Pacing traces (speed vs time or distance) taken from endurance field data present great promise for understanding the physiology of the tiring athlete because they integrate the athlete’s free response to all internal and external environmental factors under “natural” race conditions (^{1}). Indeed, recent proposals (see Tucker and Noakes [^{18}]) around the genesis of fatigue (or effort regulation) would benefit greatly from access to more diverse speed traces, and higher frequency speed traces at that, in order that nuanced features of pacing traces can be identified and explained.

Such a demand presents (at least) three challenging problems: first, field data are rarely taken at high frequency (split times typically constitute 10% of total race distance at best, e.g., Lambert et al. [^{12}]); second, even if high-frequency data were available, traditional curve fitting techniques used by the literature on low-frequency data are inappropriate for high-frequency data; and third, field data are affected by a range of uncontrolled factors; most notably, speed is lost to gradient considerations. Fortunately, the wide use of Global Positioning System (GPS)–enabled devices and the general availability of topographic information mean that the first problem has been surmounted. Hence, it is to the last two problems that we direct this work.

First, we demonstrate a novel gradient detrending and smoothing methodology that can be applied to any high-frequency running speed data; second, by applying this method to two distinct case studies—Haile Gebrselassie’s world-record marathon (Berlin, September 2008) and a unique data set taken from several years of the Six Foot Track (SFT) Ultramarathon (45 km, Sydney, Australia)—we report the underlying pacing strategy of Gebrselassie’s world-record run, free from gradient influences,1 and the same for 34 distinct runs over the SFT course. At the macroscale, we find that Gebrselassie used a near-perfect “U-shaped” (symmetric) pacing strategy, whereas runners in the SFT data set seemed to use a “positive” (progressively slowing) pacing strategy, irrespective of overall running speed (^{1}). However, at the microscale, Gebrselassie and the SFT runners both demonstrate marked complexity in their pacing variation, reminiscent in both cases of periods of “certainty” and “uncertainty” suggested by Gibson et al. (^{8}). Indeed, if one takes the neural regulatory model as given (^{9}), the highly varying gradients of the SFT run seem to generate many open-ended racing sections in the minds of the runners with only approximately the last third of the ultramarathon figuring as a well-defined exercise bout.

#### METHODS

##### Data Sets

##### Marathon world record: Gebrselassie, Berlin (September 2008).

High-frequency (1 km) split times were available from an internet source,2 while altitude and distance traces were obtained from a second internet source.3 To match the speed and elevation/gradient data, a linear interpolation on the elevation data was conducted to produce elevation and gradient estimates at each distance in the speed data. The speed data were normalized by subtracting the average running speed and then dividing by the same to produce percentage deviations from average running speed.4 The combined distance, speed deviation, and gradient data set for this run formed the Gebrselassie data set (hereafter called HG).

##### SFT Ultramarathon (various years)

The SFT Ultramarathon is the fifth largest marathon in Australia, attracting over 850 entrants each year.5 Because normal competitors only receive three split times (15, 26, and 45 km) in the event, a richer source of data was sought. At the conclusion of the 2010 running of the SFT, the authors posted a request for GPS wristwatch data traces on a popular Australian online running forum.6 In all, 41 traces pertaining to the SFT were sent to the authors.

A process of filtering was then undertaken as follows:

1. Traces having no data point within a 100-m segment distance were omitted (six omissions); and

2. Remaining traces with >50% of the event lower than 2.2 m·s^{−1} (proposed run/walk threshold as reported by Epstein et al. [^{7}]) were omitted (one omission).

This process left 34 traces in the data set. Because the wristwatch devices that produced the contributed data were commonly in “smart” recording mode (recording a datum point whenever a significant change in elevation, position, or HR was detected), the resulting traces had noncomparable data points. To overcome this problem, a segmentwise partitioning of the entire course was imposed by first normalizing all traces to the nominal 45-km race distance and then calculating the average gradient, elevation, and running speed for each running trace for each of 450 (100-m) segments on the course. From this segmented and now comparable data, common elevation and gradient profiles were obtained by averaging the segment elevation data for all traces with complete nonzero elevation fields (seven omissions from this procedure) and then calculating average within-segment gradients. Finally, because the SFT Ultramarathon effectively has a mass start down a very tight, steeply descending single track in the first 1.5 km of the race and comprises a similarly tight and steep final 1.5 km, these two portions were omitted from the analysis because all competitors must either wait or change their style significantly because of congestion.

The above process gave rise to 34, 420 segmented (split) data traces comprising running speeds and elevation gradients for every 100-m of the course. The common resultant elevation and gradient profile is presented in Figures 1A and B along with an example segment speed field from one run in the data set (C).

As with the HG speed data, for each run, the speed data were normalized by subtracting the average speed for that run and dividing by the same to produce percentage deviations from the run’s average running speed. The combined 100-m normalized and segmented distance, speed deviation, and gradient data set for the remaining 34 runs is hereafter called the SFT data set.

##### Detrending for Gradient

The strategy used was first to explain the deviation in running speed for each run by variations in the gradient alone and so reveal a runner’s actual (“flat equivalent”) pacing strategy variation as the remainder. It must be stressed that this approach will only work when the gradient data are uncorrelated with the event distance data. The reason is that if it were the case that (say) the gradient trace was positively correlated with distance, then the coefficient estimates of the gradient model below would include the influence of any pacing strategy. To check for this, Pearson correlation tests were run between the gradient and distance fields in each of the SFT and HG data sets. Whereas the HG data set showed no significant correlation (*P* = 0.70, two tailed), nonzero correlation could not be rejected in the SFT data set case (*P* < 0.01, two tailed, 95% confidence interval = 0.036–0.224).

To treat the gradient–distance correlation in the SFT data set, the gradient field was first estimated by

where the gradient and cumulative distance for each segment are *gi* and *di*, respectively, *A* and *B* are coefficients, and *γi* is the error term. Then, a new corrected gradient field was generated by simply subtracting the estimated correlation term *Bdi* from *gi*,

In the case of the SFT data set, *B* = 3.48 × 10^{−4}, i.e., adjusting down gradients in the last segments of the event by at most 3.5%. Hereafter, the gradient field of the SFT data set will refer to the corrected gradient, *gc*.

Second, a scatter plot of deviations in running speed versus gradient was produced from the pooled SFT data set (Fig. 2). A broad inverted “U” or parabolic pattern emerges in the data. Intuition suggests the same because although a high positive gradient should decrease running speed because of gravitational work considerations, a high negative gradient must also require an eventual decline in running speed because of injury avoidance or eccentric muscle loading considerations (^{5}). A quadratic functional form relating running speed deviation to gradient was thus assumed.

Third, after inspection of the speed fields in the SFT data set (see example in Fig. 1C), it was obvious that the speed data showed strong first-order (positive) autocorrelation (meaning that a good estimate of the next segment speed value for a given run would be the last segment speed value). Although this type of disturbance will not bias coefficient estimates under ordinary least squares estimation, it will cause them to be inefficient (nonminimum variance). In which case, the standard procedure of Koyck (^{11}) was applied to the estimation equation, which allows for an exponentially decaying sequence of correlations between the previous segment observations and the present observation.

Taking these two considerations together, the following functional form was estimated:

where *vr*_{,}*s* is the deviation from average speed for run *r* in segment *s*, *gs* is the gradient in segment *s*, *vr*_{,}*s*_{−1} is the deviation from average speed for run *r* in the preceding segment *s* − 1, *er*_{,}*s* is the error term, and *A*–*D* are constants. Equation 3 was estimated for each run in the SFT data set and the one run in the HG data set, thus producing (in the case of the SFT data set, for example) a sequence of 419 error terms (total 100-m segments − 1) for each run detrended of the influence of the segment gradient but including any pacing strategy (event distance/duration) information.

##### Pacing Strategy Discovery.

Pacing strategy was discovered by a two-step procedure: first, the resulting data from the gradient detrending procedure (i.e., the equivalent “flat” running speed deviations from average speed) were run through the locally weighted scatter plot smoothing (LOESS) algorithm (^{6}) to illicit likely functional forms, and second, specific functional forms were fitted via a nonlinear optimization procedure.7 The advantage of LOESS is that one does not need to specify a functional form before fitting the data; instead, the algorithm fits lines to local subsets of the data to discover, piecewise, the likely underlying functional form. One can think of the procedure as a kind of smoothing approach. The LOESS procedure requires only one input: the “span” parameter (*f* in the original article). This parameter changes the breadth of the local fitting approach: a low parameter leads to more localized fitting; a higher parameter leads to more global fitting. There is no strict rule for choosing the value of the span parameter; a very low value can lead to an apparently overfitted line, whereas a very high value will lose any nuance or nonlinearity in the underlying pattern. In the present case, three span parameters were used over a suitable range: 0.25, 0.50, and 1.0. On the basis of the outcome of the LOESS approach, functional forms were selected for fitting to each data set. These estimations were conducted with either ordinary least squares or nonlinear maximum likelihood estimation procedures from within the statistical package R. A summary of the steps involved in the above method is given in algorithm 1, and the associated example R code to implement this approach is given in Supplemental Digital Content 1, http://links.lww.com/MSS/A94.

Algorithm 1: Pacing Strategy Discovery in Undulating Field Data Setup: harmonize speed (*v*), gradient (*g*), and distance (*d*) data.

Step 1: check for correlation between *g* and *d*; if Corr(*g*, *d*) > 0, then

Step 2: detrend speed for gradient

Step 3: discover pacing strategy functional form

Step 4: run appropriate model fit

#### RESULTS

##### HG data set.

Results from the LOESS and functional (quadratic) fitting procedures are given in Figure 3. Both the original (+) and gradient-corrected (•) speeds are given for comparison (some background information given in Supplemental Digital Content 2, http://links.lww.com/MSS/A95) and, as expected, show the relatively minor effect of the Berlin course gradient (−) on HG’s pacing strategy. In the case of the HG data set, all three span values used in the LOESS procedure strongly suggested the fitting of a “U-shaped” (quadratic) pacing function:

where *er*_{,}*s* is the residuals for run *r* (in this case, Haile Gebrselassie) in segment *s*, whereas *ds* is the event distance covered up to segment *s*, *μr*_{,}*s* is a new residual term, and again, *A*, *B*, and *C* are coefficients to be estimated.

##### SFT data set.

Following Lambert et al. (^{12}), a preliminary pooling of runs by average speed was performed. Specifically, the 34 runs in the SFT data set were initially sorted from the slowest to the fastest average speed and then divided into three running speed cohorts (12 runs in cohorts 1 and 2, 10 in cohort 3). The resultant “slow,” “medium,” and “fast” cohorts had average running speeds of 2.32, 2.62, and 2.98 m·s^{−1}, respectively.

Results from the LOESS procedure are reported in Figure 4. Leaving aside the top panel (see “Discussion” below), the bottom two panels suggest several consistent features of a functional form across speed cohorts: a declining overall gradient, a nonlinear functional form (a higher decline in the early parts of the event compared with the latter parts), and a possible final surge in pace. Hence, two nonlinear functional forms were fitted to the three cohort subsets: exponential and quadratic. A linear model was also fitted for comparison with the literature. Functional forms are given below.

Results from the three functional form fits to the SFT data set are presented in Figure 5, and coefficient estimates are reported in Table 1.

Figure 5 Image Tools |
Table 1 Image Tools |

#### DISCUSSION

Both the HG and SFT data set results provide novel endurance running field evidence toward the recent complex adaptive systems perspective of pacing control as advanced by Tucker and Noakes (^{18}). No similar high-frequency study exists for ultraendurance running events to our knowledge. Indeed, both the speed example SFT panel (Fig. 1C) and the HG residual speed data (Fig. 3) report higher frequency data than previously published on endurance running and show complex high-variance patterns in running speed on a range of time scales. The HG data set remarkably reveals that the fastest and slowest kilometer splits of Gebrselassie’s world-record run were the two final kilometers. Whereas previously, we might have attributed such variation at least in part to the variation in the course gradient,8 the gradient-detrending method gives confidence that the observed speed variations were true pacing movements by Gebrselassie in need of theoretical explanation. We note that such variation in high-frequency speed (and/or HR) data has previously been reported for endurance cycling events (^{2,9,14}).

Second, the method applied to the HG data set reveals that despite the very high variability in segment speeds, the general (gradient free) macroscopic “fast–slow–fast” pacing strategy used by Gebrselassie was remarkably symmetric. Indeed, if one calculates the minima from the fitted quadratic curve, the turn point is found to be precisely 21.19 km, less than half of 1% deviation from the halfway point of the event. Significantly, the present finding extends the macroscopic fast–slow–fast approach discovered for other world-record performances on the track up to 10,000 m (^{15,17}). Such findings prompt deep questions around the adaptive control mechanism at play in human endurance efforts.

Moving to the SFT data set, we first observe that “brute” functional form fitting to high-frequency pacing data can miss important features of pacing strategy. For example, if one was to apply the foregoing method of the literature and simply choose (or allow a fitting algorithm to choose) a parametric functional form and take this to the data, significant features of pacing would be lost. Suppose that a linear (only) method were taken to the data (bottom panel of Fig. 5), we might conclude that a) all runners slow down (in the range of 10%–14%) and b) there is no significant pacing variation among running speed cohorts (in contrast to the main finding of Lambert et al. [^{12}]). Or suppose instead that a nonlinear parametric fitting approach were conducted (top panels, Fig. 5), we might add to our previous conclusions that c) slowing occurs predominantly in the early half of the event, and thereafter, pacing steadies. On the other hand, with the LOESS nonparametric fitting procedure (Fig. 4), significant features are added to this description: d) all cohorts exhibit an “end-spurt” phenomenon (compare Albertus et al. [^{3}]), e) all cohorts exhibit a sinusoidal increase–decrease–increase start to the race (compare 0.25 LOESS trace for HG in Fig. 4) such as that found for cyclists (^{16}), and f) the faster two cohorts retain a higher applied pacing strategy in the downhill section between uphill efforts (shaded gray in Fig. 4) than the slow cohort.

Clearly, a full explanation of features a–f and in particular features e–f is beyond the scope of this work. Instead, we leave questions for others to consider, which we feel this work urges: how could Gebrselassie run a near-perfect symmetric pacing strategy on a macroscale while exhibiting a high variance in 1-km segment pacing on a microscale? Or, sharpening the point, is it possible for an athlete to run a symmetric macro pacing strategy without such a high microscale variance? Why do the SFT runs—regardless of pacing cohort—exhibit such “complex” pacing, even after detrending for the significant variance in on-course gradient? Furthermore, why does the general pattern of the SFT pacing seem broadly fixed across speed cohorts? Tentatively, we observe that the Berlin marathon (on road, paced, elite, city environment) sits at one end of the “field” information spectrum, whereas the SFT marathon (off road, unpaced, amateur, bush environment) sits very close to the other. As such, it would seem that recent work on the importance of beliefs and information on athletic pacing could provide a fruitful line of inquiry (^{3,4,13}). As such, could the rises and falls in pacing exhibited by SFT runners be the product of facing so-called open-loop subevents (^{10})? Or generally, could the erratic microscale pacing of HG and the variance of the SFT runs be explained by successive periods of certainty and uncertainty as suggested by Gibson et al. (^{8})? In any case, the above discussion can provide only conjectures and ideas in the absence of more high-frequency field and laboratory data.

This work was wholly supported by the respective institutions of the authors; no additional funding source was associated with the work.

No funding from external sources was received in the preparation of this work.

The authors report no conflict of interest.

The publication of results in the present work does not in any way indicate endorsement by the American College of Sports Medicine.