Mathematical Model

i-Race Mathematical Model: Step 1

The i-Race Performance Index is all about comparing your actual race performance with the theoretical human frontier of running ability for your parameters. Let us understand what that means in simple English!

Age

Consider the 10K road race. There is a current world record of 27:07, set by Micah Kogo of Kenya on 1-Apr-2007 in Brunssum, Netherlands. This is the fastest human performance achieved or the so-called Human Frontier for 10K. However, that was set by Micah when he was 20 years 302 days old.

As the age of the runner increases, his ability to run fast reduces. So we should regard 27:07 as the 10K Human Frontier not for all ages but only for the age of 20.85 years. If there were a runner as gifted and as great as Micah Kogo, but was 60 years of age, how fast would he have clocked that particular race? That would be the 10K Human Frontier for 60 years of age.

To model the Human Frontier change with age, the best way is to take the age-graded world records for 10K races. Fitting a curve for the 10K world records from age 14 to 85, one can get a fair estimate of 10K Human Frontier.

Gender

It would incorrect to compare the women’s performance with men’s. Typically, the world records for women are 7% – 8% slower than those for men. Thus, the 10K Human Frontier for Women would be different than that for men. But will it be uniformly slower for all ages?

The world records data shows that men show a slow linear decline up to the age of 70 and a steep drop after that. On the contrary, women have a faster but steady decline after the age of 40. Thus, age slows down men and women differently. Obviously, we need to use different equations to calculate the 10K human ability for men and women of varying ages. Thus, we would have different Human Frontiers for 10K Men and 10K Women.

Racing Distance

The next question is whether the equations for the Human Frontier stay the same for distances other than 10K, i.e. is the 5K Human Frontier for men and women just a simple multiple of 10K Human Frontier? Obviously, the 5K Human Frontier will not be just half of 10K Human Frontier. The world record for 5K is 12:37, which is 2.15 times faster than the 10K world record, not 2 times. However, is the 5K Human Frontier uniformly 2.15 times faster than the 10K Human Frontier, for both genders (men and women) and for all ages? Unfortunately, No.

The world records data show that older men and women handle longer distances better than the shorter ones. The slowdown with age is more prominent for shorter races such as 5K. For example, the women’s marathon world records for the ages 25 and 50 are 2:19:41 and 2:31:05, respectively. The latter is 8.2% slower than the former. On the other hand, the women’s 5K world records for the ages 25 and 50 are 14:28 and 17:10, respectively. The latter is 18.7% slower than the former. Obviously, the degradation of performance with age changes differently for different race distances.

Interestingly, the 5K world record for the age of 49 years is by Monica Joyce at 16:02, a full 1 min 8 seconds faster than the 5K world record for the age of 50 years, which is just a 10.8% degradation, if you consider that Monica Joyce was of 49 years 313 days when she set it. That is almost 50 years in age.

Thus, taking simple ratios is not enough to calculate the Human Frontier for different race distance. One will also need to ‘smoothen’ the data for such fluctuations in the world records. All of this gives rise to a very complicated calculation for the Human Frontier as a function of age, gender and racing distance.

Equation for Step 1

We used the men’s and women’s world records data for various running distances (5K, 10K, half-marathon, full-marathon) for ages from 6 years to 90 years. This data was smoothened using the Stochastic Frontier Function. By fitting this smoothened data, we get a very complex and long mathematical function that gives us the ability to theoretically predict the world record for any distance between 5K and full marathon, and any age between 14 and 70 (there was too much scatter - or inaccuracy - for ages outside of this range), and for men or women.

Note that for ‘unusual’ distances and age combination (say, age 48, female, 7.283 km), there will never be any world record noted. But our model can give this number: 24:32 minutes! Our model predicts results within 3% of existing world records when we feed the actual data for age, gender and race distance, which is a phenomenal accuracy, if you consider that a mathematical equation can never capture the so-called Human Spirit. There will always be a 35-year old Haile Gabrasselasie racing to a marathon world record of 2:03:58, when the conventional wisdom expects a 25-year old Samuel Wanjiru to do so. Even our model fails to predict Haile’s achievement. As per our model, theoretically, a 27-year old should own the marathon world record. But then, there is only one Haile Gabrasselasie. Unfortunately for our model, and for Samuel Wanjiru, one Haile is one too many.

i-Race Mathematical Model: Step 2

Historically, it appears that the God reserved Olympic Gold Medals in shorter races for taller runners and vice versa. If we have a world class runner, he would always be a marathon straggler simply because he is 6 feet 6 inches tall. We need to adjust for this limitation, too.

Height

As the race distance increases, the best athletes for those distances are shorter in stature. This trend appears in women, too. Perhaps, there is an optimal height for each race distance.

Weight

The weight of the runner is a clear handicap as the runner has to ‘carry’ more weight across the distance. For every kg increase in the weight, the marathon finish time increases by a few minutes for the same runner!

Equation for Step 2

Unfortunately, no one has ever recorded heights and weights of the runners who set the world records at different distances and ages. So we cannot use the approach of Step 1 above.

We took the data from some separate studies of world-class (but unfortunately, not the Best in the World) runners. The average height of the best runners for various race distances was plotted and an optimal height was calculated as a function of race distance.

The weight is supposed to be a function of height. Theoretically, the weight should scale as a cube of height. If you take a 3 feet tall child and ‘stretch’ it to twice its dimensions, the child will be 6 feet tall but even the width as well as the depth (front-to-back distance) of the child would double, too. So the new 6 feet tall ‘child’ will be 2 x 2 x 2 = 8 times more in mass and weight.

In reality, certain body parts don’t scale with such doubling of height. For example, the heads of children are much bigger proportionate to their bodies. So as the child doubles in height, its weight does not increase 2 x 2 x 2 times. Years ago, some scientists said that it would increase 2 x 2 or 4 times. Hence, they coined a term, Body Mass Index (BMI), which should stay constant, independent of such scaling. And so it is widely assumed that the ratio of the weight to the square of the height should stay constant, when such scaling happens. Unfortunately, if one looks at the real-life data, the weight increases neither as square nor as cube of the height. In reality, this exponent is somewhere in between 2 and 3, almost 2.6.

Thus, we can adjust the slowdown in the runner of sub-optimal height by adjusting the Human Frontier for that height vis-à-vis the optimal height for that race distance. For example, the ideal height for a 10K race is 1.70 meters or 5 feet 6.5 inches. If you have a runner of 1.9 meters height, his slowdown would be (1.9 / 1.7) ^ 2.6 = 1.335 or 33.5% compared to the optimal runner. Of course, in reality, a part of that gets reflected, and thereby adjusted, in the actual weight of the runner.

The ideal or optimal weight is a tricky issue. An obese individual and a body-builder may have exactly the same weight and height. But their racing abilities may differ. However, we decided to focus on the Human Frontier and stipulated that the world record holder runner will have an optimal body height and weight.

We modeled that the Human Frontier will drop in inverse proportion as the weight increases above the optimal weight. We further postulated, and this is the only weak assumption in the model, that if the runner’s weight is below the optimal weight, the Human Frontier stays the same, i.e. the runner does not speed up. However, considering that there exists an ideal weight for a race distance, apparently if the runner’s weight is below that, he or she runs slower, not at the same speed. We have no way to model this slowdown. Luckily, 95% of our i-Race participants are on higher side of the optimal weight!

Finally, the question still remains whether the optimal weight and height structure stays the same or similar for different ages. Since there is no data on this, neither we nor anyone in the world knows that. Hence, the i-Race Mathematical Model does not make omnibus claims about the human racing ability. Instead, it specifies:

Given a runner’s age, gender, weight, height and race distance, the i-Race Mathematical Model predicts the world record for that race distance that can be set by the ‘Human Frontier’ (an optimal human being of the runner’s specified age, gender, weight and height, if he or she ever existed).

Note the word ‘Optimal’ is different from the word ‘Ideal’. ‘Ideal’ is an absolute, time-independent level. We human beings may never reach the humanity’s ideal abilities. ‘Optimal’ is a relative term; the current world records are held by ‘optimal’ human beings. And as the new world records are set, the definition of ‘optimal’ will keep improving, and our equation will keep updating.

Your ability to win the i-Race will be determined by how close in percentage terms you can finish compared to that theoretical ‘optimal’ human being, of your age, gender, weight and height, if he / she ever existed!