Learning curves of elite car racers

Abstract

This work is focused on racing cars driver’s training. Nine different tracks are considered and six drivers. Each driver drives on every track and performs consecutive trial sessions on each track; each session is made of various laps, and lap times are fitted using an exponential model, yielding an estimate of the initial performance, the learning constant, and the asymptotic performance. According to results, the learning curve varies significantly among pilots and among tracks; all pilots reach their session asymptotic performance in less than nine laps. The asymptotic performance in consecutive trial sessions improves significantly, and it is strongly correlated to the initial session performance (r²> 0.99). As a conclusion, it is more profitable to perform separated sessions made of few laps (less than 10) rather than performing a smaller number of longer sessions. Whenever the initial lap time stops decreasing systematically, trial sessions should end because the asymptotic performance is not likely to improve further.

Keywords

Motor racing performance analysis racing car drivers sport analytics

Introduction

This article is focused on elite car drivers training and it aims to gain some new insight into this sport, considering the paucity of data reported in literature,¹ where this specific type of training has not yet been examined.

As well known, each learning process is highly specific for each individual and this aspect has attracted the attention of many researchers since the work by Ackerman et al.² Considering car racing, for example, the learning process is expected to vary according to driver’s ability, technical aspects concerning the car, and the track complexity. The performance improvement produced by repetition over time is usually described through a learning curve. Its most typical pattern shows a rapid improvement at the beginning and slower improvements in the following trials, as the learning curve gets closer to a plateau. Three main parameters are of interest in describing the learning curve³: the initial performance (‘starting point’), the asymptotic performance, and how quickly the performance improves (inversely proportional to ‘the learning constant’). The fitting of a learning curve allows the estimation of these parameters, comparisons between different subjects learning abilities^4–7 and between training protocols to be performed.^5,8,9 The ability to identify where an individual is on a learning curve can bring a significant benefit to training protocols since that can establish if and to what extent further training can be beneficial and if training should be more conveniently split into various sessions, interspaced by resting times.⁹

This study concerning elite car racers aims to understand if and to what extent learning parameters are significantly affected by varying factors such as the driver and the training track,⁶ and if there is a correlation among the learning constant, the asymptotic performance, and the initial performance.⁴ The final aim is to optimize training programmes, establishing, for example, the ideal duration of trial sessions in relation to each pilot’s learning constant, and to track complexity.

Method

Tests description

This study is an observational study on previously acquired data, during trial sessions preceding ‘formula Renault 2.0’ competitions.

Six adult male car racers have been involved in this study aged 17–24. They had been running for at least one year in Formula Renault 2.0 and for three years overall, including previous karting experience. They all had prior experiences on the analysed tracks (gained years before) but they had not been racing on each specific track for at least six months. All racers were asked to read and sign an informed consent sheet.

The racers were actually running on ‘formula Renault 2.0’ open-wheelers cars. Each pilot p performed 4–8 trial sessions s on each track t. The interval between consecutive trial sessions (s_i,s_i+1) was equal to 1 h. During each session, only one pilot was running on the track. Nine different tracks have been considered (Figure 1), with different lengths and difficulty levels. Each of the six drivers drove on each of the nine tracks.

Figure 1.

Shape of nine racing tracks (a) to (i).

Trial sessions where the pilot experienced mechanical problems have been discarded.

Learning model

The learning process was modelled by an exponential decay function.¹⁰ The most common curves used to fit the learning curve are the exponential, the hyperbolic, and the power functions.⁴ During first trials the exponential model proved to give a good fit (R²> 0.90). In fact, the lap time decay usually showed a relative rapid rate at the beginning until reaching a phase in which much lower or even no further improvement has been observed, and a plateau has been reached, unless other processes like fatigue take place.

Considering one session s of pilot p on track t, the performance at lap l p_l_,s_(p,t) can be calculated from three main parameters³: the initial performance value p_i_,s_(p,t), the asymptotic value p_a_,s_(p,t), and the learning constant λ_s_(p,t) (whose unit is the number of laps), according to the following formula

p_{l, s (p, t)} = p_{a, s (p, t)} + (p_{i, s (p, t)} - p_{a, s (p, t)}) e^{\frac{l}{λ_{s (p, t)}}}

(1)

Learning curves have been fitted by Matlab software, with the implementation of trust-region algorithm for nonlinear least squares minimization.¹¹ Each session lasted at least 10 cycles.

The coefficient of determination R² was used to check how well the three learning parameters captured the learning process.

Statistical analyses of data

All fitting curve parameters were collected and sorted by track, pilot, and session. Three vectors were obtained for each track/pilot couple: the initial performance vector P_i_(t,p) collected first lap times ordered session by session, the components of the asymptotic performance vector P_a_(t,p) are the asymptotic lap times, the third vector Λ_(t,p) included learning constants.

The retention ability of each pilot on a given circuit was tested comparing the asymptotic performance of a given session (p_a_,s_(t,p)) to the initial performance of the immediately following session (p_i,s+1_(t,p)).

A further statistical inquiry concerned the significance of trends measured on consecutive trial sessions to establish if, for a given track and pilot couple, the first lap time or the asymptotic performance was going to improve systematically, session by session. An interpolating line was fitted to vectors P_i(t,p) and P_a(t,p), and trends significance has been quantified by means of a Page test.¹² A nonparametric test has been chosen because the existence of a monotonic trend was being tested without specific hypotheses on the trend itself (linear, exponential, etc.) and on the error distribution. Page test has been chosen because it was suitable to be applied on repeated measures. A significant (p < 0.05) negative trend of the initial performance or of the asymptotic performance suggested the pilot was taking benefit from his training.

The variability of learning curve parameters among various drivers and tracks has been tested through an extensive multivariate analysis of variance (MANOVA). The response variables are the initial performance of the first trial session, the learning constant, and the asymptotic performances of the last trial session. MANOVA was executed on all response variables simultaneously, using Pillai’s trace as statistics since this is thought to be the most robust estimator.¹³ The independent variables are the pilot (six levels) and the track (nine levels).

The authors tried also to establish if better asymptotic times were systematically related to better initial performances or faster/lower learning rates: therefore they calculated the correlation between fitting curves parameters by Pearson correlation coefficient.^14,15 Vectors P_i(t,p), P_a(t,p), Λ_(t,p) coming from all (t,p) couples were concatenated in order to perform this analysis, obtaining vectors P_i, P_a, Λ, respectively.

Results

The correlation coefficients of fitting learning curves remained almost constant across all trials sessions, drivers, and tracks, keeping always above 90% with few exceptions (five out of 86 sessions), where very inconsistent patterns were recorded: the respective data have been discarded in the following.

The average pilot performances have been calculated in order to get some clues on differences among tracks. More in detail, Figure 2 reports the patterns of the average initial performance for the first session p_i,1(t) and of the average asymptotic performance for the last session p_a,Last(t). Given a certain track, the variations among pilots ranged from 0.28 to 16.19% for the initial performance and from 0.33 to 6.16% of the mean value for the asymptotic performance.

Figure 2.

Average pilot performances versus track.

The initial performance

The retention ability of each pilot was initially assessed comparing the first lap time of a given session (s > 1) with the asymptotic performance of the preceding session (s − 1). This comparison produced a negative evaluation in most cases (96,5%), demonstrating that pilots, when starting a new session, needed to perform some laps before recovering the ability gained in the previous session. A further analysis was performed looking at first lap times vector P_i(t,p). The slope of the interpolating trend is negative for most pilot/track couples (with some exceptions on track f); however, according to a Page test,¹² this trend is not significant. There are two explanations. First, given a track/pilot couple, the sample size that is the number of trial sessions is usually small (ten at maximum) so it is hard for a statistics to be significant. Second, the initial lap time decrement is not consistent among all session data (Figure 3).

Figure 3.

Lap time plots for five consecutive trial sessions; circled points mark the initial performance.

The asymptotic performance

The asymptotic performance improved session by session. The significance of this trend has been statistically proven by means of a Page test,¹² and it is higher than 0.99. This result is interesting because it demonstrates the usefulness of performing many different sessions or, in other words, of having resting times between trials. This allows the pilot to have some rest and to reorganize his ideas in order to gain a better asymptotic performance in the following session. On the contrary, there is no benefit in prolonging a given session beyond 10 laps. In fact, the session asymptotic performance has been certainly reached at this lap number, as demonstrated in the following paragraph.

The learning curve

MANOVA has been performed in order to assess if the learning curves were significantly different among pilots and tracks. Results are reported in Table 1: both the track and the pilot do have a significant influence on the learning curve (p < 2 × 10⁻⁴).

Table 1.

MANOVA results concerning learning curve.

Source	Pillai’s trace	df	df₁	df₂	F	Prob >F
Model	1.50	13	39	120	1.50	<2.0 × 10⁻⁶
Pilot	0.88	5	15	120	1.75	<2.0 × 10⁻⁴
Track	1.09	8	24	120	2.88	<2.0 × 10⁻⁴
Error		40
Total		53

MANOVA: multivariate analysis of variance.

With specific reference to the learning constant, 80% of measured values range from 0.33 and 1.97 (Figure 4): this means that, given, a certain trial session, pilots usually achieve their asymptotic performance in 2–9 laps (this would result in l/λ ratio above 4.6 in equation (1), and the respective exponential function would reach 0.99, meaning that the training has produced 99% of its benefit). The tracks where the learning constant was highest are tracks (2) and (7) in Figure 1, while the most demanding tracks in terms of learning constants are tracks (f) and (h). Figure 5 reports the experimental data and the interpolating model.

Figure 4.

Cumulative frequency of learning constants.

Figure 5.

(a) Learning constant versus track and pilot, (b) experimentally measured learning constant versus pilot for track reported in Figure 1(a), (c) experimentally measured learning constant versus track for pilot #1, and (d) details about the interpolating linear model.

Correlations

The analysis of correlations has been reported in Table 2; the asymptotic performance does not depend on the learning constant, while it is strongly related to the initial performance. The data suggest that time laps cannot be improved more than a fixed amount, session by session.

Table 2.

Pearson coefficients of correlation.

	λ (lap)	P_i	P_a
λ (lap)	1	0.123	0.111
P_i	0.123	1	0.996
P_a	0.111	0.996	1

Discussion

Learning curves in racing have not been analysed systematically in literature; however, there is a significant amount of data regarding trial sessions in car racings, and their analysis can provide some clues on learning processes in general, and the best training strategy for this particular task. The main drawback of data coming from actual trial sessions is that they come from drivers having different experience, and they lack consistency. The number of loops per trial may vary as well as the delay between trial sessions is not the exact same. This shortcoming is partially compensated by the large quantity of data.

The exponential function proved to be able to fit the data excellently. It has been chosen because it is based on three parameters whose psychological meaning is straightforward: the initial performance, the time constant of learning, and the maximum achievable performance. Other models have been tested (power model and rational model). The rational model proved to be more or less equivalent with reference to goodness of fit, but its parameters were harder to be interpreted, and underwent larger variations, even within the same subject/track couple. The exponential function reproduces one typical aspect of learning, that is the performance increment gets smaller as the number of performed trials increases.^16–18

Participants started at different levels, learned at different rates, and achieved different asymptotic performance. This observation agrees with ﬁndings from other studies which found reliable individual differences, though having considered different tasks and having used different learning models.^4,9,19

This study emphasizes the importance of spacing practice intervals in order to reach a higher asymptotic performance; this result agrees with findings from other authors²⁰ who demonstrated that pauses are essential in order to gain the maximum benefit from practice. They hypothesized that delay may allow for consolidation of learning, possibly reflecting plastic changes in motor cortical representations of the skill.^20,21 A pattern similar to the one reported in Figure 3 has been observed in learning curves regarding other, very different tasks⁹: session by session there is a discontinuity consisting in a sharp deterioration of performance; this discontinuity is quickly recovered, after a few trials, and a new, better asymptotic performance is reached. Another relevant aspect is that if the learning of knowledge and skill acquisition is done over more days, there are more sleep opportunities, and these are crucial to place knowledge or skill into long-term memory.^22,23

Authors who analysed much simpler tasks found a much better retention ability since the first performance of each session was more or less equal to the best performance of the preceding session.⁹ However, these same authors emphasized that different components of a motor skill are learned and retained in different ways. More explicit components of a motor task, such as accuracy are better retained, while the more purely motoric components of the task are likely to require ongoing practice to be maintained since they rely on complex sensorimotor integration capabilities.

The statistically signiﬁcant covariances between the initial session performance and the asymptotic performance indicate that those pilots who performed well since the beginning of the trial session also had a higher asymptotic learning performance. This datum can be useful in order to foresee the asymptotic pilots ranking on a given track. Considering that all pilots had had similar previous experiences, the performance in the first lap of the first trial session is likely to be mainly related to factors such as the pilot driving ability and the car set-up.

Learning constants and asymptotic performances are not correlated. Drivers having shown shorter learning constants do not necessarily result in better performances or vice versa. This is due to different learning abilities being tied to different cortical and subcortical regions.⁹

The learning curve changes significantly among tracks and among drivers, demonstrating each circuit requires specific abilities which are different among pilots.

Practical applications

This study emphasizes the importance of spacing practice intervals in order to reach a higher asymptotic performance. Given a certain trial session, pilots usually achieve their asymptotic performance in 2–9 laps; further improvement requires taking a break.

Beginning a new trial session is worthy as long as the initial lap time of each session keeps decreasing, session by session.

Driver learning on a given track can be characterized by its specific learning curve. The knowledge of the learning constant allows performing an estimation of the optimum number of laps per session while is not indicative of the final driver performance.

The correlation between trial sessions performance and racing performance is not straightforward. During racing more complex phenomena take place due to presence of other cars on the track, tyres wear, greater stress, etc.

This study has considered and analysed the only lap time as an output variable, while other measures both physical (steering wheel angle, butterfly valve opening, etc.) and related to driver’s physiology (heart rate, body temperature, etc.), could give further insight into his behaviour.^1,6

Conclusion

This work has been focused on the study of learning process of elite car racers in trial sessions.

Exponential curves have been used to interpolate lap time, session by session. This model has been proved to be adequate and to provide significant parameters that are the learning constant, the initial performance, and the asymptotic performance.

Each track/driver couple is characterized by a specific learning curve; however, the learning rate is not correlated to the driver’s best performance.

Trial sessions should not last more than 10 laps since the performance is not likely to be further improved on longer sessions.

The asymptotic performance improves significantly on consecutive sessions as long as the driver is able to reduce the initial lap time, improving his retention ability.

To the best of the authors’ knowledge, learning of this complex task has not been analysed in detail previously; therefore, the article has produced some new ideas.

Footnotes

Declaration of Conflicting Interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) received no financial support for the research, authorship, and/or publication of this article.

References

Potkanowicz

Mendel

. The case for driver science in motorsport: a review and recommendations. Sport Med 2013; 43: 565–574.

Ackerman

Sternberg

Glaser

. Learning and individual differences: advances in theory and research, New York: W.H. Freeman, 1989.

Meredith

Tisak

. Latent curve analysis. Psychometrika 1990; 55: 107–122.

Rast

Zimprich

. Individual differences and reliability of paired associates learning in younger and older adults. Psychol Aging 2009; 24: 1001–1006.

Feldman

Cao

Andalib

. A method to characterize the learning curve for performance of a fundamental laparoscopic simulator task: defining ‘learning plateau’ and ‘learning rate’. Surgery 2009; 146: 381–386.

Ronen

Yair

. The adaptation period to a driving simulator. Transp Res Part F Traffic Psychol Behav 2013; 18: 94–106.

Sahami

Sayed

. How drivers adapt to drive in driving simulator, and what is the impact of practice scenario on the research? Transp Res Part F Traffic Psychol Behav 2013; 16: 41–52.

De Groot

Centeno Ricote

De Winter

JCF

. The effect of tire grip on learning driving skill and driving style: a driving simulator study. Transp Res Part F Traffic Psychol Behav 2013; 15: 413–426.

Savion-Lemieux

Penhune

. The effects of practice and delay on motor skill learning and retention. Exp Brain Res 2005; 161: 423–431.

10.

Heathcote

Brown

Mewhort

. The power law repealed: the case for an exponential law of practice. Psychon Bull Rev 2000; 7: 185–207.

11.

Coleman TF and Li Y. An interior trust region approach for nonlinear minimization subject to bounds. SIAM J Optim 1996; 6: 418–445.

12.

Page EB. Ordered hypotheses for multiple treatments: a significance test for linear ranks. JASA 1963; 58: 216–230.

13.

Rencher AC and Christensen WF. Methods of multivariate analysis. Wiley, 2012.

14.

Kuzma JW and Bohnenblust SE. Basic statistics for the health sciences. Hoboken, New Jersey: McGraw-Hill, 2005.

15.

Taylor

. Interpretation of the correlation coefficient: a basic review. J Diagn Med Sonogr 1990; 6: 35–39.

16.

Karni

Meyer

Rey-Hipolito

. The acquisition of skilled motor performance: fast and slow experience-driven changes in primary motor cortex. Proc Natl Acad Sci USA 1998; 95: 861–868.

17.

Korman

Raz

Flash

. Multiple shifts in the representation of a motor sequence during the acquisition of skilled performance. Proc Natl Acad Sci USA 2003; 100: 12492–12497.

18.

Schmidt RA and Lee TD. Motor control and learning: a behavioral emphasis. Champaign, IL: Human Kinetics, 2011.

19.

Zhang Z, Davis HP, Salthouse TA, et al. Correlates of individual, and age-related, differences in short-term learning. Learn Individ Differ 2007; 17: 231–240.

20.

Bourne

Archer

. Time continuously on target as a function of distribution of practice. J Exp Psychol 1956; 51: 25–33.

21.

Shea

Lai

Black

. Spacing practice sessions across days benefits the learning of motor skills. Hum Mov Sci 2000; 19: 737–760.

22.

Rasch

Born

. About sleep’s role in memory. Physiol Rev 2013; 93: 681–766.

23.

Walker

Stickgold

. It’s practice, with sleep, that makes perfect: implications of sleep-dependent learning and plasticity for skill performance. Clin Sports Med 2005; 24: 301–317.