RESEARCH ARTICLE
Prediction and Quantification of Individual
Athletic Performance of Runners
Duncan A. J. Blythe1,2☯¤a*, Franz J. Király3☯¤b*
1 African Institute for Mathematical Sciences, Bagamoyo, Tanzania, 2 Bernstein Centre for Computational
Neuroscience, Berlin, Germany, 3 Department of Statistical Science, University College London, London,
United Kingdom
☯ These authors contributed equally to this work.
¤a Current address: African Institute of Mathematical Sciences, Tanzania, P.O. Box 176, Alpha Zulu,
Chunguuni Street (off Indian Street), Mwambao—Bagamoyo, Pwani—Tanzania
¤b Current address: Department of Statistical Science, University College London, Gower Street, London
WC1E 6BT, United Kingdom
* duncan@aims.ac.tz (DAJB); f.kiraly@ucl.ac.uk (FJK)
Abstract
We present a novel, quantitative view on the human athletic performance of individual run-
ners. We obtain a predictor for running performance, a parsimonious model and a training
state summary consisting of three numbers by application of modern validation techniques
and recent advances in machine learning to the thepowerof10 database of British runners’
performances (164,746 individuals, 1,417,432 performances). Our predictor achieves an
average prediction error (out-of-sample) of e.g. 3.6 min on elite Marathon performances and
0.3 seconds on 100 metres performances, and a lower error than the state-of-the-art in per-
formance prediction (30% improvement, RMSE) over a range of distances. We are also the
first to report on a systematic comparison of predictors for running performance. Our model
has three parameters per runner, and three components which are the same for all runners.
The first component of the model corresponds to a power law with exponent dependent on
the runner which achieves a better goodness-of-fit than known power laws in the study of
running. Many documented phenomena in quantitative sports science, such as the form of
scoring tables, the success of existing prediction methods including Riegel’s formula, the
Purdy points scheme, the power law for world records performances and the broken power
law for world record speeds may be explained on the basis of our findings in a unified way.
We provide strong evidence that the three parameters per runner are related to physiologi-
cal and behavioural parameters, such as training state, event specialization and age, which
allows us to derive novel physiological hypotheses relating to athletic performance. We con-
jecture on this basis that our findings will be vital in exercise physiology, race planning, the
study of aging and training regime design.
PLOS ONE | DOI:10.1371/journal.pone.0157257
June 23, 2016
1 / 16
a11111
OPEN ACCESS
Citation: Blythe DAJ, Király FJ (2016) Prediction and
Quantification of Individual Athletic Performance of
Runners. PLoS ONE 11(6): e0157257. doi:10.1371/
journal.pone.0157257
Editor: Nir Eynon, Victoria University, AUSTRALIA
Received: February 26, 2016
Accepted: May 26, 2016
Published: June 23, 2016
Copyright: © 2016 Blythe, Király. This is an open
access article distributed under the terms of the
Creative Commons Attribution License, which permits
unrestricted use, distribution, and reproduction in any
medium, provided the original author and source are
credited.
Data Availability Statement: Data are available from
https://figshare.com/articles/thepowerof10/3408202
and https://figshare.com/articles/Ful_code_to_
Prediction_and_Quantification_of_Individual_
Athletic_Performance_of_Runners_/3408250.
Funding: DAJB was supported by a grant from the
German Research Foundation, research training
group GRK 1589/1 “Sensory Computation in Neural
Systems.” FJK was partially supported by
Mathematisches Forschungsinstitut Oberwolfach
(MFO). This research was partially carried out at
MFO with the support of FJK’s Oberwolfach Leibniz
Fellowship.
Introduction
Performance prediction and modeling are cornerstones of sports medicine, essential in training
and assessment of athletes with implications beyond sport, for example in the understanding
of aging, muscle physiology, and the study of the cardiovascular system. Existing research on
running performance focuses either on (A) explaining world records [1–6], (B) equivalent
scoring [7, 8], or (C) modelling of individual physiology [9–16]. Currently, however, there is
no parsimonious model which simultaneously explains individual physiology (C) and collec-
tive performance (A,B) of runners.
We present such a model, a non-linear low-rank model derived from a database of UK run-
ners. It levers an individual power law which explains the power laws known to apply to world
records, and which allows us to derive runner-individual training parameters from prior per-
formance data. Performance predictions obtained using our approach are the most accurate to
date, with an average prediction error of under 4 minutes (2% rel.MAE and 3% rel.RMSE out-
of-sample) for elite performances. We anticipate that our framework will allow researchers to
leverage existing insights in the study of world record performances and sports medicine for an
improved understanding of human physiology.
Our work builds on the three major research strands in prediction and modeling of running
performance, which we briefly summarize:
(A) Power law models of performance posit a power law dependence t = c  sα between the
time elapsed running t and the distance s, for constants c and α. This is equivalent to assuming
a linear dependence log t = α log s + log c of log-time on log-distance. Power law models have
been known to describe world record performances across sports for over a century [17], and
have been applied extensively to running performance [1–6]. These power laws have been
applied to prediction by practitioners: the Riegel formula [18] predicts performance by fitting c
to each runner and fixing α = 1.06 (derived from world-record performances). The power law
approach has the benefit of modelling performances in a scientifically parsimonious way.
(B) Scoring tables, such as those of the international association of athletics federations
(IAAF), render performances over disparate distances comparable by presenting them on a
single scale. These tables have been published by sports associations for almost a century [19]
and catalogue, rather than model, performances of equivalent standard. Performance predic-
tions may be obtained from scoring tables by forecasting a time with the same score as an exist-
ing attempt, as implemented in the Purdy Points scheme [7, 8]. The scoring table approach has
the benefit of describing performances in an empirically accurate way.
(C) Explicit modeling of performance related physiology is an active subfield of sports sci-
ence. Several physiological parameters are known to be related to athletic performance; these
include maximal oxygen uptake ( _VO2-max) and critical speed (speed at _VO2-max) [9, 10],
blood lactate concentration, and the anaerobic threshold [11, 20]. Physiological parameters
may be used (C.i) to make direct predictions when clinical measurements are available [12, 13,
21], or (C.ii) to obtain theoretical models describing physiological processes [14–16, 22]. These
approaches have the beneﬁt of explaining performances physiologically.
All three approaches (A), (B), (C) have appealing properties, as explained above, but none
provides a complete treatment of running performance prediction: (A) individual perfor-
mances do not follow the parsimonious power law perfectly; (B) the empirically accurate scor-
ing tables do not provide a simple interpretable relationship. Neither (A) nor (B) can deal with
the fact that runners may differ from one another in multiple ways. The clinical measurements
in (C.i) are informative but usually available only for a few select runners, typically at most a
few dozen (as opposed to the 164,746 considered in our study). The interpretable models in (C.
ii) are usually designed not with the aim of predicting performance but to explain physiology
Prediction and Quantification of Individual Athletic Performance of Runners
PLOS ONE | DOI:10.1371/journal.pone.0157257
June 23, 2016
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Competing Interests: The authors have declared
that no competing interests exist.
or to estimate physiological parameters from performances; thus these methods are not directly
applicable to running performance prediction without additional work.
The approach we present unifies the desirable properties of (A), (B) and (C), while avoiding
the aforementioned shortcomings. We obtain (A) a parsimonious model for individual athletic
performance that is (B) empirically derived from a large database of UK runners. It yields the
best performance predictions to date (2% average error for elite runners on all events, average
error 3.6 min for Marathon 0.3 seconds for 100 metres) and (C) unveils hidden descriptors for
individuals which we find to be related to physiological and behavioural characteristics.
Our approach bases predictions on Local Matrix Completion (LMC), a machine learning
technique which posits the existence of a small number of explanatory variables which describe
the performance of individual runners. Application of LMC to a database of runners allows us,
in a second step, to derive a parsimonious physiological model describing the running perfor-
mance of individual runners. We discover that a three number-summary for each individual
explains performance over the full range of distances from 100m to the Marathon. The three-
number-summary relates to: (1) the endurance of a runner, (2) the relative balance between
speed and endurance, and (3) specialization over middle distances. The first number explains
most of the individual differences over distances greater than 800m, and may be interpreted as
the exponent of an individual power law for each runner, which holds with remarkably high
precision, on average. The other two numbers describe individual, non-linear corrections to
this individual power law. Vitally, we show that the individual power law with its non-linear
corrections reflects the data more accurately than the power law for world records. We antici-
pate that the individual power law and three-number summary will allow for exact quantitative
assessment in the science of running and related sports.
Materials and Methods
Local Matrix Completion and the Low-Rank Model
It is well known that world records over distinct distances are held by distinct runners—no one
single runner holds all running world records. Since world record data obey an approximate
power law (see above), this implies that the individual performance of each runner deviates
from this power law. The left top panel of Fig 1 displays world records and the corresponding
individual performances of world record holders in logarithmic coordinates—an exact power
law would follow a straight line. The world records align closely to a straight line, while individ-
uals deviate non-linearly. Notable is also the kink in the world records which causes them to
deviate from an exact straight line, yielding a “broken power law” for world records [5].
Any model for individual performances must model this individual, non-linear variation,
and will, optimally, explain the broken power law observed for world records as an epiphenom-
enon of such variation over individuals. In the following paragraphs we explain how the LMC
scheme captures individual variation in a typical scenario.
Consider three runners (taken from the database) as shown in the top-right panel of Fig 1.
The 1500m performance of the green runner is not known and is to be predicted. All three run-
ners, green, blue and red, achieve similar performance over 800m. Any classical method for
performance prediction which only takes this information into account will predict that green
performs similarly over 1500m to blue and red. However, this is unrealistic, since it does not
take into account event specialization: looking at the 400m performance, we see that red is
slowest over short distances, followed by blue and then by green. Thus red is more of an endur-
ance type runner than blue, and blue is more of a speed type runner (sprinter) than red; green
specializes to a greater extent in speed than both red and blue. Using this additional informa-
tion leads to the more realistic prediction that green will be out-performed by red and blue
Prediction and Quantification of Individual Athletic Performance of Runners
PLOS ONE | DOI:10.1371/journal.pone.0157257
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over 1500m. Supplementary analysis (IV) in S1 Supplement validates that this phenomenon
illustrated in the example is prevalent throughout the data set.
LMC is a quantitative method for taking into account this event specialization. A schematic
overview of the simplest variant is displayed in the bottom panel of Fig 1: to predict an event
for a runner (figure: 1500m for green) we find a 3-by-3-pattern of performances, denoted by A,
with exactly one missing entry—this means the two other runners (figure: red and blue) have
attempted similar events and their data are available. Explanation of the green runner’s curve
by the red and the blue is mathematically modelled by demanding that the data of the green
runner is given as a weighted sum of the data of the red and the blue; i.e., more mathematically,
the green row is a linear combination of the blue and the red row. A classical result in matrix
algebra implies that the green row is a linear combination of red and blue whenever the deter-
minant of A, a polynomial function in the entries of A, vanishes; i.e., det(A) = 0.
A prediction is made by solving the Eq det(A) = 0 for “?”. To increase accuracy, candidate
solutions from multiple 3-by-3-patterns (obtained from many triples of runners) are averaged
in a way that minimizes the expected error in approximation. We shall consider variants of the
algorithm which use n-by-n-patterns, n corresponding to the complexity of the model (we later
show n = 4 to be optimal). See the methods appendix for an exact description of the algorithm
used.
Fig 1. Central phenomenon: non-linear deviation from the power law in individuals. Top left:
performances of world record holders and a selection of random runners. Curves labelled by runners are their
known best performances (y-axis) at that event (x-axis). Black crosses are world record performances.
Individual performances deviate non-linearly from the world record power law. Top right: a good model should
take into account specialization, illustration by example. Hypothetical performance curves of three runners,
green, red and blue are shown, the task is to predict green on 1500m from all other performances. Dotted
green lines are predictions. State-of-art methods such as Riegel or Purdy predict green performance on
1500m close to blue and red; a realistic predictor for 1500m performance of green—such as LMC—will
predict that green is outperformed by red and blue on 1500m; since blue and red being worse on 400m
indicates that out of the three runners, green specializes most on shorter distances. Bottom: using local
matrix completion as a mathematical prediction principle by filling in an entry in a (3 × 3) sub-pattern.
Schematic illustration of the algorithm.
doi:10.1371/journal.pone.0157257.g001
Prediction and Quantification of Individual Athletic Performance of Runners
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The LMC prediction scheme is an instance of the more general local low-rank matrix com-
pletion framework introduced in [23], here applied to performances in the form of a numerical
table (or matrix) with columns corresponding to events and rows to runners. The cited frame-
work is the first matrix completion algorithm which allows prediction of single missing entries
as opposed to all entries. While matrix completion has proved vital in predicting consumer
behaviour and recommender systems, the results in the present study show that existing
approaches which predict all entries at once cannot cope with the non-uniform missingness
and the noise associated with performance prediction in the same way as LMC can (see find-
ings and supplemental analysis II.a in S1 Supplement). See the methods appendix for more
details of the method and an exact description.
In a second step, we use the LMC scheme to fill in all missing performances (over all events
considered—100m, 200m etc.) and obtain a parsimonious low-rank model—we remark that
first filling in the entries with LMC and only then fitting the model is crucial due to the fact
that data are non-uniformly missing (see supplemental analysis II.a in S1 Supplement). The
low-rank model explains individual running times t in terms of distance s and has the form:
log t ¼ l1f1ðsÞ þ l2f2ðsÞ þ    þ lrfrðsÞ;
ð1Þ
with components f1, f2, . . ., fr that are the same for every runner, and coefﬁcients λ1, λ2, . . ., λr
which summarize the runner under consideration. The number of components and coefﬁcients
r is known as the rank of the model and measures its complexity. The Riegel power law is a
very special case, demanding that log t = 1.06log s + c; that is, a rank 2 model with λ1 = 1.06 for
every runner, f1(s) = log s, and a runner-speciﬁc constant λ2 f2(s) = c. Our analyses will show
that the best model has rank r = 3 (meaning above we consider patterns or matrices of size n ×
n = 4 since above n = r + 1). This means that the model has r = three universal components
f1(s), f2(s), f3(s), and every runner is described by their individual three-coefﬁcient-summary
λ1, λ2, λ3. Remarkably, we ﬁnd that f1(s) = log s (for a suitable unit of distance/time, see supple-
mental analysis II.b in S1 Supplement), yielding an individual power law; the corresponding
coefﬁcient λ1 thus has the natural interpretation as an individual power law exponent.
Table 1 contains the exact form for the components f1, f2, f3 in our model; they are also dis-
played in Fig 2 top left. More details on how to obtain components and coefficients can be
found in the methods section, “obtaining the low-rank components and coefficients”, and in
supplementary analysis II.b in S1 Supplement.
Data Set, Analyses and Model Validation
The basis for our analyses is the online database www.thepowerof10.info, which catalogues
British individuals’ performances achieved in officially ratified athletics competitions since
1954. The excerpt we consider contains performances between 1954 and August 3, 2013. Our
study does not use performances prior to 1954 since the database does not contain perfor-
mances dating prior to 1954. It contains (after error removal) records of 164,746 individuals of
both genders, ranging from the amateur to the elite, young to old, comprising a total of
1,417,432 individual performances over 10 different distances: 100m, 200m, 400m, 800m,
1500m, the Mile, 5km, 10km, Half-Marathon, Marathon. All British records over the distances
considered are contained in the dataset; the 95th percentile for the 100m, 1500m and Marathon
are 15.9, 6:06.5 and 6:15:34, respectively. As performances for the two genders distribute differ-
ently, we present only results on the subset of 101,775 male runners in the main corpus of the
manuscript; female runners and further subgroup analyses are considered in the supplemen-
tary results. The data set is available upon request, subject to approval by British Athletics. Full
code and data for our analyses can be obtained from [24, 25].
Prediction and Quantification of Individual Athletic Performance of Runners
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Adhering to state-of-the-art statistical practice (see [26–29]), all prediction methods are val-
idated out-of-sample, i.e., by using only a subset of the data for estimation of parameters (train-
ing set) and computing the error on predictions made for a distinct subset (validation or test
set). As error measures, we use the root mean squared error (RMSE) and the mean absolute
error (MAE), estimated by leave-one-out validation for 1000 single performances omitted at
random.
We would like to stress that out-of-sample prediction error is the correct way to evaluate
the quality of prediction, as opposed to merely reporting goodness-of-fit in-sample, since out-
putting an estimate for an instance that the method has already seen does not qualify as
prediction.
More details on the data set and our validation setup can be found in the supplementary
material.
Table 1. The three components of the low-rank model of Eq (1).
s
100m
200m
400m
800m
1500m
Mile
5km
10km
HM
Mar
f1
2.254
2.875
3.574
4.305
4.964
5.049
6.179
6.844
7.555
8.243
f2
0.4473
0.4721
0.5265
0.3045
0.0798
0.0806
-0.1597
-0.1983
-0.2279
-0.2785
f3
-0.1750
-0.2004
-0.1145
0.2224
0.3263
0.3092
0.3157
0.2717
-0.1153
-0.6912
v
0.1291
0.1647
0.2047
0.2466
0.2843
0.2892
0.3539
0.3920
0.4327
0.4721
An entry in the rows i = 1,2,3 is fi(s), where s is the column header; HM is the half-Marathon, Mar is the Marathon. The components are obtained as
described in methods, “obtaining the low-rank components and coefﬁcients”. v is the raw singular vector described there from which f1 is obtained by
rescaling. v, f2, f3 are displayed in Fig 2 top left with standard error tubes per entry. The entries for v have, on average, an estimated standard error of
0.005, the entries for f2 have, on average, an estimated standard error of 0.02, and the entries for f3 have, on average, an estimated standard error of
0.04.
doi:10.1371/journal.pone.0157257.t001
Fig 2. The three components of the low-rank model, and explanation of the world record data. Left: the components displayed (unit norm, log-time vs
log-distance). Tubes around the components are one standard deviation, estimated by the bootstrap. The first component is an exact power law (straight line
in log-log coordinates); the last two components are non-linear, describing transitions at around 800m and 10km. Middle: Comparison of first component and
world record to the exact power law (log-speed vs log-distance). Right: Least-squares fit of rank 1-3 models to the world record data (log-speed vs log-
distance).
doi:10.1371/journal.pone.0157257.g002
Prediction and Quantification of Individual Athletic Performance of Runners
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Results
(I) Prediction accuracy. We evaluate prediction accuracy of ten methods, including our pro-
posed method, LMC. We include, as naive baselines: (1.a) imputing the event mean, (1.b)
imputing the average of the k-nearest neighbours; as representative of the state-of-the-art in
quantitative sports science: (2.a) the Riegel formula, (2.b) a power law predictor with exponent
estimated from the data, which is the same for all runners, (2.c) a power law predictor with
exponent estimated from the data, with one exponent per runner, (2.d) the Purdy points
scheme [7]; as representatives for the state-of-the-art in matrix completion: (3.a) imputation by
expectation maximization on a multivariate Gaussian [30] (3.b) nuclear norm minimization
[31, 32].
We instantiate our low-rank local matrix completion (LMC) in two variants: (4.a) rank 1,
and (4.b) rank 2.
Methods (1.a), (1.b), (2.a), (2.b), (2.d), (4.a) require at least one observed performance per
runner, methods (2.c), (4.b) require at least two observed performances in distinct events. Meth-
ods (3.a), (3.b) will return a result for any number of observed performances (including zero).
Prediction accuracy is therefore measured by evaluating the RMSE and MAE out-of-sample on
the runners who have attempted at least three distances, so that the two necessary performances
remain to calculate the prediction when one is removed for leave-one-out validation. Prediction
is further restricted to the best 95-percentile of runners (measured by performance in the best
event) to reduce the effect of outliers. Whenever the method demands that the predicting events
need to be specified, the events which are closest in log-distance to the event to be predicted are
taken. The accuracy of predicting time (normalized w.r.t. the event mean), log-time, and speed
are measured. We repeat this validation setup for the year of best performance and a random cal-
endar year. Moreover, for completeness and comparison we treat 2 additional cases: the top 25%
of runners and runners who have attempted at least 4 events, each in log time. More details on
methods and validation are presented in the methods appendix.
The results are displayed in Table 2 (RMSE of log-time prediction) and supplementary
Table B in S1 Supplement (MAE of log-time prediction), S4 (rel.RMSE of time prediction) and
S5 (rel. MAE of time prediction). Of all benchmarks, k-nearest neighbours (1.b), Purdy points
(2.d) and Expectation Maximization (3.a) perform best. LMC rank 2 substantially outperforms
k-nearest neighbours, Purdy points and Expectation Maximization (two-sided Wilcoxon
signed-rank test significant on the validation samples of absolute prediction errors; p2.0e-8
on top 95% in log-time and p1.4e-11 for top 25% in log-time); rank 1 outperforms Purdy
points on the year of best performance data (p3.0e-3) for the best runners, and is on a par on
runners up to the 95th percentile. Both rank 1 and 2 outperform the power law models
(p1.1e-42), the improvement in RMSE of LMC rank 2 over the power law models reaches
over 50% for data from the fastest 25% of runners.
(II) The rank (number of components) of the model. Paragraph (I) establishes that LMC
is the best method for prediction. LMC assumes a fixed number of prototypical runners, viz.
the rank r, which is the complexity parameter of the model. We establish the optimal rank by
comparing prediction accuracy of LMC with various ranks. The rank r algorithm requires r
attempted events for prediction, thus r + 1 observed events are needed for validation. Table F
in S1 Supplement displays prediction accuracies for LMC ranks r = 1 to r = 4, on the runners
who have attempted k > r events, for all k  5. The data is restricted to the top 25% in the year
of best performance in order to obtain a high signal to noise ratio. We observe that rank 3 out-
performs all other ranks, when applicable; rank 2 aways outperforms rank 1 (both p1e-4).
We also find that the improvement of rank 2 over rank 1 depends on the event predicted:
improvement is 26.3% for short distances (100m, 200m), 29.3% for middle distances (400m,
Prediction and Quantification of Individual Athletic Performance of Runners
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Table 2. Out-of-sample RMSE for prediction methods on different data setups.
Generic Baselines
State of art Performance Predictors
State of art Matrix
Completion
Proposed Method:
LMC
evaluation percentiles no.events
data type
r.mean
k-NN
individual
power law
riegel
power law
purdy
nuclear
norm
EM
LMC rank
1
LMC rank
2
log time
0-95
3
best
0.1308
± 0.0032
0.0618
± 0.0027
0.1033
± 0.0042
0.0982
± 0.0046
0.0973
± 0.0046
0.0610
± 0.0031
0.3909
± 0.0457
0.0566
± 0.0028
0.0586
± 0.0026
0.0515
± 0.0027
normalized
0-95
3
best
0.1364
± 0.0044
0.0716
± 0.0046
0.1067
± 0.0048
0.1059
± 0.0066
0.1050
± 0.0065
0.0684
± 0.0043
0.1900
± 0.0120
0.0634
± 0.0045
0.0643
± 0.0038
0.0576
± 0.0039
speed
0-95
3
best
0.6655
± 0.0147
0.3057
± 0.0146
0.6096
± 0.0245
0.5467
± 0.0251
0.5415
± 0.0243
0.3077
± 0.0176
26.6210
± 11.4828
0.2922
± 0.0165
0.3123
± 0.0149
0.2530
± 0.0129
log time
0-95
3
random
0.1380
± 0.0032
0.0544
± 0.0025
0.0931
± 0.0035
0.0931
± 0.0038
0.0919
± 0.0038
0.0591
± 0.0028
0.4416
± 0.0428
0.0561
± 0.0031
0.0567
± 0.0027
0.0471
± 0.0024
normalized
0-95
3
random
0.1450
± 0.0043
0.0623
± 0.0037
0.0951
± 0.0039
0.1011
± 0.0048
0.0998
± 0.0046
0.0682
± 0.0038
0.2046
± 0.0117
0.0634
± 0.0039
0.0640
± 0.0037
0.0538
± 0.0035
speed
0-95
3
random
0.6935
± 0.0143
0.2585
± 0.0117
0.5917
± 0.0312
0.5052
± 0.0176
0.4979
± 0.0167
0.2835
± 0.0137
24.7206
± 10.9157
0.2801
± 0.0196
0.2863
± 0.0120
0.2261
± 0.0105
log time
0-95
4
best
0.1268
± 0.0032
0.0735
± 0.0030
0.0777
± 0.0024
0.0819
± 0.0032
0.0822
± 0.0032
0.0581
± 0.0023
0.1779
± 0.0199
0.0529
± 0.0024
0.0536
± 0.0021
0.0467
± 0.0022
log time
0-25
3
best
0.0557
± 0.0015
0.0416
± 0.0014
0.0806
± 0.0031
0.0683
± 0.0026
0.0720
± 0.0026
0.0411
± 0.0012
0.3008
± 0.0275
0.0383
± 0.0013
0.0411
± 0.0014
0.0306
± 0.0011
Predicted performance is of the 95 and 25 top percentiles of male runners who attempted 3 or 4 events resp., in their best year and a random calendar year. Standard errors are
bootstrap estimates over 1000 repetitions. Compared method classes are (1) generic baselines, (2) state-of-the-art in performance prediction, (3) state-of-the-art in matrix
completion, (4) local matrix completion (columns). Methods are (1.a) r.mean: predicting the mean of all runners (1.b) k-NN: predicting the nearest neighbor. (2.a) riegel: Riegel’s
formula (2.b) power law: power law with free exponent and coefﬁcient. Exponent is the same for all runners. (2.c) ind.power law: power law with free exponent and coefﬁcient. (2.d)
purdy: Purdy points scheme (3.a) EM: expectation maximization (3.b) nuclear norm: nuclear norm minimization (4.a) LMC with rank 1 (4.b) LMC with rank 2. Data setup is speciﬁed
by (i) evaluation: what is predicted. log-time = natural logarithm of time in seconds, normalized = time relative to mean performance, speed = average speed in meters per seconds,
(ii) percentiles: selected percentile range of runners, (iii) no.events tried = sub-set of runners who have attempted at least that number of different events, (iv) data type: collation
mode of performance matrix; best = 1 year around best performance, random = random 2 year period. LMC rank 2 signiﬁcantly outperforms all competitors in either setting.
doi:10.1371/journal.pone.0157257.t002
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800m, 1500m), 12.8% for the mile to half-marathon, and 3.1% for the Marathon (all significant
at p = 1e-3 level) (see Fig A in S1 Supplement). These results indicate that inter-runner variabil-
ity is greater for short and middle distances than for Marathon.
(III) The three components of the model. The findings in (II) imply that the best low-rank
model assumes 3 components. To estimate the components (fi in Eq (1)) we impute all missing
entries in the data matrix of the top 25% runners who have attempted 4 events and compute its
singular value decomposition (SVD) [33]. From the SVD, the exact form of components may
be directly obtained as the right singular vectors (in a least-squares sense, and up to scaling, see
supplemental analysis II.b in S1 Supplement). We obtain three components in log-time coordi-
nates, which are displayed in the left hand panel of Fig 2. The first component for log-time pre-
diction is linear (i.e., f1(s) / log s in Eq (1)) to a high degree of precision (R2 = 0.9997) and
corresponds to an individual power law, applying distinctly to each runner. The second and
third components are non-linear; the second component decreases over short sprints and
increases over the remainder, and the third component resembles a parabola with extremum
positioned around the middle distances.
In speed coordinates, the first, individual power law component does not display the “bro-
ken power law” behaviour of the world records (rank 1 component: goodness-of-fit for linear
model R2 = 0.99; world-record data: R2 = 0.93). Deviations from an exact line can be explained
by the second and third component (Fig 2 middle).
The three components together explain the world record data and its “broken power law”
far more accurately than a simple linear power law trend—with the rank 3 model fitting the
world records almost exactly (Fig 2 right).
(IV) The three runner-specific coefficients. The three summary coefficients for each run-
ner (λ1, λ2, λ3 in Eq (1)) are obtained from the entries of the left singular vectors (see methods
appendix). Since all three coefficients summarize the runner, we refer to them collectively as
the three-number-summary.
(IV.i) Fig 3 displays scatter plots and Spearman correlations between the coefficients and
performance over the full range of distances. The individual exponent correlates with perfor-
mance over distances greater than 800m. The second coefficient correlates positively with per-
formance over short distances and displays a non-linear association with performance over
Fig 3. Matrix scatter plot of the three-number-summary vs performance. For each of the scores in the
three-number-summary (rows) and each event distance (columns), the plot matrix shows: a scatter plot of
performances (time) vs the coefficient score of the top 25% (on the best event) runners who have attempted
at least 4 events. Each scatter plot in the matrix is colored on a continuous color scale according to the
absolute value of the scatter sample’s Spearman rank correlation (red = 0, green = 1).
doi:10.1371/journal.pone.0157257.g003
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middle distances. The third coefficient correlates with performance over middle distances. (All
correlations are significant at p1.0e-4; t-distribution approximation to the distribution of
Spearman’s correlation coefficient.) The associations for all three coefficients are non-linear,
with the notable exception of the individual exponent on distances exceeding 800m.
(IV.ii) Fig 4 top displays the three-number-summary for the top 95% runners in the data-
base. The runners separate into (at least) four classes, which are associated with the runner’s
preferred distance. A qualitative transition can be observed over middle distances. Three-num-
ber-summaries of world class runners (not all in the UK runners database), computed from
their personal bests, are listed in Table 3; they and also shown as highlighted points in Fig 4 top
right. The elite runners trace a frontier around the population: all elite runners are subject to a
low individual exponent. A hypothetical runner holding all the world records is also shown in
Fig 4 top right, obtaining an individual exponent which comes close to the world record expo-
nent estimated by Riegel [3] (1.08 for elite runners, 1.06 for senior runners).
(IV.iii) Fig 4 bottom left shows that a low individual exponent correlates positively with per-
formance in a runner’s preferred event. The individual exponents are higher on average
(median = 1.12; 5th, 95th percentiles = 1.10, 1.15) than the world record exponents estimated
by Riegel.
Fig 4. Scatter plots exploring the three number summary. Top left and right: 3D scatter plot of three-
number-summaries of runners in the data set, colored by preferred distance and shown from two angles. A
negative value for the second score is a indicates that the runner is a sprinter, a positive value an endurance
runner. In the top right panel, the summaries of the elite runners Usain Bolt (world record holder, 100m,
200m), Mo Farah (world beater over distances between 1500m and 10km), Haile Gabrselassie (former world
record holder from 5km to Marathon) and Takahiro Sunada (100km world record holder) are shown;
summaries are estimated from their personal bests. For comparison we also display the hypothetical data of
a runner who holds all world records. Bottom left: preferred distance vs individual exponents, color is
percentile on preferred distance. Bottom right: age vs. exponent, colored by preferred distance.
doi:10.1371/journal.pone.0157257.g004
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(IV.iv) Fig 4 bottom right shows that in cross-section, the individual exponent decreases
with age until 20 years, and subsequently increases. (All correlations significant at p1.0e-4; t-
distribution approximation to the distribution of Spearman’s correlation coefficient.)
(V) Phase transitions. We observe two transitions in behaviour between short and long dis-
tances. The data exhibit a phase transition around 800m: the second component exhibits a
kink and the third component makes a zero transition (Fig 2); the association of the first two
scores with performance shifts from the second to the first score (Fig 3). The data also exhibits
a transition around 5000m. We find that for distances shorter than 5000m, holding the event
performance constant and increasing the standard of shorter events leads to a decrease in the
predicted standard of longer events and vice versa. On the other hand for distances greater
than 5000m this behaviour reverses; holding the event performance constant, and increasing
the standard of shorter events leads to an increase in the predicted standard of longer events.
See supplementary analysis IV in S1 Supplement for details.
(VI) Universality over subgroups. Qualitatively and quantitatively similar results to the
above can be deduced for female runners, and subgroups stratified by age or training standard;
LMC remains an accurate predictor, and the low-rank model has similar form. See supplemen-
tal analysis II.c in S1 Supplement.
Discussion
We have presented the most accurate existing predictor for running performance to date—
local low-rank matrix completion (finding I); its predictive power confirms the validity of a
three-component model (finding II) that offers a parsimonious explanation for many known
phenomena in the quantitative science of running, including answers to some of the major
open questions of the field. More precisely, we establish:
The individual power law. In log-time coordinates, the first component of our physiologi-
cal model is linear with high accuracy, yielding an individual power law (finding III). This is a
novel and rather surprising finding, since, although world-record performances are known to
obey a power law [1–6], there is no reason to suppose a-priori that the performance of individ-
uals is governed by a power law. Striking is that the power-law derived is considerably more
accurate when considered in log-distance—log-speed coordinates than the power-law which
applies to world-record data. This parsimony a-posteriori unifies (A) the parsimony of the
Table 3. Estimated three-number-summary (λi) for a selection of elite runners.
runner
Specialization
Individual
Exponent (λ1)
Score 2 (λ2)
Score 3 (λ3)
Usain Bolt
Sprints
1.11
-0.367
0.081
Mo Farah
Middle-Long
1.08
0.033
-0.076
Haile Gabrselassie
Long
1.08
0.114
-0.056
Galen Rupp
Long
1.08
0.104
-0.040
Seb Coe
Middle
1.09
-0.085
-0.036
Takahiro Sunada
Ultra-Long
1.09
0.138
-0.010
Paula Radcliffe
Long (Female)
1.10
0.189
0.025
The scores λ1, λ2, λ3 are as in Eq (1). Since component 1 is a power law (see the top-left of Fig 2), λ1 may
be interpreted as an individual exponent. See the bottom right panel of Fig 4 for a scatter plot of the
runners in the database.
doi:10.1371/journal.pone.0157257.t003
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power law with the (B) empirical correctness of scoring tables. To what extent this individual
power law is exact is to be determined in future studies.
An explanation of the world record data. The broken power law [5] of world record data
can be seen as a consequence of the individual power law and the non-linearity in the second
and third component (finding III) of our low-rank model. The breakage point in the world rec-
ords can be explained by the differing contributions in the non-linear components of the dis-
tinct individuals holding the world records. Savaglio and Carbone [5] hypothesize that the
breakpoint in the log-speed—log-distance slope of world-record data, which occurs between
short and long distance events, is due to a transition in the physiology required between short
and long-distance events. Our analyses indeed show that their exist breakpoints, manifested in
the second and third components of the low-rank model. However our findings show that the
claim that there is a universal physiological transition manifesting itself in the differing slopes
of short and long-distance world-record data is unwarranted. Runners who exhibit small values
for the 2nd and 3rd numbers in their three number summaries will exhibit performances close
to log-log with little or no transition; this is because the first component of the model is much
closer to scale-free (log-log straight line) than world-record data. Some runners will indeed dis-
play an upward kink in their average speed as is the case with world-record data. Other runners
will exhibit transitions corresponding to a quicker fall off in average speed rather than faster,
i.e. a downwards kink. Thus the validity of the three component model points to a far more
complex description and diversity of average speed than world record data suggest.
Crucially both the power law and the broken power law on world record data can be under-
stood as epiphenomena of the individual power law and its non-linear corrections.
Universality of our model. The low-rank model remains unchanged when considering dif-
ferent subgroups of runners, stratified by gender, age, or calendar year; only the individual
three-number-summaries change (finding VI). This shows the low-rank model to be universal
for running.
The three-number-summary reflects a runner’s training state. Our predictive validation
implies that the number of components of our model is three (finding II), which yields three
numbers describing the training state of a given runner (finding IV). The most important sum-
mary is the individual exponent for the individual power law which describes overall perfor-
mances for distances longer than 400m (IV.iii). The second coefficient describes whether the
runner has greater endurance (positive) or speed (negative) and predicts performances over
the sprint distances, the third describes specialization over middle distances (negative) vs. short
and long distances (positive). All three numbers together clearly separate the runners into four
clusters, which fall into two clusters of short-distance runners and one cluster of middle-and
long-distance runners respectively (IV.i). Our analysis provides strong evidence that the three-
number-summary captures physiological and/or social/behavioural characteristics of the run-
ners, e.g., training state, specialization, and which distance a runner chooses to attempt. While
the data set does not allow us to separate these potential influences or to make statements
about cause and effect, we conjecture that combining the three-number-summary with specific
experimental paradigms will lead to a clarification; further, we conjecture that a combination
of the three-number-summary with additional data, e.g. training logs, high-frequency training
measurements or clinical parameters, will lead to a better understanding of (C) existing physio-
logical models.
Some novel physiological insights can be deduced from leveraging our model on the UK
runners database:
• We find that the individual exponent correlates with performances over distances greater
than 400m and especially long distances above 5km (finding III). We also find that LMC is
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most effective for the longer-sprints and middle distances; the improvement of the higher
rank over the rank 1 version is lowest over the marathon distance (supplemental analysis I.c
in S1 Supplement). This indicates that the variability in performances on long distances may
to a large extent be explained by a single factor, which may imply that there is only one way
to be a fast marathoner. On the other hand since we find that the rank-2 and 3 versions far
outperform the rank-1 version over middle distances, this may be interpreted in terms of
some runners using a high maximum velocity to coast whereas other runners using greater
endurance to run closer to their maximum speed for the duration of the race; if the type of
running (coasting vs. endurance) is a physiological correlate to the specialization summary
(as hypothesized above), it could imply that the “one way” corresponds to possessing a high
level of endurance—as opposed to being able to coast relative to a high maximum speed. In
any case, the low-rank model predicts that a marathoner who is not close to world class over
10km is unlikely to be a world class marathoner.
• The phase transitions which we observe (finding V) provide additional observational evi-
dence for a transition in the complexity of the physiology underlying performance between
long and short distances. This finding is bolstered by the difference we observe between the
increase in performance of the rank 2 predictor over the rank 1 predictor for short/middle
distances over long distances. Notice, however, that this is quite different evidence than the
kink in the power-law of world-record speeds [5], which we argued above does not necessar-
ily imply the presence of transitions at the level of the individual runner. Our results may
have implications for existing hypotheses and findings in sports science on the differences in
physiological determinants of long and short distance running respectively. These include
differences in the muscle fibre types contributing to performance (type I vs. type II) [34, 35],
whether the race length demands energy primarily from aerobic or anaerobic metabolism
[20, 36], which energy systems are mobilized (glycolysis vs. lipolysis) [37, 38] and whether
the race terminates before the onset of a _VO2 slow component [39, 40]. We conjecture that
the combination of our methodology with experiments will shed further light on these
differences.
• An open question in the physiology of aging is whether sprinting power or endurance capa-
bilities diminish faster with age. Our analysis provides cross-sectional evidence that training
standard decreases with age, and specialization shifts away from endurance: a larger expo-
nent is correlated with worse performance on endurance type events (finding IV.i), and
exponents increase, in cross-section, with age (finding IV.iv). This confirms observations of
Rittweger et al. [41] on masters world-record data. There are multiple possible explanations
for this, for example longitudinal changes in specialization, or selection bias due to the dis-
tances older runners prefer; our model renders these hypotheses amenable to quantitative
validation.
• We find that there are a number of high-standard runners who attempt distances different
from their inferred best distance; most notably a cluster of young runners (<25 yrs.) who
run short distances (mostly in accordance with legal limitations of participation), and a clus-
ter of older runners (>40 yrs.) who run long distances, but who we predict would perform
better on longer resp. shorter distances. Moreover, the third component of our model implies
the existence of runners with very strong specialization in their best event; there are indeed
high profile examples of such runners, such as Zersenay Tadese, who holds the half-mara-
thon world best performance (58:23) but has as yet to produce a marathon performance even
close to this in quality (best performance, 2:10:41).
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We also anticipate that our framework will prove fruitful in equipping the practioner with
new methods for prediction and quantification:
• Individual predictions are crucial in race planning, especially for predicting a target perfor-
mance for events such as the Marathon for which months of preparation are needed; the abil-
ity to accurately select a realistic target speed could potentially make the difference between a
runner achieving a personal best performance and “hitting the wall” or at worst dropping out
of the race from exhaustion.
N.B.: We would like to stress that using a prediction as part of marathon preparation without
professional support may lead to injury and is entirely at the risk of the user.
• Predictions and the three-number-summary yield a concise description of the runner’s spe-
cialization and training state and are thus of immediate use in training assessment and plan-
ning, for example in determining the potential effect of a training scheme or finding the
optimal event(s) for which to train.
N.B.: We would like to stress that our study is not able to assign a conclusive meaning to the
three-number summary, due to the limitations of the data set; therefore decisions should not be
based on a hypothesized interpretation without consideration.
• The presented framework allows, in principle, for the derivation of novel and more accurate
scoring schemes, including scoring tables for any type of population.
N.B.: We would like to stress that the form of the derived scoring tables may depend on the
selection of the data from which they are derived.
• Predictions for elite runners allow for a more precise estimation of quotas and betting risk.
For example, we predict that a fair race between Mo Farah and Usain Bolt is over 492m (374-
594m with 95% confidence), Chris Lemaitre and Adam Gemili have the calibre to run 43.5
(±1.3) and 43.2 (±1.3) resp. seconds over 400m. Kenenisa Bekele is capable, in a training
state where he can achieve his personal bests over 5km, 10km and the half-marathon, of a
2:00:36 marathon (±3.6 mins).
N.B.: We would like to stress that such predictions need to be taken with much caution, as they
are only correct insofar as our model extends, from the top 25% of UK runners (who success-
fully participated in official events), to the very extremes of human performance.
We further conjecture that the physiological laws we have validated for running will be
immediately transferable to any sport where a power law has been observed on the collective
level, such as swimming, cycling, and horse racing.
Supporting Information
S1 Supplement. Additional analyses and method details with corresponding figures and
tables.
(PDF)
Acknowledgments
DAJB was supported by a grant from the German Research Foundation, research training
group GRK 1589/1 “Sensory Computation in Neural Systems”. FJK was partially supported by
Mathematisches Forschungsinstitut Oberwolfach (MFO). This research was partially carried
out at MFO with the support of FJK’s Oberwolfach Leibniz Fellowship. We thank thepo-
werof10.info for permission to use their database for this paper, Ryota Tomioka for providing
us with his code for matrix completion via nuclear norm minimization, and for advice on its
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use, Louis Theran for advice regarding the implementation of local matrix completion in
higher ranks. We thank Denis Bafounta, Renato Canova, Tim Grose, Florian Lorenz, Klaus-
Robert Müller and Franz Wölfle for remarks, and discussion of the concepts and results pre-
sented in the manuscript.
Author Contributions
Conceived and designed the experiments: DAJB FJK. Performed the experiments: DAJB. Ana-
lyzed the data: DAJB FJK. Contributed reagents/materials/analysis tools: DAJB FJK. Wrote the
paper: DAJB FJK. Conceived the LMC algorithm in higher ranks: FJK. Adapted the LMC algo-
rithm and designed its concrete implementation for performance prediction: DAJB.
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Prediction and Quantification of Individual Athletic Performance of Runners
PLOS ONE | DOI:10.1371/journal.pone.0157257
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Prediction and Quantification of Individual Athletic Performance of Runners.
Blythe, Duncan A J,Király, Franz J