# A new average speed metric

## Introduction

The most commonly used metric for average speed of a bike ride (or other activity) is vague, arbitrary and often gives a very poor indication of effort. Here we propose an elegant alternative that is a better indication of effort and is very easily computed.

## Average Speed

Often in cycling we are interested in some measure of our average speed over a ride or part thereof. This simplest way to do this is divide the distance travelled by the elapsed time. This is rarely used because it it so influenced by breaks in the riding due to red lights, getting more fluids (or getting rid of fluids), rests, lunch, etc.

## Average active speed

Average "active" speed is a much more commonly used metric. It is the total distance travelled divided by the total active time. The notion of active time is rather vague. Typically there is some speed threshold and if the speed is below that threshold it is generally not considered active time. There may also be time threshold in the transition between active and inactive time, so if the speed is low only briefly it is still considered active time. Different hardware and software combinations on bike computers and activity tracking software have different ways of computing average active speed. Any precise definition has arbitrary thresholds, making it an unappealing hack for those who like elegance.

Average active speed is also a frustratingly poor measure of effort when the speed is uneven. Although time when you are stopped is (mostly) discounted, moving very slowly (but above the threshold speed) for a relatively short distance can have a significant effect on the average. For example, if you do a flat ride at constant 30kph for 30km (1 hour) but also include 500m at 5kph (6 minutes), your average active speed drops to 27.73kph (for simplicity we ignore accelleration and decelleration). Riding 30.5km at 27.73kph is significantly less effort. Similarly, average active speed greatly underestimates effort when there are hills. Suppose you grind up a climb at 12.5kph for 40 minutes (8.3km) then zoom back down at 50kph for 10 minutes (again, ignoring accelleration). The average active speed (and average speed) metric gives your climbing speed four times the weight of your descending speed because climbing takes four times as long, resulting in a 20kph average.

## Positional average speed

Average speed can be viewed as taking the speed each second (or suitably small time interval) of the ride and taking the average. More technically, if we plot a curve of the speed versus time, the average speed is the area under the curve divided by the total time. Average active speed is the same except inactive periods are ignored. Our proposed metric, "positional average speed", can be viewed as taking the speed each metre (or suitably small distance interval) and taking the average. More technically, if we plot a curve of the speed versus distance travelled, the positional average speed is the area under the curve divided by the total distance. When the speed is zero, no distance is covered so stops are naturally ignored by this metric without the need for arbitrary thresholds. It is also a better measure of effort than average active speed. In the flat ride example given above, the positional average speed is 29.51kph. For the hilly ride, equal weight is given to the climbing and descending speed, giving an average of 31.25kph. It's hard to measure overall effort without additional information such as power output, but it seems positional average speed gives a significantly better indication than average active speed. If speed does not vary at all, average speed and positional average speed are identical, otherwise positional average speed is stricty greater.

## Computing average speed metrics

For any metric, it is advantageous if it can be computed in "real time", updating a display on a bike computer, using minimal storage and computational resources. Data such as position and time are typically sampled at frequent intervals and speed is computed from these (using some method that attempts to overcome the significant noise in GPS position data) or other inputs. Average speed can be computed by regularly reading the speed and using just two accumulators: the sum of all speed readings so far and the number of speed readings. The average speed is the former divided by the latter. For average active speed, inactive periods can be ignored (with some suitable simple calculation of inactive). Positional average speed can also be computed using two accumulators: the sum of all the speed readings and the the sum of the squares of all the speed readings. The positional average speed is the latter divided by the former. Technically, such sampling methods are essentially a form of numerical integration, which results in an approximation to the true value. However, the position and speed data contains significant noise, so approximation is inevitable.

## Experimentation with gpx files

We have written a simple script that uses `gpsbabel` to read gpx files and compute speed data, which is then used to compute the average speed metrics. The most difficult challenge is finding an appropriate way of smoothing the noisey gpx data to the computed speeds are reasonably accurate. We use the "distance" parameter of `gpsbabel` to filter out data points that are close together. It seems that most of the bike computer hardware and software systems use proprietary methods to smooth the data in some way.

Lee