Fit a curve to data points x = f(y)

Hello,

I have data points represented by red stars on a graph (Fig 1, attached), and I would like to fit a curve to them. The curve does not have to be accurate -- it just needs to serve as a visual guide. I have drawn a sketch below to show what I would like. The other curves on the graph are by-products from the code I pulled this image from, and are not relevant to this problem. How can I fit a curve to the red data points in MATLAB?

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I have tried using polyfit, smoothingspline, pchip and other curve-fitting tools, but all of them connect the wrong data points together, since this graph is actually x = f(y). In MS Excel, if I switch the axes so the curve becomes a one-to-one function where y = f(x), I can easily fit a quadratic curve to the points.

I am currently plotting each point indivdually. I can plot the points with a line, but it is not very smooth (see Fig 2, attached). I guess I could just increase the number of data points, but that increases computation time too much. Any suggestions?

Please find below some things I have already tried. The red data points are saved in 1 x 11 doubles p_tn (x values) and p_Tn (y values).

Thank you so much! PLEASE let me know if you need more information, clarification, or data.

%Attempt 1 - Polyfit
[p,s,mu] = polyfit(p_Tn,p_tn,3);
[Y,delta] = polyval(p,p_Tn,s,mu)
X = linspace(0,0.05,length(Y));
plot(Y,X,'k-')

%Attempt 2 - Non parametric fitting
xq = linspace(0,0.05,100);
p = pchip(p_tn',p_Tn',xq);
pp = ppval(p,xq);
plot(xq,pp);

%Attempt 3 - Smoothingspline
f = fit(p_tn',p_Tn','smoothingspline');
plot(f)

%Attempt 4 - Split upper and lower half of data points and use polyfit
for i = 1:length(p_tn)-1

if p_tn(i) > p_tn(i+1)
    x1(i) = p_tn(i) ;
    y1(i) = p_Tn(i) ;
elseif p_tn(i) < p_tn(i+1)   
    x2(i) = p_tn(i) ;
    y2(i) = p_Tn(i) ;
end
end

p1 = polyfit(x1,y1,2);
p2 = polyfit(x2,y2,2);

xf1 = linspace(min(x1),0.05);
xf2 = linspace(min(x2),0.05);

f1 = polyval(p1,xf1);
f2 = polyval(p2,xf2);

plot(x1,y1,x2,y2)
plot(xf1,f1,'r--')
plot(xf2,f2,'r')

[dpb]

Make it easy for somebody to help you...attach the data points

[Sarah Armstrong]

Of course! I have p_tn (x values) and p_Tn (y values) attached in a .mat file. Thanks!

Answer 1

[darova]

Look HERE and HERE

[Sarah Armstrong]

Using your suggestion on this question, I wrote the following and produced this graph...Can I refine this at all? I think this is the closest so far.

figure(2)
% parameter - curve length
t = [0; cumsum(sqrt(diff(p_tn').^2+diff(p_Tn').^2))];
    % refine parameter
t1 = linspace(0,t(end));
xt = spline(t,p_tn,t1);
yt = spline(t,p_Tn,t1);
plot(p_tn,p_Tn,'o-r')                % original data
hold on
plot(xt,yt)                    % interpolated data
hold off

[darova]

Can I refine this at all?

What do you mean?

Maybe polyfit will be better for this

load('sarah_data_points.mat')

x = p_tn;
y = p_Tn;

figure(2)
% parameter - curve length
t = [0 cumsum(sqrt(diff(x).^2+diff(x).^2))];
    % refine parameter
t1 = linspace(0,t(end));
px = polyfit(t,x,2);
py = polyfit(t,y,2);
x1 = polyval(px,t1);
y1 = polyval(py,t1);
plot(x,y,'o-r')                % original data
hold on
plot(x1,y1)                    % interpolated data
hold off

[darova]

Another attempt:

• make X data the scale as Y
• make interpolation
• return original scale to X data

load('sarah_data_points.mat')
x = p_tn;
y = p_Tn;
xscale = (max(y)-min(y)) / (max(x)-min(x));
x = xscale*x;
% parameter - curve length
t = [0 cumsum(sqrt(diff(x).^2+diff(y).^2))];
    % refine parameter
t1 = linspace(0,t(end));
x1 = spline(t,x,t1);
y1 = spline(t,y,t1);

plot(x/scale,y,'o-r')                % original data
hold on
plot(x1/scale,y1)                    % interpolated data
hold off

[Sarah Armstrong]

Wow, this is beautiful! The sum of the squares and normalizing the values made all the difference -- I never would have guessed. Thank you so much!

I want to apply a linear polyfit between data points that have identical x or y values to straighten out the curve between the first and last couple of data points. Is this possible with a for loop? .

[darova]

DOn't know why you need loop? I believe separating data into groups should work!

Or maybe interpolate all data with spline and interp1 and fill with NaN areas you don't want

ind = y(3) < y1 & y1 < y(end-1);        % indices between ends

Experiment to get the result you want

[Sarah Armstrong]

Awesome, I will play around with this instead of using a loop. I am also thinking about trying polar co-ordinates to see if that can refine it as well, but for now the curve is accurate enough as-is! Thank you for your help.

Answer 2

[Ted Shultz]

Because you are looking to make a function of “y” rather than “x”, you would need to flip the X and Y when you solve the equation (this is equivalent to when you rotated in excel).

Try something like this:

p = polyfit(y,x,n)
y1 = linspace(0,4*pi);
x1 = polyval(p,y1);
figure
plot(x,y,'o')