In evaluating Froome we must wind the clock back to August of 2011 and his breakthrough performance at the Vuelta. On a 20% section of the famous Angliru with 2km to go, Froome was released from his domestique duties, but it was too little too late. Since that day, though, we’ve witnessed twelve climbs where Froome has performed at a very high level.

Using the pVAM approach, there are two ways in which we can analyse Froome’s performances. Firstly, we can compare Froome’s efforts against the historical baseline which we have already established (updated with Giro 2013 results that overall lowered the baseline). Secondly, we can estimate an equation which will explain Froome’s VAM in terms of gradient, altitude, and climb length.

Method One: Froome v Historical GT Baseline 2008-2013

The latest equation for estimating pVAM is:

pVAM = 2885.17 + 416.825 ln(Gradient) – 0.0620 Vclimb – 0.0880 Altitude

Compared with the equation before the Giro:

pVAM = 2912.14 + 426.293 ln(gradient) – 0.0711 Vclimb – 0.0836 Altitude

After lacklustre performances in the Giro the changes to the constant and altitude coefficient have shifted in the direction of slower pVAMs. However, the gradient factor has reduced, which in most cases will offset the decrease in the constant, and the Vclimb is now subtracting less from pVAM. Before and after examples are listed in the table below, and the change is minor (the pre-Giro equation is 0.1% faster on average).

The average residual for Froome for the twelve climbs mentioned above is 0.805%. On average he performs close to one per cent better than the historical baseline predicts. One important observation is that overall, performances on GT climbs (green) are better than those on climbs in shorter stage races (red).[ref]Some caution is required due to the small sample size dominated by two positive outliers.[/ref]

Such a result is unsurprising given that riders will only want to hit their best form in the three-week events. Additionally, Froome may only push himself to his absolute limit when the big occasion calls for it (although we should note that he held himself back for Wiggins last year). So far this season Froome has out-climbed his rivals comfortably, giving the appearance that on those days he left some gas in the tank. The counterpoint, however, is that during the weeklong stage races there should be less accumulated fatigue.

We can use Froome’s average residual to transform baseline pVAMs into “Froome-pVAMs” simply by multiplying pVAM by 1.0805. In this way FpVAM gives us the VAMs predicted specifically for Froome for the upcoming Tour de France.

Predicted VAMs for the 2013 Tour de France:

Method Two: Froome Regression Analysis

Alternatively, taking Froome’s previous performances, we can run a regression to define an equation which fits his performances alone: [ref]In this case, the length of the climb (Vclimb) is not a statistically significant determinant of VAM. This result is not unusual for the small sample size; it may also indicate some inconsistencies between performances. Overall, the model still fits well.[/ref]

pVAM = 3412.06  + 604.343 ln(Gradient) – 0.0855 Vclimb – 0.0971 Altitude

Applying this equation to the climbing statistics above, we can produce Froome-specific pVAMs.

These results are quite remarkable in that there is little difference between the approaches.  Indeed, they both predict exactly the same VAM for Alpe d’Huez. Quite surprising is that a VAM of 1619 on Alpe d’Huez corresponds to a time of 2488 seconds, or 41’28”.[ref]Time (s) = (Vclimb/VAM)*3600[/ref]

In historical terms this time is quite slow for a Tour de France favourite, although similar to the ascents of Sanchez (41’24”) and Contador (41’33”) in 2011. Contador’s performance came after two hours of maximum effort and at the end of an attempted Giro-Tour double. If Contador can reach his 2011 form, can he match or even beat Froome? The data suggests yes, possibly.

Remember that a pVAM is only what is expected from an average performance of a top-3 rider over a three-week tour. Given the importance of Alpe d’Huez to this year’s race, it means we would expect an above-average performance. Froome’s entire season is geared towards the final week of the Tour – if he has another level, this is where we will see it.

How fast can Froome go? Breaking the 40-minute barrier requires a VAM of 1679, or 4.5% greater than (baseline) pVAM. Historically, such a level is not unattainable, but in a presumably cleaner era it would require several factors being very favourable. Firstly, Froome has to find top form.  For a chance of a sub-40’ time we will need to see a reasonable positive residual (all things being equal) on an earlier climb in the Tour. Secondly, the conditions need to be favourable – no rain, no extreme temperatures, no headwind. Thirdly, Sky (or another team) needs to set a strong pace from the bottom of the climb; it is rare to see strong positive residuals without a big leadout. Finally, the most important factor is competition. It’s highly unlikely that we will see a maximal effort from Froome if the general classification is already decided or there are no climbers to lay down a challenge. I believe that a high 39 minute time is possible if these factors align, however, if only one or two are favourable we may see a time around 41 minutes, or 1-2.5% greater than pVAM. Given the importance of the stage/race, I would be surprised to see a winning time significantly slower than the 41 minutes 28 seconds predicted for Froome.

Alpe d’Huez is not the only climb of historical value in this year’s race. Bonascre (Ax-3-Domaines) and Ventoux are both used regularly in the Tour. The predicted VAM for Froome is very similar to the performance of Contador and Schleck in 2010. It should be noted that Contador and Schleck were performing track stands marking one another, with Menchov and Sanchez 14 seconds quicker on the day. The same climb has been used in 2001, 2003, and 2005, and the time we predict for 2013 would be close to a top 5 in those years (although in 2005 the race had already exploded on the previous climb). For Ventoux, the predicted time is around 50’17” which is more than a minute slower than Contador and Schleck in 2009. The remaining three climbs are all being used in a finishing position at the Tour for the first time, so there is no historical reference point available.

As the predicted performances of Froome on the above climbs are not remarkable by historical standards, there is room for the other general classification contenders to find an extra gear for this year’s race. Froome too will need to lift it a notch if he wants to continue his domination; otherwise the battle in the mountains may be closer than has previously been anticipated.

Rigoberto Uran climbs into contention at Montasio (photo courtesy SHIFT Active Media and the RCS Sport cycling press office)

At the beginning of the Giro it seemed as though Stage 10 would be the first day where a clear road order would emerge. The climb of Montasio was going to give us a chance to see which of the favourites were up to the task of challenging for overall honours.  However, an exciting first week provided us with a good indication of what would happen on the first mountaintop finish of the race.

Bradley Wiggins found himself in difficulty descending in the wet on several stages, and gained only a small amount of time on his rivals in the individual time trial. It is unclear if Wiggins’ difficulties were a sign of form, illness, or injuries due to crashing on Stage 7.

Defending champion Ryder Hesjedal began the race in impressive style, with an unexpected attack on just the third day of racing. Optimism turned to worry as he lost significant time to all major rivals during Saturday’s time trial, and was dropped from a group of thirty the following day.

Of the three main favourites at the beginning of the race, it is the man in Pink, Vincenzo Nibali who has been most impressive. At no time has Nibali looked in difficulty, and his ITT performance makes him a clear favourite to win the general classification.

The first mountain challenge proved to be a continuation of the order which had established during the first week. Hesjedal had been dropped on Sunday, but no one could have predicted what happened on Stage 10. The Canadian lost contact with the main group on the penultimate climb of the day, and was not seen again until the finish. This is no minor struggle with form; Hesjedal is clearly not on the same level as he was in 2012. For the sake of our analysis, Hesjedal is out of the picture.

Early on the final climb, Rigoberto Uran attacked out of the main group. The initial reaction from the bunch was subdued and a genuine reaction would not come until the closing kilometres. Wiggins did not make the final selection which formed on the steepest part of the climb. Uran won the stage, with Nibali the strongest of the main favourites at 31 seconds. Wiggins finished more than one minute behind his teammate.

The climb began in earnest from the 10.61km to go mark, with 859m of vertical gain at a gradient of 8.09%[1]. From this point it took Uran 31 minutes and 15 seconds to reach the finish line.

Earlier in May I used a data set of Hesjedal, Nibali and Wiggins to predict the time it would take for the favourites to complete the Giro’s climbs:

Uran was only ten seconds slower than predicted, with Nibali and Wiggins slower by 41” and 1’18” respectively. In terms of the percentage deviation from predicted VAM, Nibali was -2.2% and Wiggins -4%.

The performance of Uran is more impressive than the results indicate, given that he spent more than three quarters of the climb alone at the front without the advantage of drafting. This is Wiggins’ worst climbing performance in the last 12 months (excluding his Trentino mechanical). In the 2012 Tour, the lowest residual was -0.9% for both Wiggins and Nibali. Nibali may have held something back, content to just mark his rivals. The leader of the race also had a minor mechanical issue in the final kilometre which may have cost him a few seconds. In any case, the two percent difference is well within the range we would expect. Wiggins’ on the other hand was clearly performing at his limit; his lacklustre showing can only be explained by illness or injury. If that is not the case, his form is simply not as good as it was last year.

One result is far from enough to make general conclusions about the condition of riders. All riders were marginally slower than expected and this may be more a reflection of the tactics on the Giro’s first MTF than a definitive example of the limits of the race’s overall contenders. We look forward to the weekend, where summit finishes at Jafferau and Telegraphe-Galibier should provide further evidence of how well the best riders are climbing.

Editor’s note:  We’re very pleased with the way the pVAM equation is performing so far in predicting climbing performances, and are hopeful it will give us the ability to do a much more insightful analysis when comparing riders’ efforts than anything previously seen. Look for a new analysis piece here on Cyclismas after each climbing stage. By the end of the Giro we hope that readers will have a better understanding of Scott’s pVAM method. By the time the TDF rolls around hopefully it becomes the go-to source for performance analysis. A huge thanks to Scott Richards and Mike Puchowicz, better known as Doc @Veloclinic, for all their work on this.

There can be only one. (original image courtesy The Daily Mail)

While time trialling will shape the outcome of this Giro, it is the climbs where the riders will battle shoulder to shoulder for the right to don pink in Brescia. Of the contenders, it is the trio above who have performed at a Grand Tour winning level in the past 12 months. As such they provide the best benchmark of what to expect from the top climbers in the Giro.

In total, 15 observations from the 12 months since the beginning of the Giro last year were suitable for analysis. This includes Hesjedal’s five performances in the 2012 Giro, eight for Wiggins (Dauphine 2012, Tour 2012, Catalunya 2013 and Trentino 2013) and seven for Nibali (Tour 2012, Tirreno-Adriatico 2013 and Trentino 2013). Wiggins and Nibali climbed together six times: four times they finished with the same time, once they were separated by five seconds (I have used the average) and for the most recent observation in Trentino I have used Nibali’s time only due to the mechanical problems suffered by Wiggins.

The OLS estimation of these performances is surprisingly consistent.[ref]

¹ I was prepared for the model to be ineffective with this data given that we are treating three different riders as the same, comparing their performances at different races, and the data set is relatively small. [/ref]

This suggests already that the differences between the three are not great. That is obvious in the case of Wiggins and Nibali, given their head-to-head results, but it shows that Hesjedal’s performances on a completely different race schedule are comparable as well. For a more detailed comparison the percentage residual from each observation are plotted below.

Based on these results, the data suggests that on average, Hesjedal’s performances in the Giro last year were not as strong those from Wiggins and Nibali since then. Taking the differences at face value, it would not seem possible for Hesjedal to succeed against the others on the climbs of this year’s race. However, there are factors from the 2012 Giro which may explain the margin.

Firstly, in most cases a large proportion of the climb was ridden at a reserved tempo. With the race not being “on” from the bottom, it is less likely that the overall climb time will represent the best performance they were capable of on that day. In contrast, most days of climbing for Wiggins/Nibali have seen Sky set a strong tempo from the beginning.

Secondly, there are factors unique to each day which may be the reason for difference. On the Cervinia and Pian dei Resinelli climbs there was some rain about, and the climbs to Pampeago and the top of Stelvio were at the end of brutal stages. Pampeago, where Hesjedal stamped his authority on the race, is the only performance of Ryder’s which compares favourably to Wiggins and Nibali. There were reports of a tailwind on that day but this may only be a counterweight against the 3500m-plus of categorised climbing before the final climb. Focusing only on this result we would expect Hesjedal to put up a strong fight this month.

Going a step further,  the pVAM method can be used to make  general predictions of the time it will take to complete the climbs in this year’s race. By applying the model of the three riders to the characteristics of the mountains in the 2013 Giro we can calculate a pVAM, and subsequently, predicted time (pTime).

The 90% confidence intervals are included in order to provide a range which the actual time should fall into. The range may seem very large, but this ensures that any observation outside the confidence interval is only likely to occur under extraordinary circumstances such as unexpected tactics or extreme weather.  As Telegraphe-Galibier and Tre Croci-Tre Cime are closely connected climbs on the same stages, it is like that there will be a trade-off with a higher performance on the first mountain resulting in a lower performance on the second. For the remaining three climbs, under normal conditions the expected actual observations to fall within a few percent either side of the prediction.

For a rider outside of these three favourites to win the race overall, they would have to climb (on average), faster than the predicted times. Over the course of the three weeks, the analysis will be updated  with actual observations for a real time look at how the race is unfolding.

Who will it be? (photo courtesy Giro D’Italia via SHIFT Active Media)

Thanks to Veloclinic for help with this piece.

“Initially my aim was to test the hypothesis that the metrics of cycling performance are not relevant to discussions on the subject. Arguments are often based on the observation that a single performance of a rider was faster/slower than that of a given cyclist in the past. Whilst factually correct, this type of reasoning paints an incomplete picture. We should compare performances against all of those in the past, but this requires a new approach. This analysis is the culmination of a quest to devise a more satisfactory framework for comparing performance data. I considered several paths to achieving my goal, but it was guidance from Veloclinic which helped formalise the single method presented below.”

~ Scott Richards

At the pinnacle of stage racing are the three Grand Tours. Each year we look to these races for the strongest man over three weeks of climbing and time trialling. Although riders compete over 21 days, the race is ultimately decided by a handful of key moments where the difference in performance is measured in minutes. Ideally, we would investigate and see why one moment was more defining than others; why one climb shattered the favourites while another saw a group riding to the top.

Or more broadly, we would be able to settle the debate over which historical champions performed the best, and know how this season’s stock compares against those gone by. Lastly, in a sport where doping has often been the rule rather than the exception, we would continuously analyse individual performances to keep light shed on evolving doping/anti-doping developments. Ideally, we would quantify cycling performance objectively, account for the ever changing routes and climbs, and do this with nothing more than publicly available course route and the climbing time data. In this article, I propose a new method that finally has the potential to achieve exactly this ideal.

Previous attempts to compare performance have considered average speed for the general classification winner – but only a fraction of overall time is determined by the individual performance capabilities of the winner. Only when contenders are racing flat out in time trials or up decisive climbs do we see leaders emerge from the peloton.  Unfortunately as observers, it is not possible to quantify time trial performance because course design and wind conditions introduce too much error for an across-the-board approach.  On the other hand, when the gradient and length of a climb are enough to dwarf other factors, speed (more specifically, vertical speed) becomes a reasonable and easily measured indicator of performance. Thus, for the sake of our method, we need to focus on the racing when the road goes up. Although several techniques exist for analysing climbing performance, they cannot be suitably applied in all instances. Below I will review the shortcomings of currently available approaches. Then I will describe how I addressed these issues while still using easily obtained data. Finally, I will demonstrate the new method and how it can be used to more accurately compare cycling performances.

Measurements and Existing Techniques

Each mountain has a unique vertical climb and gradient. It is the beauty of cycling that we see racing in different circumstances, but it makes things difficult to compare when climbs are not used repetitiously. However, if we measure the time it takes to complete a climb of a certain height and gradient we have enough variables to more closely investigate performances.

Michele Ferrari pioneered the commonly used VAM (Vertical Ascent Meters) technique, which measures climbing speed in terms of vertical metres per hour:

VAM (m/hr) = Vertical Climb (m) / time (hr)

Unfortunately VAM does not  adjust for the gradient of a climb. Gradient affects speed due to the nature of power losses. Air resistance will be an important factor in (vertical) climb speed on gradual inclines, but this effect lessens as the slope increases. A performance on a 7% gradient would result in a higher VAM than the identical human performance on a 6% gradient (and so on). So if we were to compare climbing performance using VAM, there would be a strong bias towards climbs of high gradients. Ferrari  attempted to correct for this issue by developing an equation taking gradient into account:

Relative Power (W/kg) = VAM / (Gradient Factor * 100)

where Gradient factor = 2+ (% gradient / 10)

With this additional calculation we can begin to look across climbs of different gradients. Yet there are two problems which remain. Firstly, Ferrari’s method relies on a “fudge factor” based on his own observations. The data set he used and the precise relationship between VAM and gradient remains unknown.

Secondly, the relative power formula does not account for exertion time. Intuitively, the longer the performance, the lower the average speed will be. In cycling we expect longer climbs to be slower in performance terms than shorter ones. Thus, even the best current methods are likely to be reliable only if considering performances of comparable length. A 40 minute effort should be related to other exertions of similar time, and not a 15 minute climb. This approach makes it difficult to compare performances where time varies significantly, limiting the scope of any analysis using current methods.

Similarly, other factors which may influence VAM still cannot be accounted for such as:

• Wind and weather conditions
• Altitude
• Road surface
• Stage/race design and difficulty
• Tactics (including drafting)

These factors may be less significant than gradient and exertion time, but the fact remains that they will vary from one climb to the next.

Proposed Method

The proposed method involves a two-step process:

• 1. An equation, derived from the performances of a historical baseline group that corrects for gradient and vertical meters, is used to calculate a predicted VAM (pVAM) for the climb of interest.

pVAM = a₀ + a₁ vclimb + a₂ ln(gradient)

Where a₀, a₁ and a₂ are the parameters to be estimated

• 2. Then pVAM is subtracted from the actual VAM (aVAM) measured for the current climb. aVAM minus pVAM gives a residual which is the difference between the actual and the predicted VAMs. When this residual is divided by pVAM the result tell us in percentage terms how much faster or slower the aVAM was than pVAM.

% Residual =100 x (aVAM-pVAM)/pVAM

Essentially, the first step takes the data from a baseline group and uses it to predict how this baseline group would perform on the climb at hand. Since the equation corrects for the gradient and height of the climb it can be used for any major Grand Tour climb. The second step then tells us if the current performance of interest was faster or slower than expected, and by what percentage.

Derivation of the pVAM Equation

Data

The first step was to identify the most appropriate baseline data set from which to build the model. I chose to compare only the best riders in each race by considering the average performance (on each climb) of the top three in the final General Classification.[ref]There are some exceptions where a non-podium rider would better fit the aim of using the highest performers overall.[/ref] The selection identifies best General Classification riders overall but not necessarily the top performers on every individual climb. Sometimes the quickest up a mountain may not be in overall contention for the race. Not riding for overall general classification allows one to focus their entire three weeks on a single climb – it would be unfair to compare stage hunters to those coping with the daily efforts fighting for the GC. In summary, what I want to compare is the climbing performances of the absolute best (in performance terms) overall General Classification riders in each Grand Tour.

Similarly, the data includes predominantly finishing climbs (with a small number of exceptions), as it’s at this point where the cyclists are closest to their threshold. Climbs a long way from the finish are ridden according to the tactics chosen on the day and are far less of a race to the top than finishing climbs. Non-final climbs have only been included (where data is available) if they appeared to be ridden close to threshold.

The timeframe considered is the last five seasons of racing. Specifically, it is expected that with the introduction of the biopassport in 2008, this time period will be a more realistic baseline that is not as skewed by the effects of doping.

Regression Analysis (Ordinary Least Squares Estimation of Parameters)

With the most appropriate data set identified, I chose a regression analysis to build a mathematical model that describes the data. A regression analysis works by taking real data and fitting a line that best approximates the trends within the data.

As described above, the only way for us as observers to measure a performance is through climbing speed (VAM). However, since VAM is affected by gradient we need to account for it in our measure. Given that our data set includes random variation it was necessary to use a statistical estimation (OLS) to determine the values of these main parameters.

By adjusting for gradient we can deliver a metric which can be used to compare climbing performance. In this sense, the foremost determinant of VAM is the slope of the climb:

VAM = f[Gradient]

To reduce the margin of difference between gradients, they will be taken in log form. Looking at gradient alone, we could establish a rough estimate of expected VAM.

To advance the model even further, the next most significant variable can be added as well:

VAM = f[Gradient,Time]

Considering both gradient and time takes the model one step further than any of the existing approaches. A regression analysis using these two variables should provide an indication of expected climb speed for any given gradient and exertion time. However, exertion time is a result of performance, and not just a physical characteristic of a climb. We cannot effectively isolate the human performance factor if the expected VAM is dependent on the performance itself.[ref]Expected VAM will differ from one performance to the next based on completion time. A faster performance will result in higher expected VAM than a slower performance on the same climb. This would downplay the magnitude to which a performance actually was faster (or slower).[/ref] Expected VAM should be devised solely from the physical variables of the day (not variables of performance). In order to overcome this problem, the factors which influence completion time need to be considered. Vertical height climbed is by far the most important variable in the determination of climb time (R² > 0.97). Therefore, accounting for time can be achieved by using its primary determinant as a proxy:

VAM = f[Gradient,Vertical Climb]

At this point, we have a model which estimates a predicted VAM based on logs of the gradient and metres climbed (referred to in equations as vclimb).Predicted values (pVAM) are therefore the expected performance based on the baseline performances from 2008-2012.

Calculation of Percent Residuals

Now that the predicted VAM is adjusted for these two variables, we can use predicted VAM to normalize the human performance factor to our baseline data set. We do so by comparing the actual VAM recorded for the climb of interest to the pVAM value predicted by the model. The concept is illustrated in the figure below.

In this figure the results of the pVAM equation were plotted for each individual climb as a solid black line. This pVAM line represents the performance that would be expected based on the physical characteristics of the climb and serves as a performance baseline. The green squares are the actual VAM recorded for each climb. The difference between the pVAM and aVAM (the difference between the black line and the green point) is the residual which is the amount by which the actual performance was faster or slower than the expected baseline.

Finally, the percent residual 100 x (actual – predicted)/predicted is calculated to normalize the performance metric to the baseline. In this way, our performance metric is now comparable across climbs of different lengths and gradients. A positive percent residual indicates that a performance was faster than the period baseline while a negative percent residual indicates that a performance was slower than the baseline period.

Results

For 66 observations 2008-2012, vertical climb and gradient estimate VAM with an adjusted-R² of 0.64. Therefore, the majority of variation in VAM can be explained by these two variables. The predicted VAM (pVAM) equation is as follows:

pVAM = 2938.5 – 0.1124 vclimb + 476.45 ln(gradient)

To illustrate the process I will calculate pVAM for a popular climb in the Tour de France, Plateau de Bonascre (Ax 3 Domaines) of 664m and 7.46%:

pVAM = 2938.5 – 0.1124(664) + 476.45 ln(0.0746)

→ pVAM = 1627

This climb was used last in 2010, taking 1439 seconds to complete for an actual VAM of 1661 m/hr. Subtracting actual from predicted, this leaves a residual of 34 – this means the actual performance was faster than expected by just over 2%. This residual is against the norm for the baseline period. Percent residuals can then be compared against one another to provide a better idea of relative performance.

Looking at the results, there are periods where performance is consistently higher or lower than predicted. In these cases we can safely assume the actual human performance was above or below the normal. At other points, there is a fluctuation in the residual from one climb to the next – this is where variables we haven’t accounted for are influencing climbing speed. Thus in the latter case it is harder to say if actual performance is higher or lower than expected.

If we group the results into their respective seasons, these random fluctuations should balance one another out.

The range of the averages is less than 4%. This is remarkably small and confirms that over a season, the random fluctuations level out.

The average for 2009 is far greater than any other year; in that sense it is the year of the highest performances. This is not a huge surprise given what we know. In the Tour that year, Alberto Contador recorded what is seen as one of the fastest climbs in history. Multiple climbs in the Giro were ridden ferociously as Di Luca and Pellizotti tried to reduce the margin created by Menchov against the clock. Additionally, the Italian tour that year featured a couple of uncharacteristically “easy” mountain stages in the run in to MTFs.

Averages for the other years are closer to the normal: 2008 and 2011 were slower, whilst 2010 and 2012 are not significantly different from the average. These results cannot be explained as easily as those of 2009, and would require closer examination. It could be down to natural variation, with no reasonable explanation. Some riders may simply be better than others. Alternatively, the results could help describe the constant struggle between forces of doping and anti-doping.

Quality Checks

Problems would arise if residuals were biased to one end of the gradient or vertical climb spectrum. The charts above show this not to be the case; the residuals are uncorrelated with the dependent variables. As such, we can compare results from different climbs/gradients and compare the residuals across the entire sample.

Conclusions

Existing approaches for measuring cycling performance are not robust enough to be useful in a historical analysis. Currently, the only suitable methods are the VAM and relative power calculations used to determine climbing speed. In considering the role of exertion time, the method I have outlined above can more effectively isolate the human performance factor in climbing speed. This approach can be applied to large sets of data in order to reveal more precisely where performance has differed from average. The most reasonable application of the above analysis would be to detect trends in performance across years. Additionally, it may be useful as a tool to compare performances between races or even analyse the effect of tactics and fatigue on individual climbs within a Grand Tour.