Supplement: Prediction and Quantiﬁcation of Individual Athletic
Performance of Runners
Duncan A.J. Blythe ∗ 1,2 and Franz J. Király † 3
1African Institute for Mathematical Sciences, Bagamoyo, Tanzania
2 Bernstein Center for Computational Neuroscience, Berlin, Germany
3 Department of Statistical Science, University College London, United Kingdom
Methods
The following provides a guideline for reproducing the results. Raw and pre-processed data in MATLAB and
CSV formats is available upon request, subject to approval by British Athletics. Complete and documented
source code of algorithms and analyses as well as data can be obtained from [1, 2].
Data Source
The basis for our analyses is the online database www.thepowerof10.info, which catalogues British individ-
uals’ performances achieved in oﬃcially ratiﬁed athletics competitions since 1954, including Olympic athletic
events (ﬁeld and non-ﬁeld events), non-Olympic athletic events, cross country events and road races of all
distances.
With the permission of British Athletics, we obtained an excerpt of the database by automated querying
of the freely accessible parts of www.thepowerof10.info, restricted to ten types of running events: 100m,
200m, 400m, 800m, 1500m, the Mile, 5000m (track and road races), 10000m (track and road races), Half-
Marathon and Marathon.
Other types of running events were available but excluded from the present
analyses; the reasons for exclusion were a smaller total of attempts (e.g. 3000m), a diﬀerent population of
runners (e.g. 3000m is mainly attempted by younger runners), and varying conditions (steeplechase/ hurdles
and cross-country races).
The data set consists of two tables: athletes.csv, containing records of individual runners, with ﬁelds:
runner ID, gender, date of birth; and events.csv, containing records of individual attempts on running
events until August 3, 2013, with ﬁelds: runner ID, event type, date of the attempt, and performance in
seconds.
Data Cleaning
Our excerpt of the database contains (after error and duplication removal) records of 164,746 individuals
of both genders, ranging from the amateur to the elite, young to old, and a total of 1,410,789 individual
performances for 10 diﬀerent types of events (see previous section).
Gender is available for all runners in the database (101,775 male, 62,971 female). The dates of birth
of 114,168 runners are missing (recorded as January 1, 1900 in athletes.csv due to particulars of the
automated querying); the date of birth of six runners is set to missing due to a recorded age at recorded
attempts of eight years or less.
For the above runners, a total of 1,417,476 attempts was recorded, out of which 1,410,789 remained in
the data set after cleaning: 192,947 over 100m, 194,107 over 200m, 109,430 over 400m, 239,666 over 800m,
∗duncan.blythe@bccn-berlin.de
†f.kiraly@ucl.ac.uk
1
176,284 over 1500m, 6,590 at the Mile distance, 96,793 over 5000m (the track and road races), 161,504 over
10000m (on the track and road races), 140,446 for the Half-Marathon and 93,033 for the Marathon. 6,643
duplicate events were removed, and a number of 44 events whose reported performances are better than the
oﬃcial world records of their time, or extremely slow. Dates of the attempt were set to missing for 225 of
the attempts that recorded January 1, 1901, and one of the attempts that recorded August 20, 2038.
Data Preprocessing
The events and athletes data sets are collated into (10×164, 746)-tables/matrices of performances, where the
10 columns correspond to events and the 164, 746 rows to individual runners. Rows are indexed increasingly
by runner ID, columns by the type of event. Each entry of the table/matrix contains one performance (in
seconds) of the runner by which the row is indexed, at the event by which the column is indexed, or a missing
value. If the entry contains a performance, the date of that performance is stored as meta-information.
We consider two diﬀerent modes of collation, yielding one table/matrix of performances of size (10 ×
164, 746) each.
In the ﬁrst mode, which in Tables 1 ﬀ. is referenced as “best”, one proceeds as follows. First, for each
individual runner, one ﬁnds the best event of each individual, measured by population percentile. Then, for
each type of event which was attempted by that runner within a year before or after that best event, the
best performance for that type of event is entered into the table. If a certain event was not attempted in
this period, it is recorded as missing.
For the second mode of collation, which in Tables 1 ﬀ. is referenced as “random”, one proceeds as follows.
First, for each individual runner, a calendar year is uniformly randomly selected among the calendar years
in which that runner has attempted at least one event. Then, for each type of event which was attempted
by that runner within the selected calendar year, the best performance for that type of event is entered into
the table. If a certain event was not attempted in the selected calendar year, it is recorded as missing.
The ﬁrst collation mode ensures that the data is of high quality: runners are close to optimal ﬁtness,
since their best performance was achieved in this time period. Moreover, since ﬁtness was at a high level, it
is plausible that the number of injuries incurred was low leading to multiple attempts being made at events
– this will lead to higher data quality; indeed it can be observed that the number of attempts per event is
higher in this period, eﬀectively decreasing the inﬂuence of noise and the chance that outliers are present
after collation.
The second collation mode is used to check whether and, if so how strongly, the results depend on the
runners being close to optimal ﬁtness.
In both cases choosing a narrow time frame ensures that performances are relevant to one another for
prediction.
Runner-Speciﬁc Summary Statistics
For each given runner, several summaries are computed based on the collated matrix.
Performance percentiles are computed for each event which a runner attempts in relation to the other
runners’ performances on the same event. These column-wise event-speciﬁc percentiles, yield a percentile
matrix with the same ﬁlling pattern (pattern of missing entries) as the collated matrix.
The preferred distance for a given runner is the geometric mean of the attempted events’ distances.
That is, if s1, . . . , sm are the distances for the events which the runner has attempted, then ˜s = (s1 · s2 · . . . ·
sm)1/m is the preferred distance.
The training standard for a given runner is the mean of all performance percentiles in the corresponding
row.
The no. events for a given runner is the number of events attempted by a runner in the time period of
the data considered (best or random).
Note that the percentiles yield a mostly physiological description; the preferred distance is a behavioural
summary since it describes the type of events the runner attempts. The training standard combines both
physiological and behavioural characteristics.
Percentiles and training standard depend on the collated matrix. When we consider genders, age or
runners who have attempted more than a certain no.event these summary statistics are calculated separately
2
for the subgroup. However, within subgroup these values depend on the entire collated submatrix for that
subgroup.
Outlier Removal
Outliers are removed from the data in both collated matrices.
An outlier score for each runner/row is
obtained as the diﬀerence of maximum and minimum of all performance percentile of the runner. The ﬁve
percent rows/runners with the highest outlier score are removed from the matrix.
Prediction: Evaluation and Validation
Prediction accuracy is evaluated on row-sub-samples of the collated matrices, deﬁned by (a) a potential
subgroup, e.g., given by age or gender, (b) degrees-of-freedom constraints in the prediction methods that
require a certain number of entries per row, and (c) a certain range of performance percentiles of runners.
The row-sub-samples referred to in the main text and in Tables 1 ﬀ. are obtained by (a) retaining all
rows/runners in the subgroup speciﬁed by gender, or age in the best event, (b) retaining all rows/runners
with at least no. events or more entries non-missing, and discarding all rows/runners with strictly less
than no. events entries non-missing, then (c) retaining all runners in a certain percentile range.
The
percentiles referred to in (c) are computed as follows: ﬁrst, for each column, in the data retained after step
(b), percentiles are computed. Then, for each row/runner, the best of these percentiles is selected as the
score over which the overall percentiles are taken.
The accuracy of prediction is principally measured empirically in terms of out-of-sample root mean
squared error (RMSE) and mean absolute error (MAE), with RMSE, MAE, and standard deviations es-
timated from the empirical sample of residuals obtained in 1000 iterations of leave-one-out validation. In
selected analyses we measure error in terms of relative RMSE (rRMSE),
1
N
P
i

predictor(i)−predicted(i)
predicted(i)
2
and
relative MAE (deﬁned analogously) (rMAE).
Given the row-sub-sample matrix obtained from (a), (b), (c), prediction and thus we perform leave-one-
out validation in two ways: (i) predicting the left-out entry from potentially all remaining entries. In this
scenario, the prediction method may have access to the performances of the runner in question which lie in
the future of the event to be predicted, though only performances of other events are available; (ii) predicting
the left-out entry from all remaining entries of other runners, but only from those events of the runner in
question that lie in the past of the event to be predicted. In this task, temporal causality is preserved on
the level of the single runner for whom prediction is done; though information about other runners’ results
that lie in the future of the event to be predicted may be used.
The third option (iii) where predictions are made only from past events has not been studied due to
the size of the data set which makes collation of the data set for every single prediction per method and
group a computationally extensive task, and due to the potential group-wise sampling bias which would be
introduced, skewing the measures of prediction-quality—the population of runners on the older attempts is
diﬀerent in many respects from the more recent attempts. We further argue that in the absence of such
technical issues, evaluation as in (ii) would be equivalent to (iii); since the performances of two randomly
picked runners, no matter how they are related temporally, may be reasonably modelled as statistically
independent; positing the contrary would be equivalent to postulating that any given runner’s performance
is very likely to be directly inﬂuenced by a large number of other runners’ performance history, which is an
assumption which is scientiﬁcally implausible. Given the above, due to equivalence of (ii) and (iii), and the
issues occurring in (iii) exclusively, we can conclude that (ii) is preferrable over (iii) from a scientiﬁc and
statistical viewpoint.
Prediction: Target Outcomes
The principal target outcome for the prediction is “performance”, which we present to the prediction methods
in three distinct parameterisations. This corresponds to passing not the raw performance matrices obtained
in the section “Data Pre-processing” to the prediction methods, but re-parameterized variants where the
non-missing entries undergo a univariate variable transform. The three parameterizations of performance
considered in our experiments are the following:
3
(a) normalized: performance as the time in which the given runner (row) completes the event in question
(column), divided by the average time in which the event in question (column) is completed in the sub-
sample;
(b) log-time: performance as the natural logarithm of time in seconds in which the given runner (row)
completes the event in question (column);
(c) speed: performance as the average speed in meters per second, with which the given runner (row)
completes the event in question (column).
The words in italics indicate which parameterisation is referred to in Table 1. The error measures, RMSE
and MAE, are evaluated in the same parameterisation in which prediction is performed. We do not evaluate
performance directly in un-normalized time units, as in this representation performances between 100m and
the Marathon span 4 orders of magnitude (base-10), which would skew the measures of goodness heavily
towards accuracy over the Marathon.
Unless stated otherwise, predictions are made in the same parameterisation on which the models are
learnt.
Prediction: Models and Algorithms
In the experiments, a variety of prediction methods are used to perform prediction from the performance
data, given as described in “Prediction: Target Outcomes”, evaluated by the measures as described in the
section “Prediction: Evaluation and Validation”.
In the code available for download, each method is encapsulated as a routine which predicts a missing entry
when given the (training entries in the) performance matrix. The methods can be roughly divided in four
classes: (1) naive baselines, (2) representatives of the state-of-the-art in prediction of running performance,
(3) representatives of the state-of-the-art in matrix completion, and (4) our proposed method and its variants.
The naive baselines are:
(1.a) mean: predicting the the mean over all performances for the same event within the subgroup considered.
(1.b) k-NN: k-nearest neighbours prediction. The parameter k is obtained as the minimizer of out-of-sample
RMSE on ﬁve groups of 50 randomly chosen validation data points from the training set, from among
k = 1, k = 5, and k = 20.
The representatives of the state-of-the-art in predicting running performance are:
(2.a) Riegel: The Riegel power law formula with exponent 1.06.
(2.b) power law: A power law
predictor, as per the Riegel formula, but with the exponent estimated from the data. The exponent is the
same for all runners and estimated as the minimizer of the residual sum of squares. (2.c) ind.power law:
A power law predictor, as per the Riegel formula, but with the exponent estimated from the data. The
exponent may be diﬀerent for each runner and is estimated as the minimizer of the residual sum of squares.
(2.d) Purdy: Prediction by calculation of equivalent performances using the Purdy points scheme [3]. Purdy
points are calculated by using the measurements given by the Portugese scoring tables which estimate the
maximum velocity for a given distance in a straight line, and adjust for the cost of traversing curves and
the time required to reach race velocity. The performance with the same number of points as the predicting
event is imputed.
The representatives of the state-of-the-art in matrix completion are:
(3.a) EM: Expectation maximization algorithm assuming a multivariate Gaussian model for the rows
of the performance matrix in log-time parameterisation.
Missing entries are initialized by the mean of
each column. The updates are terminated when the percent increase in log-likelihood is less than 0.1%.
For a review of the EM-algorithm see [4].
(3.b) Nuclear Norm: Matrix completion via nuclear norm
minimization [5, 6].
The variants of our proposed method are as follows:
(4.a-d) LMC rank r: local matrix completion for the low-rank model, with rank r = 1, 2, 3, 4. (4.a) is
LMC rank 1, (4.b) is LMC rank 2, and so on.
Our algorithm follows the local/entry-wise matrix completion paradigm in [7]. It extends the rank 1 local
matrix completion method described in [8] to arbitrary ranks.
Our implementation uses: determinants of size (r + 1 × r + 1) as the only circuits; the weighted variance
minimization principle in [8]; the linear approximation for the circuit variance outlined in the appendix of [9];
modelling circuits as independent for the co-variance approximation.
4
We further restrict to circuits supported on the event to be predicted and the r log-distance closest events.
For the convenience of the reader, we describe the exact way in which the local matrix completion
principle is instantiated, in the section “Prediction: Local Matrix Completion” below
In the supplementary experiments we also investigate two aggregate predictors to study the potential
beneﬁt of using other lengths for prediction:
(5.a) bagged power law: bagging the power law predictor with estimated coeﬃcient (2.b) by a weighted
average of predictions obtained from diﬀerent events. The weighting procedure is described below. (5.b)
bagged LMC rank 2: estimation by LMC rank 2 where determinants can be supported at any three events,
not only on the closest ones (as in line 1 of Algorithm 1 below). The ﬁnal, bagged predictor is obtained as a
weighted average of LMC rank 2 running on diﬀerent triples of events. The weighting procedure is described
below.
The averaging weights for (5.a) and (5.b) are both obtained from the Gaussian radial basis function
kernel exp
Obtaining the Low-Rank Components and Coeﬃcients
We obtain three low-rank components f1, . . . , f3 and corresponding coeﬃcients λ1, . . . , λ3 for each runner
by considering the data in log-time coordinates. Each component fi is a vector of length 10, with entries
corresponding to events. Each coeﬃcient is a scalar, potentially diﬀerent for each runner.
To obtain the components and coeﬃcients, we consider the data matrix for the speciﬁc target outcome,
sub-sampled to contain the runners who have attempted four or more events and the top 25% percentiles,
as described in “Prediction: Evaluation and Validation”. In this data matrix, all missing values are imputed
using the rank 3 local matrix completion algorithm, as described in (4.c) of “Prediction: Models and Algo-
rithms”, to obtain a complete data matrix M. For this matrix, the singular value decomposition M = USV ⊤
is computed, see [10].
We take the components f2, f3 to be the the 2-th and 3-rd right singular vectors, which are the 2-nd and
3-rd column of V . The component f1 is a re-scaled version of the 1-st column v of V , such that f1(s) ≈ log s,
where the natural logarithm is taken. More precisely, f1 := β−1v, where the re-scaling factor β is obtained
as the ordinary least-squares regression coeﬃcient of the linear explanatory model v(s) = β log s + c, where
s ranges over the ten event distances, which is β = 0.0572. A more detailed study of v and the regression
coeﬃcient can be found in supplementary experiment (II.b).
The three-number-summary referenced in the main corpus of the manuscript is obtained as follows: for
the k-th runner we obtain from the left singular vector the entries Ukj. The second and third score of the
three-number-summary are obtained as λ2 = Uk2 and λ3 = Uk3. The individual exponent is λ1 = β · Uj1.
The singular value decomposition has the property that the fi and λj are guaranteed to be least-squares
estimators for the components and the coeﬃcients in a projection sense.
Computation of standard error and signiﬁcance
Standard errors for the singular vectors (components of the model of Equation 1) are computed via inde-
pendent bootstrap sub-sampling on the rows of the data set (runners).
Standard errors for prediction accuracies are obtained by bootstrapping of the predicted performances
(1000 per experiment). A method is considered to perform signiﬁcantly better than another when error
regions at the 95% conﬁdence level (= mean over repetitions ± 1.96 standard errors) do not intersect.
Predictions and three-number-summary for elite runners
Performance predictions and three-number-summaries for the selected elite runners in Table 1 and Figure 4
are obtained from their personal best times. The relative standard error of the predicted performances is
estimated to be the same as the relative RMSE of predicting time, as reported in Table 1.
Calculating a fair race
Here we describe the procedure for calculating a fair racing distance with error bars between two runners:
runner 1 and runner 2. We ﬁrst calculate predictions for all events. Provided that runner 1 is quicker on
some events and runner 2 is quicker on others, then calculating a fair race is feasible. If runner 1 is quicker on
shorter events then runner 2 is typically quicker on all longer events beyond a certain distance. In that case,
we can ﬁnd the shortest race si whereby runner 2 is predicted to be quicker; then a fair race lies between si
and si−1. The performance curves in log-time vs. log-distance of both runners will be locally approximately
linear. We thus interpolate the performance curves between log(si) and log(si−1)—the crossing point gives
the position of a fair race in log-coordinates. We obtain conﬁdence intervals by repeating this procedure
after sampling data points around the estimated performances with standard deviation equal to the RMSE
(see Table 1) on the top 25% of runners in log-time.
6
Supplementary Analyses
This appendix contains a series of additional experiments supplementing those in the main corpus. It con-
tains the following ﬁndings:
(I) Validation of the LMC prediction framework.
(I.a) Evaluation in terms of MAE. The results in terms of MAE are qualitatively similar to those in
RMSE; smaller MAEs indicate the presence of outliers.
(I.b) Evaluation in terms of time prediction. The results are qualitatively similar to measuring pre-
diction accuracy in RMSE and MAE of log-time. LMC rank 2 has an average error of approximately 2%
when predicting the top 25% of male runners.
(I.c) Prediction for individual events. LMC outperforms the other predictors on each type of event.
The beneﬁt of higher rank is greatest for middle distances.
(I.d) Stability w.r.t. the unit measuring performance. LMC performs equally well in predicting (per-
formance in time units) when performances are presented in log-time or time normalized by event average.
Speed is worse when the rank 2 predictor is used.
(I.e) Stability w.r.t. the events used in prediction. LMC performs equally well when predicting from
the closest-distance events and when using a bagged version which uses all observed events for prediction.
(I.f) Stability w.r.t. the event predicted. LMC performs well both when the predicted event is close
to those observed and when the predicted event is further from those observed, in terms of event distance.
(I.g) Temporal independence of performances. There are negligible diﬀerences between predictions
made only from past events and predictions made from all available events (in the training set).
(I.h) Run-time comparisons. LMC is by orders of magnitude the fastest among the matrix completion
methods.
(II) Validation of the low-rank model.
(II.a) Synthetic validation. In a synthetic low-rank model of athletic performance that is a proxy to the
real data, the singular components of the model can be correctly recovered by the exact same procedure as
on the real data.
(II.b) The individual power law component, and the distance/time unit. The ﬁrst singular com-
ponent can be explained by a linear model in log-distance (R-square 0.9997) with slope β = 0.0572 ± 0.0003
and intercept c = −0.136 ± 0.003.
(II.c) Universality in sub-groups. Quality of prediction, the low-rank model, its rank, and the singular
components remain mostly unchanged when considering subgroups male/female, older/younger, elite/amateur.
(III) Exploration of the low-rank model.
(III.a) Further exploration of the three-number-summary. The three number summary also corre-
lates with specialization and training standard.
(III.b) Preferred distance vs optimal distance. Most but not all runners prefer to attend the event
at which they are predicted to perform best. A notable number of younger runners prefer distances shorter
than optimal, and some older runners prefer distances longer than optimal.
(IV) Pivoting and phase transitions. The pivoting phenomenon described in Figure 1, right panel,
is found in the data for any three close-by distances up to the Mile, with anti-correlation between the shorter
and the longer distance. Above 5000m, a change in the shorter of the three distances positively correlates
with a change in the longer distance.
(I.a) Evaluation in terms of MAE. Table A reports on the goodness of prediction methods in terms
of MAE. Compared with the RMSE (Table 1, the MAE tend to be smaller than the RMSE, indicating the
presence of outliers. The relative prediction-accuracy of methods when compared to each other is qualita-
tively the same.
(I.b) Evaluation in terms of time prediction. Tables C and D report on the prediction accuracy of
the methods tested in terms the relative RMSE and MAE of predicting time. Relative measures are chosen
7
to avoid bias towards the longer events. The results are qualitatively and quantitatively very similar to the
log-time results in Tables 1 and A; this can be explained that mathematically the RMSE and MAE of a
logarithm approximate the relative RMSE and MAE well for small values.
(I.c) Individual Events. Prediction accuracy of LMC rank 1 and rank 2 on the ten diﬀerent events is
displayed in Figure A. The reported prediction accuracy is out-of-sample RMSE in predicting log-time, on
the top 25 percentiles of Male runners who have attempted 3 or more events, of events in their best year
of performance. The reported RMSE for a given event is the mean over 1000 random prediction samples,
standard errors are estimated by the bootstrap.
The relative improvement of rank 2 over rank 1 tends to be greater for shorter distances below the Mile. This
is in accordance with observation (IV.i) which indicates that the individual exponent is the best descriptor
among the three-number summary for longer events, above the Mile.
(I.d) Stability w.r.t. the measure of performance. In the main experiment, the LMC model is learnt on
the same measure of performance (log-time, speed, normalized) which is predicted. We investigate whether
the measure of performance on which the model is learnt inﬂuences the prediction by learning the LMC model
on either measure and comparing all predictions using the log-time measure. Table G displays prediction
accuracy when the model is learnt in any one of the measures of performance. Here we check the eﬀect of
calibration in one coordinates system and testing in another. The reported goodness is out-of-sample RMSE
of predicting log-time, on the top 25 percentiles of Male runners who have attempted 3 or more events,
of events in their best year of performance. The reported RMSE for a given event is the mean over 1000
random prediction samples, standard errors are estimated by the bootstrap.
We ﬁnd that there is no signiﬁcant diﬀerence in prediction goodness when learning the model in log-time
coordinates or normalized time coordinates. Learning the model in speed coordinates leads to a signiﬁcantly
better prediction than log-time or normalized time when LMC rank 1 is applied, but to a worse prediction
with LMC rank 2. As overall prediction with LMC rank 2 is better, log-time or normalized time are the
preferable units for predicting performance.
(I.e) Stability w.r.t. the event predicted.
We consider here the eﬀect of the ratio between the predicted event and the closest predictor. For data
of the best 25% of Males in the year of best performance (best), we compute the log-ratio of the closest
predicting distance and the predicted distance for Purdy Points, the power law formula and LMC rank 2.
See Figure B, where this log ratio is plotted against error. The results show that LMC is far more robust to
error for predicting distances far from the predicted distance.
(I.f) Stability w.r.t. the events used in prediction. We compare whether we can improve predic-
tion by using all events a runner has attempted, by using one of the aggregate predictors (5.a) bagged
power law or (5.b) bagged LMC rank 2. The kernel width γ for the aggregate predictors is chosen from
−0.001, −0.01, −0.1, −1, −10 as the minimizer of out-of-sample RMSE on ﬁve groups of 50 randomly chosen
validation data points from the training set. The validation setting is the same as in the main prediction
experiment.
Results are displayed in Table H. We ﬁnd that prediction accuracy of (2.b) power law and (5.a) bagged
power law is not signiﬁcantly diﬀerent, nor is (4.b) LMC rank 2 signiﬁcantly diﬀerent from (5.b) bagged
LMC rank 2 (both p > 0.05; Wilcoxon signed-rank on the absolute residuals). Even though the kernel width
selected is in the majority of cases σ = −1 and not σ = −10, the incorporation of all events does not lead to
an improvement in prediction accuracy in our aggregation scheme. We ﬁnd there is no signiﬁcant diﬀerence
(p > 0.05; Wilcoxon signed-rank on the absolute errors) between the bagged and vanilla LMC for the top
95% of runners. This demonstrates that the relevance of closer events for prediction may be learnt from the
data. The same holds for the bagged version of the power law formula.
(I.g) Temporal independence of performances. We check here whether the results are aﬀected by using
only temporally prior attempts in predicting a runner’s performance, see section “Prediction: Evaluation and
Validation” in “Methods”. To this end, we compute out-of-sample RMSEs when predictions are made only
from those events.
8
Table B reports out-of-sample RMSE of predicting log-time, on the top 25 percentiles of Male runners
who have attempted 3 or more events, of events in their best year of performance. The reported RMSE
for a given event is the mean over 1000 random prediction samples, standard errors are estimated by the
bootstrap.
The results are qualitatively similar to those of Table 1 where all events are used in prediction.
(I.h) Run-time comparisons. We compare the run-time cost of a single prediction for the three matrix
completion methods LMC, nuclear norm minimiziation, and EM. The other (non-matrix completion) meth-
ods are fast or depend only negligibly on the matrix size. We measure run time of LMC rank 3 for completion
of a single entry for matrices of 28, 29, . . . , 213 runners, generated as described in (II.a). This is repeated
100 times. For a fair comparison, the nuclear norm minimization algorithm is run with a hyper-parameter
already pre-selected by cross validation. The results are displayed in Figure C; LMC is faster by orders of
magnitude than nuclear norm and EM and is very robust to the size of the matrix. The reason computation
speeds up over the smallest matrix sizes is that 4 × 4 minors, which are required for rank 3 estimation are
not available, thus the algorithm must attempt all ranks lower than 3 to ﬁnd suﬃciently many minors.
(II.a) Synthetic validation. To validate the assumption of a low-rank generative model, we investigate
prediction accuracy and recovery of singular vectors in a synthetic model of athletic performance.
Synthetic data for a given number of runners is generated as follows:
For each runner, a three-number summary (λ1, λ2, λ3) is generated independently from a Gaussian dis-
tribution with the same mean and variance as the three-number-summaries measured on the real data and
with uncorrelated entries.
Matrices of performances are generated from the model
log(t) = λ1f1(s) + λ2f2(s) + λ3f3(s) + η(s)
(1)
where f1, f2, f3 are the three components estimated from the real data and η(s) is a stationary zero-mean
Gaussian white noise process with adjustable variance.
We take the components estimated in log-time
coordinates from the top 25% of male runners who have attempted at least 4 events as the three components
of the model. The distances s are the same ten distances as encountered in the real data. In each experiment
the standard deviation of η(s) is set to Std(η) = 0.01, which was shown to be plausible in the previous
section.
Accuracy of prediction: We synthetically generate a matrix of 1000 runners according to the model
of Equation (1), taking as distances the same distances measured on the real data. Missing entries are
randomized according to two schemes: (a) 6 (out of 10) uniformly random missing entries per row/runner.
(b) per row/runner, four in terms of distance-consecutive entries are non-missing, uniformly at random.
We then apply LMC rank 2 and nuclear norm minimization for prediction. This setup is repeated 100
times for ten diﬀerent standard deviations of η between 0.01 and 0.1. The results are displayed in Figure D.
LMC performance outperforms nuclear norm; LMC performance is also robust to the pattern of miss-
ingness, while nuclear norm minimization is negatively aﬀected by clustering in the rows. RMSE of LMC
approaches zero with small noise variance, while RMSE of nuclear norm minimization does not.
Comparing the performances with Table 1, an assumption of a noise variance of Std(η) = 0.01 seems
plausible. The performance of nuclear norm on the real data is explained by a mix of the sampling schemes
(a) and (b).
Recovery of model components. We synthetically generate a matrix which has a size and pattern of
observed entries identical to the matrix of top 25% of male runners who have attempted at least 4 events in
their best year. We set Std(η) = 0.01, which was shown to be plausible in the previous section.
We then complete all missing entries of the matrix using LMC rank 3. After this initial step we estimate
singular components using SVD, exactly as on the real data.
Conﬁdence intervals are estimated by a
bootstrap on the rows with 100 iterations.
The results are displayed in Figure E.
We observe that the ﬁrst two singular components are recovered almost exactly, while the third is a
slightly deformed. This is due to the smaller singular value of the third component.
9
(II.b) The individual power law component, and the distance/time unit. We examine linearity of
the ﬁrst singular vector v, as listed in Table 1 and as described in methods section “Obtaining the Low-Rank
Components and Coeﬃcients”. In an ordinary least squares regression model explaining v by log s and an
intercept, we ﬁnd that v ≈ β log s + c with an R-squared of 0.9997 (Table I), where the scaling factor is
β = 0.0572 ± 0.0003 and the intercept is c = −0.136 ± 0.003. The intercept corresponds to a choice of
unit, the scaling factor to a choice of basis for the logarithm. Thus re-scaling v with β−1, that is, setting
f1 := β−1v in the low-rank model, and re-scaling the ﬁrst individual coeﬃcient with β, corresponds to the
choice of the natural basis.
The residuals of the the linear model appear to be plausibly explained by the second and third singular com-
ponent (Table I), though the small number of ﬁtting nodes which is 10 does not allow a for an assessment
that is more than qualitative.
(II.c) Universality in sub-groups. We repeat the methodology for component estimation described above
and obtain the three components in the following sub-groups: female runners, older runners (> 30 years),
and amateur runners (25-95 percentile range of training standard). Male runners were considered in the
main corpus. For female and older runners, we restrict to the top 95% percentiles of the respective groups
for estimation.
Figure F displays the estimated components of the low-rank model. The individual power law is found
to be unchanged in all groups considered. The second and third component vary between the groups but
resemble the components for the male runners. The empirical variance of the second and third component is
higher, which may be explained by a slightly reduced consistency in performance, or a reduction in sample
size. Whether there is a genuine diﬀerence in form or whether the variation is explained by diﬀerent three-
number-summaries in the subgroups cannot be answered from the dataset considered.
Table F displays the prediction results in the three subgroups. Prediction accuracy is similar but slightly
worse when compared to the male runners. Again this may be explained by reduced consistency in the
subgroups’ performances.
(III.a) Further exploration of the three-number-summary. Scatter plots of preferred distance and
training standard against the runners’ three-number-summaries are displayed in Figure G. The training
standard correlates predominantly with the individual exponent (score 1); score 1 vs. standard—r = −0.89
(p ≤ 0.001); score 2 vs. standard—r = 0.22 (p ≤ 0.001); score 3 vs. standard—r = 0.031 (p = 0.07); all
correlations are Spearman correlations with signiﬁcance computed using a t-distribution approximation to
the correlation coeﬃcient under the null. On the other hand preferred distance is associated with all three
numbers in the summary, especially the second; score 1 vs. log(specialization)—r = 0.29 (p ≤ 0.001); score
2 vs. log(specialization)—r = −0.58 (p ≤ 0.001); score 3 vs. log(specialization)—r = −0.14 (p =≤ 0.001);
The association between the third score and specialization is non-linear with an optimal value around the
middle distances. We stress that low correlation does not imply low predictive power; the whole summary
should be considered as a whole, and the LMC predictor is non-linear. Also, we observe that correlations
increase when considering only performances over certain distances, see Figure 2.
(III.b) Preferred event vs best event. For the top 95% male runners who have attempted 3 or more
events, we use LMC rank 2 to compute which percentile they would achieve in each event. We then determine
the distance of the event at which they would achieve the best percentile, to which we will refer as the “optimal
distance”. Figure H shows for each runner the diﬀerence between their preferred and optimal distance.
It can be observed that the large majority of runners prefer to attempt events in the vicinity of their
optimal event. There is a group of young runners who attempt events which are shorter than the predicted
optimal distance, and a group of old runners attempting events which are longer than optimal. One may
hypothesize that both groups could be explained by social phenomena: young runners usually start to train
on shorter distances, regardless of their potential over long distances. Older runners may be biased to at-
tempting endurance type events.
(IV) Pivoting and phase transitions. We look more closely at the pivoting phenomenon illustrated in
Figure 1 top right, and the phase transition discussed in observation (V). We consider the top 25% of male
runners who have attempted at least 3 events, in their best year.
10
We compute 10 performances of equivalent standard by using LMC rank 1 in log-time coordinates,
by setting a benchmark performance over the marathon and sequentially predicting each lower distance
(marathon predicts HM, HM predicts 10km etc.). This yields equivalent benchmark performances t1, . . . , t10.
We then consider triples of consecutive distances si−1, si, si+1 (excluding the Mile since close in distance
to the 1500m) and study the pivoting behaviour on the data set, by performing the analogous prediction
displayed in Figure 1.
More speciﬁcally, for each triple, we predict the performance on the distance si+1 using LMC rank 2,
from the performances over the distances si−1 and si. The prediction is performed in two ways, once with
and once without perturbation of the benchmark performance at si−1, which we then compare. Intuitively,
this corresponds to comparing the red to the green curve in Figure 1. In mathematical terms:
1. We obtain a prediction bti+1 for the distance si+1 from the benchmark performances ti, ti−1 and consider
this as the unperturbed prediction, and
2. We obtain a prediction bti+1 + δ(ϵ) for the distance si+1 from the benchmark performance ti on si
and the perturbed performance (1 + ϵ)ti−1 on the distance si−1, considering this as the perturbed
prediction.
We record these estimates for ϵ = −0.1, 0.09, . . . , 0, 0.01, . . . , 0.1 and calculate the relative change of the
perturbed prediction with respect to the unperturbed, which is δi(ϵ)/bti. The results are displayed in Figure I.
We ﬁnd that for pivot distances si shorter than 5km, a slower performance on the shorter distance si−2
leads to a faster performance over the longer distance si, insofar as this is predicted by the rank 2 predictor.
On the other hand we ﬁnd that for pivot distances greater than or equal to 5km, a faster performance over
the shorter distance also implies a faster performance over the longer distance.
References
[1] Blythe DAJ, Király FJ. Full data to “Prediction and Quantiﬁcation of Individual Athletic Performance of
Runners"; 2016.
DOI: 10.6084/m9.ﬁgshare.3408202.v1.
Available from: https://figshare.com/articles/
thepowerof10/3408202.
[2] Blythe DAJ, Király FJ. Full code to “Prediction and Quantiﬁcation of Individual Athletic Performance of Run-
ners"; 2016.
DOI: 10.6084/m9.ﬁgshare.3408250.v1.
Available from: https://figshare.com/articles/Ful_
code_to_Prediction_and_Quantification_of_Individual_Athletic_Performance_of_Runners_/3408250.
[3] Purdy JG. Computer generated track and ﬁeld scoring tables: II. Theoretical foundation and development of a
model. Medicine and science in sports. 1974;7(2):111–115.
[4] Bishop CM, et al. Pattern recognition and machine learning. vol. 4. springer New York; 2006.
[5] Candès EJ, Recht B. Exact matrix completion via convex optimization. Foundations of Computational mathe-
matics. 2009;9(6):717–772.
[6] Tomioka R, Hayashi K, Kashima H. On the extension of trace norm to tensors. In: NIPS Workshop on Tensors,
Kernels, and Machine Learning; 2010. p. 7.
[7] Király FJ, Theran L, Tomioka R. The algebraic combinatorial approach for low-rank matrix completion. Journal
of Machine Learning Research. 2015;.
[8] Király FJ, Theran L. Obtaining error-minimizing estimates and universal entry-wise error bounds for low-rank
matrix completion. NIPS 2013. 2013;.
[9] Blythe DA, Theran L, Kiraly F.
Algebraic-Combinatorial Methods for Low-Rank Matrix Completion with
Application to Athletic Performance Prediction. arXiv preprint arXiv:14062864. 2014;.
[10] Golub GH, Reinsch C.
Singular value decomposition and least squares solutions.
Numerische Mathematik.
1970;14(5):403–420.
200m
800m
Mile
10km
Mar
0
0.005
0.01
0.015
0.02
0.025
0.03
0.035
0.04
0.045
0.05
log(distance)
RMSE
Accuracy by event rank 1 vs. rank 2
 
 
rank 1
rank 2
Figure A: The ﬁgure displays the results of prediction by event for the top 25% of male runners who attended ≥ 3 events in their
year of best performance. For each event the prediction accuracy of LMC rank 1 (blue) is compared to prediction accuracy in
rank 2 (red). RMSE is displayed on the y-axis against distance on the x-axis; the error bars extend two standard deviations of the
bootstrapped RMSE either side of the RMSE.
0
1
2
3
0
0.02
0.04
0.06
0.08
0.1
0.12
Absolute Relative Error
LMC r2 absolute relative errors
log(ratio) [DV]
0
1
2
3
power−law absolute relative errors
log(ratio) [DV]
0
1
2
3
Purdy absolute relative errors
log(ratio) [DV]
Figure B: The ﬁgure displays the absolute log ratio in distance predicted and predicting distance vs. absolute relative error per
runner. In each case the log ratio in distance is displayed on the x-axis and the absolute errors of single data points of the y-axis.
We see that LMC rank 2 is particularly robust for large ratios in comparison to the power law and Purdy Points. Data is taken
from the top 25% of male runners with no. events≥ 3 in the best year.
7
8
9
10
11
12
13
14
0.1
1
10
run time [secs.]
log(no.athletes)
Run Time
 
 
Nuclear Norm
LMC
EM algorithm
Figure C: The ﬁgure displays mean run-times for the 3 matrix completion algorithms tested in the paper: Nuclear Norm, EM
and LMC (rank 3). Run-times (y-axis) are recorded for completing a single entry in a matrix of size indicated by the x-axis. The
averages are over 100 repetitions, standard errors are estimated by the bootstrap.
0
0.02
0.04
0.06
0.08
0.1
0.04
0.06
0.08
0.1
0.12
0.14
0.16
0.18
0.2
RMSE in log−time
Uniform Missing
 
 
LMC rank 2
Nuclear Norm
0
0.02
0.04
0.06
0.08
0.1
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
Clustered Missing Entries
noise−level−nl [log(seconds)]
Figure D: LMC and Nuclear Norm prediction accuracy on the synthetic low-rank data. x-axis denotes the noise level (standard
deviation of additive noise in log-time coordinates); y-axis is out-of-sample RMSE predicting log-time. Left: prediction performance
when (a) the missing entries in each ros are uniform. Right: prediction performance when (b) the observed entries are consecutive.
Error bars are one standard deviation, estimated by the bootstrap.
100m
1500m
5km
Mar
−0.8
−0.6
−0.4
−0.2
0
0.2
0.4
0.6
Singular Components Estimated
 from Complete Performances
100m
1500m
5km
Mar
−0.8
−0.6
−0.4
−0.2
0
0.2
0.4
0.6
log(time) [log(seconds)]
Singular Components Estimated
 from Incomplete Performances
 
 
Singular Component 1
Singular Component 2
Singular Component 3
distance
Figure E: Accuracy of singular component estimation with missing data on synthetic model of performance. x-axis is distance,
y-axis is components in log-time. Left: singular components of data generated according to Equation 1 with all data present. Right:
singular components of data generated according to Equation 1 with missing entries estimated with LMC rank 3; the observation
pattern and number of runners is identical to the real data. The tubes denote one standard deviation estimated by the bootstrap.
100m
1km
10km
Mar
−1
−0.8
−0.6
−0.4
−0.2
0
0.2
0.4
0.6
0.8
Older Runners (30−80yrs) component scores
log(time) [log(seconds)]
Distances
100m
1km
10km
Mar
−1
−0.8
−0.6
−0.4
−0.2
0
0.2
0.4
0.6
0.8
1
Amateur Runners component scores
log(time) [log(seconds)]
Distances
100m
1km
10km
Mar
−0.8
−0.6
−0.4
−0.2
0
0.2
0.4
0.6
Female Runners component scores
log(time) [log(seconds)]
Distances
Figure F: The three components of the low-rank model in subgroups. Left: for older runners. Middle: for amateur runners =
best event below 25th percentile. Right: for female runners. Tubes around the components are one standard deviation, estimated
by the bootstrap. The components are the analogous components for the subgroups described as computed in the left-hand panel
of Figure 2.
Figure G: Scatter plots of training standard vs. three-number-summary (top) and preferred distance vs. three-number-summary.
In each case the individual exponents, 2nd and 3rd scores (λ2, λ3) are displayed on the y-axis and the log-preferred distance and
training standard on the x-axis.
Figure H: Diﬀerence of preferred distance and optimal distance, versus age of the runner, colored by specialization distance. Most
runners prefer the distance they are predicted to be best at. There is a mismatch of best and preferred for a group of younger
runners who have greater potential over longer distances, and for a group of older runners who’s potential is maximized over shorter
distances than attempted.
200m
400m
800m
1500m
5km
10km
HM
−0.1
−0.05
0
0.05
0.1
f(ε)/ti
distance
Short Performance peturbations
 
 
(1−ε)ti−2>ti−2
(1−ε)ti−2<ti−2
Figure I: Pivot phenomenon in the low-rank model. The ﬁgure quantiﬁes the strength and sign of pivoting as in Figure 1, top
right, at diﬀerent middle distances si (x-axis). The computations are based on equivalent log-time performances ti−1, ti, ti+1 at
consecutive triples si−1, si, si+1 of distances. The y-coordinate indicates the signed relative change of the LMC rank 2 prediction of
ti+1 from ti−1 and ti changes, when ti is ﬁxed and ti−1 undergoes a relative change of 1%, 2%, . . . , 10% (red curves, line thickness
is proportional to change), or −1%, −2%, . . . , −10% (blue curves, line thickness is proportional to change). For example, the largest
peak corresponds to a middle distance of si = 400m. When predicting 800m from 400m and 200m, the predicted log-time ti+1 (=
800m performance) decreases by 8% when ti−1 (= 200m performance) is increased by 10% while ti (= 400m performance) is kept
constant.
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
0.1054
0.0421
0.0696
0.0661
0.0654
0.0423
0.1282
0.0387
0.0402
0.0336
±0.0025
±0.0014
±0.0024
±0.0023
±0.0023
±0.0014
±0.0115
±0.0013
±0.0014
±0.0012
normalized
0-95
3
best
0.1062
0.0441
0.0700
0.0681
0.0674
0.0441
0.0907
0.0400
0.0413
0.0347
±0.0027
±0.0018
±0.0026
±0.0026
±0.0025
±0.0017
±0.0051
±0.0016
±0.0016
±0.0014
speed
0-95
3
best
0.5463
0.2118
0.3989
0.3640
0.3600
0.2197
2.2846
0.2023
0.2130
0.1716
±0.0122
±0.0067
±0.0145
±0.0130
±0.0126
±0.0070
±0.8163
±0.0065
±0.0072
±0.0058
log time
0-95
3
random
0.1119
0.0373
0.0655
0.0656
0.0646
0.0411
0.1501
0.0376
0.0385
0.0320
±0.0026
±0.0012
±0.0021
±0.0021
±0.0021
±0.0014
±0.0131
±0.0014
±0.0013
±0.0011
normalized
0-95
3
random
0.1136
0.0397
0.0660
0.0686
0.0676
0.0438
0.1013
0.0395
0.0405
0.0338
±0.0029
±0.0016
±0.0021
±0.0025
±0.0023
±0.0016
±0.0055
±0.0016
±0.0015
±0.0013
speed
0-95
3
random
0.5727
0.1825
0.3795
0.3552
0.3497
0.2066
2.5636
0.1913
0.1995
0.1607
±0.0126
±0.0060
±0.0145
±0.0114
±0.0112
±0.0061
±0.7841
±0.0065
±0.0065
±0.0052
log time
0-95
4
best
0.1014
0.0514
0.0557
0.0551
0.0553
0.0406
0.0716
0.0350
0.0366
0.0310
±0.0024
±0.0016
±0.0017
±0.0020
±0.0020
±0.0013
±0.0051
±0.0013
±0.0013
±0.0010
log time
0-25
3
best
0.0424
0.0294
0.0559
0.0479
0.0507
0.0310
0.0970
0.0282
0.0300
0.0221
±0.0012
±0.0009
±0.0019
±0.0015
±0.0016
±0.0008
±0.0092
±0.0008
±0.0009
±0.0007
Table A: Exactly the same table as Table 1 but mean absolute errors reported.
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
0.1398
0.0637
0.1065
0.0100
0.0991
0.0639
0.3624
0.0574
0.0622
0.0545
±0.0066
±0.0057
±0.0054
±0.0063
±0.0063
±0.0066
±0.0849
±0.0060
±0.0054
±0.0050
normalized
0-95
3
best
0.1483
0.0792
0.1103
0.1051
0.1042
0.0724
0.1769
0.0658
0.0694
0.0620
±0.0097
±0.0104
±0.0068
±0.0066
±0.0068
±0.0097
±0.0222
±0.0096
±0.0078
±0.0086
speed
0-95
3
best
0.7153
0.3308
0.6553
0.5827
0.5772
0.3383
19.2009
0.3067
0.3410
0.2918
±0.0349
±0.0348
±0.0356
±0.0368
±0.0385
±0.0440
±9.9799
±0.0393
±0.0310
±0.0321
log time
0-95
3
random
0.1380
0.0544
0.0931
0.0931
0.0919
0.0591
0.4416
0.0561
0.0567
0.0471
±0.0032
±0.0027
±0.0035
±0.0039
±0.0038
±0.0027
±0.0435
±0.0031
±0.0027
±0.0023
normalized
0-95
3
random
0.1450
0.0623
0.0951
0.1011
0.0998
0.0682
0.2046
0.0634
0.0640
0.0538
±0.0044
±0.0037
±0.0038
±0.0049
±0.0049
±0.0039
±0.0124
±0.0041
±0.0038
±0.0033
speed
0-95
3
random
0.6935
0.2585
0.5917
0.5052
0.4979
0.2835
24.7206
0.2801
0.2863
0.2261
±0.0147
±0.0121
±0.0329
±0.0171
±0.0167
±0.0134
±10.7164
±0.0199
±0.0121
±0.0112
log time
0-95
4
best
0.1368
0.0763
0.0823
0.0859
0.0862
0.0620
0.2371
0.0608
0.0599
0.0531
±0.0075
±0.0060
±0.0042
±0.0060
±0.0059
±0.0038
±0.0423
±0.0064
±0.0041
±0.0040
log time
0-25
3
best
0.0539
0.0425
0.0810
0.0675
0.0710
0.0412
0.2479
0.0358
0.0417
0.0318
±0.0027
±0.0030
±0.0056
±0.0050
±0.0051
±0.0026
±0.0600
±0.0022
±0.0030
±0.0022
Table B: Prediction only from events which are earlier in time than the performance to be predicted. The table shows out-of-
sample RMSE for performance prediction methods on diﬀerent data setups. Predicted performance is of the 25 top percentiles of
male runners, in their best year. Standard errors are bootstrap estimates over 1000 repetitions. Legend is as in Table 1.
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
time
0-95
3
best
0.1295
0.0627
0.0959
0.0973
0.0964
0.0596
0.1785
0.0560
0.0569
0.0499
±0.0027
±0.0027
±0.0035
±0.0064
±0.0065
±0.0025
±0.0105
±0.0028
±0.0023
±0.0024
time
0-95
3
random
0.1357
0.0535
0.0874
0.0907
0.0895
0.0585
0.1961
0.0544
0.0550
0.0461
±0.0029
±0.0022
±0.0028
±0.0031
±0.0031
±0.0026
±0.0116
±0.0025
±0.0022
±0.0020
time
0-95
4
best
0.1232
0.0745
0.0750
0.0782
0.0785
0.0566
0.1167
0.0525
0.0522
0.0455
±0.0025
±0.0031
±0.0021
±0.0027
±0.0027
±0.0021
±0.0084
±0.0029
±0.0019
±0.0019
time
0-25
3
best
0.0559
0.0422
0.0760
0.0668
0.0704
0.0406
0.1579
0.0377
0.0402
0.0302
±0.0015
±0.0016
±0.0025
±0.0022
±0.0023
±0.0012
±0.0113
±0.0012
±0.0014
±0.0001
Table C: Exactly the same table as Table 1 but relative root mean squared errors reported in terms of time. Models are learnt on
the performances in log-time.
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
time
0-95
3
best
0.1057
0.0424
0.0669
0.0654
0.0647
0.0420
0.0876
0.0384
0.0397
0.0333
±0.0023
±0.0015
±0.0022
±0.0023
±0.0024
±0.0014
±0.0048
±0.0013
±0.0013
±0.0012
time
0-95
3
random
0.1116
0.0372
0.0635
0.0651
0.0642
0.0410
0.0980
0.0373
0.0381
0.0318
±0.0024
±0.0012
±0.0018
±0.0019
±0.0020
±0.0013
±0.0055
±0.0013
±0.0013
±0.0011
time
0-95
4
best
0.1006
0.0519
0.0547
0.0540
0.0543
0.0401
0.0605
0.0348
0.0362
0.0307
±0.0023
±0.0016
±0.0016
±0.0018
±0.0018
±0.0013
±0.0032
±0.0013
±0.0012
±0.0011
time
0-25
3
best
0.0425
0.0296
0.0542
0.0476
0.0504
0.0308
0.0688
0.0280
0.0297
0.0220
±0.0011
±0.0001
±0.0017
±0.0015
±0.0016
±0.0008
±0.0046
±0.0008
±0.0008
±0.0007
Table D: Exactly the same table as Table 1 but relative mean absolute errors reported in terms of time. Models are learnt on the
performances in log-time.
no events.
r1
r2
r3
r4
3
0.0411
0.0306
—
—
±0.0014
±0.0011
4
0.0446
0.0328
0.0309
—
±0.0016
±0.0013
±0.0012
5
0.0518
0.0408
0.0400
0.0408
±0.0032
±0.0033
±0.0034
±0.0036
Table E: Determination of the true rank of the model. Table displays out-of-sample RMSE for predicting performance with LMC
rank 1-4 (columns) Predicted performance is of the 25 top percentiles of male runners, in their best year, who have attempted at
least the number of events indicated by the row. The model is learnt on performances in log-time coordinates. Standard errors are
bootstrap estimates over 1000 repetitions. The entries where no. events ≥ rank are empty, as LMC rank r needs r + 1 attempted
events for leave-one-out-validation. Prediction with LMC rank 3 is always better or equally good compared to using a diﬀerent
rank, in terms of out-of-sample prediction accuracy.
subgroup
RMSE
Amateur
0.0305
±0.0002
Female
0.0305
±0.0003
Old
0.0326
±0.0003
Table F: Prediction in three diﬀerent subgroups: amateur runners, female runners, older runners. Table displays out-of-sample
RMSE for predicting performance with LMC rank 2.
rank
log time
speed
normalized
1
0.0410
0.0376
0.0399
±0.0014
±0.0011
±0.0013
2
0.0304
0.0315
0.0305
±0.0011
±0.0011
±0.0001
Table G: Eﬀect of performance measure in which the LMC model is learnt. The model is learnt on three diﬀerent measures of
performance: log-time, time normalized by event mean, speed (columns).
The table shows out-of-sample RMSE for predicting
log-time performance with LMC rank 1,2. Standard errors are bootstrap estimates over 1000 repetitions. Performance is of the 25
top percentiles of male runners, in their best year of performance.
percentiles
no.event
bagged LMC r2
bagged power-law
LMC r2
power-law
0-25
3
0.0310
0.0654
0.0308
0.0666
±0.0011
±0.0025
±0.0011
±0.0025
0-95
3
0.0529
0.0898
0.0512
0.0948
±0.0031
±0.0040
±0.0028
±0.0039
0-95
4
0.0480
0.0762
0.0467
0.0825
±0.0034
±0.0029
±0.0021
±0.0030
Table H: Comparison of prediction using all distances, to prediction using only closest distances. Table displayes out-of-sample
RMSE of predicting log-time, for (5.a) the bagged power law and (5.b) the bagged LMC rank 2 predictor, compared with the un-
bagged variants, (2.b) and (4.b). Predicted performance is of the 25 top percentiles of male runners, in their best year. Standard
errors are bootstrap estimates over 1000 repetitions. The results of the bagging predictors are very similar to the unbagged one.
variables
β
β2
β3
c
model 1
log s
0.0572 ± 0.0003
−0.136 ± 0.003
model 2
log s, f2
0.0547 ± 0.0007
−0.017 ± 0.004
−0.115 ± 0.006
model 3
log s, f2, f3
0.0554 ± 0.0007
−0.013 ± 0.004
0.002 ± 0.001
−0.120 ± 0.006
t1
p(X > |t1|)
t2
p(X > |t2|)
t3
p(X > |t3|)
tc
p(X > |tc|)
model 1
168
1.7e-15
-51
2.3e-11
model 2
81
1.1e-12
-3.9
5.9e-3
-21
1.5e-7
model 3
80
2.5e-10
-3.0
2.5e-2
1.8
0.13
-21
7.1e-7
F
P (X > F )
RSE
R-squared
model 1
2.8e+4
1.7e-15
0.0020
0.9997
model 2
3.9e+4
6.6e-15
0.0012
0.9999
model 3
3.4e+4
4.4e-13
0.0011
0.9999
Table I: Explaining the ﬁrst singular component, v. The following explanatory linear models are ﬁtted: v explained by β log s + c
(model 1); v explained by β log s + β2f2 + c (model 2); v explained by β log s + β2f2 + β3f3 + c. The β, β2, β3 are the estimated
coeﬃcients, ± one standard error. t1, t2, t3 are the t-statistics of β, β2, β3; tc is the t-statistic of c. The F-statistic of the respective
model is F , RSE is the residual standard error.

Prediction and Quantification of Individual Athletic Performance of Runners.
Blythe, Duncan A J,Király, Franz J