [18,19] or "Metabolic Power" [8,20].

At present, no distinct and automatically applicable algorithm has been established that can compare the locomotor profiles of individual players during matches. Arbitrary time intervals are currently used to analyze player performance during sports games (e.g., 5 min for high intensity) [21], but there is no approach that describes these requirements in an holistic and robust way. We therefore propose a derivation of locomotor analysis that enables the direct comparison and performance diagnostics of whole match plays from computerized data preparation based on the principle of power-to-duration profiles as formerly shown for bicycling [22,23]. Beyond such graphic depictions, we further aimed to select and evaluate a numerical fitting model for the description of these types of profiles.

Following approaches introduced to derive maximum power outputs from discontinuous endurance competitions such as bicycling or mountain bike racing, we previously argued that competitors would present their individual power maxima at least once during a cycling competition. Clustering all maximum power outputs measured by force-sensitive cranks to the specified time domains, individual maximum “power profiles” or envelope-curves were generated, characterizing the individual performances of the related cyclists [22].

The aim of this study was to evaluate the relationship between effective power output (horizontal movement velocity, HSpeed) and the time duration of movement (tdur) during soccer games. Thus, this study should evaluate whether there is a general and unique function that describes the relationship between HSpeed and tdur in soccer. Such a general function describing the performance/power output of soccer players would be a powerful tool that could be used to improve the individual assessment of the distribution of metabolic and conditional preconditions in soccer. As a result, this kind of working model would allow analysis of "chaotic" data from soccer matches and would make it much easier for coaches and athletes to interpret individual performances.

Materials and methods


Thirty-eight healthy members of three soccer teams participated in this study. The first team (team A) played in the 5th German men’s amateur soccer league ("Oberliga"; 13 male participants: age 24.4 ± 2.4 y, body weight 78.6 ± 4.2 kg, height 181 ± 5 cm), the second team (team B) in the 4th German women's soccer league ("Oberliga"; 12 female participants: age 18.7 ± 3.3 y, weight 59.6 ± 6.9 kg, height 167 ± 2 cm), and the third team (team C) in the 8th German men’s amateur soccer league ("Kreisliga"; 13 male participants, age 22.6 ± 3.4 y, weight 75.3 ± 6.7 kg, height 182 ± 4 cm). The mean age, weight and height of team B was significantly (p < 0.01) lower than those of teams A and C. The positions played by the study participants are listed in Table 1.

The study protocol was approved by the ethics committee of Albert-Ludwigs-University Freiburg in accordance with the latest revision of the Declaration of Helsinki. All participants gave their written informed consent following full disclosure of the study protocol and procedures.

Motion analysis


Each participant wore a wearable tracking device (miCoach Elite Team System, adidas, Herzogenaurach, Germany) during the games, which was fixed into position on the back between the shoulder blades. The sensor location was chosen to maximize the validity of the recordings without hiding the athlete's body center, to ensure a sufficient connection with the global positioning system (GPS), and to minimize disturbances to the player during activity. The device contained a 10-Hz (10 samples per second) global positioning system (GPS) combined with an inertial measurement unit.

Data acquisition.

The devices were applied 45 min before kick-off and were worn until 30 min after the end of each game by all field players except the goal keepers. Data recorded before kick-off and after the final whistle were discarded prior to analysis. All other data, including movements during the half-time break and during any game interruptions by the referee during regular play time, remained within the analysis. Games from players with < 45 min play time were excluded from the calculations of the speed to movement duration relationships.


First, timed tracking points from the GPS were converted to distances and filtered through a Kalman filter [24] running at 33 Hz, converting the player-centric frame of reference of the data to a fixed frame of reference with respect to the earth’s surface (axes: East, North and Up). The outputs from the filter algorithm were then used to report and save the filtered location, HSpeed, and accumulated running distances at 10 Hz.

To analyze the basic relationship between tdur and HSpeed for each player and game, we applied so-called "moving medians". Medians were preferred over (moving) arithmetic means, because medians often give superior results due to their stability against outliers [25]. Hence, a sequence of moving medians of HSpeed against the course of play time (t) were computed (Eqs 1 and 2): (1) where (Eq 2) (2) T is the total time recorded in each experiment, and tdur is an ordered set of 21 unique time durations (tdurj) determining the window sizes for the median calculations (Eq 3): (3)

These procedures resulted in 21 distinct time-ordered sets of median-smoothed HSpeed, one for every time duration domain tdurj and for every player and game. Algorithms proposed by Härdle and Steiger [26] were used to increase computational efficiency and to optimize program running times for these computations.

From every median smoothed set , unique maximum values for every tdurj were selected, resulting in distinctive profiles of maximum HSpeed () for any specific time duration tdurj of the mentioned set (Eq 4)): (4)

Here, nj is the size of the related median-smoothed data set. For every player and every game, the resulting set contains 21 values, one for each member of the ordered set tdurj.

Soccer-specific movement characteristics might not only depend on maximum moving speeds, but also on agility [27], which is potentially represented by dynamic changes in HSpeed over time. Therefore, we additionally calculated moving medians [28] weighted with the absolute values of concurrent horizontal acceleration (|HAccel|). Thus, the value pairs (HSpeed(ti), |HAccel|(ti)) in each of the time windows described in Eqs 1 to 3 were transformed into ascending order by HSpeed(ti). In each set, the weighted median corresponds to the element HSpeed(tk)j that satisfies the following two conditions (Eqs 5 and 6): (5) and (6)

The starting indices s are thereby set at t (titdur j / 2) with i = 1, ⋯, nj. Analogously to Eq 4, the profiles for the maximum weighted medians of HSpeed (HSpeed_weighted) against tdur were calculated from the given sets (Eq 7): (7)

As a result, for each player and game, distinct profiles with values for maximal achieved weighted median HSpeed were given to characterize weighted locomotor performance during their games.

Nonlinear fittings.

To evaluate for regularity and to estimate function parameters for HSpeed against tdur, seven nonlinear functions were chosen based on their principal applicability (s. S1 Appendix), as they had logistic or sigmoidal decay characteristics. Prior to the fitting procedure, tdur was log-transformed for better clarity. For the optimization procedure, Newton-Raphson's method [29] was applied using a stop limit of 300 iterations aimed at 1.0 · 10-6 for the relative gradient. The Akaike Information Criterion (AICc) [30] was used as a relevant evaluation criterion to judge fitting quality. Other than the F-statistic, AICc is independent in the comparison of models with a different number of parameters.

Statistical analysis.

Statistical analyses, model fittings and evaluations were performed using JMP Pro Version 13.1.0 (SAS Institute Inc., Cary, NC, USA). Data are presented as means ± standard deviations unless otherwise indicated. For statistical calculations, a critical significance level of α = 0.05 was assumed. Group comparison considering more than two groups was performed using a nonparametric one-way analysis of variance (Steel-Dwass all-pairs rank-order statistics for non-normally distributed data and data that failed the equal variance test). Where repeated measures occurred in group comparisons (two or more games were analyzed for most of the players), we applied a general linear mixed model [31,32] with participant, gender and total play time as random factors to evaluate for significant differences of the curve fitting parameter results.


In total, 82 soccer game participations (≥ 30 min) of the 38 players from 3 teams (A, B, C) were analyzed. Total play times, and absolute and relative running distances per minute are given in Table 2. There were no statistically significant differences between the teams in these results.

Table 2. Minutes of game participation, total and relative running distances grouped by team.

The quantile statistics of maximum HSpeed versus tdur of all game participations (Fig 1) already represent a characteristically descending sigmoidal shape (S-curve shape with a distinct inflection point) of the relationship.

Fig 1. HSpeed results overview.

Quantile statistics (boxplots) for the maximal horizontal moving speeds (Hspeed, blue) and HSpeed weighted by acceleration (red) for the 21 calculated time durations (tdur, from Eq 3).

During the evaluation of seven formally matching fitting equations for the individual relationships between the log-transformed time duration (Log(tdur)) and the related fastest horizontal moving speeds (HSpeed), a five-parameter generalized logistic model (adopted from Richards' growth function [33], Eq. 8, S1 Appendix) revealed by far the best fitting quality. Fitting details for all equations are shown in Table 3. Based on individual game participations, root-mean-square error (RMSE) was as low as 0.10 m/s for the best fitting equation without any obvious bias; in contrast, bias was evident for all other models evaluated (Fig 2).

Table 3. Overall evaluation of seven models tested to represent the relationship between maximum horizontal moving speed (HSpeed; y) and time duration of movement (tdur; x) for every participant and soccer game tested in the study.

Fig 2. Residuals of the seven evaluated fitting equations.

The differences between the measured and the predicted values against the mean between both are shown. Only the Richards' Logistic 5P equation (Eq. 8, S1 Appendix) does not display a systematic deviation from the measurements. For further descriptions of the applied equations, see the S1 Appendix.

The range of parameter values for Eq. 8, S1 Appendix regarding each player's position, sex or team are given in Table 4. Remarkably, neither decreasing rate (ea) nor power (f) showed any significant differences when comparing factors of sex, team or position, while inflection point (eb), lower (c) and upper asymptotes (d) differed depending on position, sex, and team. In particular, Central Defenders and Central Midfielders showed significantly lower values for top speed ("Sprint excess", d) than did all other game positions. Similarly, Central Midfielders showed significantly higher values for c, or long term running speed ("Critical speed"), while Central Defenders had the lowest values for c. Fig 3A–3E gives a graphical overview of the fitted functions for all participants ordered by their position, gender, or team.

Table 4. Mean ± SD values for the parameters found with fittings of Eq. 8, S1 Appendix) to the relationship between time duration of movement (tdur) and maximum horizontal moving speed (HSpeedmax) for all players.

Fig 3. Groups of power-profile curve fittings.

(A) All fitting results. (B) Fitting results grouped by gender (blue: male, red: female). (C) Fitting results grouped by team (red: Team A, green: Team B, blue: Team C). (D) Fitting results of all female participants, grouped by position (red: Central Defenders, green: Central Midfielders, blue: External Midfielder, orange: Full-Backs, cyan: Strikers). (E) Fitting results of all male participants, grouped by position (red: Central Defenders, green: Central Midfielders, blue: External Midfielder, orange: Full-Backs, cyan: Strikers). HSpeed was log-transformed for better clarity in all cases. The close-ups in (D) and (E) show the upper and the lower part of the fittings, respectively.

To better reflect the match-specific endurance requirements of soccer players, we also applied profile fittings using Eq. 8 (S1 Appendix) to HSpeed after weighting the median values by each participant’s acceleration values (Eq 7). The fitting results for this alternative approach are shown in Table 5. We assumed that weighing by acceleration would lead to better representations of the discontinuous locomotor events that occur during soccer matches. However, this type of analysis did not further discriminate players by position, sex or team.

Table 5. Mean ± SD values for the parameters found when Eq. 8 (S1 Appendix) was used to model the relationship between time duration of movement (tdur) and the weighted maxima of horizontal moving speed (HSpeed_weightedmax, Eq 7) for all players.

As we used repeated measurements from several players, we performed a standard least squares analysis with the player's identity as a random effect. This analysis yielded r2-values (in decreasing order) for the upper asymptote (d) r2 = 0.79 (weighted by acceleration r2 = 0.75), lower asymptote (c) r2 = 0.64 (weighted by acceleration r2 = 0.57), inflection point (b) r2 = 0.57 (weighted by acceleration r2 = 0.11), decrease rate (a) r2 = 0.29 (weighted by acceleration r2 = 0.09), and power (f) r2 = 0.02 (weighted by acceleration r2 = 0.11). Furthermore, the correlation analysis shown in Fig 4 illustrates statistical independence between the first two variables, c and d.

Fig 4. Independence of the lower against the upper asymptotes.

A linear (Pearson) correlation between parameters c and d from all individual measurements is given (r2 = 0.04).


Main findings

In this study, we showed for the first time that a sigmoidal decaying function describes the numerical relationship between horizontal moving speed and the log-transformed time duration of running for soccer players during a match. A five-parameter logistic decay function derived from the so-called Richard's Logistic equation fit this relationship with almost zero systematic bias. Using this function, the upper and lower asymptotes and the points of inflection could discriminate the players’ characteristics and positions. The remaining parameters "power" (f) and "decrease rate" (a) of HSpeed were not different between participants and tests, meaning that there was a common and reliable curve shape in every case.

Methodological aspects

We used commercially available GPS sensors with only limited technical documentation in this study. The reliability of GPS-measured velocities has been reported to depend on many factors including sampling frequency and, especially, moving velocity (increased velocity reduces reliability) [34]. The manufacturers state that an integrated Kalman-type filter [24,35] and sensor-fusion algorithms using an accelerometer, a magnetometer, and a gyroscope "to adjust the reference frame of the sensor to the horizontal Earth's surface with a frequency of 10Hz" are used for the sensors [36]. An internal validation against a video analysis system (Immotio, Amsterdam, NL) resulted in mean differences of only 0.1 to 0.3 m·s-1 at the 95% confidence level [36], indicating excellent validity of the applied sensor system. Also, the particular profiles analyzed in this study yielded physiologically credible results. For example, for 30 m, the fastest sprints ranged from 3.46 s to 5.91 s (flying), which are overall plausible results for the tested selection of players and also compare favorably to the reference data of other researchers [37,38]. The fitting procedure applied further minimizes noise from the measurements of HSpeed, because separate deviating data points are adjusted to the curve's function.

Maximum acceleration values measured by wearable devices can yield erroneous results due to passive impacts or tackles to the body [39]. However, such collision-related impulses only last for fractions of a second, and seldom affect standing or slowly moving players (the only conditions that could cause positive, forward-directed acceleration). From the raw accelerometry data, a maximum positive outlier value within a time duration of 0.3 sec was determined at 8.6 m·s-2 (mean: 6.2 ± 0.9 m·s-2). At 1.0 sec, for comparison, this value decreased markedly to an overall maximum of 7.1 m·s-2 (mean: 5.1 ± 0.7 m·s-2, p < 0.0001). This is why low-pass filtering is recommended to improve the validity of accelerometers in team sports [5]. However, HSpeed profiles are most likely not affected by this phenomenon of passive impact, because acceleration must be effective for a longer time to induce the maximum HSpeed values that were observed in this study. Otherwise, reliability of acceleration data from trunk mounted systems is generally poor, especially at higher levels of acceleration [5]. Thus, weighting the profiles with acceleration did not improve the characterization of the players in this study (Tables 4 and 5). The search for alternative means of determining the factor agility is a promising area for future research.

Isolated tests of sprint running performance tend to have high variability and low reliability [40]. However, maximum sprint speeds are important factors in the functional assessment of players, and are specifically targeted during training interventions in soccer [41,42]. Similar arguments can be used for mid- and long-term running endurance performance [43,44]. The methodology that we describe here allows direct assessment of these performances. With each reiteration during a player's season, particularly at the highest intensity as is achieved during competitions, the characterization of the player will be more reliable. Thus, the approach presented here can lead to much more comprehensive information about individual performance than other conventional performance testing methods that use out-of-competition sprint time analysis or laboratory exercise testing.

Curve shape of HSpeed against tdur

The relationship between maximum power output and time has been discussed primarily in the field of occupational physiology. At first, Monod and Scherrer proposed the term "critical power [] from the notions of maximum work and maximum time of work" [45]. This 'critical' power (Wlim) is one parameter of several alternative non-linear functions that are all characterized by an asymptotic approximation of power output at [45] or below [46] tlim = 0. Since that introduction, intense and diverse discussions have taken place regarding the best numerical interrelationship between power output and time [47], with the aim of achieving interpretable parameters for performance diagnostics and participant characterization, especially in endurance sports [4850]. It turns out that experimental approaches to obtaining complete power-to-duration profiles are a challenging task: several maximum exercise tests in the relevant time domains are needed to form the basis for comprehensive curve fitting. This makes routine applications of power-to-duration analyses virtually impossible, and might be the major reason why modeling approaches have been developed to predict the power duration relationship from ordinary test procedures [51,52].

Other studies have determined a 'critical speed' from running instead of work measurements [53,54] based on the assumption that running velocity is linearly correlated to energy expenditure. This is (approximately) true when running at steady speeds with small accelerations [55]. However, our study used data from intermittent, not steady, running. That is why the lower asymptotes (“Critical speed”) are at very slow speeds (2.0 to 2.5 m·s-1), although most of the players reached their metabolic limits. To compensate for this, we tried the weighting by acceleration, which caused interval type variations of HSpeed to be weighted higher. In spite of this, the sigmoidal characteristics of the performance/duration relationship remained the same (s. Fig 1). Further investigation is needed to determine whether the model that fit our data would also apply to continuous running events over short intervals.

Profiles with intermittent maximal sprinting performances might be mathematically much more complex and hypothetically not describable with a simple or bi- exponential function. In addition to pure energy expenditure, neuromuscular, power-related and coordinative factors determine maximum running speeds over short [56] to ultra-short (< 2 sec) time durations [5,57]. Most importantly, these fast ultra-short sprints could be a prerequisite for successful participation in team sports [27,58,59].

Furthermore, our observations are in line with previous theoretical discussion by Bundle and Weyand [56] who claimed that there were specific limitations to all-out sprint exercise performances below 60-s duration due to these performances were determined by not only metabolic energy availability but also by musculoskeletal force application. The upper asymptote and flattening of the performances below 60 s, reported for the first time in this paper, are probably related to “non-metabolic” performance-limiting factors, as proposed by Bundle and Weyand.


This study shows for the first time empirical proof that the relationship between maximum running speed and tdur is better represented with a sigmoidal curve than with a simple exponential function. This type of curve shape was found in every game participation in this study, with no exceptions. A five-parameter function (s. S1 Appendix, Eq. 8; Fig 5) was sufficient to fit every profile without any bias. Furthermore, this sigmoidal curve shape varied individually in terms of upper ("Sprint excess") and lower ("Critical speed") asymptotes, and by tdur of the inflection point (staying power at maximum sprint speed). The other two parameters of this function, power and decreasing rate, did not vary individually, emphasizing the ubiquitousness of the described curve shape. Further work will be needed to validate the model by the modification of selective parameters over time and after certain types of training.

Fig 5. Graphical description of the fitting parameters.

The five parameters within the favored fitting model for the relationship between HSpeed and tdur (Eq. 8, S1 Appendix). Small open circle symbols (o) are from real results for one participant in one game.


We thank all the volunteers who participated in the study. We also thank Mr. Prof. Dr. Ulrich Abel for his help with the mathematical notation.

Author Contributions

  1. Conceptualization: KR HM CH AG.
  2. Data curation: HM MR.
  3. Formal analysis: KR.
  4. Funding acquisition: AG HM MR.
  5. Investigation: HM MR KR.
  6. Methodology: KR HM MR CH.
  7. Project administration: HM MR.
  8. Resources: KR AG.
  9. Software: KR.
  10. Supervision: KR AG.
  11. Validation: HM MR CH AG.
  12. Visualization: KR.
  13. Writing – original draft: KR.
  14. Writing – review & editing: KR HM MR CH AG.


  1. 1. Carling C, Bloomfield J, Nelsen L, Reilly T. The Role of Motion Analysis in Elite Soccer. Sports Med. Springer International Publishing; 2008;38: 839–862.
  2. 2. Chambers R, Gabbett TJ, Cole MH, Beard A. The Use of Wearable Microsensors to Quantify Sport-Specific Movements. Sports Med. 2015;45: 1065–1081. pmid:25834998
  3. 3. Hausler J, Halaki M, Orr R. Application of Global Positioning System and Microsensor Technology in Competitive Rugby League Match-Play: A Systematic Review and Meta-analysis. Sports Med. Springer International Publishing; 2015;46: 559–588. pmid:26714810
  4. 4. Yang C-C, Hsu Y-L. A review of accelerometry-based wearable motion detectors for physical activity monitoring. Sensors. Molecular Diversity Preservation International; 2010;10: 7772–7788. pmid:22163626
  5. 5. Wundersitz DWT, Gastin PB, Robertson S, Davey PC, Netto KJ. Validation of a Trunk-mounted Accelerometer to Measure Peak Impacts during Team Sport Movements. Int J Sports Med. © Georg Thieme Verlag KG; 2015;36: 742–746. pmid:25806591
  6. 6. Rampinini E, Coutts AJ, Castagna C, Sassi R, Impellizzeri FM. Variation in top level soccer match performance. Int J Sports Med. 2007;28: 1018–1024. pmid:17497575
  7. 7. Di Salvo V, W G, G A, P T, B D. Analysis of high intensity activity in Premier League soccer. Int J Sports Med. 2009;30: 205–212. pmid:19214939
  8. 8. Buchheit M, Manouvrier C, Cassirame J, Morin JB. Monitoring Locomotor Load in Soccer: Is Metabolic Power, Powerful? Int J Sports Med. 2015;36: 1149–1155. pmid:26393813
  9. 9. Barris S, Button C. A Review of Vision-Based Motion Analysis in Sport. Sports Med. Springer International Publishing; 2008;38: 1025–1043. pmid:19026019
  10. 10. Faude O, Koch T, Meyer T. Straight sprinting is the most frequent action in goal situations in professional football. Journal of Sports Sciences. 2012;30: 625–631. pmid:22394328
  11. 11. Mohr M, Krustrup P, Bangsbo J. Match performance of high-standard soccer players with special reference to development of fatigue. Journal of Sports Sciences. Taylor & Francis; 2011;21: 519–528. pmid:12848386
  12. 12. Bangsbo J, Nørregaard L, Thorsø F. Activity profile of competition soccer. Can J Sport Sci. 1991;16: 110–116. pmid:1647856
  13. 13. Schimpchen J, Skorski S, Nopp S, Meyer T. Are “classical” tests of repeated-sprint ability in football externally valid? A new approach to determine in-game sprinting behaviour in elite football players. J Sports Sci. 2015;34: 519–526. pmid:26580089
  14. 14. Varley MC, Aughey RJ. Acceleration Profiles in Elite Australian Soccer. Int J Sports Med. © Georg Thieme Verlag KG; 2012;:–. pmid:22895869
  15. 15. Wundersitz DWT, Josman C, Gupta R, Netto KJ, Gastin PB, Robertson S. Classification of team sport activities using a single wearable tracking device. J Biomech. 2015;48: 3975–3981. pmid:26472301
  16. 16. Aughey RJ. Applications of GPS technologies to field sports. Int J Sports Physiol Perform. 2011;6: 295–310. pmid:21911856
  17. 17. Carling C. Interpreting physical performance in professional soccer match-play: should we be more pragmatic in our approach? Sports Med. Springer International Publishing; 2013;43: 655–663. pmid:23661303
  18. 18. Boyd LJ, Ball K, Aughey RJ. The reliability of MinimaxX accelerometers for measuring physical activity in Australian football. Int J Sports Physiol Perform. 2011;6: 311–321. pmid:21911857
  19. 19. Dalen T, Jørgen I, Gertjan E, Geir Havard H, Ulrik W. Player Load, Acceleration, and Deceleration During Forty-Five Competitive Matches of Elite Soccer. Journal of Strength and Conditioning Research. 2016;30: 351–359. pmid:26057190
  20. 20. Osgnach C, Poser S, Bernardini R, Rinaldo R, Pietro Enrico di Prampero. Energy cost and metabolic power in elite soccer: a new match analysis approach. Med Sci Sports Exerc. 2009;42: 170–178.
  21. 21. Bradley PS, Sheldon W, Wooster B, Olsen P, Boanas P, Krustrup P. High-intensity running in English FA Premier League soccer matches. J Sports Sci. 2009;27: 159–168. pmid:19153866
  22. 22. Quod MJ, Martin DT, Martin JC, Laursen PB. The power profile predicts road cycling MMP. Int J Sports Med. © Georg Thieme Verlag KG Stuttgart · New York; 2010;31: 397–401. pmid:20301046
  23. 23. Pinot J, Grappe F. The record power profile to assess performance in elite cyclists. Int J Sports Med. © Georg Thieme Verlag KG Stuttgart · New York; 2011;32: 839–844. pmid:22052032
  24. 24. Kalman RE. A New Approach to Linear Filtering and Prediction Problems. J Basic Eng. American Society of Mechanical Engineers; 1960;82: 35–45.
  25. 25. Dixon WJ. Processing Data for Outliers. Biometrics. 1953;9: 74.
  26. 26. Härdle W, Steiger W. Algorithm AS 296: Optimal median smoothing. Appl Statist. 1995;44: 258–268.
  27. 27. Paul DJ, Gabbett TJ, Nassis GP. Agility in Team Sports: Testing, Training and Factors Affecting Performance. Sports Med. Springer International Publishing; 2015;46: 421–442. pmid:26670456
  28. 28. Edgeworth FY. XXII. On a new method of reducing observations relating to several quantities. Philosophical Magazine Series 5. 1888;25: 184–191.
  29. 29. Ypma TJ. Historical Development of the Newton–Raphson Method. SIAM Rev. Society for Industrial and Applied Mathematics; 2012;37: 531–551.
  30. 30. Akaike H. A new look at the statistical model identification. IEEE Transactions on Automatic Control. IEEE; 1974;19: 716–723.
  31. 31. Cnaan A, Laird NM, Slasor P. Using the general linear mixed model to analyse unbalanced repeated measures and longitudinal data. Stat Med. 1997;16: 2349–2380. pmid:9351170
  32. 32. Wilkinson M, Akenhead R. Violation of Statistical Assumptions in a Recent Publication? Int J Sports Med. © Georg Thieme Verlag KG; 2013;34: 281–281. pmid:23420193
  33. 33. Richards FJ. A Flexible Growth Function for Empirical Use. Oxford University Press; 1959;10: 290–301.
  34. 34. Cummins C, Orr R, O’Connor H, West C. Global Positioning Systems (GPS) and Microtechnology Sensors in Team Sports: A Systematic Review. Sports Med. 2013;43: 1025–1042. pmid:23812857
  35. 35. Gilgien M, Spörri J, Limpach P, Geiger A, Müller E. The Effect of Different Global Navigation Satellite System Methods on Positioning Accuracy in Elite Alpine Skiing. Sensors. Multidisciplinary Digital Publishing Institute; 2014;14: 18433–18453. pmid:25285461
  36. 36. Armitage R. Validation of the Team Elite System. Herzogenaurach; 2016. pp. 1–31.
  37. 37. CRONIN JB, HANSEN KT. Strength and power predictors of sports speed. Journal of Strength and Conditioning Research. 2005;19: 349–357. pmid:15903374
  38. 38. Wisloff U, Castagna C, Helgerud J, Jones R, Hoff J. Strong correlation of maximal squat strength with sprint performance and vertical jump height in elite soccer players. Br J Sports Med. BMJ Group; 2004;38: 285–288. pmid:15155427
  39. 39. Gabbett TJ, Jenkins DG, Abernethy B. Physical demands of professional rugby league training and competition using microtechnology. J Sci Med Sport. 2012;15: 80–86. pmid:21820959
  40. 40. Haugen T, Buchheit M. Sprint Running Performance Monitoring: Methodological and Practical Considerations. Sports Med. Springer International Publishing; 2015;46: 641–656. pmid:26660758
  41. 41. Faculty of Health and Sport Sciences, University of Agder, Kristiansand, Norway., Haugen TA, Tønnessen E, Hisdal J, Seiler S. The Role and Development of Sprinting Speed in Soccer. Int J Sports Physiol Perform. 2014;9: 432–441. pmid:23982902
  42. 42. Vescovi JD. Sprint speed characteristics of high-level American female soccer players: Female Athletes in Motion (FAiM) Study. Journal of Science and Medicine in Sport. 2012;15: 474–478. pmid:22516691
  43. 43. Helgerud J, Engen LC, Wisloff U, Hoff J. Aerobic endurance training improves soccer performance. Med Sci Sports Exerc. 2001;33: 1925–1931. pmid:11689745
  44. 44. Hoppe MW, Slomka M, Baumgart C, Weber H, Freiwald J. Match Running Performance and Success Across a Season in German Bundesliga Soccer Teams. Int J Sports Med. © Georg Thieme Verlag KG; 2015;36: 563–566. pmid:25760152
  45. 45. Monod H, Scherrer J. THE WORK CAPACITY OF A SYNERGIC MUSCULAR GROUP. Ergonomics. Taylor & Francis Group; 1965;8: 329–338.
  46. 46. Morton RH. A 3-parameter critical power model. Ergonomics. Taylor & Francis Group; 1996;39: 611–619. pmid:8854981
  47. 47. Billat LV, Koralsztein JP, Morton RH. Time in human endurance models. From empirical models to physiological models. Sports Med. 1999;27: 359–379. pmid:10418072
  48. 48. de Lucas RD, de Souza KM, Costa VP, Grossl T, Guglielmo LGA. Time to exhaustion at and above critical power in trained cyclists: The relationship between heavy and severe intensity domains. Science & Sports. 2013;28: e9–e14.
  49. 49. Vautier JF, Vandewalle H, Arabi H, Monod H. Critical power as an endurance index. Appl Ergon. 1995;26: 117–121. pmid:15677008
  50. 50. Morton RH, Hodgson DJ. The relationship between power output and endurance: a brief review. Eur J Appl Physiol. 1995;73: 491–502.
  51. 51. Broxterman RM, Ade CJ, Poole DC, Harms CA, Barstow TJ. A single test for the determination of parameters of the speed–time relationship for running. Respiratory Physiology & Neurobiology. 2013;185: 380–385. pmid:22981969
  52. 52. Dekerle J, Vanhatalo A, Burnley M. Determination of critical power from a single test. Science & Sports. 2008;23: 231–238.
  53. 53. Carter H, Dekerle J. Metabolic stress at cycling critical power vs. running critical speed. Science & Sports. 2013;29: 51–54.
  54. 54. Smith CGM, Jones AM. The relationship between critical velocity, maximal lactate steady-state velocity and lactate turnpoint velocity in runners. Eur J Appl Physiol. 2001;85: 19–26. pmid:11513315
  55. 55. Cavagna GA, Kaneko M. Mechanical work and efficiency in level walking and running. J Physiol. Wiley-Blackwell; 1977;268: 467–481.
  56. 56. Bundle MW, Weyand PG. Sprint exercise performance: does metabolic power matter? Exerc Sport Sci Rev. 2012;40: 174–182. pmid:22732427
  57. 57. Gabbett TJ, Wiig H, Spencer M. Repeated high-intensity running and sprinting in elite women's soccer competition. Int J Sports Physiol Perform. 2013;8: 130–138. pmid:22868237
  58. 58. Carling C, Bradley P, McCall A, Dupont G. Match-to-match variability in high-speed running activity in a professional soccer team. J Sports Sci. 2016;: 1–9. pmid:27144879
  59. 59. Bangsbo J. The physiology of soccer—with special reference to intense intermittent exercise. Acta Physiol Scand Suppl. 1994;619: 1–155. pmid:8059610