TLDR: I analyzed my Strava records of 20K-30K runs in the past two years and conclude that there are 2.3-3.5% loss of speed per 1 degree Celsius increase of temperature, on condition that the **Heart Rate Reserve (HRR) stays constant**.

## My data

My first trail race was in Mar 2021. My structured training & tracking started at end of 2020/ beginning of 2021. To ensure the data consistency and also cover different time of year, I back date an integral number of years.

I first start from May 21, 2021, which is 2 integral years to today.

Here are key definitions:

- HR efficiency is the average speed divided by HRR.
- The average speed here is the equivalent distance divided by time.
- We convert every 100m ascent to 1000m distance to obtain equivalent distance.

HR efficiency captures a relative performance impact, and makes this study different from most other works that mainly look at maximal/ sub-maximal performance based on race data. Our concept of HR efficiency is very like the "performance condition" termed by some watches. In the following discussion, we will use HR efficiency and performance condition interchangeably.

Following is the scatter plot of average temperature and HR efficiency.

We can see a rough trend of declining performance when temperature increases. Turning this into boxplot to viauzlize the distribution of performance condition.

One glitch is that, statistically the HR efficiency becomes lower when temperature drops below 25C (we'd expect it to be higher).

One key factor not in above 2 dimensional analysis is equivalent distance. When temperature is lower, we tend to do more long distance, e.g. HK100 is in Jan/ Feb. We can first get an idea of distance impact on HR efficiency below.

It is expected that longer total distance results in lower overall HR efficiency, mainly due to fatigue/ overuse muscles.

We can limit the study distance range to 20K-30K to workout the quantitative impact. This distance window is a sensible choice as I had run the range in both summer and winter. There are also adequate data points for regression.

The following boxplot makes more sense. Although there is some randomness, we can visually tell the negative correlation between HR efficiency and temperature.

Fitting the data to a line, we get coefficients: `array([-0.02270856, 5.18841753])`

.

Given our definition, that means `HREfficiency = Speed / HRR = -0.02270856 * T + 5.18841753`

.

The coefficient before T is what we are looking for. The result can be interpreted as: there is -2.27% loss of speed with 1 Celsius increase of temperature when HRR stays the same.

The key constraint here is a constant HRR. When temperature increases, the increase of HRR is a combination of different factors:

- Body needs to sweat more to control temperature. Faster HR helps to circulate the heat.
- Higher temperature may impact enzymes efficiency, and body uses HR to compensate.
- Water loss results in less blood pumped per heart beat, so higher HR is needed for same energy output level.

In my experience, when I run at the same speed with same feeling, my HR can be 10 BPM more in summer than in winter. The implication is that, for performance purpose, I can use 160 BPM for a race in summer and feel similarly difficult as when I use 150 BPM in winter. In the studies of race data, it is likely that people naturally used higher HR to compensate the weather condition.

So the key question is, if I want to keep a constant feeling (not necessary constant HR), what is the impact of temperature?

Feeling is very personal and relies on exercise/ sense to be able to tell the difference. I don't have enough data points; actually, I have not systematically logged "Relative Effort" in Strava.

Without this feeling data, the most close thing I can make is to tabulate pace versus temperature, so I can use this table to see if I am in "normal" status. I will keep the following table with me and look up a pace and temperature occasionally during exercises. If the HR is not too much higher than the forecast, I can keep going. Or else, I shall stop and exit.

The analysis so far uses my past two years data. Another major variable is that my training paid off and results in higher HR efficiency later on. Towards this end, we repeat the above analysis using most recent one year data.

The coefficients are `array([-0.03513679, 5.88428578])`

.

The tabulared result is below:

One can see this new table has more blue cells. For example, when temperature is 15C, I used to run 5:00 pace with 142 BPM HR; now I can run 5:00 pace with 134 HR. This is the positive result from hard training.

However, we note that the coefficient is -3.5%, which implies a sharper drop of HR efficiency. As sports performance grows, the body is more sensitive to heat imapct.

## Other researches

My initial thought was to search the Internet but found only vaguely useful references.

ChatGPT helped to give some summarised idea that every 1 degree of Celsius results in 2-3%/ 2.0%/ 1.76% from some literatures. There are no exact bibliography given no matter how I asked.

After some manual search, I find a study [^2] of 46 years Boston Marathon results revealed a 2.5 seconds/ KM pace loss per 1 degree Celsius increase in temperature. The limitation is apparent: Boston Marathon is an elite event and the race happens at cool dates by Hong Kong standard.

Since the testing environment, running distance and runner's profiles are different, those tests may not be very comparable.

Or a more fundamental question: is the impact of temperature on running speed linear?

Some literature [^1] suggests the performance decrease in quadratic relation to temperature, as shown below.

Note that 5-25 degrees Marathons are well studied. There are rarely races in very high temperature condition; Or the race data was not so well maintained. Another key factor is that **trail running** we care more requires one to load 1-2KG of water even with adequate supply. The increase in weight load further impacts running performance.

For my upcoming activity, I need to forecast for a temperature of 28-30C. My personal condition is also different from other runners, so I ended up doing this analysis myself.

## Nuances

Talking about performance impact of temperature without proper context is misleading. We note some nuances of this work and some similar quantitative analysis:

- Course type: the ascent v.s. distance combination. Ascent implies different workload across runners. (aka 100m ascent = X distance?)
- Total distance: the longer distance, the more negative impact.
- HRR: does this work refer to maximal/ submaximal (i.e. during race) performance? Or does it refer to casual exercises?
- Support type: does the runner carry water (trails) or nothing (marathon).
- Hormones in the air of race venue can also incude higher HR even when the power output and feeling stay the same.

## References

[^1]: Nour El Helou ,Muriel Tafflet,Geoffroy Berthelot,Julien Tolaini,Andy Marc,Marion Guillaume,Christophe Hausswirth,Jean-François Toussaint, 2012, Impact of Environmental Parameters on Marathon Running Performance. [^2]: Beat Knechtle ,Stefania Di Gangi,Christoph Alexander Rüst,Elias Villiger,Thomas Rosemann,Pantelis Theo Nikolaidis, Published: March 8, 2019, The role of weather conditions on running performance in the Boston Marathon from 1972 to 2018.