Data treatment: How to use exponential regression types of functions to approximate to a specific model (tobler's hiking function)

I have a code which i read a gpx file and obtain the values of Latitude, Longitude and Elevation in matrix A.

route = gpxread('Patrulha_Quinta_das_Lameirinhas_relogio.gpx');
A=[route.Latitude;route.Longitude;route.Elevation];
A=A';

Then i made a filter to consider the consecutive 10 to 10 seconds. This was to remove possible outliers.

B=A(1:10:end,:);
tempo=length(B);

in my project i want to study the behaviour of velocity with different types of terrain and different intervals of slope.

Distance, velocity and slope were calculated by this for cicle

for i=1:length(B)
    if i<length(B)
        deltaLat(i,:)=(B(i+1,1)-B(i,1))*60*1852; % diferença das latitudes
        deltaLon(i,:)=(B(i+1,2)-B(i,2))*60*1852; % diferença das longitudes
        deltaZ(i,:)=B(i+1,3)-B(i,3);
    end
end
x=deltaLat;
y=deltaLon;
z=deltaZ;
distancia=sqrt(x.^2+y.^2+z.^2);
% total distance travelled
sum(distancia);                     
% velocity in each instance (10 to 10 seconds)
velocidade=distancia/10;
% mean velocity of all track
velocidade_media=sum(distancia)/length(A);
% slope (%)
declive=(z./distancia)*100;

Then i organized my data velocidade(velocity) and declive(slope) data in 4 intervals of slope: >10%, 0-10%, -10% - 0%, < -10% in oder to calculate the mean of velocity in each interval of slope.

R=[velocidade,declive];


v_mean_10_plus= mean(velocidade(declive > 10));
v_mean_0_10 = mean(velocidade(declive <= 10 & declive >= 0));
v_mean_10_0 = mean(velocidade(declive < 0 & declive >= -10));
v_mean_10_minus = mean(velocidade(declive < -10));

figure
dec_0_plus=declive(declive>=0);
dec_0_minor=declive(declive<0);
vel_0_plusP=velocidade(declive>=0);
vel_0_minor=velocidade(declive<0);

My question is how can I treat my raw data in order to get to a semehant version of Tobler's Hiking Function (exponential model) (Figure below). I've tried a few options like polyfit, fit, but it doesnt work for my project.

I appreciate your consideration.

[Rúben Ricardo]

Can anyone help me out with this matter?

I would apreciate it very much.

Answer 1

[William Rose]

@Rúben Ricardo,

You have

deltaLat(i,:)=(B(i+1,1)-B(i,1))*60*1852; % diferença das latitudes
deltaLon(i,:)=(B(i+1,2)-B(i,2))*60*1852; % diferença das longitudes

The equation for deltaLat convert the latitude difference from degres of latitude to meters, using the conversion factor

1 degee of latitude=60*1852 meters.

This is correct. However, for deltaLon, the correct conversion is

1 degee of latitude=60*1852*cos(Lat) meters.

Quinta Das Lameirinhas is at latitude 39.33 N, therefore cos(Lat)=0.7735. Therefore deltaLon should be

deltaLon(i,:)=(B(i+1,2)-B(i,2))*60*1852*0.7735; % diferença das longitudes

To make a plot like Tobler's model:

Find the mean slope for the difference sets of data:

d_mean_10_plus= mean(declive(declive > 10));
d_mean_0_10 = mean(declive(declive <= 10 & declive >= 0));
d_mean_10_0 = mean(declive(declive < 0 & declive >= -10));
d_mean_10_minus = mean(declive(declive < -10));

Combine the mean slopes into a vector, and combine the mean velocities into a vector. This will make plotting easier.

x=[d_mean_10_plus, d_mean_0_10, d_mean_10_0, d_mean_10_minus];
y=[v_mean_10_plus, v_mean_0_10, v_mean_10_0, v_mean_10_minus];

Plot the data:

plot(x,y,'rs');
grid on; xlabel('Declive'); ylabel('Velocidade');

See how that works. Good luck.

[Rúben Ricardo]

It works ! Thanks a lot man!

[William Rose]

@Rúben Ricardo, You are welcome.

I was thinking, while running on hills this weekend, that it would be interesting to harvest data from STrava to estimate the rlationship between speed and slope. There is a lot of publicly available data on Strava. Someone who is good at harvesting that data could do a lot with it.

See this 2019 report in the N.Y. Times. The authors harvested public data from Strava for more than one million marathons and half marathons, run between 2014 and 2018. They used the information to do a really excellent analysis of the benefits of new running shoes. When you have massive amounts of data you can do a much better analysis. The analysis would also require skill with geographic informaiton systems, to get accurate elevations for the masured Lat and Long. That is ncessary because the vertical value returned by most GPS devices is several times less accurate that the horixzontal fix. Therefore you would want to get the elevation for each point by reference to a national map database. In the USA, there is accurate public elevation data for the entire country on a grid of 1/3 arcsecond (about 10 m spacing). You could to a 2D interpolation to the exact measured GPS Lat & Long. There is elevation data on a 1 m grid for much of the USA.

[Rúben Ricardo]

That's super interesting! In my project I gathered data both from GoPro cameras and Garmin watches to analyse the relationship between slope and types of terrain with velocity. But in my case is something really specific. Im trying to build a Matrix with n dimensions to implement on a military agent-based simulator which we consider different types of equipment the soldier transports, different types of terrain, slope, if he is patroling during the day or at night, etc.

But anyway thanks a lot! Possibly in a near future we could to do something like that!

[William Rose]

@Rúben Ricardo, thnak you for describing your work. It sounds interesting. Good luck!