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1
Where Bolt and Bekele meet: the analytical basis of
running performance estimates
Wim Westera,
CELSTEC - Centre for Learning Sciences and Technologies
Open University of the Netherlands
NL-PO Box 2960
6401 DL Heerlen
The Netherlands
Telephone: +31-45-5762408
Wim.westera@ou.nl
Abstract
This paper proposes a self-contained analytical model for the prediction of individual running
performances. The model uses two personal bests for calibration, and then allows the
prediction of the athlete’s personal bests for any other distance. It is based on a simple, first
order estimate of the way lap time increments with total distance. Also, the model accounts
for delays that occur during start-up. It therefore covers a wide range of events including
endurance and sprinting distances. The model is validated with empirical data of a variety of
world class and sub top athletes. Outcomes display valid and reliable predictions with
inaccuracies typically around 1%. It greatly outperforms existing models (typically 3% or
higher). Importantly, the model is transparent, since it is based on theoretical principles rather
than arbitrariness and negotiation. It is self-contained, easy to use and affordable, because it
does not require any physiological or biomechanical tests to be carried out first. Also, the
model displays a universal validity, as the results suggest its applicability for any speed and
distance related sports event, including running, speed skating and swimming.
Author biography
Wim Westera is a full professor at the Open University of the Netherlands. His background is
in physics and digital media for learning. He is Director of the Learning Media Research
Programme of the Centre for Learning Sciences and Technologies. He is a master athlete and
racing cyclist.
Introduction
Human performance is strongly dependent on the time span that the performance has to be
delivered. Intense efforts that require high degrees of physical or mental power can only be
made for a short period of time. Reversely, for tasks that require prolonged efforts, the
internal economy of endurance reinforces effort levels well below maximum capacity.
In athletics it is well-established that average running speeds go down at longer distances.
Indeed, the average speed of top athletes in a 10,000 m event is some 20% below the average
speed in an 800 m race. This decay effect is obviously linked with the fundamental
restrictions to human power and energy resources, as well as the underlying physiological
processes and biomechanical constraints. Personal characteristics of the athlete (i.e. length,
weight, muscle fibre type, lung capacity, anaerobic power) make a big difference in the
amount of decay: sprinters tend to display severe decays of speed at longer distances, whereas
for long distance runners the decays are less pronounced.
2
Various researchers have been working on the relationships between human performance and
the duration of the efforts, in order to make predictions for individual athletes and devise
personalised training schedules for these. For many decades it is known that endurance
performance is directly related with maximal aerobic power 1. Measurement of maximum
aerobic running speed, or speed at maximal oxygen uptake can be used for predicting
performances across a wide range of running distances 2. At shorter distances, however,
predictions are highly unreliable because unknown dependences of performance and maximal
aerobic power. Bundle, Hoyt and Weyand 3 proposed a model that combined the
physiological limits of anaerobic and aerobic power for predicting performances in both the
sprinting and mid-range distances (ranging from a few seconds to a few minutes). They
suggest a simple exponential relationship between speed and run duration, and incorporate the
different time scales of available anaerobic and aerobic power in relation to time. However,
accuracies reported are poor: the predictions deviate on average well above 3 % from realised
performances. An additional disadvantage is that accurate tests for assessing the athlete’s
speeds at maximal aerobic and anaerobic power are required. Likewise, progressive tables
based on statistical processing of large numbers of performance data are known to be
inaccurate and unreliable. The International Association of Athletics Federations (IAAF)
provides and uses such scoring tables for several purposes: to evaluate scores in team
competitions over multiple events, to determine result scores of performances for all event
world rankings, or to produce national, school or club rankings 4. The official IAAF-
committee that is watching over the validity of the table, regularly needs to make
modifications to the scoring tables because of apparent irregularities in the relationship
between results and assigned points. But even in the latest version of the scoring tables
inconsistencies and suspect data can easily be tracked. For instance, the men’s world records
on 100 m, 200 m, 400 m, 800 m, 1,500 m, 3,000 m, 5,000 m, 10,000 m, and marathon,
respectively, yield the following array of scores: 1374, 1356, 1300, 1321, 1302, 1299, 1294,
1295, 1293. Unfortunately, these scores fail to be equal, or even be close to each other;
instead the scores suggest a unjustified bias toward shorter distances. Similarly, empirical
scoring tables for decathlon and heptathlon are criticised for their inaccuracies, producing
questionable rankings and records 5. Historical bias and frequent changes seem to demonstrate
the arbitrariness and opportunism of performance alignments 6. Harder7 takes a different
approach by considering population fractions achieving a certain performance level.
Calibration of the fractions between different events enables statistical mapping for inter-sport
comparisons. Although Harder’s tables deviate from the IAAF-tables, they correlate very well
with these, and thus display the same inaccuracies. These statistical approaches have two
things in common. First, they reflect a population-based average representing a mixture of
many different human features and conditions, which may severely affect their applicability
for individuals. Second, they are phenomenological in kind and do not rely on an underlying
theory that would improve our understanding of the mechanisms involved.
Rather than an empirical or a statistical model, this paper presents an analytical model that
allows accurate predictions without requiring any physiological or biomechanical test data.
Instead, the model uses two personal bests for calibration, and then it allows predicting an
athlete’s hypothetical personal bests for any other distance. The model is based on a first
order estimate of the way lap time (which is equivalent with reciprocal speed) increments
with total distance. Also, the model accounts for delays that occur during start-up. This way
the model covers the entire range including endurance and sprinting distances.
As first step, the basic model will be explained. Second, the basic model is validated using a
variety of empirical data. Third, the basic model will be extended with a mechanism that
accounts for start-up bias. Finally, the paper will present various practical applications.
3
Analytical model
Let t be the running time of an athlete required for a total distance s. What we are actually
looking for is a mathematical formula which expresses the relationship between running time
t and distance s. As an intermediate step we introduce the average lap time L (this corresponds
with reciprocal running speed), which is given by:
s
t
L= . (1)
The starting point of the model is that an infinitesimal increment ds of the running distance s
would produce an increase dL of the average lap time, while accounting for the effect that dL
gradually fades out at longer distances.
As a first order estimate the lap time increment dL is assumed to be reciprocally proportional
to the distance s, yielding
s
ds
dL ⋅=
α
, (2)
where is a constant.
Integration over s gives a simple logarithmic expression
)ln(
β
α
s
L⋅= , (3)
where is a constant. The first order approximation reflected in equations (1) - (3) is not only
theoretically grounded. Empirical evidence of its appropriateness can be found by using some
existing data. This is done in figure 1 which displays a plot of lap time L against total distance
s for men’s track running world records. The data suggest the validity of a linear relationship
in accordance with equation (3), even though data of different athletes were used.
4
Figure 1. Growth of lap times with distance for men’s track running world records.
Substituting equation (1) into equation (3) produces running time t as a function of total
distance s:
)ln()(
β
α
s
sst ⋅⋅= . (4)
The unknown coefficients and can be resolved from equation (4) by applying a twofold
calibration procedure. It requires the availability of two data pairs: [s1, t1] and [s2, t2],
respectively. This would mean using two personal records on different distances. The
outcomes of this procedure are:
)ln(
2
1
2
2
1
1
s
s
s
t
s
t−
=
α
(5)
and
)1(
)ln(
)ln()ln(
21
12
2
1
1
st
st
s
s
s
⋅
⋅
−
−=
β
. (6)
5
Inserting the values of
and ln(
)
into equation (4) enables us to produce prognoses of
running times for any distance. The two distances used for the calibration should produce a
sufficiently wide interval: Using someone’s personal records at 800 m and 3,000 m would
probably make a sound basis for forecasting the person’s achievable 1,500 m performance
(reliable interpolation), but are likely not to produce reliable forecasts of marathon
performance (unreliable extrapolation).
Model validation
For assessing the empirical validity of the method covered by equation (4), the following
procedure is used:
1.
Performance data of a 4 different groups of athletes are user:
-World class male athletes
-World class female athletes
-Committed male sub top athletes
-Committed female sub top athletes
2.
An additional requirement is that each of the selected athletes cover a minimum of
three distances. This is because two personal records are needed for calibration cf.
equations (5) and (6), and the third one for checking the prediction based on these.
This prediction is calculated via equation (4) and the outcome can easily be compared
with the real personal record.
3.
Only track events will be accounted for, because road events may be subject to
inaccuracies of the traversed distance due to uncontrolled bends and corners.
4.
Sprinting distances will be omitted for the moment, because of the disturbing effects
of reaction time delays and the time needed for acceleration from the starting blocks.
Later on we will go into corrections for these disturbances.
5.
The validation is preferably based on interpolation rather than extrapolation, because
of better accuracies.
6.
For each personal record prediction of an athlete the outcome is compared with the
athlete’s real personal record. The accuracy of the prediction is expressed as a
percentage of the deviation relative to the existing record.
7.
As an alternative yardstick a simple linear interpolation procedure is used. Linear
interpolation (and extrapolation) uses the following equation as a replacement for
equation (4):
)(
)(
)(
12
12
1
1
tt
ss
ss
tt −⋅
−
−
+= . (7)
This alternative procedure will be used as a reference for assessing the added value of
the logarithmic model of equation (4).
The outcomes of this validation procedure are summarised in the tables below. Table 1
presents the calculations of a sample of 8 male world class athletes. The personal record data
were collected from the all time performances lists of IAAF
8
.
Logarithmic model Linear interpolation Athlete Distance
(m)
Personal
best
(IAAF)
Predicted
personal
best
Deviation Predicted
personal
best
Deviation
1500 3:26.00 3:26.00 3:26.00 H.E.G.
3000 7:23.09 7:20.87 0.005 7:27.82 -0.011
6
5000 12:50.24 12:50.24 12:50.24
1500 3:26.34 3:26.34 3:26.34
3000 7:33.15 7:24.26 0.020 7:31.86 0.003
B.L.
5000 12:59.22 12:59.22 12:59.22
3000 7:25.79 7:25.79 7:25.79
5000 12:37.35 12:42.41 -0.007 12:49.14 -0.016
K.B.
10000 26:17.53 26:17.53 26:17.53
1500 3:33.73 3:33.73 3:33.73
3000 7:25.09 7:24.77 0.001 7:35.32 -0.023
5000 12:39.36 12:42.53 -0.004 12:57.44 -0.024
H.G.
10000 26:22.75 26:22.75 26:22.75
800 1:44.05 1:44.05 1:44.05
1500 3:29.14 3:32.61 -0.017 3:38.53 -0.045
R.R.
3000 7:43.85 7:43.85 7:43.85
1500 3:29.46 3:29.46 3:29.46
3000 7:20.67 7:20.18 0.001 7:25.29 -0.010
D.K.
5000 12:39.74 12:39.74 12:39.74
3000 7:45.44 7:45.44 7:45.44
5000 13:13.06 13:15.85 -0.004 13:22.83 -0.012
K.M.
10000 27:26.29 27:26.29 27:26.29
3000 7:44.40 7:44.40 7:44.40
5000 13:21.90 13:18.03 0.005 13:26.36 -0.006
J.H.
10000 27:41.25 27:41.25 27:41.25
1500 3:38.83 3:38.83 3:38.83
3000 7:52.50 7:44.61 0.017 8:01.04 -0.018
5000 13:36.10 13:27.44 0.011 13:50.66 -0.018
K.L.
10000 28:24.70 28:24.70 28:24.70
Overall
deviation 0.008 0.017
Table 1. Logarithmic and linear performance predictions for male world class athletes.
World class athletes provide an important sample because top athletes are usually well-
prepared and perform near the limits of human capability. For each of the athletes 3 or
occasionally 4 official personal bests are listed in the third column. Outer distances (shortest
and longest) have been used for the calibration, viz. the calculation of
α
and
β
according to
equations (5) and (6). These parameters were used to make predictions for the intermediate
events, both by the logarithmic model and the linear model, covered by equation (4) and
equation (7), respectively. The predictions of the logarithmic model are quite close to the
official personal bests. The average deviation (minus signs neglected) is only 0.8% (0.008, cf.
bottom row of table 1). The average deviation of the linear model is 1.7%. Likewise, table 2
displays the results for a sample of 6 female world class athletes, excelling at multiple
distances.
Logarithmic model Linear interpolation Athlete Distance
(m)
Personal
best
(IAAF)
Predicted
personal
best
Deviation Predicted
personal
best
Deviation
3000 8:22.20 8:22.20 8:22.20
5000 14:29.11 14:23.96 0.006 14:33.31 -0.005
P.R.
10000 30:01.09 30:01.09 30:01.09
3000 8:28.66 8:28.66 8:28.66
5000 14:12.88 14:45.20 -0.038 14:58.19 -0.053
V.C.
10000 31:12.00 31:12.00 31:12.00
3000 8:24.51 8:24.51 8:24.51
5000 14:12.88 14:25.78 -0.015 14:34.42 -0.025
M.D.
10000 29:59.20 29:59.20 29:59.20
7
3000 8:29.55 8:29.55 8:29.55
5000 14:11.15 14:29.65 -0.022 14:36.72 -0.030
T.D.
10000 29:54.66 29:54.66 29:54.66
800 1:57.80 1:57.80 1:57.80
1500 3:56.18 3:56.87 -0.003 4:06.78 -0.045
3000 8:28.87 8:29.01 0.000 8:43.17 -0.028
M.Y.J.
5000 14:51.68 14:51.68 14:51.68
800 2:02.49 2:02.49 2:02.49
1500 3:56.91 4:01.36 -0.019 4:08.61 -0.049
3000 8:25.56 8:28.52 -0.006 8:38.87 -0.026
T.T.
5000 14:39.22 14:39.22 14:39.22
Overall
deviation 0.014 0.033
Table 2. Logarithmic and linear performance predictions for female world class athletes.
The calculations for female top athletes show similar results as those for top class males, be it
that deviations are somewhat higher (1.4%). Again the logarithmic model clearly outperforms
the linear model. A general comment on the sample of top athletes would be that there are
only very few athletes that perform at world class level at multiple distances. Top athletes
tend to specialise in one or two events. Although they might occasionally participate in a third
or even fourth event, this is often not taken as serious as their specialism, nor is there training
optimised for it. For this reason, the validation procedure is extended to a male group and a
female group of amateur runners, under the condition that these amateur runners are well-
trained and highly ambitious sportsmen rather than leisure joggers. Table 3 lists the
calculation for a sample of male sub top runners.
Logarithmic model Linear interpolation Athlete Distance
(m)
Personal
best Predicted
personal
best
Deviation Predicted
personal
best
Deviation
800 1:56.82 1:56.82 1:56.82
1500 3:54.80 3:53.83 0.004 4:09.20 -0.061
3000 8:16.22 8:20.30 -0.008 8:52.86 -0.074
5000 14:28.61 14:33.92 -0.006 15:11.08 -0.049
E.G.
10000 30:56.62 30:56.62 30:56.62
800 1:57.20 1:57.20 1:57.20
1500 3:53.90 3:54.12 -0.001 4:09.03 -0.065
3000 8:19.50 8:19.91 -0.001 8:51.52 -0.064
5000 14:22.28 14:32.10 -0.011 15:08.17 -0.053
M.V.
10000 30:49.80 30:49.80 30:49.80
800 1:58.30 1:58.30 1:58.30
1500 3:59.51 3:54.80 0.020 4:08.29 -0.037
3000 8:25.60 8:18.25 0.015 8:46.83 -0.042
5000 14:28.50 14:25.61 0.003 14:58.22 -0.034
J.V.
10000 30:26.70 30:26.70 30:26.70
1500 4:00.51 4:00.51 4:00.51
3000 8:48.02 8:32.92 0.029 8:52.37 -0.008
5000 15:05.50 14:54.04 0.013 15:21.52 -0.018
H.K.
10000 31:34.40 31:34.40 31:34.40
1500 4:00.70 4:00.70 4:00.70
3000 8:26.70 8:27.96 -0.002 8:44.16 -0.034
5000 14:45.50 14:39.23 0.007 15:02.12 -0.019
H.D.
10000 30:47:00 30:47:00 30:47:00
800 1:46.40 1:46.40 1:46.40
1500 3:46.10 3:45.62 0.002 4:01.80 -0.069
S.H.
3000 8:25.90 8:28.83 -0.006 8:51.94 -0.051
8
5000 15:18.80 15:18.80 15:18.80
Overall
deviation 0.009 0.046
Table 3. Logarithmic and linear performance predictions for male sub-top athletes.
The personal record data in table 3 originate from the all time best lists of a local athletics
association
9
. Average deviation of the logarithmic prediction is 0.9%, against 4.6% for the
linear prediction. Similar results (0.9% versus 5.0 %) hold for sub top females, cf. table 4.
Logarithmic model Linear interpolation Athlete Distance
(m)
Personal
best Predicted
personal
best
Deviation Predicted
personal
best
Deviation
800 2:08.49 2:08.49 2:08.49
1500 4:28.71 4:27.74 0.004 4:44.36 -0.058
3000 9:44.00 9:54.63 -0.018 10:18.36 -0.059
J.W.
5000 17:43.70 17:43.70 17:43.70
800 2:16.00 2:16.00 2:16.00
1500 4:37.47 4:29.50 0.029 4:44.55 -0.026
3000 9:38.50 9:30.98 0.013 10:02.88 -0.042
5000 16:38.60 16:30.90 0.008 17:7.31 -0.029
J.B.
10000 34:48.40 34:48.40 34:48.40
800 2:13.50 2:13.50 2:13.50
1500 4:30.60 4:29.82 0.003 4:50.08 -0.072
3000 9:43.00 9:42.68 0.001 10:25.61 -0.073
C.S.
10000 36:31.40 36:31.40 36:31.40
800 2:17.80 2:17.80 2:17.80
1500 4:43.40 4:43.81 -0.001 4:59.57 -0.057
3000 10:25.60 10:23.70 0.003 10:46.21 -0.033
J.M.
5000 18:28.40 18:28.40 18:28.40
Overall
deviation 0.009 0.050
Table 4. Logarithmic and linear performance predictions for female sub-top athletes.
From the above it can be concluded that the logarithmic model produces far more accurate
predictions than the linear model. The overall average deviation of the presented cases is
found to be 0,9% for the logarithmic model, against 3.7% for the linear model. Compared
with existing models and tables, the logarithmic model produces far better accuracies.
Compensating for start-up bias
For the purpose of validation, only distances of 800 m and longer were used to avoid the
effect of time delays incurred during the early phase of the event, when the athlete has to
accelerate from standstill to cruise speed. For shorter distances, however, start-up delays will
strongly confound the outcomes of predictions. Various researches carried out biomechanical
studies of the running start-up process
10
, but these are mostly concerned with the dynamics of
body angle, required metabolic power and the techniques for start-up optimisation. For
evaluating the impact of start-up empirical data of sprinting events should be analysed.
Because of ongoing specialisation, however, only very few world class sprinters excel in
three sprinting disciplines (100-200-400). Yet, some exceptions could be found and a number
of these are presented in table 5 , together with some amateur examples
(8,9)
.
Athlete Distance Personal Logarithmic model Linear interpolation
9
(m) best Predicted
personal
best
Deviation Predicted
personal
best
Deviation
100 0:11.10 0:11.10 0:11.10
200 0:22.21 0:23.42 -0.055 0:23.83 -0.073
I.S.
(female)
400 0:49.29 0:49.29 0:49.29
100 0:10.83 0:10.83 0:10.83
200 0:21.71 0:22.73 -0.047 0:23.09 -0.063
M.K.
(female)
400 0:47.60 0:47.60 0:47.60
100 0:10.96 0:10.96 0:10.96
200 0:21.99 0:23.02 -0.047 0:23.39 -0.064
M.J.P
(female)
400 0:48.25 0:48.25 0:48.25
100 0:10.30 0:10.30 0:10.30
200 0:20.40 0:21.78 -0.067 0:22.17 -0.087
H.M.
(male)
400 0:45.90 0:45.90 0:45.90
100 0:10.10 0:10.10 0:10.10
200 0:19.83 0:21.23 -0.070 0:21.57 -0.088
T.S.
(male)
400 0:44.50 0:44.50 0:44.50
100 0:10.96 0:10.96 0:10.96
200 0:22.30 0:23.11 -0.036 0:23.50 -0.054
E.M.
(male)
400 0:48.58 0:48.58 0:48.58
100 0:10.79 0:10.79 0:10.79
200 0:21.41 0:22.57 -0.054 0:22.89 -0.069
L.B. (male)
400 0:47.10 0:47.10 0:47.10
100 0:12.44 0:12.44 0:12.44
200 0:26.04 0:27.68 -0.063 0:28.61 -0.099
N.K.
(female)
400 1:00.94 1:00.94 1:00.94
100 0:13.05 0:13.05 0:13.05
200 0:26.46 0:27.14 -0.026 0:28.18 -0.065
I.W.
(female)
400 0:59.02 0:59.02 0:59.02
Overall
deviation 0.054 0.074
Table 5. Logarithmic and linear performance predictions for sprinters.
The resulting prediction have poor matches with the realised personal bests. On average the
predictions deviates 5.4 % for the logarithmic model and 7.4 % for the linear model. These
deviations are substantially higher than those of the previous tables with endurance athletes.
Also, most deviations are negative which is in accordance with the neglect of acceleration
losses. For preserving the predictive quality of our model at lower distances this start-up bias
should be corrected for. Estimates for such correction can be given by analysing the sprinting
start-up process in more detail.
For sprinting distances the effects of acceleration at start-up cannot be ignored. In contrast
with longer distances, time losses during the acceleration phase of sprinting cause the average
speed <v> of the event to drop well below the athlete’s cruise speed v
c
, which is the steady
state speed achieved after completing the acceleration. Since any acceleration was ignored in
the model of equation (4), the model essentially assumes cruise speed v
c
, while average speed
<v> has been inserted so far. For longer distances this doesn’t make any difference, because
v
c
and <v> are nearly the same, but for sprinting relative large differences will occur. To
eliminate the start-up effects, we should be replacing average speed <v> with cruise speed v
c
in equation (4). This means that acceleration losses can be accounted for by replacing the
event distance s with a extended value s* that is given by:
10
tv
v
v
tsts
c
c
⋅=
><
⋅= )()(* . (8)
Thus, using s*(t) in equation (4) rather than s(t) would compensate for acceleration losses
since it virtually accounts for a flying start and the associated extra meters that have to be
traversed within the same time span t.
As a next step we need to establish practical values for cruise speed vc to be substituted in
equation (8). For this we will use some split times available for various world top sprinters.
Table 6 lists a sample of cumulative 100 m splits (11,12):
Ben
Johnson
1998
Carl
Lewis
1988
Maurice
Green
1999
Maurice
Green
2001
Tim
Montg.
2002
Asafa
Powell
2005
Usain
Bolt
2009
Usain
Bolt
2009
Average
10 1.83 1.89 1.86 1.83 1.89 1.89 1.85 1.89 1.87
20 2.87 2.96 2.89 2.83 2.92 2.91 2.87 2.88 2.89
30 3.80 3.9 3.81 3.75 3.83 3.83 3.78 3.78 3.81
40 4.66 4.79 4.69 4.64 4.70 4.69 4.65 4.64 4.68
50 5.50 5.65 5.57 5.50 5.54 5.54 5.50 5.47 5.53
60 6.33 6.48 6.40 6.33 6.37 6.39 6.32 6.29 6.36
70 7.17 7.33 7.23 7.16 7.21 7.23 7.14 7.10 7.20
80 8.02 8.18 8.09 8.02 8.05 8.07 7.96 7.92 8.04
90 8.89 9.04 8.94 8.91 8.90 8.92 8.79 8.75 8.89
100 9.79 9.92 9.79 9.82 9.78 9.77 9.69 9.58 9.77
vc 11.69 11.63 11.71 11.53 11.76 11.78 11.84 12.07 11.75
vc ·t 114.4 115.3 114.6 113.2 115.1 115.1 114.8 115.6 114.8
Table 6. World class 100 m split times.
From these splits it can be derived that steady state cruise speed vc is readily achieved after
some 30 meters. The table allows us to determine cruise speeds vc and s* for each sprinter by
using
)30()100(
30100
tt
v
c
−
−
= . (9)
Resulting values for vc and s* ( =vc.t) are also given in table 6, as well as their averages. So,
according to the column at the extreme right, running a 100 m event in 9.77 s (average speed
100/9.77=10.24 m/s) is technically equivalent with running 114.8 m at cruise speed 11.75
m/s.
For other distances, unfortunately, only few split recordings are available. There is some
incidental data of Usain Bolts 2009 world record at 200 m available though, which yield
accumulative 50 m splits of 5.60, 9.92, 14.44 and 19,19 seconds, respectively 13. From these
data it follows that for this 200 m event vc=11.04 m/s. Before using this single outcome for
our formulas, a slight correction should be carried out though, since it follows from table 6
that Bolt’s 100 m performance deviates substantially from average performance. Accounting
for the same (relative) deviation at 200 m reduces the 200 m reference value to vc= 10,96 m/s.
So, now we have two reference data of vc (vc= 11.75 m/s for 100 m, and vc= 10.96 m/s for
200m, respectively) which can be used for compensating acceleration losses. Although our
calibration value of vc at 200 m may not be very accurate because it is only based on a single
11
athlete’s performance at one event, it will be used for the time being, since recording new
split data is beyond the scope of this study. Note, however, that any modifications of the value
vc would only induce second order effects, since it would concern a correction of a correction.
For being able the apply the correction two things still have to be sorted out. First, having
used split times of world class sprinters, raises the question if the outcomes also hold for
amateurs. Second, having calibration points for v
c
at 100m and 200 m leaves the question how
to estimate the values of vc at other distances.
With respect to the first question, it is important to note that we actually need the value of vc.t.
The available 100 m split data show no significant relationship of the product vc.t with time t
(which acts as an indicator for performance). Note that vc and t counterbalance each other:
when cruise speed vc drops, performance goes down, which means that time t goes up. As a
first approximation we assume that the value of s* ( =vc.t) doesn’t change significantly with
performance t.
With respect to the second question it is important to note that the disturbing effects of start-
up will gradually disappear at longer distances. In fact, the difference between cruise speed vc
and average speed <v> will gradually approach zero at longer distances. Assuming
exponential decay of this correction with distance s yields:
s
ev
c
v
⋅
−
⋅>=<−
δ
γ
, (10)
where
and
are constants. After substitution of the calibration values for vc at 100 m and
200 m, we obtain
=21.3 and
=0.00365. Figure 2 displays the resulting graph for vc.t,
representing the extra distance (s*-s), which compensates acceleration losses, versus distance
s.
12
Figure 2. Fictitious extra distance for compensating acceleration losses, against event distance
s.
Now that we are able to correct for start-up delays with the help of equation (10) or the curve
in figure 2, it is interesting to see the effects of this on the sprinting predictions of table 5.
Table 7 shows the outcomes of the recalculation of sprinting performances of table 5, while
taking into account start-up effects.
Logarithmic model Linear interpolation Athlete Distance
(m)
Personal
best
(IAAF)
Predicted
personal
best
Deviation Predicted
personal
best
Deviation
100 0:11.10 0:11.10 0:11.10
200 0:22.21 0:22.86 -0.029 0:23.67 -0.066
I.S.
(female)
400 0:49.29 0:49.29 0:49.29
100 0:10.83 0:10.83 0:10.83
200 0:21.71 0:22.18 -0.022 0:22.93 -0.056
M.K.
(female)
400 0:47.60 0:47.60 0:47.60
100 0:10.96 0:10.96 0:10.96
200 0:21.99 0:22.47 -0.022 0:23.23 -0.056
M.J.P
(female)
400 0:48.25 0:48.25 0:48.25
100 0:10.30 0:10.30 0:10.30
200 0:20.40 0:21.25 -0.042 0:22.01 -0.079
H.M.
(male)
400 0:45.90 0:45.90 0:45.90
100 0:10.10 0:10.10 0:10.10
200 0:19.83 0:20.71 -0.045 0:21.42 -0.080
T.S.
(male)
400 0:44.50 0:44.50 0:44.50
100 0:10.96 0:10.96 0:10.96
200 0:22.30 0:22.55 -0.011 0:23.34 -0.047
E.M.
(male)
400 0:48.58 0:48.58 0:48.58
L.B. 100 0:10.79 0:10.79 0:10.79
13
200 0:21.41 0:22.02 -0.028 0:22.74 -0.062 (male)
400 0:47.10 0:47.10 0:47.10
100 0:12.44 0:12.44 0:12.44
200 0:26.04 0:27.04 -0.038 0:28.40 -0.091
N.K.
(female)
400 1:00.94 1:00.94 1:00.94
100 0:13.05 0:13.05 0:13.05
200 0:26.46 0:27.14 -0.026 0:28.18 -0.065
I.W.
(female)
400 0:59.02 0:59.02 0:59.02
Overall
deviation 0.029 0.067
Table 7. Performance predictions for sprinters, corrected for start-up losses.
In all cases the deviations between predicted and realised performances go down. The average
deviation for the whole sprinters’ sample goes down from 5.4 to 2.9% for the logarithmic
model, and from 7.4 to 6.7% for the linear model. The logarithmic model still greatly
outperforms the linear interpolation model. So, by accounting for start-up losses, the accuracy
of the approach has improved substantially.
In sum, the general prediction procedure now reads as follows:
1. Establish two sound personal bests: s1, t1, and s2, t2.
2. Replace all distances s1 and s2 with s1* and s2*, respectively by using
s
ettsts
⋅
−
⋅⋅+=
δ
γ
)()(*
, (11)
with
=21.3 and
=0.00365 (estimation of these parameters might be improved by
extended recording of split times).
3. Calculate the personal coefficients and through
)
*
*
ln(
**
2
1
2
2
1
1
s
s
s
t
s
t−
=
α
(12)
and
)
*
*
1(
)
*
*
ln(
*)ln()ln(
21
12
2
1
1
st
st
s
s
s
⋅
⋅
−
−=
β
. (13)
4. Choose a distance s for predicting time t.
5. Replace distance s with s*, using equation (11).
6. Calculate predicted time t by using:
)
*
ln(**)(
β
α
s
sst ⋅⋅= . (14)
Application
Now that the model compensates for acceleration losses it can be used across a wide range of
distances. To do the calculations, a computer programme might be devised, a preliminary
version of which is available on the web 14. Users enter their two personal bests required for
calibration and enter one or more distances for which they receive their prophesised times.
The logarithmic linearity of equation (3) also offers the opportunity of a simple graphical
representation of the model. This is displayed in figure (3). While the vertical axis denotes lap
time and the horizontal axis covers the logarithmic scale of total distance of the event, the
14
performances of an individual athlete are given by a unique straight line. For reasons of
convenience, performance times (derived from the product of lap time and distance) for each
event are projected at the appropriate co-ordinates.
Figure 3. Graphical representation of the logarithmic model.
The yellow line (labelled “Example”) is the performance curve of a fictitious athlete with
personal bests of 0:55 at 400 m and 40:00 at 10,000 m. These two calibration points are
indicated with yellow squares. The intersections with the ordinates provide predicted
outcomes at the various distances: these are 0:10.89 at 100 m, 0:24.08 at 200 m, 2:06.57 at
800 m, 4:27.57 at 1,500 m, 10:02.65 at 3,000 m, 18:07,40 at 5,000 m, 1:32:55 at the half
marathon and 3:21:41 at the marathon. The graphic representation also explains that the
accuracy will go up when the two calibration events are chosen on the opposite ends of the
horizontal scale. When these calibration events are too close to each other, the required
extrapolation rather than interpolation will increase the inaccuracies. The dark lines in figure
3 are the performance curves of selected world class athletes who cover multiple distances:
Kenenisa Bekele (K.B.), Hicham El Guerrouj (H.E.G.), Paula Radcliffe (P.R.), respectively.
The calibration values used are marked with dots. According to the model Kenenisa Bekele is
supposed to break the marathon world record (2:00:07). Also, he will be able to run 400 m in
52.20 s, which is only slightly below his 10,000 finish. Paula Radcliffe seems to have a
“weak” 10,000 m record: the prognosis is 29:49 (against 30:01.09 personal best). Note that
the curves of these two athletes display about the same slope, indicating the same type of
decay that is probably distinctive for long distance runners. The curve of Hicham El Guerrouj
shows a steeper slope, which can be attributed to the higher cruise speeds that he is capable of
in the mid-range distances. One of the things that turns out is that Hicham El Guerrouj would
be capable of running 10,000 m in 27:17. The performance curve of Usain Bolt (U.B.) is also
displayed in figure 3. Here, his personal bests at 100 m and 400 m are used, although the latter
(45.28) dates from 2007 which is well before his breakthrough as a world class sprinter;
indeed the predicted 200 m time of 20.41 sec is far behind his actual record. Nevertheless,
15
using the 100 m and 200 m events for calibration doesn’t make much sense exactly because of
the inaccuracies due to reinforced extrapolation. Still, the predictions at the longer distances
don’t make much sense here because of the extreme extrapolations they would require.
An interesting question would be at what distance world class sprinter Usain Bolt would
compare with long distance runner Kenenisa Bekele. With the model it can be calculated that
Bolt would meet Bekele at 1431 m. This corresponds with the horizontal co-ordinate of the
intersection of the two performance curves in figure 3 (although graphical estimation is less
accurate). A pre-assumption of this prediction is that both calibration records are of equal
quality. This is probably not the case. When Bolt would be able to update his 400 m best time,
his curve will go up at the right hand side, so he might be able to make it until 1,500 m or
more.
In principle, the approach is not limited to running events. It may also be applicable for
similar events in other disciplines, like speed skating and swimming. Table 8 presents the
outcomes for a small sample of world class speed skaters (15,16).
Logarithmic model Skater Distance
(m)
Personal
best (ISU) Predicted
personal
best
Deviation
1500 1:51.79 1:51.79
3000 3:53.34 3:55.59 -0.010
Cindy
Klassen
(female) 5000 6:48.79 6:48.79
1500 1:51.79 1:51.79
3000 3:55.83 3:54.49 0.006
Martine
Sablikova
(female) 5000 6:45.61 6:45.61
1500 1:41.80 1:41.80
5000 6:10.49 6:08.16 0.006
Shani
Davis
(male) 10000 13:05.94 13:5.94
1500 1:43.99 1:43.99
5000 6:03.32 6:03.19 0.000
Sven
Kramer
(male) 10000 12:41.69 12:41.69
Overall
deviation 0.005
Table 8. Logarithmic predictions for speed skating.
For easy estimation of start-up corrections for each speed skating event, the split times of the
associated world records have been used. Outcomes are promising, showing average
inaccuracy of 0.5% for the sample.
Likewise, table 9 shows the results for a sample of world class swimmers 17.
Logarithmic model Swimmer Distance
(m)
Personal
best (ISU) Predicted
personal
best
Deviation
200 1:55.52 1:55.52
400 4:02.13 3:59.90 0.009
800 8:18.80 8:17.53 0.003
Laure
Manaudou
(female)
1500 16:03.01 16:3.01
200 1:44.06 1:44.06
400 3:40.08 3:38.85 0.006
Ian
Thorpe
(male) 800 7:39.16 7:39.16
16
200 1:45.61 1:45.61
400 3:42.51 3:38.79 0.017
800 7:38.65 7:32.71 0.013
Grant
Hackett
(male)
1500 14:34.56 14:34.56
Overall
deviation
0.009
Table 9. Logarithmic predictions for swimming.
Because of the low speeds in swimming and for reasons of simplicity start-up corrections
have been neglected here ( =0). Nevertheless, predictive power for this sample of swimmers
is around 1 %. Note that the eventual boosting effects of turning points (every 50 m) have not
been taken into account.
Conclusion
The calculations presented in this paper provide strong evidence that the proposed model
produces valid and reliable predictions. Inaccuracies are typically around 1%. Therefore the
model greatly outperforms existing models (typically 3% or higher). Specific accuracies of
the calculations, however, are quite dependent on the conditions for interpolation or
extrapolation. Besides its unchallenged accuracy, the model has some additional advantages.
Importantly, the model is transparent, since it is based on theoretical principles rather than
arbitrariness and negotiation. Furthermore it is self-contained, easy to use and affordable,
because it does not require any physiological or biomechanical tests to be carried out first: it
just uses two personal bests for individual self–calibration. Since the model compensates for
start-up delays it is valid across a wide range of events, including sprinting, mid range and
long distance running. Finally, the model displays a universal validity: it seems to be highly
applicable for any speed and distance related sports event, including running, speed skating
and swimming.
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