The Journey To 13 Miles: A Monte Carlo Story
In early September of 2024, I decided to start training to be able to run 13 miles. The aim was to achieve this goal by March 31st. More details on why I chose this goal are in this article's “Setting Goals” section. I used to run 2–3 miles regularly, but it had been a couple of months since I had been running. This meant finding out what my baseline was. Then, figuring out at what rate do I need to increase my running distance to get to running 13 miles by March 31st.
The Average
Let us look at it from a naive perspective, using the average rate of increase. My first run when I started this journey was on September 12th. It was for a distance of 2.14 miles. This lays out the variables as follows -
Starting Distance = 2.14 Miles
Ending Distance = 13 Miles
Days Available (Mar 31 — Sep 12) = 200
Simple math here would tell us that I will have to increase at a steady rate of X(calculated below) daily to get to my desired outcome. We can calculate this X as
(Ending Distance — Starting Distance)/Days
= (13–2.14)/200
= 0.0543 miles/day
So, if I increase my running distance by 0.0543 miles per day, I will hit my target. Simple!
The Reality
Anyone who has attempted anything like this will tell you that you don't increase your distance by a precise amount every day. You don't even run every day. Too many days of running in a week can cause fatigue, leading to injuries. Over the past month or so, I have discovered other factors that have caused disruptions -
- Random injuries — I wasn't able to run for a few days as I picked up a backache while playing cricket (as I do on Sundays)
- Work — I had a series of classes in the morning, and that disrupted my running schedule (I only run in the mornings since, in Florida, where I live, running in the afternoon or evening is essentially a death sentence)
- Natural Disasters — Hurricanes forecasted to come through the area disrupted some of my run days. At least one of the days, it looked like I was as likely to fly as run if I stepped outside.
- General fatigue and off-days — Some days, when I set out for a longer run, for whatever reason, my body did not respond in a way that allowed for a long run.
These interruptions have resulted in the following chart tracking my Max Run Distance
The progress has not been a simple, steady increase. The mileage increase has been in spurts. Let's compare the progress to where the average said I should have been. Today is October 17th, and my maximum distance currently sits at 4.39 miles. Doing the math using the average, I should currently be at 4.0405 miles. That looks good. I am definitely on track. Does this mean I can be 100% confident that I will hit my goal? What about all the curveballs that reality has been throwing my way? Will they change the likelihood of a positive outcome?
The Return of Monte Carlo
If you are one of the folks that has been to this blog before, I am sure you have heard of Monte Carlo simulations. If you have not, the first 7 minutes of the video below provides a good overview —
We can take our past data and run it through a Monte Carlo engine to get an idea of our likelihood of success. Why don't we do that?
Below is a chart of how much my run distance has increased by, every day. The dates with 0 means either I did not run that day or the distance I ran was below my current maximum.
We can use this data to sample for all future dates. Tomorrow is October 18th. I can assume it is going to lead to an increase from any random day on this chart. Let us say that the random selection is September 25th, the increase in that date was 0. I can randomly select for October 19th as well. Let us say the randomly selected date from the past is September 30th. That would mean an increase of 0.23 miles. If I keep doing this for dates all the way up to March 31st, I would get one single Monte Carlo simulation. The sum of all the random samplings in the Monte Carlo simulations, when added to the current maximum distance, would tell me what that simulation expects my run distance to be on March 31st.
The Results
We can run these simulations over and over again. I ran them 1,000 times. Each simulation came up with a different result. They ranged anywhere from my maximum run distance on March 31st being 9.04 miles to 21.24 miles. Great, now we can apply all the fun Monte Carlo interpretation techniques we love. The histogram for results is shown below.
When we do some more math on these, we can get the percentiles for these results —
Based on these results, I am 50% likely to get to a max distance of 14.455 miles or more. I am 95% likely to get to 11.909 miles or more.
But, what about my target of 13 miles? Well, we can simply count what percentage of the results are above 13 miles. It turns out that there are 81.60% of the results where the total is at or above 13 miles.
Based on these numbers, I should be a little more than 80% confident that I can hit the goal that I have set.
What Now (Continuous Forecasting)?
These results are great news as they stand. I know a few things that are going to happen in the near future. We have holidays coming up, which might mean some merriment and family time, which might slow me down a bit. There is also a week-long vacation coming up in December. Through those, though, I intend to keep making progress. Just because I intend to do so might not mean it will necessarily happen. I will need to forecast again after those events. Luckily, I have all the simulations set up. As I enter new up-to-date data, the simulations run again and give me new, updated information about my likelihood of meeting my goal.
Why even wait for major events to happen? I can update the simulations with the latest data on a daily basis and continuously find out if I need to push myself a bit harder to maintain a high probability of hitting my goal. At all times, I have an idea of the risk inherent in my goal. Instead of flying blind, assuming a 100% certainty, I can make adjustments to manage that risk.
