Marathon Race Planning: A Case-Based Reasoning Approach

1 Introduction

This work aims to help marathoners to achieve a new personal best (PB) by using case-based reasoning [Smyth, 2007; de M´antaras et al., 2005; Bridge et al., 2005] to address two problems faced by runners: (a) predicting an achievable finish-time; and (b) recommending a tailored pacing plan to help the runner achieve this time.

While health applications of AI are not new [Peek et al., 2015; Buchanan and Shortliffe, 1984; Bichindaritz et al., 2008; Wiesner and Pfeifer, 2014; 2010], always-on smartphones and wearables are creating an even greater opportunity for novel preventative, proactive, and personalised interventions [Ohlin and Olsson, 2015; Geleijnse et al., 2011]. The sports world has embraced the data-centric world of sensors [Campbell et al., 2008] and mobile apps, as athletes explore the power of data to optimise training and performance [Lewis, 2004; Kelly et al., 2012; Leijdekkers and Gay, 2015; M¨oller et al., 2011; Hermens et al., 2014].

We focus on helping recreational runners to plan their race strategy. A key concept is pace, which is measured in minutes per mile/km - the inverse of speed - so that higher values correspond to slower speeds and vice versa [Trubee, 2011; Foster et al., 1994; Deaner, 2006; Haney Jr, 2010]. There are three basic pacing strategies. 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, running it faster than the first. Many elites and disciplined runners aim for even or slightly negative-splits [Abbiss and Laursen, 2008]. Recreational runners typically run positivesplits, slowing during the second-half of the race, and sometimes even hitting the dreaded wall [Foster et al., 1994].

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 and pacing: the finish-time they aim for; whether they opt for positive, negative or even splits; whether they avoid hitting the wall etc. This is where we believe there is a significant opportunity to support marathon runners, by advising them on a suitable target finish-time, and by providing them with a concrete race-plan, one that is personalized to their ability and tailored to a specific marathon course.

2 ProblemDefinition

Our objective is to support a runner who has completed at least one previous marathon and so has a race record to serve as a starting point.

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 a finishtime 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 &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 8x5km stages or segments (0-5km, 5km-10km, ..., 35km-40km), plus a final 2.2km segment (40km-42.2km). A race-plan, or pacing plan, consists of a sequence of average paces (measured in minutes per km) for segments. For example, Figure 1(a) illustrates a race-record for a runner who completed a marathon in 4 hours and 13 minutes. The pacing profile shows relative paces - that is segment paces relative to the runner's mean race pace - and indicates a positivesplit. They started their race (the first 5km and 10km segments) 10% faster than their average pace, before slowing in the second-half, to finish 7% slower than their average pace in the final segment. We wish to generate tailored pacing plans, so that runners can benefit from a plan that is suitable for their goal time, their personal fitness level and ability, and that reflects the peculiarities of a given marathon course.

3 Using CBR to Achieve a Personal Best

One insight of this work is to use a CBR approach to predict suitable target finish-times and recommend pacing plans, based on the runner's own race experience and the experiences of similar runners. 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 Figure 1(a). Each case contains two race records; see Figure 1(b). One of these race records corresponds to a non personal best (nPB) race, the other corresponds 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 Figure 1(c).

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 Equation 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 Figure 1(b) we highlight 3 race records for r (for marathons m1,m2,m3).

The race record with the best finish-time is designated the personal best, and it is paired with the remaining n - 1 non personal best records, producing n - 1 cases. As per Figure 1(b), r's best race is m2, with a finish-time of 236 minutes. This is paired with the two nPB records (m1 and m3) to produce two cases, c(r,m1,m2) and c(r,m3,m2) 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.

Next, we filter on the basis of gender, only considering cases for retrieval if they have the same gender as the query runner, because physiological differences between men and women have a material impact on marathon performance.

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 Euclideanbased 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

Algorithm 1: Outline CBR Algorithm. Data: Given: q, query race record; CB, case base; k, number of cases to be retrieved; t, finish-time threshold. Result: pb, predicted finish-time; pn, recommended pacing profile. begin C = {c ϵ CB : T ime(q) - t < T ime(c) < T ime(q) + t} C = {c ϵ C : c.gender == q.gender} if len(C) ≥ k then R = sortk(sim(q, c) ∀ c ϵ C) pb = predict(q, R) pn = recommend(q, R) return pb, pn elsereturn None end end

achieve a faster personal best on the same marathon course. We test three prediction approaches, with the predicted PB finish-times weighted based on the relative difference between the query runner's finish-time and the corresponding nPB finish-time of a retrieved case; see Equation 2.

Best PB. In Equation 3 the predicted time is the weighted PB finish-time of the single fastest retrieved case, Cbest.

Mean PB. Our second prediction approach calculates the weighted mean of the PB finish-times of the retrieved cases, as in Equation 4.

Even PB. Even PB is based on the idea that more even pacing is better than more varied pacing. By measuring the coefficient of variation (CoV ) of the relative paces in the PB pacing profiles, we can select the one (Ceven) with the lowest CoV value. The PB time of this Ceven case is used as the predicted time for q.

3.4 Pacing Recommendation

We 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 uses the relative pacing from Cbest and maps its relative paces to the average pace for the predicted PB time for the query runner. For example, if the predicted PB time is for 232 minutes, indicating an average pace of 5 minutes 30 second per km, and the PB profile in Cbest calls for a first 5km that is 5% faster than average pace, then the generated pacing profile for q will advise running the first 5km at just over 5 minutes and 13 seconds per km.

Mean Profile generates a new pacing plan based on the mean relative segment paces of the PB profiles from the k retrieved cases.

Even Profile generates a pacing plan from the PB profile of Ceven, the case with the most even pacing.

4 Evaluation

We evaluate our approach using 12,968 cases from runners of the London Marathon (2011 - 2016), who have run at least

3 races. We use 10-fold cross-validation 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% for prediction and recommendation, repeating this 10 times and averaging the results. The nPB part of each test case is used as a query and the PB part is held back to evaluate the prediction and pacing plan recommendation.

We generate predictions/recommendations using the three CBR approaches (Best, Mean, Even) described earlier. 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.

4.1 Prediction Error and Profile Similarity vs. k

We begin with prediction accuracy and recommendation quality versus k, the number of cases retrieved; see Figures 2(a) & (b). Our strategies behave differently for increasing k. Mean produces the lowest errors and benefits from increasing values of k, up to k ≥ 10; Mean achieves an error of ≈ 4.5%, for all runners, and as low as 4% for women. In contrast, Even produces predictions with an average error of ≈ 6% regardless of k, while the accuracy of Best deteriorates with k. The problem for Best is that more retrieved cases lead to faster, best finish-times as predictions. So Best tends to predict ibreasingly ambitious PB times as k increases, times that have not been achieved by our test runners.

For pacing similarity - our proxy for recommendation quality - we see a similar, albeit inverted pattern in Figure 2(b). Note too how the prediction accuracy (and profile similarity) for all strategies is better for women than for men, regardless of k. In short, women run more evenly paced (predictable) races [Trubee, 2011; Deaner, 2006]. Our results suggest that this also extends to the matter of predicting a personal best time and recommending a bespoke pacing plan.

4.2 On the Influence of Ability

It is also useful to consider the relationship between accuracy/quality and a runner's ability level, in terms of finishtimes. These prediction error and profile similarity results, for different finish-times, are presented in Figures 2(c) & (d); 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.

The similarity of recommended profiles to the actual PB profiles falls as finish-times increase; see Figure 2(d). The Mean strategy continues to perform better than Even and Best and women enjoy more similar pacing profiles than men.

5 Conclusions

We have summarised a CBR solution to help marathon runners achieve a personal best. The results indicate that the Mean CBR approach is capable of accurately predicting personal best finish-times and of recommending high-quality pacing plans. A more comprehensive treatment and additional research can be found in [Smyth and Cunningham, 2017a] and [Smyth and Cunningham, 2017b].

Acknowledgments

This work is supported by Science Foundation Ireland under grant number SFI/12/RC/2289 and is an abridged version of a paper titled "Running with Cases: A CBR Approach to Running your Best Marathon", from the 25th International Conference on Case-Based Reasoning.