Improved prediction of hiking speeds using a data driven approach

Andrew Wood1*, William Mackaness2, T. Ian Simpson1, J. Douglas Armstrong1

1 School of Informatics, University of Edinburgh, Edinburgh, UK

2 School of Geoscience, University of Edinburgh, Edinburgh, UK

*<EMAIL_ADDRESS>
## Abstract

Hikers and hillwalkers typically use the gradient in the direction of travel
(walking slope) as the main variable in established methods for predicting
walking time (via the walking speed) along a route. Research into fell-running
has suggested further variables which may improve speed algorithms in this
context; the gradient of the terrain (hill slope) and the level of terrain
obstruction. Recent improvements in data availability, as well as widespread
use of GPS tracking now make it possible to explore these variables in a
walking speed model at a sufficient scale to test statistical significance. We
tested various established models used to predict walking speed against public
GPS data from almost 88,000 km of UK walking / hiking tracks. Tracks were
filtered to remove breaks and non-walking sections. A new generalised linear
model (GLM) was then used to predict walking speeds. Key differences between
the GLM and established rules were that the GLM considered the gradient of the
terrain (hill slope) irrespective of walking slope, as well as the terrain
type and level of terrain obstruction in off-road travel. All of these factors
were shown to be highly significant, and this is supported by a lower root-
mean-square-error compared to existing functions. We also observed an increase
in RMSE between the GLM and established methods as hill slope increases,
further supporting the importance of this variable.

## Introduction

Knowing how fast people are able to walk between locations is critical
information in many situations. In hiking and hillwalking scenarios, this
information is vital for safety reasons. If you are leaving in the morning for
a hike then it is good practice to provide an estimated return time such that
emergency services can be contacted if you get into difficulty and do not
return [1]. An inaccurate estimate for how long a route will take could lead
to unnecessary callouts, or delay a callout in a situation where every minute
is important. Furthermore, in circumstances where a hiker has gone missing, an
accurate measure of walking speed can help to restrict a potential search area
around a last known location. Finally, when out on a hike there are situations
where hikers may be deciding whether to follow a footpath, or take a more
direct cross-country route. Accurate estimates of the walking speed and time
for both scenarios are required to be able to select the optimal route.

There are a multitude of factors which can impact the walking speed and time
predictions for a route [2], although these can generally be split into two
categories [3, 4]. The first category covers the individual effects which
depend on who precisely is undertaking the walk, and when they are doing it.
These effects include group size (larger groups often walk slower), age or
fitness of participants, and weather conditions, as well as the aim of the
walk (afternoon stroll vs. specific hike). The second category covers the
fixed effects which will affect all individuals who attempt the same route.
These include how steep the terrain is and whether the route is paved, along a
track or in wild country.

Most of the individual effects cannot be modelled without considerable prior
knowledge about the person who is planning a route. Therefore, most existing
hiking route planners calculate the walking speed solely based on the terrain,
and this is presented as the average time (or time range) it takes to complete
a hike. It is then left up to the individual to tune the predicted time for a
hike given their knowledge about personal ability and circumstances.

Formulae of varying complexity have been proposed to estimate human walking
speed and time along a projected path. A popular early method that is still
widely used was put forward by Naismith [5] which calculates walking time
under normal conditions as:

> “an hour for every three miles on the map, with an additional hour for every
> 2,000 feet of ascent.”

This approximates to a walking speed of 5 km/h with 10 minutes added on for
every 100 m of ascent. This was later adjusted by Aitken [6], who introduced a
reduced base movement speed of 4 km/h on surfaces which are not paths or
roads. Naismith’s rule is still used today by Scout groups and other casual
hikers due to the ease of calculating walking time by hand using a paper map.
However, despite the widespread use, Naismith’s rule does have a well-known
limitation; namely that the predicted speed does not change when descending a
hill, regardless of the gradient.

An alternative hiking function proposed by Tobler [7], has become more popular
in recent research and other situations where speeds do not need to be
calculated by hand:

$W=6*exp(-3.5|S+0.05|),$

where

> W = velocity (km/h)
> S = gradient of slope.

Like Naismith’s rule, this gives a speed of 5 km/h on flat ground, with a
maximum speed of 6 km/h on a mild descent (around 3 degrees). In a similar
manner to Aitken’s correction, a factor of 0.6 is applied to the calculated
speed for all off-road travel. Tobler’s function avoids the issues seen in
Naismith’s rule when descending slopes, but it predicts a sharp peak in
walking speed on mild descents, which may be unrealistic. The formulae
discussed here are directly compared in Fig 1.

Fig 1: Existing functions used to calculate walking speed. Naismith’s rule
[5], Tobler’s hiking function [7] and Campbell et al.’s function [8] plotted
as predicted walking speed in km/h against the slope in the direction of
travel (walking slope) in degrees where positive is uphill. For Naismith’s
function and Tobler’s function, on and off-path versions are shown.

Other studies have also looked at providing alternative methods to calculate
walking speeds [9, 10, 11], but all continue to use walking slope as the main
variable to determine walking speed (with various multiplicative factors
applied for off-road travel).

When exploring speeds of fell-runners, Arnet [12] suggested that movement
velocity may be dependent on three factors: obstruction (with different
factors applied depending on the kind of obstruction), ascent in the run
direction (walking slope) and slope of the terrain (hill slope). The actual
values used in Arnet’s calculations cannot be directly applied to walking
speeds as they were based on orienteering championships where participants
were running.

Experience tells us that traversing on a steep hill (while maintaining
constant elevation) is more difficult than traversing flat ground. However,
the existing methods estimate the same walking speed for both situations.
Similarly, high levels of terrain obstruction in off-road areas (such as a
thick gorse bush) are much more difficult to walk through than empty fields.
The simple multipliers for off-road travel in Aitken’s correction and Tobler’s
function do not provide any further distinction between two such regions.

Wood and Schmidtlein [13], took all three of Arnet’s factors into account, and
looked at evacuating citizens in the event of a hurricane. They applied
Tobler’s function to both the hill slopes and walking slopes, and calculated
the terrain obstruction coefficients based on energy usage rather than walking
speed (using [14]). They accepted that these were likely not the correct
values, but were unable to find any better alternatives. Campbell, Dennison,
and Butler [15] conducted a study using lidar data to explore the effects of
ground roughness and vegetation density on firefighter evacuation speeds, but
they did not consider the hill slope separately.

All of the studies mentioned above utilised relatively small sample sizes.
However, the rise in use of global navigation satellite systems (GNSS), more
frequently referred to as GPS tracking, means that a data-driven approach to
modelling walking speed is now possible, which provides two main benefits.
Firstly, it is possible to access GPS tracks from a wide variety of regions
and terrains. Secondly, each track can easily be broken down into individual
sections, enabling specific route features to be investigated at much higher
spatio-temporal resolution. This has been explored in recent work [16, 8],
however the crowdsourced nature of these studies meant that data collection
was not controlled, and thus that the data could not be assumed to consist
wholly of walking or hiking tracks. In [16], data from hikes, jogs and runs
was processed together, resulting in a very wide range of movement speed
estimates. Campbell et al. attempted to overcome this in [8] by only
considering data points with a speed between 0.2 m/s and 5 m/s (and the
resulting model is shown in Fig 1). However, 5 m/s (18 km/h) is much higher
than the maximum predicted speeds from existing methods (such as Naismith’s
rule), so it is likely some non-walking data remained. Furthermore, applying a
blanket 0.2 m/s minimum speed may well overlook valid datapoints recorded by
particularly slow individuals, or in especially difficult regions. Finally,
although these studies had the benefit of using large sample sizes, they both
looked solely at the effect of the walking slope on speed, and did not explore
additional variables.

Here we used a data-driven approach to explore the impact of all three factors
discussed by Arnet on walking speeds. These are the walking slope, the hill
slope and the terrain obstruction. We aimed to use these factors to develop a
model for the walking speed for an average individual. As with the existing
methods, this model did not seek to model individual effects, and would still
require tuning based on personal ability or conditions.

## Materials and methods

### Data set, cleaning and key assumptions

Full details of the various datasets used in this study are provided in S1
Supporting Information.. Further, a detailed description of the data filtering
processes, and choices/assumptions made during data processing are described
in S2 Supporting Information.

In summary, GPS tracks were obtained for hikes in the UK from Hikr.org [17]
and OpenStreetMap (OSM) [18]. Elevation and walking slope values were
calculated and added to every GPS point using data from the Ordnance Survey
Terrain 5 Digital Terrain Map (DTM), which provides elevation data at 5 m
intervals across the whole of the UK [19]. Hill slope values were found using
the quadratic surface method [20, 21]. Each data point was then classified as
on a paved road, on an unpaved road, or off road, determined by searching a 50
m radius around each point in an OSM Road dataset [22]. Paved and unpaved road
classification was determined using [23], with the unpaved road values being
‘path’, ‘bridleway’ and ‘track’.

Terrain obstruction information was calculated using lidar datasets [24, 25,
26], as the difference in values between a Digital Surface Map (DSM) and
Digital Terrain Map (DTM). This meant that any physical feature which
protruded from the ground was regarded as an obstruction. We had access to
lidar data at 2 m resolution covering large areas of England and Wales, but
the coverage was not complete. Of our off-road data ($\sim$2,900 km, spread
across over 1,200 tracks), over 2,000 km had lidar data available. Exploration
of the lidar data (see S5 Supporting Information) showed that there was a
clear drop in walking speeds once the height of an obstruction was greater
than 10 cm, beyond which the speed was relatively constant. We used this
information to classify points into heavy obstruction (>10 cm) or light
obstruction ($<=10$ cm) for modelling purposes.

Visual inspection of the tracks showed that a large number contained long
breaks which could impact the accuracy of a walking speed model. Fig 2 shows
examples of regions where breaks are visible in a GPS track, and the process
developed to identify these regions is outlined in Algorithm 1.

Fig 2: A GPS track where 3 breaks can be identified by finding point clusters.
Clusters of points can form on a GPS track when a break is taken during a
hike. By identifying these clusters as potential breaks we are able to remove
most break periods from the datasets used for our analysis of walking speeds.
For full details of these and other data filtering methods see S2 Supporting
Information. Background images from OpenStreetMap and OpenStreetMap Foundation
[27], visualised using QGIS [28]. Algorithm 1 Breakfinding process for a GPX
track segment

1:Breakpoint_list $=\emptyset$

2:Find the median distance ($r_{median}$) and speed ($s_{median}$) of the
segment

3:for point ($p_{i}$) in segment do

4: Calculate travel direction quadrant and point angle

5: Calculate break likelihood using the point speed and angle

6: if speed == 0 or distance >1 km or duration >3 minutes then

7: Breakpoint_list += $p_{i}$

8: end if

9: if speed >10 km/h and duration($p_{i-1}$) >3 minutes then

10: Breakpoint_list += $p_{i}$

11: end if

12:end for

13:for point ($p$) in segment do

14: if Neighbourhood of $p$ is a cluster ($C$) then $\triangleright$ See Defs
1 & 2, S2 Supporting Information

15: for point ($p_{c}$) in $C$ do

16: if Neighbourhood of $p_{c}$ is a new cluster ($C_{n}$) then

17: $C=C\cap C_{n}$

18: end if

19: end for

20: Remove points at the ends of the cluster with low break likelihood

21: Add ‘missing’ points to the cluster (to make a continuous run of points)
to form a Potential Break (B*)

22: if less than half the points in B* have low break likelihood and there is
travel in opposite quadrants (Q1 & 3 or Q2 & 4) then

23: Breakpoint_list += B*

24: end if

25: end if

26:end for

Where the datapoints in the original GPS track were under 50 m in length, they
were merged together to minimise the effects of errors in the GPS location
values. While doing this, the resulting distance was the sum of all distances
in the constituent GPS points, so may be longer than the straight line
distance between co-ordinates. Similarly, both hill and walking slope values,
as well as obstruction height, were calculated as the weighted average of
constituent points, weighted by point duration.

While the Hikr dataset consisted of tracks which were tagged as a walk or
hike, within some of these there were segments where it was clear that the
participant was driving to or from the hike location, based on the observed
speeds. The OSM data, on the other hand, was not filtered by transport type.
There were a large number of tracks which were clearly from faster modes of
transport, as their speed was implausible for a hiker. A process to remove
these non-walking tracks and segments was created, whereby the known Hikr
walking segments were used to create filtering bounds of plausible walking
speeds, which could then be applied to the remainder of the dataset. This
process is summarised in Algorithm 2.

Algorithm 2 Filtering process for GPS data from Hikr and OpenStreetMap

1:Remove duplicate segments (containing sections with identical start
location, end location, start time and duration)

2:Remove all segments with median speed >10 km/h

3:Remove all breaks with duration >30 seconds

4:Remove all breaks containing points with speed >10 km/h or distance >1 km

5:Merge remaining points into sections at least 50 m in length.

6:Recursively remove points with speed >10 km/h adjacent to a break, or the
end of the track

7:

8:if Hikr data then

9: if segment mean speed >10 km/h then

10: remove segment

11: end if

12: Calculate filtering bounds $\triangleright$ Eq (2) - (5), S2 Supporting
Information

13:else

14: Identify Key Points $\triangleright$ see S2 Supporting Information

15: Remove single datapoints between Key Points

16: Remove points where median speed between consecutive key points >Eq (2)

17: while segment length is not consistent do

18: Remove points with speed >10 km/h adjacent to a break, or the end of the
track

19: if segment median speed >Eq (2) or segment minimum speed >Eq (3) or
segment upper quartile speed >Eq (4) or segment upper whisker speed <Eq (5) or
segment duretion <2.5 minutes then

20: Remove segment

21: end if

22: end while

23:end if

24:

25:Combine all segments into a single dataset

26:Remove the fastest and slowest 0.5% of the data

Following this, a decision was made to remove data from tracks found in
Scotland. Lidar data covering the walking tracks was necessary to model the
terrain obstruction, and was not sufficiently available in Scotland at the
time of the study. Furthermore, analysis showed that that walking speeds in
Scotland were at the extreme end of what is seen throughout the rest of the UK
(see S4 Supporting Information.). Including this data without also including a
corresponding extreme dataset where lidar data is available may result in
incorrect modelling. All OSM track segments which took place within Scotland
were excluded from further processing. Similarly Hikr tracks which were tagged
as taking place in Scotland, and which fully took place in Scotland were
excluded.

Our final modelling dataset consisted of 7,636 GPS tracks from England and
Wales, with over 1.4 million individual data points and almost 88,000 km of
travel. Each datapoint represented approximately 50-100 m of travel, and
contained:

  * •

Start coordinate

  * •

End coordinate

  * •

Start time

  * •

Duration

  * •

Distance

  * •

Speed

  * •

Elevation

  * •

Walking slope

  * •

Hill slope

  * •

On-road flag

  * •

Paved road flag (if on-road)

  * •

Obstruction data available flag (if off-road)

  * •

Heavy obstruction flag (if off-road and obstruction data available)

### Modelling

#### Model Formulation

Pilot studies were conducted to identify an appropriate model framework, using
tracks within Scotland (see S3 Supporting Information.). Generalised linear
model (GLM) and generalised additive model (GAM) approaches were explored, and
within both we looked at the relationship between the walking and hill slopes,
and the walking speed, with a small number of prior assumptions. As it is more
challenging to walk on steeper slopes, for both the hill and walking slope
components we knew that the walking speed should be a decreasing function of
the magnitude of slope (with some allowance for faster walking speeds on mild
descents). Models which failed to predict this were removed under the
assumption that the data were overfitted. Furthermore, previous work [11, 29,
30, 31] has identified the existence of a critical gradient; the angle at
which it is faster to zig-zag up a hill, rather than ascend directly. This
occurs at a walking slope of around 15 – 21 degrees, so models which failed to
predict the critical gradient occurring below 21 degrees were removed.

10-fold cross-validation was used to compare the remaining model parameters,
looking at R-squared values, root-mean-squared error (RMSE) and mean absolute
error. Where multiple models performed equally well, the simplest model was
selected for ease of interpretabilty and real-world application. The selected
model type was a Generalised Linear Model (GLM). Models were implemented using
R version 3.6.1 [32].

#### Terrain Types

Each of the three road types (paved road, unpaved road, off-road) was included
in the model, both as factor variables, and as interaction terms with each of
the slope variables.

Before adding terrain obstruction data to the model, we checked that there was
no systematic difference between the walking speeds in regions where we had
lidar data, and regions where we did not (see S5 Supporting Information). Thus
our findings in regions where lidar data was available could be extended to
those where it was unavailable. Factor variables were then added to the model
for each obstruction level (heavy, light or unknown obstruction).

#### Statistical Analysis

Variables within the model were tested for significance using the Wald test,
which allows us to account for correlation between points within the same
track (coeftest function within lmtest package in R).

To measure the impact of our model, we compared walking speed predictions of
our model against those of Naismith’s, Tobler’s and Campbell et al.’s models.
Four different metrics were compared; the average percentage error, mean
squared error (MSE), root-mean squared error (RMSE) and R squared value. These
were explored when looking at both individual 50 m track sections, as well as
predicted walking times for tracks as a whole. Finally, we isolated the off-
road track sections in order to assess the improvement of our model at
predicting walking speeds for off-path travel.

### Code Availability

Documented code written for this study is available online in a Github
repository AndrewWood94/PhDThesis, and is licensed under the terms of the GNU
General Public License v3.0.

## Results

We started by assembling a dataset of hikes derived from approximately 20,000
public GPS tracks. These tracks recorded a variety of transport methods and
required significant filtering. This process included iterative data cleaning
to remove erroneous or non-walking data and identify/remove breaks (e.g. Fig
2) to give us a final usable dataset containing 7,636 GPS tracks, with over
1.4 million individual data points and covering almost 88,000 km of travel in
the U.K. Each data point represents at least 50 m of travel (with a mean
distance of 60.3 m), and the breakdown of the data by slope angle and terrain
type is shown in Table 1. Previous research has found that most walking takes
place on low walking slopes [33], and this is evidenced by our data ($\sim$98%
of our data was from walking slopes of under 10 degrees).

Table 1: Total distance of data under different terrain conditions (km) | Hill Slope (degrees)  | $\mathbf{|Walking\ Slope|\ (degrees)}$
---|---|---
| 0-10 | 10-20 | >20  | 0-10 | 10-20 | >20
Paved road  | 62159.1 | 7841.2 | 2081.9  | 70726.5 | 1277.3 | 78.4
Unpaved road  | 9996.9 | 2210.3 | 700.7  | 12421.7 | 460.0 | 26.2
Off Road (obstruction unknown)  | 773.5 | 114.2 | 17.8  | 871.7 | 31.7 | 2.0
Off Road (light obstruction)  | 1282.9 | 150.1 | 23.8  | 1424.6 | 30.6 | 1.7
Off Road (heavy obstruction)  | 428.7 | 105.2 | 28.5  | 543.5 | 18.5 | 0.4

Our curated hike dataset allowed us to create a data-driven model which we can
directly compare with existing walking speed algorithms. The model formulation
was selected using a small-scale exploratory study which considered data from
Scotland (see S3 Supporting Information.). In this exploratory study, multiple
different model types were explored which could fit the data, and which
matched existing knowledge about walking speeds. Cross-validation methods
showed that there was very little difference in performance of the best
models, therefore the final model was a Generalised Linear Model (GLM), which
was chosen as it was the simplest of those tested (we had no evidence that a
more complex model would be superior). This choice also meant that our model
was both easy to interpret, and simple to apply to future work.

This final GLM model included all three of the variables suggested by Arnet
[12]:

$v=exp(a+b\phi+c\theta+d\theta^{2})$ (1)

where

> $v=\text{walking speed (km/h)}$
> $\phi=\text{hill slope angle (degrees)}$
> $\theta=\text{walking slope angle (degrees)}$

Terrain obstruction level was included as a factor variable, while we
considered the road types as both factor variables and interaction terms. Not
all terms had a significant effect on all variables; we therefore created a
model with all possible terms, and removed them one at a time (in order of
least significance) until all remaining terms were significant to at least 95%
confidence level (using Wald test). The final values for a, b, c and d are
given in Table 2 for each of the terrain obstruction levels and road types.
The critical gradient for this model is between 14 – 16 degrees when walking
uphill and -16 – -18 degrees when walking downhill (depending on road and
obstruction conditions), which is in line with previous findings.

Table 2: Final walking speed model variable coefficients | $a$ | $b$ | $c$ | $d$
---|---|---|---|---
Paved road  | 1.580 | -0.00389 | -0.00726 | -0.00218
Unpaved road  | 1.580 | -0.00389 | -0.00965 | -0.00248
Off-road (obstruction unknown)  | 1.536 | -0.00731 | -0.00965 | -0.00187
Off-road (light obstruction)  | 1.580 | -0.00731 | -0.00965 | -0.00187
Off-road (heavy obstruction)  | 1.443 | -0.00731 | -0.00965 | -0.00187

Fig 3 shows the predicted walking speeds under different conditions. The
importance of including both the hill slope and terrain obstruction variables
can be clearly seen when looking at the Off Road Light Obstruction speed
predictions. When directly ascending or descending a slope, the walking speed
is comparable to walking on a road. However, when traversing a slope while off
road, the walking speed is comparable to traversing a slope of double the
gradient while on a road or path. Similarly, comparing the walking speed
predictions of Off Road Light Obstruction and Off Road Heavy Obstruction
reveals that just 10 cm of vegetation (our cutoff point for heavy obstruction)
can reduce the walking speed by more than 0.5 km/h.

Fig 3: Walking speed predictions under different terrain conditions. When: (A)
travelling directly up or down hills of varying slope (walking slope), (B)
traversing across hills of varying slope (hill slope).

Fig 4 shows the same walking speed predictions as Fig 3, alongside the
confidence interval for the mean walking speed for each terrain type. In the
low-slope regions where most walking occurs, our model fits closely with the
mean data confidence intervals. Our model does deviate from the confidence
interval in some areas, particularly in high-slope and off-road regions.
However, these are also the areas where we have the least amount of data (see
Table 1). In Fig 4J the confidence interval for the mean would suggest that it
is faster to walk on hill slopes of 30 degrees than hill slopes of 10 degrees.
We have less than 30 km of data recorded in heavy obstruction regions on hill
slopes of over 20 degrees, and less than 20 km of this had a walking slope
magnitude of under 5 degrees (indicating that the slope was being traversed).
Further, even within this range, the data is skewed towards the lower hill
slope values. This lack of data explains the widening confidence interval, and
counter-intuitive observations and we suggest that a targeted study would be
required to collect more data in this region.

Fig 4: Walking speed predictions under different terrain conditions. When:
(A,C,E,G,I) travelling directly up or down hills of varying slope (walking
slope), (B,D,F,H,J) traversing across hills of varying slope (hill slope).
Also shown in each plot is the 95% confidence interval of the mean value of
the walking speed for the terrain type, calculated at 5 degree intervals,
using data bins with a width of 10 degrees. Note that the confidence intervals
were calculated using only data which is within 5 degrees of directly
ascending (A,C,E,G,I) or traversing (B,D,F,H,J) the slope.

Fig 5 compares the Paved Road and Off Road Heavy Obstruction speed predictions
from our model against the existing functions from Naismith, Tobler and
Campbell et al. When looking at the walking slope, the largest areas of
deviation between our model and Naismith’s rule occurs when descending a
slope, as Naismith’s rule does not predict a reduced speed in this scenario.
For both Tobler’s and Campbell et al.’s functions, the shape of the walking
slope component is relatively similar to our new model, with the main
distinction being the peak predicted speed on flat ground. None of the
existing functions account for the hill slope, which leads to large
disparities when predicting the walking speed for slope traversals. A further
example of this can be seen in S6 Supporting Information, which shows the
walking speeds for a simulated off-road route which encounters the full range
of hill and walking slopes.

Fig 5: Comparison of new model and existing hiking functions. Predicted
walking speeds of the new model, Naismith’s rule, Tobler’s function and
Campbell et al.’s function when: (A, C, E) travelling directly up or down
hills of varying slope (walking slope), (B, D, F) traversing across hills of
varying slope (hill slope).

When comparing the performances of each of the models (Table 3), the predicted
speeds for individual 50 m sections had a lower RMSE and percentage error, and
a higher R squared value using our new model than in the existing ones. The
R-squared value is still very low, however we suggest that this is due to the
variability within the data. We have previously acknowledged that there are
many individual effects which can impact the walking speed, and which we did
not attempt to capture in our model. Instead it captures the general trend of
the walking speed for an average individual under average conditions, and does
this better than existing models (evidenced by the improved RMSE).

Table 3: Comparison of new model against existing methods to calculate walking speeds. | New Model | Naismith | Tobler | Campbell
---|---|---|---|---
Average % error | 23.68 | 26.36 | 26.17 | 25.33
MSE | 1.20 | 1.61 | 1.53 | 1.58
RMSE | 1.10 | 1.27 | 1.24 | 1.26
R2 | 0.09 | -0.22 | -0.16 | -0.19

To isolate the impact of each of the slope variables, we filtered the results
to look at the data where a slope was being directly climbed or traversed.
Figs 6A, B and 7A, B show the RMSE and mean residuals for each of the models,
for data which was within 5 degrees of directly climbing (A) or traversing (B)
hills of varying slope. From this we can clearly see that Naismith’s rule
consistently overestimates walking speeds when descending a slope, and
underestimates speeds when climbing a slope. When ascending or descending a
slope, the RMSE of our GLM is similar to that of Tobler’s hiking function.
However, one of the main areas where we see an improvement using our model is
on slight declines. Tobler’s hiking function suggests that walking speed
increases on mild descents up to a maximum of 6 km/h. It is clear from Fig 6A,
that Tobler’s function overestimates the walking speed in this region.
Campbell et al.’s function has a slightly lower RMSE value than our new model
on the steepest walking slopes, however it underestimates the walking speeds
on flat ground and mild slopes; the regions where most walking occurs.
Improved walking speed predictions in this region therefore have the greatest
impact in real-world situations. Within this region our model consistently has
a lower RMSE than the existing functions, and a mean residual error close to 0
km/h.

Fig 6: Comparing RMSE values for the new model, Naismith’s rule, Tobler’s
function and Campbell et al.’s function. When: (A) travelling directly up or
down hills of varying slope (all data, walking slope), (B) traversing across
hills of varying slope (all data, hill slope), (C) travelling directly up or
down hills of varying slope (off-road data only, walking slope), (D)
traversing across hills of varying slope (off-road data only, hill slope).
Campbell et al.’s function does not provide off-road speed estimates, so was
not included in the off-road data comparisons. Fig 7: Comparing mean residual
values for the new model, Naismith’s rule, Tobler’s function and Campbell et
al.’s function. When: (A) travelling directly up or down hills of varying
slope (all data, walking slope), (B) traversing across hills of varying slope
(all data, hill slope), (C) travelling directly up or down hills of varying
slope (off-road data only, walking slope), (D) traversing across hills of
varying slope (off-road data only, hill slope). Campbell et al.’s function
does not provide off-road speed estimates, so was not included in the off-road
data comparisons.

We also see an improvement in RMSE when using our model to predict speeds for
hill traversals (Fig 6B). We can note from Fig 7B that both Naismith’s rule
and Tobler’s hiking function consistently overestimate the walking speed when
traversing a slope, as they do not take into account the impact that the hill
slope has on reducing walking speeds. The performance of Campbell et al’s
model improves as the hill slope increases, although we suggest this is more
due to it underestimating the speed on shallow slopes. We do see that the
average error in our model increases as the hill slope increases, but we
believe that this is due to limited volumes of data at high hill slopes
($\sim$0.5% of our data occurs on hill slopes steeper than 40 degrees).

As well as looking at the overall performance of our new model, we looked to
explore how well our model performed in off-road conditions, compared to the
off-road adjustments for the existing functions (Naismith’s reduced base speed
of 4 km/h, and Tobler’s correction factor of 0.6). Figs 6C, D and 7C, D show
the RMSE and mean residuals, only considering data which was recorded in off-
road conditions. From Figs 6C and 7C it is clear that Tobler’s function
consistently underestimates the walking speed when off-road. The factor of 0.6
is a larger reduction in walking speed than is observed in practice. As we
found when looking at our data as a whole, Naismith’s rule underestimates the
walking speed when climbing a slope and overestimates when descending a slope.
Our new model does not suffer from these problems, with both a lower RMSE and
lower absolute mean residual value across all walking slopes. Both of these
existing models also consistently underestimate walking speeds when traversing
a slope, unlike our new model which has a mean residual of less than 0.4 km/h
on slopes of up to 35 degrees. The error in predictions of our new model does
increase as the hill slope increases, though the RMSE is generally lower than
seen in the existing models. On the steepest hill slopes our model appears to
perform less well than the existing ones, though only 0.2% of our off-road
data occurred on a hill slope steeper than 40 degrees.

Although we have shown an improvement in walking speed predictions over short
sections of routes, this did not translate to similar results when looking at
predicted walking times for routes as a whole. Our model and all of the
existing models which we have explored here had an average percentage error of
13.5% - 15.5% when predicting the time taken for a complete route. However,
based on the errors seen in Figs 6 and 7, we believe that this is a result of
errors cancelling out over the course of a hike. For example while ascending a
hill, Naismith’s rule will underestimate the walking speed (and thus
overestimate the walking time), but it will then overestimate the walking
speed on the subsequent descent, leading to a relatively accurate total time
estimate. The results here suggest that Naismith’s rule, and other existing
functions, are still a good rule of thumb to calculate route times as a whole,
but time estimates for individual sections of a route will be less accurate
than when using the new model found here.

## Discussion

We have developed a model for walking speed which is very robust, due the
large volume of data (88,000 km) used to build it, and which correlates with
the data over a wider range of conditions than commonly used formulae. Data
from tracks confirms that each of the walking slope, the hill slope and the
terrain type or obstruction are significant factors in determining walking
speeds. The model improves on existing methods to predict walking speeds (Figs
6 & 7). We have also shown the specific improvement that our new model has on
predicting walking speeds in off-road conditions, compared to the simple off-
road speed reductions used by existing models. The existing methods to
calculate walking speeds require tuning for use in real-world scenarios, as
there are many factors which can affect an individual’s walking speed beyond
the slope and obstruction level (such as the weather, fitness level or age)
[2, 3, 4]. The model presented here requires the same tuning as these existing
methods but provides more a more accurate population average walking speed
across a wide range of terrain and slope conditions.

Our results confirm that Naismith’s rule (Fig 1) is still a good rule-of-thumb
to use when estimating the total walking time for a route, especially in
situations where the calculation must be done by hand. However, the findings
here can be used as an addition to Naismith’s rule; it is likely that (under
Naismith’s rule) the predicted ascent time will be overestimated and the
predicted descent time will be underestimated. It is not uncommon for hikers
to contact one another when they reach the summit of a hill, and provide an
estimated arrival time back at the campsite. Knowing that the descent will
likely take longer than estimated by Naismith’s rule will result in more
accurate arrival estimations being given. Similarly, the knowledge of how the
hill slope reduces walking speeds, or that just 10 cm of vegetation can reduce
walking speeds by up to 0.6 km/h may well affect route choices made when out
on a walk. For example, if a hiker is following a footpath, but can see from
their map that the path forms a large curve then they can use our findings to
decide whether it will be faster to travel off-road and cut the corner. On
flat terrain with heavy levels of obstruction, our model suggests that such a
short cut will be faster if the distance covered on the path is more than 15%
longer than the off-road distance. Speed is not the only factor which would
affect this decision, as safety and navigability are also important variables,
but these results can help people make more informed choices when on a hike.

The benefit of using crowdsourced GPS data to build our model is also a
limitation of the approach, as we did not have control over data collection.
This meant that models were unable to account for any bias in our data such as
group size, ability and composition, or other potential variables such as
weather conditions, as factors in determining walking speed (although we would
expect the volume of data to cause most of these effects to average out).

Unlike previous work [8], we did not use fixed values to classify breaks and
non-walking or hiking tracks. Instead we developed filters based on the
attributes of known walking data (see S2 Supporting Information). The methods
used to filter the datasets were blinded to the outcome of the model
generation, the choice of filtering methods will have had an impact on the
dataset and subsequent model and no ground truth was available against which
to test our assumptions.

Our method of calculating the terrain obstruction value was relatively crude,
looking only at the obstruction height at each GPS point. While this did prove
to be successful, and we observed a clear difference in walking speeds between
areas of light and heavy obstruction (see S5 Supporting Information), the
inaccuracies present within GPS data may have led to some erroneous
obstruction measurements, for example in a field sparsely populated with
trees. In future, efforts should be made to refine this approach, such as
considering the average obstruction level over a wider area around each point.

A further limitation of our data came when we looked to classify points into
paved roads, unpaved roads or off-road. A combination of GPS drift and map
error means that there is significant uncertainty and so we had to use a
search radius around each data point to identify potential roads. We suspect
that we were likely overclassifying tracks on roads. While our model appears
to be robust to this overclassification (due to the volumes of correctly
classified data used), the overclassification left us with a reduced number of
off-road datapoints from which to predict off-road travel speeds.

Furthermore, the use of crowdsourced data meant that all of our data came from
‘walkable’ regions by definition. When including the terrain obstruction
variable, we were unable to determine if there are levels of terrain
obstruction which makes walking impossible. Similarly, the vast majority of
the data was collected on shallow hill- and walking slopes, leading to a
sparcity of data in steeper areas. While this does mean that we can be very
confident about our walking speed predictions in less steep regions (where
most walking occurs), it is unclear whether the lack of data on steeper
regions is a result of steep slopes being relatively rare, or that they cannot
be easily navigated, so hikers chose an alternate path. As described above we
had to make a number of assumptions regarding data filtering and processing
including model selection, and other choices may give different results. To
support anyone who wants to challenge or test these assumptions, or try
different models, we have made all our code available on Github. Further, all
of the data sources used are detailed in S1 Supporting Information. and the
filters/assumptions we used to clean the data are fully detailed in S2
Supporting Information.

## Conclusion

Widely used algorithms (e.g. Naismith’s rule) for estimating walking/hiking
speed are simple to understand, very easy to calculate but are based on
limited observations. Here we curated a dataset of almost 88,000 km of walking
and hiking data. We found that the existing algorithms perform quite well
against the dataset but they tend to overestimate ascent time, underestimate
descent time and most ignore terrain obstruction and hill slope both of which
we found to be significant factors. We used the data to derive a new model
that takes into account these variables. We demonstrated that the model
provides more accurate walking speeds than the existing methods in all
scenarios, and particularly in off-road regions. By providing improved walking
speed predictions in these off-road regions, we have enabled more accurate
calculations of the fastest route to or from any given location, which could
save minutes in an emergency situation where every second is important.

## Acknowledgments

Preprocessing of the GPX files made use of the resources provided by the
Edinburgh Compute and Data Facility (ECDF) [34].

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## Supporting information

##### S1 Supporting Information.

Data sources table

##### S2 Supporting Information

Data acquisition and preparation

##### S3 Supporting Information.

Exploratory data modelling study.

##### S4 Supporting Information.

Exploring the differences between Scotland and the rest of the UK.

##### S5 Supporting Information

Exploring the impact of terrain obstruction.

##### S6 Supporting Information

Comparison of walking speed changes while crossing a simulated off-road
terrain region.

## S1 Supporting Information. Data sources table

Table 4: Summary of data sources used during this work

Data Type |  Data Source |  Download Date |  Notes
---|---|---|---
Hikr GPS data |  Hikr.org |  01-07-2021 |  Within the UK data1, only tracks which took place within Scotland2 were used for exploratory study (see S3 Supporting Information.).
OpenStreetMap GPS data |  OpenStreetMap.org |  01-07-2021 |  Accessed using planet.gpx regional extracts3
Ordnance Survey elevation data |  Ordnace Survey Terrain 5 DTM |  05-08-2021 |  Accessed using EDINA Digimap Ordnance Survey Service4
OpenStreetMap road data |  OpenStreetMap.org |  04-08-2021 |  Accessed using planet.osm regional extracts5
England lidar data |  National LIDAR Programme |  16-09-2021 |  2m resolution data was used and accessed using EDINA LIDAR Digimap Service4
Wales lidar data |  LIDAR terrain and surfaces models Wales |  16-09-2021 |  2m resolution data was used and accessed using EDINA LIDAR Digimap Service4

Note: In regions where lidar data was available as part of both the England
and Wales lidar datasets, the data values from England were used.

1 https://www.hikr.org/region516/ped/?gps=1

2 https://www.hikr.org/region518/ped/?gps=1

3
http://zverik.openstreetmap.ru/gps/files/extracts/europe/great_britain.tar.xz

4 https://digimap.edina.ac.uk

5 http://download.geofabrik.de/europe/great-britain.html

## S2 Supporting Information. Data acquisition and preparation

### Importing GPS tracks

Approximately 20,000 tracks with GPS data were obtained, the majority of the
tracks used here were from OpenStreetMap (OSM) [18], with a second, smaller
set from Hikr.org [17]. The OSM data consisted of a GPS data dump of all
tracks on OpenStreetMap which contain points within the UK, as of April 2013
(the date of the last GPS data dump by OSM). This dataset does not contain any
information about the mode of transport being used when the track was
recorded, or the device used for the recording. The Hikr dataset contained all
of the Hikr reports which were uploaded from July 2009 to July 2021. Although
only relatively small quantities of Hikr data are available, all of the tracks
used were explicitly tagged as hiking reports, so could be used as the basis
of a filter to determine which of the OSM tracks contained walking data (see
Data Filtering).

Each GPS track is made up of track segments, where each track segment is a
continuous run of points; a point is a data pair containing the device
location and time of recording. The exact nature of tracks and track segments
depends on the individual device settings being used. Some devices record
points at fixed time intervals, others may only record a new point after
travelling a sufficiently large distance away from the previous point. In
general, a new track or a new track segment is created whenever the device is
switched off or loses signal for a period, although this is not always the
case.

All track segments which were fully contained within the area covered by the
OS Terrain 5 DTM were imported (see Elevation and Slope). The GPX files were
read using a customised version of the GPX Segment Importer plugin for QGIS
[35], extended to also read elevation and terrain information (code on Github
- AndrewWood94/PhDThesis). Although a single GPX file can contain multiple
separate tracks, we considered all data within a file to be part of the same
track. Tracks within the same file were highly likely to be undertaken by the
same individual, and thus have correlated walking speeds. To account for the
variance caused by different hike locations and difficulty, each track segment
within a file was processed individually.

For each track segment, the list of points was converted to a series of
connected linestrings, with the following properties calculated and attached
to each one:

  * •

Start coordinate

  * •

End coordinate

  * •

Start time

  * •

Duration

  * •

Distance

  * •

Speed

We took a decision to exclude very short tracks (less than 250m in length of
2.5 minutes in duration as well as any tracks where the median speed was
greater than 10km/h, as it was felt they might not be a reliable
representation of a real walk.

Duplicate track segments (those which contained a section with the same start
and end coordinates, the same starting date and time, and the same duration)
were tagged, and only the first instance of each was retained. It is believed
that these duplicates were a result of either users uploading the same track
multiple times, or GPS devices recording multiple versions of the same track
with different levels of accuracy or automatic filtering. Finally, a small
number of outlier Hikr tracks were checked manually and removed as looking at
the metadata or track description revealed that the routes in question were
labelled as trailruns, and so should not be considered.

#### Elevation and Slope

The Ordnance Survey Terrain 5 Digital Terrain Map (DTM) [19] provides the
elevation over the whole of Great Britain at 5 m intervals, with an accuracy
of greater than 2.5 m RMSE. After reading in each track, the DTM was used to
calculate the following:

  * •

Elevation

  * •

Walking slope

  * •

Hill slope

Although the GPX files also contained elevation data which could be used to
calculate the walking slope, there were a number of reasons to prefer the
Ordnance Survey data:

  * •

Multiple tracks were found where the elevation was measured to the nearest
half metre or metre on the GPS device. Using this value would lead to rounding
errors in slope calculations over short distances. The Ordnance Survey values
for elevation have a higher level of precision which could alleviate these
errors.

  * •

Using OS values avoids any concern about potential discrepancies arising from
differences in calibration across GPS devices, so we could ensure that the
same location on different tracks would have the same elevation.

  * •

We were also investigating the impact of hill slope, which could not be
calculated from the GPS track data, and using the same data source for both
calculations should allow for better evaluation of interactions between the
two variables.

When first reading each data track, the elevation DTM was sampled to provide
the spot height of the start co-ordinate, and the hill slope at this point was
calculated using the quadratic surface method [20, 21]. As explored in
previous work [36], this method produces the most accurate slope estimates,
especially given the high resolution of the DTM being used for calculations.
The walking slope was calculated using the spot heights of the start and end
coordinates of each linestring, and the distance between those points.

#### Roads and Paths

Road and path data from OpenStreetMap was downloaded from GeoFabrik.de [22]
and contains the data as of August 2021. Although similar data is available
from Ordnance Survey, the OSM dataset was preferred due to it providing a more
detailed classification than Ordnance Survey, in terms of the road or path
type.

The OSM road and path data is made up of linestrings (single vector lines, as
opposed to area features which match the width of the road) and it would be
very unlikely for all of the points on a GPS route which follows a road to
fall exactly along that line, making it difficult to determine which points
should be classified as following the road or path. This problem was then
exacerbated by two factors; inaccuracies within the GPS readings, and
inaccuracies within the map data itself. This combination of potential errors
meant that we needed to decide upon a radius around each point to search for
roads or paths.

Here we classified a point as being on-road if a single feature of the OSM
road dataset was found within a 50 m radius around the point. This radius was
determined after a number of iterations.

This inclusive distance will lead to an over classification of on-road points.
Due to the relative numbers of on-road vs off-road points, we decided it would
be preferential to mis-classify points as being on a road rather than the
other way around, as this would have a much smaller impact on any resulting
models.

By consulting the descriptions of the OSM road-type definitions [23], we could
separate the on-road points into two categories, paved and unpaved. When doing
this, we assumed that the standard road type was paved. Therefore, if a point
contained multiple road types, we only considered it to be unpaved if none of
the road types detected were paved. The paved road types are the following:

  * •

Cycleway

  * •

Footway

  * •

Living_street

  * •

Motorway

  * •

Motorway_link

  * •

Pedestrian

  * •

Primary

  * •

Primary_link

  * •

Residential

  * •

Secondary

  * •

Secondary_link

  * •

Service

  * •

Steps

  * •

Tertiary

  * •

Tertiary_link

  * •

Trunk

  * •

Trunk_link

  * •

Unclassified

  * •

Unknown

  *   *

While these are unpaved:

  * •

Bridleway

  * •

Path

  * •

Track

  * •

Track_grade1

  * •

Track_grade2

  * •

Track_grade3

  * •

Track_grade4

  * •

Track_grade5

  *

#### Obstruction Height

For England and Wales, a lidar Digital Terrain Map (DTM) and Digital Surface
Map (DSM) were downloaded (see S1 Supporting Information.), both at 2 m
resolution, with an accuracy of 15 cm RMSE [24, 25, 26]. The DTM gives the
ground height above sea level every 2 m, while the DSM provides the surface
height (i.e. taking into account buildings or trees etc). For Scotland, there
is limited coverage in more rural (i.e. off-road) areas, and most of the
available data is at very high resolution (25-50 cm). This is generally
greater than the accuracy of the GPS devices used. Given the partial coverage
and resolution mis-match in the Scottish datasets, further analysis was
limited to tracks in England and Wales.

The start coordinate for each point in a GPS track was sampled in both the DTM
and DSM, and the difference between the two values taken as the level of
terrain obstruction for the point. For example, a point in a woodland may have
a DTM height of 80 m above sea level, and a DSM height of 85 m (the height at
the top of the tree canopy), giving us a terrain obstruction value of 5 m.

### Break Finding

For this work, we were interested in calculating the active components of a
route (as opposed to breaks or rests), as that is the data on which a walking
speed model should be based. Performing analysis on the data without first
removing breaks would likely result in inaccurate and lower estimates for the
movement speed. Similarly, when developing the filter to find walking routes
in the OpenStreetMap dataset we wanted to ensure that we were only considering
the active route components.

Initial inspection of the walking tracks showed that a large number contained
obvious breaks, while the device was still recording. The simplest method to
find breaks, and the first implemented, was to tag all points where there was
no movement (i.e. walking speed = 0 km/h). The next step was to tag any
individual points which represented over 1 km or 3 minutes of travel as
breaks. These points could occur in areas where the device lost signal for a
period. Although this should result in the creation of a new track segment,
there were a number of tracks where this was clearly not the case. Points with
speeds >10 km/h occurring immediately following a long (>3 minute) point were
also tagged as breaks. Situations like this occurred on a number of devices,
likely when a device automatically paused recording for a break. Once
significant movement was detected, two points were then added in quick
succession, the first at the original location with the time the movement
started again, and the second at the new position of the device. This resulted
in one high duration point with very little movement, followed by single-
second duration movement with high speed, where the device ‘caught up’ with
the correct location.

Unfortunately, this did not capture all of the break points due to GPS drift;
the distance between the measured position of the GPS device and the true
location. This error is always present, but is more obvious when stationary,
as a large number of points are recorded in the same general area, forming
clusters. Examples of these clusters can be seen in Fig 2.

Although these locations are easy to identify when the route is visualised, we
needed an automatic filter in order to remove them from our analysis. The
movement speeds calculated from these drift points can vary greatly, depending
on the sampling rate of the device and the amount of GPS drift. A drift
measurement of 6 m in a very short time (1-2 seconds) would mean a very high
speed (10-20 km/h) is found, but a series of very low speeds in a row could be
due to a break with a small amount of drift, or it could indicate a
particularly difficult (and therefore slow) section of the route.

A number of previous studies explored methods to classify GPS tracks into
activity types [37, 38, 39, 40, 41, 42, 43, 44], however none could be
directly applied to this problem. This is because they were usually trying to
identify different modes of transport, which can be clearly distinguished by
different travel speeds. However, as mentioned above, the speed measurements
caused by GPS drift can easily be in the same range as an expected walking
speed. Methods which look for clusters similar to those being investigated
here were also not useable as they generally require a known sampling rate for
the GPS device [42], or involve checking clusters against a pre-existing
database of features [43, 44]. Our data consisted of tracks created using a
wide range of devices and settings, and over a very large area, so it was not
possible to either assume a fixed sampling rate, or to pre-select features
where breaks were likely. Instead, various ideas from a number of works were
combined and adapted to find clusters of points, which were then checked to
see if they should be identified as a break.

Firstly, based on [44], a modified version of DBSCAN [45] was used to identify
clusters. Unlike in the standard algorithm, each point only looked for
neighbours whose timestamp was within 10 minutes of the point being checked.
This prevented clusters being found on routes which doubled back on themselves
or contained loops. Secondly, as sampling rates were not consistent across
devices and the tracks covered a variety of terrains, we could not assume a
fixed radius to find neighbouring points. Instead the median travel distance
for the particular track segment being investigated was used (rmedian).

> DEFINITION 1. Neighbourhood of a point: Let $\\{p_{0},p_{1},...,p_{n}\\}$ be
> points on a GPS track segment, with timestamps
> $\\{t_{0},t_{1},...,t_{n}\\}$, a median distance $r_{median}$ between
> consecutive points and a median point speed $s_{median}$. The neighbourhood
> $N_{k}$ of a point, $p_{k}$, is the set of points $p_{i}$ such that:
>
> $dist(p_{i},p_{k})<r_{median}\text{ and }|t_{i}-t_{k}|<600s$
>

Using these conditions, all points along each track segment were tested to
find point clusters.

> DEFINITION 2. Point cluster: A point cluster, C, is formed from a point
> neighbourhood, $N_{k}$, if any of the following hold:
>
>   1. 1.
>
> At least 5 non-consecutive points are found in $N_{k}$
>
>   2. 2.
>
> At least 10 consecutive points are found in $N_{k}$
>
>   3. 3.
>
> A point within $N_{k}$ has a ‘high speed’; a speed greater than
> $2*s_{median}$
>
>   4. 4.
>
> A point $p_{k}$ is immediately preceded by a ‘high speed’ point
>
>   5. 5.
>
> A point within $N_{k}$ has a ‘very low speed’; a speed less than 0.01 km/h
>
>
>

Examples of each of these conditions are shown in Fig 8A. The first two
conditions worked together to prevent finding clusters in unusually slow
sections of a route, such as a steep hill climb, but allowed for areas where
GPS drift was small and an entire cluster was contained within $N_{k}$. The
third and fourth conditions found ‘high speed’ points which occurred when a
large amount of GPS drift was measured, and could spread a cluster out over a
wider area. Without accounting for these points separately, the algorithm
would often end up registering a single break as multiple short breaks
separated by high speed movements. The fifth condition was necessary for
situations where the device was set to only record new points once a minimum
distance has been travelled from the previous location. Note that in the third
and fifth cases in definition two, the point immediately following the high-
or low-speed point was added to the cluster, even if it was not included in
$N_{k}$. Similarly, in the fourth case, the cluster included the preceding
high-speed point regardless of whether it was in $N_{k}$.

Fig 8: Demonstration of how breaks are identified from point clusters. (A) The
five scenarios in which a point cluster is identified around a point. (B)
Points in a cluster are checked to find their break likelihood, and see if a
potential break can be identified. Potential breaks are then checked to see if
a break can be formed.

Once a point cluster was identified, each point within it was tested and any
further clusters found were added to the original. This continued recursively
until every point within the cluster had been checked and no new clusters were
found. This allowed us to identify clusters (and breaks) which lasted longer
than the 10 minute Neighbourhood threshold, as each point added to a cluster
allows the search window to be extended beyond the initial limit.

Following this, further steps were taken to try and ensure that only
legitimate breaks were classified, rather than slow sections of movement.
Firstly, each point within the cluster was tested for ‘break likelihood’ using
a simple classification methodology based on the approach used by Wan and Lin
[41], which uses two variables; the point speed, $s_{i}$, and the point angle,
$\alpha_{i}$.

> DEFINITION 3. Point speed: The speed, $s_{i}$, at a point, $p_{i}$ is
> categorised as follows:
>
>   * –
>
> Low: $s_{i}<s_{median}/2$
>
>   * –
>
> Medium: $s_{median}/2<s_{i}<10m/s$
>
>   * –
>
> High: $s_{i}>10m/s$
>
>
>

> DEFINITION 4. Point angle: The point angle, $\alpha_{i}$, for a point,
> $p_{i}$, is the angle created at $p_{i}$ by the lines connecting it to
> $p_{i-1}$ and $p_{i+1}$. It is then categorised as follows:
>
>   * –
>
> Narrow: $\alpha_{i}<90^{\circ}$
>
>   * –
>
> Wide: $\alpha_{i}>=90^{\circ}$
>
>
>

As discussed in [41], normal walking is unlikely to result in point angles
below 90 degrees, unlike a stationary object subject to GPS drift which often
produces narrower point angles. The break likelihood for each point in a
cluster was found using the classifications in Table 5.

Table 5: Break likelihood classifications based on point speeds and angles | Point Speed
---|---
| Low | Medium | High
Point Angle | Narrow  | High | Medium | High
Wide  | Medium | Low | Medium

Once the break likelihoods were identified, the cluster was checked to find a
potential break. Performing this step helped to limit the sizes of breaks, as
walking points immediately preceding or following a break were often caught in
the cluster.

> DEFINITION 5. Potential Break: A potential break, $B^{\ast}$, is created by
> ordering the points in a cluster and identifying the first and last points
> with a break likelihood of medium or high. All points in the cluster between
> these points (including the points themselves) form the potential break.
>

As a break by definition implies no movement, any GPS drift in a given
direction should be cancelled out by subsequent GPS drift in the opposite
direction. For this reason, the bearing of all points within the potential
break was found and assigned a quadrant, and a break was only formed if there
was apparent motion in opposite quadrants.

$\displaystyle Q_{i}=$ $\displaystyle True\ iff$ $\displaystyle\exists\ p\in
B^{\ast}\ |\ (90*(i-1))^{\circ}<p_{bearing}<(90*i)^{\circ},$ $\displaystyle
i=(1,2,3,4)$

> DEFINITION 6. Break: A break, $B$, is created from a potential break,
> $B^{\ast}$, if both of the following hold:
>
>   * •
>
> $Q_{1}=True\ and\ Q_{3}=True,\ or\ Q_{3}=True\ and\ Q_{4}=True$.
>
>   * •
>
> Less than half the points in $B^{\ast}$ have a low break likelihood.
>
>
>

Fig 8B provides examples of two point clusters and the process to identify
breaks. Note that not every point in each cluster becomes part of the
potential break; low break likelihood points at the ends of a cluster are
excluded. After processing the second cluster, a break is not defined because
the cluster does meet the requirement of apparent travel in opposite
quadrants. While there is a substantial amount of east-to-west deviation in
the track, it consistently heads north. Regions such as this identified in GPS
tracks are more likely to indicate a period of challenging (and therefore
slow) movement.

A substantial ground truth dataset of routes where the breaks are tagged does
not exist, so we were unable to measure the accuracy of this break-finding
algorithm numerically. However, a qualitative inspection of a small number of
tracks suggested that it classified breaks well, with over-classification in
some circumstances. This was preferable to under-classification for two
reasons; firstly, as discussed below, only the longer breaks were actually
removed from calculations so small breaks found as a result of over-
classification have no impact. Secondly, we were performing analysis on very
large datasets, so we expect the analysis to be robust to the incorrect
removal of a small number of data points.

### Data Filtering

Before processing our data, we wanted to use information about walking speeds
found in the Hikr data to identify and remove non-walking tracks from the OSM
dataset. When doing this, we only wanted to compare the active component of
each of the tracks. However, not all break points should necessarily be
considered ‘inactive’. The majority of the breaks seen in the dataset were
under 30 seconds in length, and these were defined as ‘micro-breaks’. Micro-
breaks were felt to be a constituent part of the walk which most people would
have to do, such as pausing to catch your breath, and were not excluded from
further analysis. The exception to this was if the micro-break contained a
single point with over 1 km of movement, or points with speeds of over 10
km/h. Speeds over 10 km/h in a region which had been identified as a break
were very likely to be errors caused by GPS drift, and as such these points
would not be useful in representing a typical walking speed. It was felt that
30 seconds is a reasonable length such that breaks of this length or longer
would be a conscious choice by the walker to stop, rather than being necessary
for the route. After excluding long breaks, it was also decided to remove any
breaks which occurred at the start or end of a track segment.

The remaining data were merged together into continuous sections at least 50
metres in length, to remove some of the variability caused by the GPS devices
and elevation data resolution. Devices generally measure the time to the
nearest second, so rounding errors over short distances could have a large
impact on the estimated speeds, especially in combination with small
inaccuracies in the location data. By merging the data together into longer
sections the impact of these errors was greatly reduced, as any error made up
a smaller percentage of the merged section. A similar inaccuracy existed with
walking slope values. As our elevation DTM had a resolution of 5 m, a distance
of under 5 m between two consecutive points could result in a walking slope of
0 degrees, regardless of how steep the terrain is in reality. By merging the
points together, we smoothed out the steps in the data and produced a single
datapoint with a slope value closer to the true value, which will help produce
a model with more accurate walking speed predictions.

When merging data, the distance was taken as the cumulative distance across
all points, and not the direct start point to end point distance. The merged
point was classified as on a paved road if at least one of the constituent
points was classified as being on a paved road. (Similarly a point was only
classified as being off-road if none of the constituent points were on-road).
The hill slope, walking slope and average obstruction height were calculated
as the weighted average of the value at each datapoint, weighted by the
duration of each point.

Once the data were merged into 50 m sections, any section with a speed above
10 km/h which was at the start or end of a segment, or next to a break point
was considered to be part of the break. This was repeated recursively until no
more high speed points were found next to breaks.

Although all of the Hikr data was tagged as a walk or hike, there were a small
number of individual track segments with high average speeds. Upon further
inspection it was clear that these were segments, within a larger walking
track, where other modes of transport were used and should be filtered out,
therefore segments with an average speed greater than 10km/h were removed.

For each remaining track segment the median, upper quartile and maximum speed
were calculated, and statistics from these were found to use in filtering the
OpenStreetMap data to remove non-walking tracks:

$\displaystyle\begin{split}&\text{The upper quartile of the maximum Hikr
speeds}\\\ &\text{(approx. 6.0 km/h)}\end{split}$ (2)
$\displaystyle\begin{split}&\text{The median of the median Hikr speeds}\\\
&\text{(approx. 3.2 km/h)}\end{split}$ (3)
$\displaystyle\begin{split}&\text{The upper whisker of the maximum Hikr
speeds}\\\ &\text{(approx. 8.1 km/h)}\end{split}$ (4)
$\displaystyle\begin{split}&\text{The minimum of the upper quartile Hikr
speed}\\\ &\text{(approx. 2.3 km/h) }\end{split}$ (5)

The OpenStreetMap data were also merged into intervals of at least 50 m, while
ignoring short micro-breaks. Similarly to the Hikr tracks, this dataset
contained a number of tracks where there were clearly multiple movement
methods, often when the user was driving or cycling to a hike location. Unlike
in the Hikr data however, a change in transport method was often not
accompanied by the start of a new track segment. Instead, individual route
segments appeared to contain a variety of transport methods, often separated
by a number of very extreme points (likely where the device lost signal for a
period). As we did not want to remove these segments entirely if they
contained valid walking data, we used the extreme points as markers to break
the segment down into smaller sections. The ‘key points’ were defined as the
start and end points of the segment, as well as any point with a distance
greater than 500 m, a duration greater than 3 minutes, or an apparent speed
over 100 km/h (this is typically caused by a device refresh). After
identifying the key points, the following conditions were applied:

  * •

If only a single data point existed between a pair of key points it was
ignored

  * •

If the median speed between a pair of key points was greater than (2), then
all points in the range were ignored

Following this, all segments were checked and the steps outlined below were
carried out to remove unwanted data. These were repeated until no further data
was removed.

  1. 1.

If the segment contained less than 2.5 minutes or 250 m of useable data it was
removed

  2. 2.

Segments were removed if any of the following were true:

     * •

The median speed was greater than (2)

     * •

The minimum speed was greater than (3)

     * •

The upper quartile speed was greater than (4)

     * •

The upper whisker speed was less than (5)

  3. 3.

All points with a speed above 10 km/h which were at the start or end of a
segment, or next to a break point were considered part of the break.

For the remaining segments, the Hikr and OpenStreetMap data were then combined
together into a single dataset. Visual inspection of the dataset revealed that
the filters were good, but a small number of outliers remained. Therefore, the
fastest and slowest 0.5% of the merged datapoints were removed (classified as
breaks for further processing) and were not included in any modelling.

Following this a decision was made to remove data from tracks found in
Scotland. Lidar data covering the walking tracks was necessary to model the
terrain obstruction, and was not sufficiently available in Scotland at the
time of the study. Furthermore, analysis showed that that walking speeds in
Scotland were at the extreme end of what is seen throughout the rest of the UK
(see S4 Supporting Information.). Including this data without also including a
corresponding extreme dataset where lidar data is available may result in
incorrect modelling. All OSM track segments which took place within Scotland
were excluded from further processing. Similarly Hikr tracks which were tagged
as taking place in Scotland, and which fully took place in Scotland were
excluded. Note that we defined tracks in Scotland as any track segments which
were fully contained within the following OS grid squares:

HP HT HU HW HX HY HZ NA NB NC ND NF NG NH NJ NK NL NM NN NO NR NS NT NU NW NX
NY

Within the NY tile, the following tiles were excluded:

09 19 29 39 49 59 69 08 18 28 38 48 58 07 17 27 37 47 06 16 26 36

A small number of Hikr tracks which took place in the Scottish islands were
not removed as these were tagged separately on Hikr.org.

This left us with a final dataset of almost 88,000 km and over 7,600 tracks
across the UK, with a mean speed of 4.64 km/h. All of the data remaining was
assumed to consist solely of hiking or walking tracks, although there are
likely to be a number of areas where this was not the case. There is not a
large difference in speed profile when walking or cycling up a steep incline,
so further data filtering would require in-depth analysis of each individual
track. There was no objective way to remove invalid points without potentially
removing valid data as well. However, the volume of data used for modelling
should alleviate errors arising from any non-hiking track segments remaining.

## S3 Supporting Information. Exploratory data modelling study.

Data from Scotland was used for exploratory testing of different modelling
approaches. When doing this, an earlier iteration of the break finding and
data filtering process was used. Data were processed as described in S2
Supporting Information, with the following exceptions:

  * •

GPS track segments had to be fully contained in the reduced ‘Scotland’ OS grid
tiles described in S2 Supporting Information.

  * •

Track segments where the median speed was greater than 10 km/h were not
automatically removed prior to break identification.

  * •

Individual points representing 10 minutes of travel were tagged as breaks (3
minutes was used in later versions).

  * •

High speed (>10 km/h) points occurring immediately following a long (>3
minute) point were not automatically tagged as breaks.

  * •

When merging data into 50 m sections, sections under 50 m in length
immediately preceding a break or the end of the segment were also tagged as
breaks, rather than combined with the previous section.

  * •

After merging the data into 50 m sections, any section with a speed above 10
km/h which was at the start or end of a segment, or next to a break point were
only considered to be part of the break in the OSM data, not the Hikr data.

  * •

A minimum distance of 250 m of travel was included between breaks. Sections
with a distance below this were tagged as a break.

  * •

When looking for ‘key points’ to filter out non-walking track sections, the
duration required to be tagged as a ‘key point’ was 10 minutes (3 minutes was
used in later versions).

  * •

Duplicate track segments were not identified or removed.

  * •

The fastest and slowest 0.5% of the merged datapoints were not removed as
outliers.

Once the data were filtered and processed, we were able to use them to test
models for the walking speed. Two different approaches were explored in order
to model the data: a generalised linear model (GLM) and a generalised additive
model (GAM).

We know that predictions for walking speeds must be non-negative, and two
different setups were explored to achieve this: a Gaussian distribution with
log link function and a Gamma distribution with inverse link function. The GAM
approach was also deployed with both thin plate spline or cubic regression
basis functions. Investigations into different models showed that there was no
improvement to model fit beyond cubic terms in a GLM, or 7 knots in each GAM
smoothing term, so more complex models than this were not considered for
selection.

Both model types were created in R version 3.6.1:

$\displaystyle\texttt{glm}(v\sim
a\phi+b\phi^{2}+c\phi^{3}+d\theta+e\theta^{2}+f\theta^{3},\texttt{distribution})$
$\displaystyle\texttt{gam}(v\sim
s(\phi,k,\beta)+s(\theta,k,\beta),\texttt{distribution})$

where

> $v=\text{walking speed}$
> $\phi=\text{hill slope angle (degrees)}$
> $\theta=\text{walking slope angle (degrees)}$
> k = knots used in spline (up to 7)
> $\beta$ = basis function
>

Initially, 10-fold cross-validation was used to compare the model parameters,
looking at R-squared values, root-mean-squared error (RMSE) and mean absolute
error. All models produced very similar results, with no change in the RMSE to
2 decimal places, although there was a general trend of marginal improvements
as the model complexity increased. As no best model could be chosen based on
the cross-validation, each was checked in more detail. Firstly, the hill slope
component was isolated by investigating the speed when the walking slope was
zero (i.e when traversing across a slope). Intuitively, and from experience,
this should be a decreasing function; as the slope gets steeper it is harder
to traverse, so the walking speed will decrease. Models which failed to
predict this were removed under the assumption that the data were overfitted.
Following this, the walking slope component was investigated, specifically
looking at the walking speed when travelling directly up- or down-hill. By
inspection of the data, existing functions, and intuition, this should be
modelled as a roughly bell-shaped function with the peak at, or close to, 0
degrees. Any models which predicted an increase in speed as walking slope
steepness increased (from a minimum magnitude of 10 degrees) were removed.
Secondly, we know from existing work that there exists a critical gradient at
a walking slope of around 15 – 21 degrees, at which it becomes more efficient
to zig-zag up a steep hill rather than going directly uphill. Models which
failed to predict the critical gradient occurring below 21 degrees when
travelling uphill were also removed.

This resulted in 21 model configurations remaining, although it is clear from
Fig 9 that the speed predictions are very similar in most circumstances. Fig
9A shows that all of the remaining models predict very similar speeds when
traversing a slope of up to 40 degrees, after which there is more deviation in
predictions. Similarly, when travelling in the slope direction (Fig 9B), all
of the models are broadly similar on slopes up to approximately $\pm 15$
degrees. More than 96% of the data is contained within this area, and the
relative lack of data outside this region explains the divergent speed
predictions. As all of the models provided both very similar R-squared values
and very similar predictions over the vast majority of the dataset, we used
the following points to make our final selection:

  * •

It is easier to apply GLMs than GAMs to future work, as a simple formula to
predict the walking speed can be produced for application elsewhere, without
needing to recreate the model from the original data.

  * •

In general, simpler models are easier to interpret, and we had no clear
evidence that a more complex model would perform better.

Fig 9: Walking speed predictions from 21 possible models (coloured
individually) generated from the Scotland GPS dataset. (A) Walking speed
predictions for traversing across hills of varying slope, overlaid on GPS data
where walking slope is below 5 degrees. (B) Walking speed predictions for
travelling directly up or down hills of varying slope, overlaid on GPS data
where walking slope is within 5 degrees of hill slope.

## S4 Supporting Information. Exploring the differences between Scotland and
the rest of the UK.

When applying the GLM formulation separately to datasets covering Scotland and
the rest of the UK (ROUK) (Fig 10), we see faster predicted walking speeds in
the ROUK model than in the Scotland model when traversing a slope, or when
walking uphill. Before building a model on a combined dataset, we wanted to
check whether the Scotland dataset was a reasonable subset of the ROUK data.

Fig 10: Comparison of walking speed models produced using data from Scotland
and the rest of the UK. Walking speed predictions when: (A) travelling
directly up or down hills of varying slope, (B) traversing across hills of
varying slope.

To take into account the fact that the ROUK datset was much larger than the
Scotland dataset (7636 tracks vs 648), we took 100 samples of 650 tracks from
the ROUK dataset (to form sample sets of comparable size to the Scotland data)
and modelled the walking speed for each one. The results are visualised in Fig
11.

Fig 11: Comparison of walking speed models produced using data from Scotland
against 100 sampled datasets from the rest of the UK. Walking speed
predictions when: (A) travelling directly up or down hills of varying slope,
(B) traversing across hills of varying slope.

When we look at traversing a hill, it is clear that the two datasets are
distinct, as the model for Scotland is outside the range of results seen in
the ROUK sample models. Further investigations showed that there were
differences between the proportions of paved road, unpaved road and off-road
data within each set. To take this into account, we modelled these
individually, once again sampling the ROUK dataset 100 times to form sample
sets of comparable size to the Scotland dataset (sample sizes of 600 for paved
roads and 450 for unpaved and 200 tracks for off-road), and the resulting
models can be seen in Fig 12.

Fig 12: Comparison of walking speed models produced using data from Scotland
against 100 sampled datasets from the rest of the UK. Walking speed
predictions when: (A) travelling on paved roads directly up or down hills of
varying slope, (B) traversing on paved roads across hills of varying slope,
(C) travelling on unpaved roads directly up or down hills of varying slope,
(D) traversing on unpaved roads across hills of varying slope, (E) for
travelling off-road directly up or down hills of varying slope, (F) traversing
off-road across hills of varying slope.

We can clearly see now that our model for paved roads in the Scotland data is
comfortably within the range of samples of the ROUK data. It is reasonable to
suggest, therefore, that there is no difference in walking on a paved road in
Scotland compared to the rest of the UK. However, our unpaved road model and
off-raod models for Scotland lie at the extreme edge, or outside of the range
of sample models taken from the ROUK unpaved data.

Before finally determining that the two datasets were distinct, we wanted to
see if we could find another variable which would account for the lower
walking speeds seen in Scotland on both unpaved roads and when off-road. We
explored whether this elevation could be responsible for this difference, as
higher elevations are likely to have higher exposure, and be more affected by
inclement weather, leading to slower walking speeds.

There was a much greater proportion of data at high elevation (>500 m) in the
Scotland dataset than the ROUK dataset on both unpaved roads and when off-
road, while a silimar difference was not seen on paved roads (where our models
were equal) - Fig 13.

Fig 13: Comparing elevations of track sections between Scotland and the rest
of the UK. (A) Scotland paved roads. (B) ROUK paved roads. (C) Scotland
unpaved roads. (D) ROUK unpaved roads. (E) Scotland off-road. (F) ROUK off-
road.

For this reason we included elevation as a model variable, both as a
continuous variable or as a factor variable classifying all points as either
high elevation or low elevation (where high elevation consisted of all data
>500 m). However, in both cases we found this to not be a significant factor
in the model. Based on the sample data taken, we suggest that the model
formulated using the Scotland data is an extreme sample of the ROUK data,
where a greater-than-average portion of the data has been sampled from high
elevation regions. However, the high elevations themselves are not the cause
of the difference between the model coefficients.

## S5 Supporting Information. Exploring the impact of terrain obstruction.

To understand how terrain obstruction might impact the walking speed (and thus
how it should be incorporated into a model), we conducted an initial
exploration into the data. Before doing this, however, we wanted to check that
there was not a systematic difference between the walking speeds in regions
where we had lidar data, and regions where we did not. If the two regions were
not found to be different, then any findings about the effects of terrain
obstruction in regions where we had lidar data could also be applied to areas
where we didn’t have the data.

When modelling the data for the separate datasets (‘obstruction available’ vs
‘no obstruction available’), we see that the models are very similar when
ascending or descending a slope (Fig 14A). This was not the case when
traversing the slope however, as the ‘no obstruction available’ model predicts
that hill slope has a greater impact on reducing walking speed than the
‘obstruction available’ data model (Fig 14B).

Fig 14: Comparison of off-road walking speed models where obstruction data is,
or is not, available. Walking speed predictions when: (A) travelling directly
up or down hills of varying slope, (B) traversing across hills of varying
slope.

We sampled our larger (‘obstruction available’) dataset, so that we had a
similar number of tracks as in our smaller dataset, and compared models made
from more equal volumes of data. When doing this (Fig 15), we found that the
‘no obstruction available’ model is within the range of sample models for
traversing the slope, albeit at an extreme end. This is likely due to the low
volume of data which we had at high hill-slopes. (Only 50 km of data had a
hill slope greater than 15 degrees with no lidar data available, and only 130
km with lidar data available). Going forward, we assumed that the regions
where we had lidar data were representative of all off-road regions, and so
any findings could be applied to both areas.

Fig 15: Comparison of off-road walking speed models produced using a dataset
where obstruction data isn’t available against 100 sampled datasets where
obstruction data is available. Walking speed predictions when: (A) travelling
directly up or down hills of varying slope, (B) traversing across hills of
varying slope.

To explore the effects of terrain obstruction, we first looked at the range of
speeds across the different obstruction values. The data was split into 25
quantiles, and the average walking speed for each was calculated. The results
are shown in Fig 16A. This shows us two things; firstly the vast majority of
our data had very little, or no obstruction (as most of the quantile points
occur below 0.5 m of obstruction). Secondly we can see that there is a very
steep drop off in walking speed initially, and it then remains relatively
constant across obstruction levels. Our initial assumption was that walking
would be relatively easy with no, or very little obstruction, and then much
slower at obstruction values of approximately 0.5 m - 4 m when it would
involve walking through thick vegetation, before getting slightly faster again
at higher obstruction values (as you would be walking through a forest and
could walk between the trees below the canopy). The data shows this not to be
the case, although this may be a result of our data only showing us regions
where walking was possible. Due to the crowdsourced nature of our GPS tracks,
we had no data showing us the walking speed when in 2 m of thick vegetation,
as it is very unlikely that people would have chosen to walk there.

Fig 16: Binned average walking speeds across different levels of obstruction.
(A) Full range. (B) Zoomed range. Each bin contains 1/25th of the datapoints.

Fig 16B shows a close-up of the steep speed drop off, and we can see that the
average speed dropped from approximately 4.8 km/h when there was no
obstruction down to about 4 km/h once there was more than 10 cm of
obstruction. We used this information to classify all points into heavy
obstruction (>10 cm) or light obstruction ($<=10$ cm). Although the figure
suggests a gradual decrease in walking speed between 0 and 10 cm of
obstruction, we chose not to model this. Vegetation length is highly variable
throughout the year, and it is more practical to classify regions as light or
heavy obstruction when discussing walking speeds.

## S6 Supporting Information. Comparison of walking speed changes while
crossing a simulated off-road terrain region.

Fig 17: Comparison of walking speed changes while crossing a simulated off-
road terrain region. (A), (B) The simulated route (green) across the terrain.
Terrain is coloured by elevation value from low (dark) to high (light). (C)
The elevation profile of the route, (D) The walking slope profile of the
route, (E) The hill slope profile of the route. (F) Walking speed predictions
for different models as the route is traversed. For Naismith’s and Tobler’s
functions, the off-road variants of the models have been used. For the new
model, the off-road unknown obstruction coefficients have been used.