Plot change in velocity

Hi again,

I am now wanting to slightly alter my Bland-Altman curve, to be able to determine whether velocity causes larger errors.

I have my average data and would like to calcualte the change in mean angle over percentage gait cycle as follows:

delta angle = (mean data (i +1) - mean data (i - 1) ) / 0.02.

I have 101 data points, but the issue comes when I want to calculate the angular change in giat cycle percentage for the first data point.

Mean data is the average angle measured in the gait cycle and so I am wanting to take a point in the gait cycle, move to the point in front of it (+1%), then to the point behind it (-1%) to calculate the velocity, and divide by 2% of the giat cycle etc etc for the entire gait cycle.

I am having problems calculating delta angle.

I then want to plot delta angle against difference.

%% Angular velocity per gait cycle percentage   
   figure()
   for a = 0
    delta_angle(a,1) = ((mean_data(a+1)- mean_data(a-1))/0.02);
   end
   
   plot(delta_angle, mean_diff)
   xlabel('Change in mean angle/ gait cycle percentage')
   ylabel('Difference (deg)')

If this does not make sense please let me know.

Thanks in advance

Answer 1

[Mathieu NOE]

hello again

we could use gradient or the function provided below

then all vectors are of same size (101 samples) which allows you to do :

(NB : no need for the for loop)

delta_angle = gradient(mean_data)/0.02;

or

[dy, ddy] = firstsecondderivatives((1:numel(mean_data)),mean_data);
delta_angle = dy/0.02;

(I prefer the second option as it seems the end points are better handled)

% gradient of firstsecondderivatives ? 

[dy, ddy] = firstsecondderivatives((1:numel(mean_data)),mean_data);

figure(1),
plot(gradient(mean_data))
hold on 
plot(dy)
legend('gradient','function')

%%%%%%%%%%%%%%%%
function [dy, ddy] = firstsecondderivatives(x,y)

% The function calculates the first & second derivative of a function that is given by a set
% of points. The first derivatives at the first and last points are calculated by
% the 3 point forward and 3 point backward finite difference scheme respectively.
% The first derivatives at all the other points are calculated by the 2 point
% central approach.
% The second derivatives at the first and last points are calculated by
% the 4 point forward and 4 point backward finite difference scheme respectively.
% The second derivatives at all the other points are calculated by the 3 point
% central approach.

n = length (x);
dy = zeros;
ddy = zeros;
% Input variables:
% x: vector with the x the data points.
% y: vector with the f(x) data points.
% Output variable:
% dy: Vector with first derivative at each point.
% ddy: Vector with second derivative at each point.
dy(1) = (-3*y(1) + 4*y(2) - y(3)) / (2*(x(2) - x(1))); % First derivative
ddy(1) = (2*y(1) - 5*y(2) + 4*y(3) - y(4)) / (x(2) - x(1))^2; % Second derivative

for i = 2:n-1
   
	dy(i) = (y(i+1) - y(i-1)) / (x(i+1) - x(i-1));
	ddy(i) = (y(i-1) - 2*y(i) + y(i+1)) / (x(i-1) - x(i))^2;
end

	dy(n) = (y(n-2) - 4*y(n-1) + 3*y(n)) / (2*(x(n) - x(n-1)));
	ddy(n) = (-y(n-3) + 4*y(n-2) - 5*y(n-1) + 2*y(n)) / (x(n) - x(n-1))^2;

end

[alexandra ligeti]

Hi Again,

Thank you so much for your reply. If I want to show whether velocity has an influence on error in readings could I just plot (dy, diff_data)?

Because when I do that, I get a very strange curve.

Thanks for your help

[Mathieu NOE]

if I normaize both datas by their maximum abs values (so they are within +/-1 range on the y axis, that makes visual comparisons easier) I get this result

maybe we can see that there are soe coincidence about the local peaks (positive and negative) , but not really a perfect match

figure,plot(delta_angle./max(abs(delta_angle)))
hold on
plot(diff_data./max(abs(diff_data)))
legend('normalized delta angle','normalized diff data');

[Mathieu NOE]

now if I look at the absolute normalized values I get this ... yes we can "see" some ressemblance, but .... again no perfect match

figure,plot(abs(delta_angle)./max(abs(delta_angle)))
hold on
plot(abs(diff_data)./max(abs(diff_data)))
legend('normalized delta angle','normalized diff data');

[Mathieu NOE]

it's my own interpretation, but we could pick for both curves their 4 peaks and then trace the values of one set of peaks vs the other set ?

[Mathieu NOE]

here I plotted the 4 peaks belonging to both datas sets. Of course the x indexes don't match but we consider that as "non problematic" (??)

assuming this is not pure nonsense, then If I plot only the the peaks of abs(diff data) vs the peaks of Abs(dy), maybe there is a kind of almost linear trend that appears :

[dy, ddy] = firstsecondderivatives((1:numel(mean_data)),mean_data);
delta_angle = dy/0.02;

x = 1:numel(delta_angle);

% delta_angle
y1 = abs(delta_angle);
y1 = y1./max(y1);
tf1 = islocalmax(y1,'MinSeparation',10);
x1_peaks = x(tf1);
y1_peaks = y1(tf1);

% diff_data
y2 = abs(diff_data);
y2 = y2./max(y2);
tf2 = islocalmax(y2,'MinSeparation',10);
x2_peaks = x(tf2);
y2_peaks = y2(tf2);


figure,
plot(x,y1)
hold on
plot(x,y2)
plot(x1_peaks,y1_peaks,'db')
plot(x2_peaks,y2_peaks,'dr')
legend('normalized delta angle','normalized diff data','normalized delta angle peaks ','normalized diff data peaks');

% now use tf1 and tf2 logical indexes on the raw (non normalized ) data 
figure,
[xx,ind] = sort(abs(dy(tf1)));% use sort to avoid zig zags in plot 
yy = abs(diff_data(tf2));
yy = yy(ind);
plot(xx, yy,'-*')
title('Abs(diff data) vs. Abs(dy)')
xlabel('Abs(dy)')
ylabel('Abs(diff data)')