Keywords

1 Introduction

Running a marathon is hard. It depends on months of dedicated training and the discipline to follow a time-consuming and exhausting schedule. To complete a successful marathon a runner also needs an appropriate race-plan for the event itself. The runner must select a suitable target finish-time and carefully plan the pacing of their race, stage by stage, kilometer by kilometer. The aim of this work is to help marathon runners to achieve a new personal best during their next race. We do this by using case-based reasoning to solve these dual problems by: (1) predicting a best achievable finish-time; and (2) recommending a tailored pacing plan to help the runner achieve this time on race-day.

This work sits at the intersection between personal sensing, machine learning, and connected health. An explosion of wearable sensors and mobile devices has created a tsunami of personal data [1], and the promise that it can be harnessed to help people to make better decisions and live healthier and more productive lives [2, 3]. Indeed, within the case-based reasoning community there has been a long history of applying case-based, data-driven methods to a wide range of healthcare problems [4]. Recently, the world of sports and fitness has similarly embraced this data-centric vision, as teams and athletes embrace the power of data to optimise the business of sports and the training of athletes [5, 6].

In this work we focus on recreational marathon running, helping runners to plan their race strategy for optimal performance. A key concept is that of pace, measured in minutes per mile/km, so that higher pace means slower speed. There is a growing body of research that explores the various factors that influence pacing during the marathon and other endurance events. For example, the work of [7] looks at the effect of age, gender, and ability on marathon pacing to conclude that female runners are typically more disciplined (running with less pace variation) than male runners; see also [8, 9]. Then there is the runner’s pacing strategy: how the runner plans to adjust their pace during the race [10]. There are 3 basic pacing strategies, based on how the pace of a runner varies between the first and second-half of a race. For example, we say that a runner completes an even-split if their pace is even throughout the race. Running a positive-split means the second-half of the race is slower (higher pace) than the first-half of the race, whereas a negative-split means the runner speeds-up in the second-half of the race, running it faster than the first-half. Many elites and disciplined runners will aim for even or slightly negative-splits [11]. Recreational runners typically run positive-splits, slowing during the second-half of the race, sometimes significantly. And for some this means hitting the wall, referring to the sudden onset of debilitating fatigue and a near-complete loss of energy that can occur late in a race [12]. At best, this temporarily slows even the swiftest of runners; at worst it reduces them to a shuffling gait for the rest of the race.

All this is to say that running the marathon is a challenge, and the difference between a good day and a terrible day, training aside, may well come down to how carefully a runner plans their race: what finish-time to aim for; positive vs. negative vs. even splits; avoiding hitting the wall etc. This is where we believe there is a significant opportunity to support marathon runners, by advising them on a suitable a target finish-time and providing them with a concrete race-plan, one that is personalized to their ability and tailored to the marathon course.

2 Problem Definition

The problem addressed by this work is how to help a marathon runner achieve a new personal best in their next race. We will assume we are dealing with a runner who has completed at least one previous marathon and so has a race record to serve as a starting point. As a reminder, this problem involves two related tasks: the prediction of a suitable target finish-time and the recommendation of an appropriate pacing plan.

2.1 Best Achievable Finish-Time

We start by assuming our runner wants to beat their current best-time, but by how much? If they are too conservative they will chose a finish-time that does not fully test them, and may leave them disappointed if they finish too comfortably on race-day. If they are too ambitious they may select finish-time that is beyond their ability and risk sabotaging their race; aiming for an overly ambitious target time is one sure way to end up hitting the wall later in the race. The point is that selecting a best achievable finish-time is non-trivial and getting it wrong can have a disastrous effect on race-day.

2.2 Race Plans and Pacing Profiles

Given a best achievable finish-time, the next task is to devise a suitable race-plan. We will assume the marathon is divided up into \(8 \times 5\) km stages or segments (0–5 km, 5 km–10 km, ..., 35 km–40 km), plus a final 2.2 km segment (40 km–42.2 km); many big-city marathons measure times in 5 km segments and so race records typically provide these split-times. For the purpose of this work a race-plan, or pacing plan, will consist of a sequence of average paces (measured in kms per minute) for each of these race segments. For the avoidance of doubt we will refer to a pacing plan as a proposed set of paces for the runner and a pacing profile as the actual set of paces achieved by the runner in their race. For example, Fig. 1(a) illustrates a race-record for a runner who completed a marathon in 4 h and 13 min; that is an average pace of 6 mins/km. The corresponding pacing profile shows they ran a positive-split, starting their race 10% faster than their average pace during the first 5K and 10K segments of the race, and slowing in the second-half, to finish 7% slower than their average pace in the final segment.

Most conventional pacing plans are fairly simple by encouraging the runner to run an even-split. For example, if our runner wishes to run a 4-hour (240 min) marathon then this suggests an average pace of 5 min and 41 s per km for the duration of the race. Some plans may account for positive or negative splits, by advising the runner to run the first half of the race more quickly or slowly, respectively. However, in general, conventional plans are not especially tailored for the runner or the course, and so they leave considerable room for improvement, especially when it comes to achieving a new personal best. Thus, in this work we argue for a more personalized and customized approach to pacing, so that runners can benefit from a pacing plan that is suitable for their goal time, their personal fitness level and ability, and that is tailored for the peculiarities of a given marathon course.

3 Using CBR to Achieve a Personal Best in the Marathon

The key insight of this work is that we can use a CBR approach to both predict suitable target finish-times and recommend tailored pacing plans, based on the runner’s own race experience and the experiences of similar runners who have achieved personal bests on the target course. To do this we will rely on a case base of race pairs, representing a pair of race records for a single runner. Each race record contains a pacing profile and a finish-time for a completed race; see Fig. 1(b). One of these race records corresponds to a non personal best (nPB) race, the other to a personal best (PB) race; the nPB plays the part of the case description while the PB is the case solution. Given a target/query runner (q), and their own recent race record (finish-time and pacing profile), we generate a finish-time prediction and pacing plan recommendation based on the PB’s of cases that have a similar nPB to q, as summarised in Fig. 1(c).

Fig. 1.
figure 1

Races, cases, predictions, and recommendations. (a) An example race record for a runner, showing a finish-time and a pacing profile containing pacing data for each of the 5 km race segments; (b) Converting race records into cases; (c) An overview of the CBR process: given an nPB race record as a query, the system retrieves a set of k cases with similar nPB parts, and combines these to generate a personal best finish-time prediction and a pacing plan to achieve this finish-time.

Full size image

3.1 Case Generation

Each case in the case base corresponds to a single runner, r, with a nPB part and a PB part; see Eq. 1. To be represented in the case base, r must have at least two race records, and in general may have \(n>2\) race records if they have run many races; for example, in Fig. 1(b) we highlight 3 race records for r (for marathons \(m_1, m_2, m_3\)).

$$\begin{aligned} c_{ij}(r, m_i, m_j)=\Big \langle nPB_i(r, m_i), PB(r, m_j) \Big \rangle \end{aligned}$$
(1)

The race record with the fastest finish-time is considered to be the personal best, and it is paired with the remaining \(n-1\) non personal best records, producing \(n-1\) cases. As per Fig. 1(b), r’s best race is \(m_2\), with a finish-time of 236 min. This is paired with the two nPB records (\(m_1\) and \(m_3\)) to produce two cases, \(c(r, m_1, m_2)\) and \(c(r, m_3, m_2)\) as shown.

3.2 Case Retrieval

Retrieval is a three-step process, as shown in Algorithm 1. Given a query race record (q) — that is a runner, a finish-time, and a nPB pacing profile — we first filter the available cases (CB) based on their finish-times, so that we only consider cases for retrieval if their finish-times are within t minutes of the query finish-time. This ensures that we are basing our reasoning on a set of cases that are somewhat comparable in terms of performance and ability.

figure a

Next, we filter on the basis of gender, only considering cases for retrieval if they have the same gender as the query runner. The reason for this is that physiological differences between men and women have a material impact on marathon performance. On average women tend to finish about 12% slower than men [7], and thus men and women with the same approximate finish-times will be racing at very different levels of maximum effort; in relative terms, a 3-hour female finisher will be performing at a higher level of effort than a 3-hour male finisher, for example. Hence we separate male and female cases for more effective matching, during retrieval.

Finally, we perform a standard, distance-weighted kNN retrieval over the remaining candidate cases C, comparing q’s pacing profile to their nPB profiles. These pacing profiles are real-valued vectors and, for now, we use a simple Euclidean-based similarity metric for similarity assessment. We select the top k most similar as the retrieved cases, R.

3.3 Personal Best Finish-Time Prediction

Given a set of similar cases, R, we need to estimate the best achievable finish-time for q. Each case in R represents another runner with a similar nPB to q, but who has gone on to achieve a faster personal best on the same marathon course. The intuition is that since these PBs were achievable by these similar runners, then a similar PB should be achievable by the query runner.

For the purpose of this work we test three straightforward prediction approaches in which the predicted PB time of the query runner is a simple function of the personal best times of the retrieved cases. With each approach the predicted PB finish-times are weighted based on the relative difference between the query runner’s finish-time and the corresponding nPB finish-time of a retrieved case; see Eq. 2. This adapts the PB times based on whether they were achieved by similar runners with slightly slower or faster nPB times.

$$\begin{aligned} w(q,c) = \frac{q(nPB).finish}{c(nPB).finish} \end{aligned}$$
(2)

Best PB . In Eq. 3 the predicted finish-time is the weighted PB finish-time of the single fastest retrieved case, \(C_{best}\); the case with the minimum PB time. For example, if the query runner has a finish-time of 245 min, and the nPB time for the best retrieved case is a slightly faster 240 min, then the weighting factor will be 1.02 (245/240). If the PB time of this retrieved case is 232 min then the predicted PB time for q will be just over 236 min.

$$\begin{aligned} PB_{best}(q, C) = w(q, C_{best}) \bullet Time(C_{best}(PB)) \end{aligned}$$
(3)

Mean PB . Our second prediction approach calculates the weighted mean of the PB finish-times of the retrieved cases, as in Eq. 4. This will obviously tend to provide a slower predicted time than the Best PB approach above, but it will be more representative of the personal bests achieved by the similar runners.

$$\begin{aligned} PB_{mean}(q, C) = \frac{\sum _{\forall i\epsilon 1 \ldots k} w(q, C_i) \bullet Time(C_i(PB))}{k} \end{aligned}$$
(4)

Even PB . Finally, our third approach (Even PB) is based on the idea that more consistent or even pacing is better than more varied or erratic pacing. By measuring the coefficient of variation (CoV) of the relative paces in the PB pacing profile for each retrieved case, we can estimate their pacing variation, and select the one (\(C_{even}\)) with the lowest CoV value. The PB time of this \(C_{even}\) case is used as the predicted time for q.

$$\begin{aligned} PB_{even}(q, C) = w(q, C_{even}) \bullet Time(C_{even}(PB)) \end{aligned}$$
(5)

3.4 Pacing Recommendation

Given a predicted personal best finish-time it is straightforward to turn this into an average pace (mins/km) for q to run her race. However, as discussed previously, such an approach is unlikely to be successful, due to a variety of factors including ability, discipline, and course conditions. Therefore, in this section we describe how to use CBR to fulfill the requirement for a more personalized and tailored pacing plan with which to achieve this new personal best finish-time. In fact, a key advantage of CBR in this regard, compared to a more global, eager learning approaches, is precisely that it retains access to the local cases and their pacing profiles. It means that we can use the PB profiles of retrieved cases as the basis for a pacing plan for q and, in what follows, we describe 3 different approaches as companions to our 3 prediction approaches.

Best Profile. The first recommendation approach (Best Profile) reuses the PB pacing profile of the case with the best PB time, \(C_{best}\). In other words, we take the relative pacing from \(C_{best}\) and map its relative paces to the average pace for predicted PB time for the query runner. For example, if the predicted PB time is for 232 min, indicating an average pace of 5 min 30 s per km, and the PB profile in \(C_{best}\) calls for a first 5 km that is 5% faster than average pace, then the generated pacing profile for q will advise running the first 5 Km at just over 5 min and 13 s per km.

Mean Profile. The second approach (Mean Profile) generates a new pacing plan based on the mean relative segment paces of the PB profiles from the k retrieved cases. So, if, on average, the PB profiles of the retrieved cases have their runners starting 6% faster than their average pace then our Mean Profile plan for a 232-minute finish will suggest that q begins her race at 5 min and 10 s per km.

Even Profile. Finally, we can generate our pacing plan based on the PB profile of \(C_{even}\), the case with the most even pacing profile. If its profile calls for the first 5 km to be run at a pace that is just 2% faster than average race pace then the pacing profile for q will suggest a pace of 5 min and 23 s per km.

3.5 A Worked Example

Up to now we have described an approach to generating cases from pairs of race records. We have shown how these cases can be reused (according to three different strategies, Best, Mean, Even) as the basis for finish-time predictions and pacing recommendations, to help q achieve a new personal best in their marathon. By design we have kept our case representation, retrieval, and reuse strategies straightforward, to make it easy to embed this approach in a practical application for runners.

A summary example of such an application is depicted in Fig. 1(c) in which a (query) runner submits a (nPB) race record for a 248-minute finish-time with a pace profile that indicates a strong positive-split; they started out fast and completed the first-half of the race quite a bit faster than the second-half of the race. In terms of providing this information to the CBR system, it may be feasible for the runner to provide their race number and for the appropriate record to be pulled from the public race records in real-time. Large marathons such as London or Boston could readily accommodate this by integrating the service as part of their site offering.

In any event, having submitted a race record, the CBR system retrieves a set of k cases, \(c_1 \ldots c_k\) and, using one of the strategies above, generates a predicted PB time and suitable race plan for the runner. In this example the predicted PB time indicates to the runner that she should be able to achieve a personal best finish-time of 237 min, a fairly significant 11-minute improvement over her previous time. The recommended pacing plan suggests running the race with a slight positive-split, starting out a few percent faster than their target average pace of 5 min and 37 s per km, while running a slightly slower second-half of the race; it is a much more evenly paced race plan than their nPB race record. Importantly, this pacing plan provides our runner with a set of concrete paces for each of the race segments based on how other runners (with similar nPB records to the query runner) have run to a PB on the same course; in short, it provides the runner with a more personalized and tailored race plan than might otherwise be available.

4 Evaluation

To evaluate our approach we will focus separately on the tasks of finish-time prediction and profile recommendation using data from the last six years of the London Marathon (2011–2016).

4.1 Dataset

Briefly, our London dataset includes 215,575 race records for 185,143 unique runners. Each race record contains, among other features, the 5 km split-times and finish-times that we need as the basis for our case base. 37% of runners are female and just over 20% (or 37,704 unique runners) appear more than once in the dataset. This means we can construct at least 37,704 cases. However, since many of these repeat runners have only run 2 races this does limit the variation available for the identification of personal best times and profiles. Consequently, in this study we limit our data to runners who have run 3 or more races; this way we can choose a personal best race from at least 3 races and every runner will be represented by at least 2 cases. There are 5,390 such runners and on average they have run 3.4 races each, leading to a case base of 12,968 individual cases with 5,390 personal bests.

4.2 Methodology

For the purpose of this evaluation we use a standard 10-fold cross-validation methodology to test prediction accuracy and recommendation quality. Briefly, we randomly hold-out 10% of cases to act as a test-set and use the remaining 90% of cases as the basis for prediction and recommendation, repeating this 10 times and averaging the results. Thus, the nPB part of each test case is used as a query and the PB part is held back to evaluate the finish-time prediction and pacing plan recommendation.

Evaluation Metrics. To evaluate prediction accuracy we calculate the average percentage error between the predicted personal best time and the actual personal best time held back from the test case. To evaluate the quality of the recommended pacing plan we estimate the similarity between this recommended plan and actual pacing profile that was also held back; our measure of profile similarity is based on the relative difference between the segment paces of the recommended plan and the segment paces of the actual pace profile.

Test Algorithms. For each prediction/recommendation session we generate predictions/recommendations using the 3 CBR approaches (Best, Mean, Even) as described earlier. While in theory we could possibly mix-and-match between prediction and recommendation strategies, for the purpose of this study, we use a like-with-like approach and pair each prediction strategy with the corresponding recommendation strategy. Thus when we refer to the Best strategy, for example, we mean the Best prediction strategy combined with the Best recommendation strategy.

4.3 Prediction Error and Profile Similarity vs. k

To begin with we will look at prediction accuracy and recommendation quality versus k, the number of cases retrieved; see Figs. 2 (a) and (b). Both graphs contain ribbon-plots for each of the 3 test algorithms. Each ribbon-plot shows the prediction error (or profile similarity) for all runners, as the marked central lines, and separately for male and female runners, as the boundary lines. In this way we can get a sense of how gender influences prediction error and pacing profile similarity, as k varies.

Fig. 2.
figure 2

Prediction error (a) and pacing profile similarity (b) versus k for Best, Mean, and Even strategies, for all runners and men and women.

Full size image

With respect to prediction error, we can see how our strategies behave quite differently for increasing k. Mean produces the lowest errors and benefits from increasing values of k, before stabilising for \(k\ge 10\); Mean achieves an error of \(\approx \)4.5%, for all runners, and as low as 4% for women. In contrast, Even produces predictions with an average error of \(\approx \)6% regardless of k, while the accuracy of Best deteriorates steadily with increasing k. The problem for Best is that more retrieved cases generally lead to faster best finish-times to use as a prediction. So Best tends to predict more and more ambitious personal best times as k increases, times that have not been achieved by our test runners.

When it comes to pacing profile similarity — our proxy for recommendation quality, and with higher similarity equating to better quality — we see a similar, albeit inverted pattern in Fig. 2(b). The Mean strategy out-performs the others, recommending pacing plans that are increasingly similar to actual PB pacing profiles (before stabilising at \(k\approx 5\)). The profile similarity of Even recommendations is largely unaffected by k, while the Best recommendations become less and less similar to the actual PB pacing profiles as k increases; once again more cases mean faster best \(C_{Best}\)’s with pacing profiles that are increasingly different from the personal best race records of the test users.

It is also interesting to note how the prediction accuracy (and profile similarity) for all strategies is better for women than for men, regardless of k. This is something that is not so surprising in marathon running research, where recent studies have shown how female runners tend to run in a more disciplined manner than their male counterparts [7, 8]. In short, women run more evenly paced races, they are less likely to hit the wall, and so their races unfold in a more predictable fashion, compared to men’s races. Our results suggest that this also extends to the matter of predicting a personal best time and recommending a bespoke pacing plan.

4.4 On the Influence of Ability

These results speak to the utility of the Mean strategy when it comes to making accurate personal best predictions and recommending high quality pacing plans. But the difference between men and women suggests that all runners are not equal. Building on this idea, it is also useful to consider the relationship between accuracy/quality and a runner’s ability, in terms of finish-times. For example, is it easier to predict personal best times for faster or slower runners?

Fig. 3.
figure 3

Prediction error (a) and pacing profile similarity (b) vs. nPB finish-time for Best, Mean, and Even strategies, for all runners and men and women.

Full size image

These prediction error and profile similarity results, for different finish-times, are presented in Figs. 3(a) and (b); for simplicity the results have been averaged over all values of k. Clearly, finish-times have a significant impact on both prediction error and profile similarity. For example, the fastest (elite) runners benefit from very accurate personal best predictions by all three strategies, but as finish-times increase so too do prediction errors, and at different rates for different strategies. Once again the Mean strategy benefits from the most accurate predictions across all finish-times, and the difference between men and women generally persists. It is interesting too to note that error rates begin to fall again after the 300-minute mark, suggesting that the PBs of slower finishers are more predictable.

The similarity of recommended pacing profiles to the actual personal best profiles falls as finish-times increase; see Fig. 3. The Mean strategy continues to perform better than Even and Best and women enjoy more similar pacing profiles than men.

4.5 Personal Best Differences

One more factor that may influence prediction accuracy and recommendation quality is the ambitiousness of a personal best. Are very ambitious personal bests more or less difficult to predict and plan for? To test this we define the PB Difference as the relative difference between a runner’s non personal best time and their personal best time; for example, a PB difference of 0.1 indicates that the PB time of a case is 10% faster than its nPB time.

Fig. 4.
figure 4

Prediction error (a) and pacing profile similarity (b) vs. PB Difference for Best, Mean, and Even strategies, and for all runners and men and women.

Full size image

Figures 4(a) and (b) shows how prediction error and profile similarity are influenced by PB Difference. As we might expect, runners whose PB times are less ambitious (lower PB Difference values) enjoy more accurate predictions for Mean and Even) than runners with larger PB differences; see Fig. 4(a). Likewise, profile similarity is also highest for runners with lower PB Difference values; see Fig. 4(b).

Again, the Best strategy performs poorly in general but with one notable exception. If PB times are more than 12% faster than their nPB times, then the inherently ambitious Best strategy out-performs Mean and Even. That said, it is worth noting that a 12% improvement in marathon finish-time represents a very significant improvement, and is not so common in practice; in our dataset less than 20% of runners achieve such an improvement.

5 Discussion

These results point to Mean as the best strategy to use in practice, because it offers more accurate PB finish-time predictions and more similar PB pacing profiles when compared with the actual PBs of the test runners. Even also performs well but Best appears to be far too ambitious.

5.1 Callibrating Personal Best Ambitions

With all this discussion about error rates and similarities it is easy to lose sight of the practical reason for the proposed system: to help marathon runners plan for, and achieve, new personal best finish-times. And so, as we draw to a conclusion, it is worth forming a more concrete sense of what these new PB times are likely to be for runners’ nPB times. These results are presented in Fig. 5 for our most accurate strategy, Mean, as a graph of the predicted PB improvement in minutes versus the nPB time.

Fig. 5.
figure 5

Prediction PB finish-times for runners based on their non PB finish-times.

Full size image

As we might expect, the predicted PB improvement increases with finish-time; it is ‘easier’ for a 5-hour marathoner to increase their time by 20 min than it is for a 3-hour marathoner. What is particularly encouraging here is the scale of the improvements on offer. For example, a 4-hour marathoner can expect to improve their PB time by just over 20 min on average (±75 s based on the error associated with predictions for such runners). This should be very encouraging for most recreational marathoners, and the combination of these accurate predictions and sensible, tailored pacing profiles may help many enthusiastic marathoners to improve significantly in their next race.

5.2 Pushing Beyond the PB

Our various strategies have been evaluated using the actual PB races of test runners and it should be pointed out that it remains unclear whether these runners could have produced even better PBs had they received the right pacing advice when they ran these races; no doubt some did but most will have adopted fairly simply pacing strategies. Perhaps with the right pacing advice the more ambitious Even or Best predictions might be achievable on the day.

Certainly conventional marathon wisdom suggests that a more even pacing profile is better and thus in practice we may find an additional benefit to presenting runners with the Even pacing plan even though its corresponding finish-time prediction is marginally less accurate (faster) than the mean. By following this Even profile the runner may achieve a more ambitious finish-time. Testing this hypothesis requires further work but it speaks to the potential benefits of presenting Mean and Even predictions and recommendations to runners, rather than focusing exclusively on just one strategy, such as Mean.

5.3 PB Quality

Another factor worth considering is the method by which PBs are identified. Currently they are the fastest race record for a runner. They may or may not reflect a genuine PB. As part of future work we plan to explore this by using different techniques for selecting more reliable PBs. Limiting our case base to runners who have run many races is one way to do this. Another way might be to limit PBs to those race records that have more even pacing profiles. Either way these approaches will provide a way to at least partially evaluate the likely impact of PB quality.

6 Conclusions

We have proposed a case-based reasoning solution to help marathon runners achieve a personal best at their next race. It represents a novel application of CBR to an important, open problem for marathon runners, namely how to select a best achievable target-time and how to pace the race to achieve it.

We have described and evaluated three CBR variations using a large-scale, real-world dataset from the London Marathon. The results indicate that the Mean CBR approach is capable of accurately predicting personal best finish-times and of recommending high-quality pacing plans. This may prove to be extremely valuable in helping runners to calibrate their expectations and plan effectively for their next race-day. By using training data from other marathons it will be possible to produce similarly personalised predictions and tailored recommendations for different marathon courses.

As part of future work we plan to explore other factors that likely impact prediction and recommendation performance, PB quality for example, as discussed, but also the length of time between a nPB and a PB in a case for instance; if the race-pair represent races 5 years apart, for example, will this affect prediction or recommendation? There is an exciting opportunity to also include this, and related ideas, into a new generation of exercise apps to help provide users with sensible training advice and real-time race feedback. Moreover, there is considerable opportunity to further enrich the case representation, either with additional race histories and/or context information, which has the potential to further improve prediction accuracy. Finally, there is no reason why this should be limited to marathon running or even running. Similar techniques may prove valuable for shorter races, for example, or other types of events, cycling, race-walking, triathalons etc., all of which are obvious targets for future research.