ARTICLE
Human running performance from real-world
big data
Thorsten Emig
1✉ & Jussi Peltonen
2
Wearable exercise trackers provide data that encode information on individual running
performance. These data hold great potential for enhancing our understanding of the complex
interplay between training and performance. Here we demonstrate feasibility of this idea
by applying a previously validated mathematical model to real-world running activities
of ≈ 14,000 individuals with ≈ 1.6 million exercise sessions containing duration and distance,
with a total distance of ≈ 20 million km. Our model depends on two performance parameters:
an aerobic power index and an endurance index. Inclusion of endurance, which describes the
decline in sustainable power over duration, offers novel insights into performance: a highly
accurate race time prediction and the identiﬁcation of key parameters such as the lactate
threshold, commonly used in exercise physiology. Correlations between performance indices
and training volume and intensity are quantiﬁed, pointing to an optimal training. Our ﬁndings
hint at new ways to quantify and predict athletic performance under real-world conditions.
https://doi.org/10.1038/s41467-020-18737-6
OPEN
1 Université Paris-Saclay, CNRS, Laboratoire de Physique Théorique et Modèles Statistiques, 91405 Orsay, France. 2 Polar Electro Oy, Professorintie 5, 90440
Kempele, Finland. ✉email: thorsten.emig@u-psud.fr
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S
keletal evidence suggests that endurance running may have
evolved 2 million years ago1. It probably originated as a
hunting skill but has later developed to competition, dating
back to ancient Olympic Games ~720 BC2 and exercise form for
mass population. Over the years, endurance running has under-
gone substantially change. Recent decades have witnessed an ever
growing exercising population which uses wearable sensors to
bring together astonishing volumes of data for speed, distance,
heart rate, accelerations, and more3–5. For example, endurance
athletes like runners and cyclists currently upload from GPS
enabled sensors more than a billion activities per year world-
wide6. In principle, these data provide an exciting opportunity to
monitor human physiology noninvasively under real-world
conditions outside the laboratory. Measuring the physiological
response to physical activity can provide important insights for a
variety of populations ranging from elite athletes to recreational
exercisers to patients in rehabilitation7,8. However, the analysis of
big data sets of large, heterogeneous groups of individuals poses a
substantial challenge due to the quality of the data itself9,10, lack
of effective theoretical models11, and inﬂuence of environmental
factors like weather conditions12,13. The important, robust
properties of an individual’s physiology can be overshadowed by
details speciﬁc to the conditions of recording. Thus, there is a
demand for universal theoretical models that have been validated
for noise-free exercise data and can be applied under noisy real-
world conditions to derive meaningful physiological and perfor-
mance information14.
To date, exercise physiologists conventionally use laboratory
testing to determine parameters that measure ﬁtness and per-
formance potential15. A strength of laboratory testing is that it
can distinguish between cardiovascular limit, maximal rate of
oxygen consumption (VO2max), neuromuscular effects, and run-
ning economy16,17. Together VO2max and running economy
determine maximal aerobic speed, which is the slowest speed at
which VO2max occurs. Maximal aerobic speed correlates with race
speed on shorter distances but alone cannot predict race times for
longer distances such as the marathon. Exercise thresholds have
been used in exercise testing to quantify metabolism. However,
the determination of such thresholds, like the lactate threshold, in
the laboratory is somewhat limited. Typical laboratory testing is
short-lasting and does not always fully capture time and distance
dependent reduction in running economy18,19. For example, only
sparse results exist for the endurance limited fractional utilization
of maximal aerobic power (MAP) and its dependence on exercise
duration20. Moreover laboratory testing is expensive and not
available to most of the population. The undeniable fact that the
best test of running performance is an actual race and not
laboratory tests highlights the need for models speciﬁcally con-
structed to extract performance indices of an athlete from their
regular exercise performance. For these reasons, models that can
utilize data from wearable devices and turn those into meaningful
performance parameters may offer a cost effective alternative
approach to laboratory testing. However, it must stressed that this
type of approach does not elucidate the physiological and bio-
mechanical mechanisms that control performance. It is an
adjunct to the methods which are already used, providing addi-
tional insight into running and the potential training factors
inﬂuencing performance and it does not replace the insights that
we can gain from laboratory testing.
Several empirical and physiological models have been put
forward for explaining running world records in terms of a few
physiological parameters. The noted physiologist Hill empirically
proposed a hyperbola to describe the maximal power output as a
function of exercise duration21. Also a purely mechanical
approach, based on the runners equation of motion, has been
proposed22. These approaches predict that the average racing
velocity tends to be a constant value with increasing race distance
which contradicts observation. While more recent approaches
have combined physiology and observations to propose more
realistic logarithmic relations between maximal power output and
duration23, these models depend on many parameters that vary
among individuals24. Recently we have developed a universal
running model which builds on concepts in exercise physiology,
depends only a minimal set of key performance indices that are
required to predict race performance, contains no additional
individual-dependent quantities and has been validated with
running world-records14. Here, we show that it is also possible to
obtain novel insights into individual’s running performance by
applying this model to big exercise datasets.
Exercise data are a valuable source of information about
individual long-term training protocols. Endurance training leads
to a wide spectrum of physiological responses. However, in
practice, training is prescribed often only by anecdotal evidence
and personal experience. This might be due to a lack of knowl-
edge of statistically signiﬁcant correlations between the relevant
physiological parameters and training characteristics for large
groups of individuals with different ﬁtness status. Here, we
demonstrate the feasibility to extract key performance indices
from real-world running exercise data recorded with wearable
exercise trackers. We apply our method to runners during their
training season before a marathon race. Our universal running
model characterizes a runner’s performance with two indices that
measure (1) endurance (endurance index) and (2) the velocity
requiring MAP output (aerobic power index). The main aim of
our work is to demonstrate the feasibility of extracting perfor-
mance indices from real-world racing results in a big population
of runners and to use these indices to predict accurate race times
and evaluate the effect and efﬁciency of training. Our approach
represents a potentially powerful platform to enlarge dramatically
the number of tested subjects in sports science by extending
performance index acquisition from conventional laboratory
testing to real-world conditions with the aid of mathematical
modeling and wearable technology.
Results
Universal performance model. In previous work we have
developed a model that can be used to extract aerobic perfor-
mance indices from race data14. To summarize, this model
expresses exercise intensity on a relative power scale p, which
varies between zero, corresponding to basal metabolic rate, and
unity at MAP generation. MAP is expected to correspond to
maximal oxygen uptake VO2,max but this analogy needs not to be
assumed in our approach. A linear relation p(v) maps running
velocity v to relative power with p(vm) = 1 deﬁning vm as
an aerobic power index associated with MAP beyond which
anaerobic energy supply can yield p > 1 for a short time only.
Anaerobic supply contributes to maximal exercise shorter than a
crossover time tc which in our model is the longest time over
which MAP can be sustained. An important prediction of our
model is that the maximal value of the relative power p that a
runner can maintain declines logarithmically with duration, with
a rate γl, assuming that the durations are longer than tc. This
ﬁnding is in agreement with a ﬁnding of A.V. Hill who observed
this form of decline in running world records21. For more details
on our model, see the “Methods” section. Here, we use this
universal, i.e., subject independent model for human running
performance, to extract aerobic performance indices from ﬁn-
ishing times of runners worldwide by matching them with model
predictions14. The analyzed data set comes from an exercise
tracking platform that contains precise records of distance and
duration (and hence average velocity) of running activities
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of ≈19K individuals, who ran a total distance of 32M km over a
period of 3.5 years. The data were recorded by the individuals
with a GPS digital sports watch (V800, Polar Electro Oy, Oulu,
Finland)25, and uploaded to the platform. Maximal performance
of an individual was measured by the fastest ﬁnishing time for the
four most common racing distances 5000 m, 10,000 m, half-
marathon (21,097.5 m) and marathon (42,195 m) within a racing
season, which is deﬁned as the 180 days preceding the marathon
race (see “Methods” section for detection of racing activities).
The velocity corresponding to our parameter vm is difﬁcult to
measure in laboratory settings since VO2,max can be achieved
over a wide range of sub-maximal intensities because of an
upward drift of oxygen uptake with exercise duration18,19. In
general, our model can determine vm from the crossover of the
race–time–distance relation at time tc, and hence is free from
this complications. The simplest version of the model assumes a
ﬁxed time tc. Model predictions for sub-MAP performances do
not depend on this ﬁxed time since other choices lead only to
consistently renormalized values for vm and γl (which are then
no longer associated strictly with MAP but with a slightly
different power). In agreement with the application of our
model to running records on both the super- and sub-MAP
branches14 and laboratory testings26, we choose tc = 6 min in
the following. Combining running economy and the decline of
the fractional utilization of maximal power output with race
duration, the fastest time T(d) over a distance d is given by the
universal expression
TðdÞ ¼  tc
γl
d
dc
1
W1  d
dc
expð1=γlÞ
γl
h
i
for d ≥ dc ;
ð1Þ
where we deﬁned dc = vmtc, and W−1 is a real branch of
the Lambert W-function which is deﬁned as the multi-valued
inverse of the function w ! w expðwÞ27. W−1(z) is real valued
for −1/e ≤ z < 0 which is fulﬁlled for all distances d that we
consider (see the “Methods” section for more detail). Note that
T(dc) = tc, i.e., dc is the distance that can be maximally raced in
the time tc. The condition d ≥ dc is always satisﬁed for the race
distances considered here. We note that Eq. (1) is an exact
solution of our model. It can be also obtained from earlier
descriptions of the energetics of endurance running28–30 when
the fractional utilization of MAP is described by our prediction
of a slow, logarithmic decay, and a linear increase of the energy
cost of running with velocity is assumed.
The model parameters, called performance indices, quantify
different aspects of performance and provide a unique insight
into basic determinants of ﬁtness in a large population of
runners over a wide range of exercise capacities and over long
time scales. The velocity vm measures combined running
economy and MAP and is known to be a better predictor of
performance than VO2,max alone31. We deﬁne the endurance
index as El ¼ expð0:1=γlÞ, which encodes that 90% of vm can
be maintained for an extended time Eltc > tc. The pair of
performance indices vm, El is sufﬁcient to account for racing
velocity variations for distances from dc (typically one mile in
our data set) to the marathon. For example, when analyzing
consistent running records of individuals, we found strong
evidence that they follow the same universal scaling law of
Eq. (1) as running world (or national) records do, with mean
errors below 1%14. Here, our model estimates are based on an
individual’s fastest times for the four ﬁxed racing distances, 5 k,
10 k, half-marathon, and marathon. Unfortunately, we cannot
determine from the available data set if performance was
achieved during an actual racing event. For our approach
however, it is only required that the recorded performance
corresponds to the maximal effort over a given running distance
achieved during the racing season.
Exercise data. An overview of the data analysis design is provided
in Fig. 1. All available subjects and activities in the data set of the
exercise tracking platform were grouped by SID and marathon
date, combining all individual running activities during the
180 days before the marathon, deﬁning a season. For each season,
activities with the fastest time for the four ﬁxed race distances
deﬁned a racing season. We imposed the condition that each
racing season contains at least two races. If a season contained 30
or more total running activities they were deﬁned as training
season. For consistency certain data ﬁlters were applied to all
activities and races (see the “Methods” section for more detail).
Two variants of racing season were deﬁned, with the marathon
included and excluded. A total of ~25,000 racing seasons with
the marathon included and ~10,000 racing seasons without the
marathon, and ~22,000 training seasons were analyzed (see
Table 1 for a summary of the available data and performed
analyses).
Accuracy of performance prediction. For all individuals, we
estimated their performance indices vm and γl for each racing
season by matching race events to Eq. (1) by minimizing the
relative prediction error for the race times. The probability den-
sities of these indices are shown in Fig. 2. For all racing seasons
with three and more races (N = 12,309), the mean error between
model prediction and actual race time was only 2.0%. This sug-
gests that our model captures correctly determinants of aerobic
endurance
performance.
Correlations
between
performance
indices and marathon ﬁnishing times are presented in Fig. 3. To
investigate the predictive power of our model in more detail, we
applied our model also to the racing season with the marathon
performance excluded (see Fig. 4). This allowed us to estimate
the marathon ﬁnishing time from the performances on shorter
distances only. As a function of performance indices, in the most
likely parameter range the model predicted the marathon per-
formance with an overall accuracy of better than 10%. Only for
very small (or large) endurance El, estimated times tended to be
too slow (or fast) which indicates that sub-marathon distances
were raced inconsistently, leading to an under (or over) estima-
tion of El. Given all the possible uncertainties in marathon racing
that are beyond the control of this study (e.g., weather, course
proﬁle, and motivation of the athlete), our predictions for the
marathon ﬁnishing times are rather satisfying.
Maximal velocity for 1 h. Analysis of ~25,000 racing seasons
reveals a normally distributed velocity vm and an exponential decay
of the probability density for the endurance El (see Fig. 2). Inter-
estingly, VO2,max in a study on 450 elite soccer players has also been
found to obey a normal distribution32. Note that vm also measures
running economy, which varies considerably among individuals
and modulates performance24. In exercise physiology, the ability of
a runner to maintain a certain effort is often characterized in terms
of thresholds, of which a common example is lactate threshold. In
our approach, however, there is a continuous relationship between
power output and velocity, and the change of this relation with
duration appears to be a natural measure for endurance capability.
Hence, as a practical measure for endurance, we deﬁne in our
model the velocity v1hU ¼ vm½1  0:1 log ð60 min =tcÞ=log ðElÞ
that a runner can maintain for 1 h, corresponding to the maximal
fractional utilization of MAP for 1 h. While any duration
could be chosen here, we used 1 h in analogy to running coaches
deﬁning threshold velocity as the effort that can be maintained for
about 1 h33. The 1h utilization ratio p1hU = v1hU/vm had been
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estimated previously from laboratory measurements and races for a
smaller group of 18 male long distance runners to be approximately
0.82 ± 0.0534. Strikingly, our ﬁndings from the running data
for ~14,000 subjects corroborate this range without any invasive
measurements, as demonstrated in Fig. 2c. Moreover, our obser-
vation of exponentially small but ﬁnite probability for larger El
explains observed values p1hU ≈0.9 in some well trained long dis-
tance runners.
We also computed the marathon race time from our model
and compared it to the actual marathon time Tm for all racing
seasons, see Fig. 3. Our model predicts theoretical curves of
constant Tm in the plane of performance indices (shown as
dashed lines in Fig. 3a). We found that the actual race times are
ordered according to these curves. This shows that our selected
physiological
proﬁles,
computed
from
sub-marathon
and
marathon best performances, are highly correlated with Tm. It
is important to understand that the position of a marathon
performance in the parameter space is determined by all races
and hence reﬂects relative importance of the indices vm and El.
This demonstrates the crucial importance of taking into account
endurance in addition to MAP and running economy when
assessing performance of long distance runners.
Importance of endurance. Our ﬁndings demonstrate the strong
sensitivity of performance to endurance. For example, a runner
with a velocity of vm = 5 m s−1 can improve his/her marathon
time from 3 h 27 min 38 s to 2 h 53 min 8 s by doubling endur-
ance from El = 3 to El = 6 (corresponding to a change in the one-
hour utilization from 79 to 87% of VO2max), without any change
in VO2,max or running economy. We also ﬁnd that faster runners
Exercise tracking platform
Raw data
Model
Training
Observations
Subjects
(N=18,993)
completed at least one marathon
[SID, gender]
Activities
(N=2,487,037; d=32,091,664 km)
during period of 180 days before M-date,
distance ≥ 1km, velocity≤7.8m/s
[SID, M-date, time, distance]
Athletes (w/o M)
(N=6749)
at least 2 races/season
Athletes (with M)
(N=14,304)
at least 2 races/season
Race seasons (w/o M)
[SID, M-date]
Race seasons (with M)
(Nraces=64,045)
(Nraces=21,184)
[SID, M-date]
Estimate M-time
Physiol. param. (w/o M)
(NM=9714)  [vm, El]
(NM=24,858)  [vm, El]
Physiol. param. (with M)
Correlations between
Performance and Training
Training parameters
(NT=21,605)
[dtrain, ptrain, TRIMP]
Training athletes
(N=12,233)
athlete has full training season and
at least 2 races with mean model error< 5%
Training seasons (180 days before M-date)
(activities: N=1,616,004; d=19,959,214 km)
full season: 30 or more activities
[SID, M-date]
SID
Fig. 1 Flowchart of the exercise data analysis. SID: subject identiﬁer, M: marathon, M-date: date of marathon, d: total running distance, “race season”:
fastest times of an athlete for at least two of the distances 5 km, 10 km, half-marathon, and marathon (±3% to account for GPS tolerance), Nraces: total
number of races, NM: number of successful model ﬁts, NT: number of analyzed training seasons for which physiological parameters vm, El could be obtained
and predicted actual race times within a mean error below 5%, “full training season”: at least 30 activities during the 180 days before M-date.
Table 1 Summary of data sets analyzed.
Data
Available
Fit with marathon
Fit w/o marathon
Training seasonc
# Subjects
18,993
14,304
6749
12,233
# Activitiesa (distance ≥ 1 km)
2,487,037
1,616,004
Total distance [km]
32,091,664
19,959,214
Mean distance/activity [km]
12.9
12.4
# Racing eventsb
85,993
64,045
21,184
54,620
# Race/training seasons
24,858
9714
21,605
All data were collected through the PolarFlow web service48.
aAfter removal of unrealistic average velocities (faster than world record).
bDistances are 5 km, 10 km, half-marathon, and marathon depending on the model ﬁt (w or w/o marathon).
cSeasons with # runs ≥ 30.
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tend to race more consistently over all race distances than slower
runners, highlighted by the dependence of the prediction error
ΔTm on the marathon ﬁnishing time (see Fig. 4b). For example,
within our fastest group of runners with a marathon time below
160 min, the prediction error was typically less than ±2.5%. This
observation supports our explanation for the observed uncer-
tainty in the endurance parameter El.
Correlation with training. Finally, we compared physiological
proﬁles to running activities within a training season. There exist
a few studies of the relation between training volume and
intensity, improvements of aerobic ﬁtness and performance35.
For example, it has been stated that running at velocity vm might
represent an optimal stimulus for improving endurance36. There
is also evidence supporting that a relatively large percentage of
low-intensity training over a long period improves performance
during highly intense endurance events37,38. It has been argued
that running velocity at lactate threshold is the best physiological
predictor for distance running performance39.
To investigate the effect of training distance and speed, relative
to the velocity vm, we selected consistent racing seasons deﬁned
by having a mean race time prediction error below 5%. Figure 5a
shows that as the total training distance dtrain of the training
season increases, vm increases on average linearly, with a weak
saturation trend at largest dtrain. Several studies have demon-
strated an increased vm due to endurance training35. A faster
velocity vm can be achieved by a better running economy and/or
an increase in MAP. We hypothesize that longer training distance
has generated improved running economy, in agreement with
earlier observations in a group of eleven well-trained long
0.0
0.1
0.2
0.3
0.4
0.5
1
3
5
7
9
11
13
Endurance El
Probability density
b
0.0
0.1
0.2
0.3
0.4
0.5
2
3
4
5
6
7
vm [m s–1]
Probability density
a
0
1
2
3
4
5
6
0.65 0.70 0.75 0.80 0.85 0.90 0.95 1.00
Relative power p1hU = v1hU/vm
Probability density
c
Fig. 2 Probability density of model parameters. The crossover velocity vm which is the smallest velocity that elicits maximal aerobic power MAP and the
endurance El are obtained by applying our model to the fastest performances of a subject for the four distances 5 km, 10 km, Halfmarathon, and Marathon
of a racing season. For these distributions, a total of 24,858 racing seasons have been analyzed. a The velocity vm is approximately normally distributed
with a mean of 4.4 m s−1. b The probability density for the endurance El resembles an exponential decay. c The probability density for the relative power for
1h utilization (1hU) peaks at about 82% of MAP.
b
a
12
Endurance El
10
8
6
4
2
12
Endurance El
10
8
6
4
2
2
3
4
5
6
7
6:00
5:30
4:30
5:00
4:00
3:40
3:20
3:00
2:40
2:20
2:00
6:00
5:30
4:30
5:00
4:00
3:40
3:20
3:00
2:40
Velocity vm [m s–1]
2
3
4
5
6
7
Velocity vm [m s–1]
Tm [min]
330
40
30
20
10
290
250
210
170
n
Fig. 3 Correlation between performance indices and marathon race time (model estimates for 24,504 racing seasons are shown here). a Visualization
of the marathon race time Tm in the (vm, El) parameter plane. Performance indices are obtained from individual's best performances during the racing
season. Color changes from fast (magenta) to slower (blue) ﬁnishing times (see color legend for time in minutes). Parameter pairs (vm, El) along the
dashed curves yield the same marathon race time indicated at the top of the graph (in hh:mm format). b Color coded visualization of the number n of racing
seasons analyzed as function of the parameters (vm, El).
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ΔTm
< −20 %
0.06
0.04
0.02
0.00
−20 to −15 %
−15 to −10 %
−10 to −5 %
−5 to 0 %
0 to 5 %
5 to 10 %
10 to 15 %
15 to 20 %
> 20 %
Actual Tm [min]
Tm < 160
160 < Tm < 180
180 < Tm < 200
200 < Tm < 220
220 < Tm < 240
a
b
Endurance El
Probability density
12
10
8
6
4
2
2
3
4
5
6
7
–20 –15 –10 –5
0
5
10
15
20
Velocity vm [m s–1]
ΔTm [%]
6:00
5:30
4:30
5:00
4:00
3:40
3:20
3:00
2:40
Fig. 4 Estimate of Marathon race time from the racing season (for 9410 seasons). a Visualization of the relative difference ΔTm between actual and
estimated marathon race time Tm (in percent of race time) as function of crossover velocity vm and endurance El. Magenta (blue) color indicates a faster
(slower) than estimated ﬁnish. b Probability density of race time differences color coded according to groups of different race time intervals.
2.5
3.0
3.5
4.0
4.5
5.0
5.5
6.0
0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0
3
5
7
9
11
0.5
0.6
0.7
0.8
3
5
7
9
11
5
10
15
20
25
30
Training TRIMP [×103]
Endurance El
Endurance El
Training intensity ptrain = vtrain / vm
Training intensity ptrain = vtrain / vm
3.0
3.5
4.0
4.5
5.0
5.5
6.0
6.5
1000
2000
3000
dtrain [km]
vm [m s–1]
vm [m s–1]
a
b
c
d
El = 1.83 + 0.101 ptrain e5.19 ptrain
Fig. 5 Correlations between performance indices and training characteristics. We have measured the distance and time for each running activity during a
training season for a total number of 21,605 seasons. The graphs show the observed relations between performance indices (obtained from a model ﬁt to
the racing season) and different measures of training volume and intensity. The magenta line indicates the average, the gray region one standard deviation,
and the light magenta and blue shaded areas represent the standard error of the mean and standard deviation, respectively, as obtained from bootstrap
resampling with replacement (see “Methods” section for more details). a Increase of crossover velocity vm with total distance dtrain of training runs.
b Relation between crossover velocity vm and relative training intensity ptrain ¼ vtrain=vm where vtrain is the average training velocity. c Increase and
saturation of endurance El with training impulse (TRIMP). d Exponential growth of endurance El with relative training intensity.
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distance runners40. Our analysis provides a statistically signiﬁ-
cant, quantitative relation between training distance and speed at
MAP, vm, for ~22,000 training seasons. Another explanation for
this relation could be that ﬁtter runners with a larger MAP and
hence higher vm log more kilometer during their training.
Unfortunately, we could not measure vm at the beginning and the
end of the training season independently from two different
racing seasons or time trials. We also found a linear decrease of
vm with the mean relative training intensity between 50% and
about 90% of vm, as shown in Fig. 5b. Our ﬁndings can be
interpreted as faster runners train typically at lower relative
intensities which is consistent with high-intensity performance
improvement due to low-intensity training. The range of training
velocities increases with larger vm which reﬂects a wider range of
accessible intensities between minimal (jogging) and maximal
speed. For example, a runner with vm = 4 m s−1 typically (within
one standard deviation) trains between 64 and 84% of vm or
MAP, while a runner with vm = 5 m s−1 trains typically up to
66% of vm so that both runners have an almost identical upper
pace ~5 min km−1 for the majority of their runs. Slow runners
must train at a relative high intensity if they want to avoid a
transition to walking. It is important to realize that these typical
ranges do not include fast, high-intensity workouts which account
only for a small fraction of total training volume. However, high-
intensity sessions involve also resting phases that can reduce the
average velocity when timer is not stopped, potentially explaining
observed intensities below ~50% of vm.
Optimal training impulse. We found strong evidence that
combined effect of training volume and intensity, known as
TRaining IMPulse (TRIMP)41, enhances endurance only up to a
limit. Previously, it was found in recreational long distance run-
ners that individual TRIMP correlates with 5000 m and 10,000 m
track performances42. We computed TRIMP by summing the
TRIMP points of all runs of the training season. For each run,
TRIMP points were assigned according to the duration of the run
and its relative average velocity v=vm (see “Methods” section for
details). We analyzed the quantitative relation between endurance
El and total TRIMP of a training season (see Fig. 5c). We
observed an initial linear increase of El with TRIMP, a plateau
around El = 7.5 ± 2 for TRIMP ~25,000, and a statistically sig-
niﬁcant ﬁnal drop which may be due to over-training. This result
suggests that there is an optimal TRIMP per training season, and
the corresponding maximal endurance enables a close to optimal
marathon race time for a given velocity vm (see Fig. 3a). Finally,
we probed the deﬁnition of TRIMP itself to determine if it
implements the best relation between endurance and training
intensity. We found a striking agreement between the exponential
dependence of El on vtrain=vm and the original deﬁnition of
TRIMP based on the rise of blood lactate with intensity, as
demonstrated in Fig. 5d. Our ﬁndings for thousands of runners
show that relations between training mode and performance
indices that are usually only accessible by invasive and resource-
consuming laboratory testing can be obtained reliably from
running activity data.
Discussion
Recent advances in wearable sensor technology have enabled real-
time and noninvasive measurement of physiological data during
exercise. However, if we are to employ these data to better
understand interplay between exercise, performance and human
health, we must develop new models that are adapted to extract
from the raw data quantities that are most relevant for health and
performance assessment. In this work we have taken this
approach for long distance running to estimate physiological
model indices such as MAP and endurance, and examined their
correlations with training volume and intensity by analyzing
exercise data of ~14,000 marathon runners worldwide. We found
that our recent universal model for a logarithmic relation between
fractional utilization of maximal power and exercise duration14 is
crucial for going beyond previous approaches which ignored this
relation, and for deﬁning a parameter measuring endurance. This
is an important complement to physiological testing in the
laboratory where the required maximal effort is unpractical to
achieve for distances over 20 km. Indeed, our results provide
evidence of the possibility to extract precise indicators for
performance and ﬁtness status from long-duration real-world
exercise tracking data. Using automated digital exercise tracking
goes beyond previous outside-lab studies that relied often on
frequently inaccurate self-reports of exercise. The probability
distributions of the extracted performance indices show large
variances, implying that studies with only a few individuals might
produce misleading results, missing the large interindividual
variability of response to exercise.
Our work has also some limitations: For each activity, only
total distance and duration was available in the data set. This
could lead to biased estimates of the mean velocity, for example
due to periods of rest or stopping with the device timer not
stopped. For the detected correlations between performance
indices and training the direction of any cause–effect relationship
remained open: for example, training with a higher total TRIMP
might produce better endurance, but higher endurance could also
enable runners to follow training modes with a higher TRIMP. To
resolve this relationship, additional data ﬁlters need to be devel-
oped to select groups of runners with similar initial performance
which subsequently follow different training modes. However, the
observed correlations can be of practical importance. They can be
useful for estimating realistic expectations for a race for less
experienced runners from their training intensity and volume. In
addition, our observation that endurance peaks at a given training
load (TRIMP) should help preventing over-training, i.e., unpro-
ductive increase in training that can cause injury and other health
problems. It should also be stressed that real-world data always
lack the controlled environment of laboratory based testing. For
example, the energy cost of running has been measured very
accurately in laboratory conditions43–46 and the theoretical
approaches derived from these experiments have motivated the
development of our model.
Our work implies several directions for future research. The
combination of effective models and real-world exercise data
holds great potential for a change in our theoretical description
and understanding of human response to physical activity over
longer periods of time, optimal exercise dosing and training, early
injury detection and prevention, and elite athlete performance.
Approaches similar to ours could be used to develop standards
for cardiorespiratory ﬁtness based on the probability distribution
of performance indices in populations with certain characteristics.
More
detailed,
time-resolved
activity
data
for
heart
rate,
mechanical power output and others could be integrated in our
model to improve accuracy and to extract other performance
indices. Further applications of our approach include the detec-
tion of the usage of performance enhancers in professional sports,
the early identiﬁcation of talented athletes, and even the effect of
sports equipment like new running shoe technology on perfor-
mance indices47.
Methods
Exercise tracking platform. Exercise data were obtained from Polar Flow web
service48, which is an exercise tracking platform that allows users to upload various
exercise data, including running distance and velocity from GPS watches. Meta
data and activity data of users are linked anonymously through user identiﬁcation.
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Selection of subjects and activities. Users of the exercise tracking platform were
selected as subjects for this study under the conditions that they had completed a
run over the marathon distance (42,195 m) in the period between 1 Jul 2015 and 31
Dec 2018, and used the same GPS watch (Polar V800) for activity recording to
assure comparable accuracy of GPS based distance recording. We analyzed the
running data of ~19,000 individuals who completed ~2.5M activities with a total
distance of ~32M km (see Table 1 for details). For each individual all running
activities in the 180 days before a completed marathon race were grouped together
with the marathon race and the groups labeled uniquely by a subject identiﬁer
(SID) and the marathon date (M-date). Note that an individual may have have
completed multiple marathons during the studied period. For each of those groups,
labeled by the pair (SID, M-date), a race season was deﬁned as the fastest runs of all
activities over the four race distances 5km, 10km, half-marathon (21,097.5m) and
marathon (42,195 m), if distances were available. A tolerance of ±3% was allowed
in the distance selection to account for GPS inaccuracy, and average race velocities
were determined by assuming the actual race distances (which are more reliable
than GPS recordings). We applied conditions that race velocities must increase
with decreasing race distance and must be slower than current world record
velocities. Inconsistent race seasons were identiﬁed by violation of these conditions
and excluded from further analysis. Race seasons were deﬁned both with and
without the marathon race included. A valid race season must contain at least two
different race distances. For each race season with a successful performance model
ﬁt with mean race time error below 5% (see section below) a corresponding
training season was deﬁned as all running activities with a total distance ≥1000 m
in the 180 days before the marathon. Runs with apparent velocities ≥7.8 m s−1
(world record for 1000 m) were excluded. Only training seasons with 30 or more
runs were considered so that runner had trained at least once per week and training
seasons with longer interruptions were excluded.
Performance model. We mathematically describe running performance by a
minimal model based on a relative power scale14. The model is formulated in terms
of relative quantities to eliminate irrelevant, subject dependent quantities. The
nominal power expenditure P(v) that is required to run at a constant velocity v, the
so-called running economy, determines the relative power as
pðvÞ ¼ PðvÞ  Pb
Pm  Pb
¼ v
vm
;
ð2Þ
where we introduced a basal power Pb that is obtained by linearly extrapolating the
running economy to zero velocity and a crossover power Pm that we expect to be
close to the MAP associated with maximal oxygen uptake VO2max. This power Pm
deﬁnes a crossover velocity vm that is close to the velocity that permits exercise with
maximal time at MAP. For velocities v > vm the energy cost of running cannot be
determined from oxygen uptake alone due to anaerobic energy supply.
The running performance of an athlete is not only determined by p(v) (which is
ﬁxed by running economy and VO2max) but depends crucially on the average
power Pmax that can be maximally generated over a duration T over which it can be
sustained. To run at the average velocity vmax that can be maximally sustained over
the time T, the nominal power P(vmax) = Pmax(T) is required, establishing a relation
between vmax and T. It has been shown14 that Pmax(T) can be obtained from a self-
consistency relation which states that the time average of the instantaneously
utilized power Pmax(T − t) equals the sum of Pmax(T) and a supplemental power.
This supplemental power has aerobic and anaerobic contributions and accounts for
an upward shift in the power that is required to complete a run with a given
average velocity, for example, due to deteriorating running economy or muscle
fatigue. The existence of an upward shift has been observed experimentally and it is
essential since its absence would yield a duration independent Pmax, which
contradicts the fact that a given power cannot be sustained for an arbitrary
duration. The solution of the self-consistency equation yields
PmaxðTÞ ¼ Pm  Pllog T
tc
for T ≥ tc ;
ð3Þ
where Pl measures the supplemental power supply and tc is a crossover time scale
separating different anaerobic and aerobic forms of supplemental power. It can be
shown that for T < tc, Pmax is given by Eq. (3) with Pl replaced by another constant.
By inverting PmaxðTÞ and using the power–velocity relation of Eq. (2), we get the
maximal time TmaxðvÞ ¼ tc exp½ðvm  vÞ=ðγlvmÞ over which an average velocity v
can be sustained. Here, the constant γl = Pl/(Pm − Pb) measures endurance
El ¼ expð0:1=γlÞ, see main text. The shortest time T(d) for covering a distance d
follows from solving T ¼ Tmaxðv ¼ d=TÞ for T, yielding Eq. (1). It is important for
the application to a large, inhomogeneous group of subjects that this model is
universal in the sense that it only depends on three parameters vm, tc, and γl and
does not depend directly on any additional, subject-dependent parameters.
Performance data analysis. We tested whether or not meaningful performance
indices can be deduced only from the racing performance of individuals, employing
the performance model described before. For each racing season, uniquely labeled
by a pair (SID, M-date), two model parameters, vm and γl, were computed from
Eq. (1) applied to all races in the racing season. In general, the time tc must be
obtained from the crossover between anaerobic and aerobic regimes, and hence
from races that involve both means of energy supply, i.e., events with ﬁnishing time
shorter and longer than tc. Explicit comparison to racing results and laboratory testing
has shown that tc = 6 min is a good approximation on average, and this estimate was
used in our data analysis14. We numerically minimized the sum of the squared
relative differences between the actual race time and the one predicted by Eq. (1). The
nonlinear ﬁtting was based on a Levenberg–Marquardt type algorithm with multiple
starting values to minimize probability to converge only to local minimum, and with
support for lower and upper parameter bounds. Parameter bounds were chosen as 2
m s−1 ≤ vm ≤ 7 m s−1, 0.039 ≤ γl ≤ 0.135 corresponding to 2.1 ≤ El ≤ 13.014. Fits that
converged onto these bounds were excluded from further analysis.
Training data analysis. To quantify training of individuals during the 180-day
period before a marathon, we must establish measures based on duration and
distances of activities within the training season. We considered an optimal set of
three variables that measure quantity, quality, and a combination of quantity and
quality. Training volume was quantiﬁed by total running distance dtrain of a
training season. To account for possibly varying physiological adaptions during
different training modes, training intensity ptrain ¼ vtrain=vm was measured by the
average running velocity vtrain in relation to the characteristic velocity vm that was
determined for each race season independently. Finally, the overall training load
was evaluated by the TRIMP scale, which is frequently employed in exercise
physiology and the design of training. TRIMP is a measure for both volume and
intensity of exercise. We assigned to each activity of a training season a TRIMP
number using the deﬁnition TRIMP ¼ Ttrainκ1ðv=vmÞ expðκ2v=vmÞ for activity of
duration Ttrain and average velocity v with κ1 = 0.64, κ2 = 1.92 for male subjects,
and κ1 = 0.86, κ2 = 1.67 for female subjects49. The total training TRIMP number
was then obtained by summing the individual TRIMP numbers of all activities
within a training season. Usually TRIMP is deﬁned in terms of the average heart
rate reserve during exercise which is expected to be well approximated by the ratio
v=vm. We are interested in the relation between physiological model parameters vm
and El, and training variables. To measure these relations, we grouped training
variables into bins of widths Δdtrain = 300 km, Δptrain = 0.025 and ΔTRIMP = 2000.
The standard error of the mean and of the standard deviation of vm and El within
each bin was estimated by bootstrap resampling with replacement and computa-
tion of the standard deviation from 1000 bootstrap replicates.
Reporting summary. Further information on research design is available in the Nature
Research Reporting Summary linked to this article.
Data availability
The data that support the ﬁndings of this study are available from Polar Electro Oy but
restrictions apply to the availability of these data, which were used under the license
for the current study, and so are not publicly available. Data are, however, available
from the authors upon reasonable request and with permission of Polar Electro Oy
(research@polar.com).
Code availability
The code (R-script) is available from the Zenodo website https://doi.org/10.5281/
zenodo.4008806.
Received: 18 January 2020; Accepted: 8 September 2020;
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Acknowledgements
The support by Polar Electro in obtaining the exercise data from their data base is greatly
acknowledged.
Author contributions
T.E. designed the study and performed the numerical analysis. T.E. and J.P. wrote
the paper.
Competing interests
The authors declare no competing interests.
Additional information
Supplementary information is available for this paper at https://doi.org/10.1038/s41467-
020-18737-6.
Correspondence and requests for materials should be addressed to T.E.
Peer review information Nature Communications thanks Guido Ferretti and the other,
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Human running performance from real-world big data.
Emig, Thorsten,Peltonen, Jussi