How to put fitting constraints on smoothing splines

Having a curve as the one shown in the image, and knowing for sure that the peak of this curve is the blue point, we would like to reconstruct it such that it has its peak at the blue point, so

I did some fitting for it using smoothing splines through the curve fiiting toolbox, and with significantly increasing the weight of the two end points and the peak point in blue I got the following

But this has two issues:

1. The peak point of the spline is not exactly at the peak point that we want, it tries to approach it as much as possible but never be exactly at it.
2. Increasing the weight of this point made the fiited curve leave the original track of the curve. trying to solve this, again, we visualy choose some knot point and increased the weight for them and got the following, (each plot shows the results for choosing different knot point).

But again, incresing the weight of some point would make the fitted curve lose its track for other points.

So what is a better way to choose the knot points other than just incresing the weight of some of them such that the fitted curve would pass through as much points as possible from the original curve without affecting its smoothness? and how can we set its maximum point to be exactly at the point that we want. Or would there be a better technique to use in our case rather than smoothing splines?

Attached is the data I am using for the plots.

[Bruno Luong]

If you know the model has peak at the dark blue point, then clearly your data is incorrect.

Any fitting would not be accurate. This is NOT question of using spline or some other model.

[Malak]

the peak part of the green data is incorrect that is true. and we have the correct data (the blue point) which we are trying to use as a knot point to reconstruct the curve correctly. We are trying to look for a method that would force the spline to have its peak at the point that we want.

Answer 1

[Mathieu NOE]

hello

this is the poor's man suggestion, (no Curve Fitting toolbox, no spline smoothing)

try it and let me know

load data.mat
[myb,ib] = max(yb);
mxb = xb(ib);
xa = xb;
ya = yb;
xa(ib) = [];
ya(ib) = [];


% spline smoothing or simply with smoothdata ? 
xs = xa;
ys = smoothdata(ya,'lowess',9);

% non linear y  shift ? 
% trying to create a y shift based on x distance from peak point (exp
% distribution)
dx = abs(xs-mxb);
dy = abs(myb-max(ys));
y_shift = dy*exp(-dx.^2/2000);
yss = ys + y_shift;


figure(1);
plot(ya,xa,'og');
hold on 
plot(myb,mxb,'*b','Markersize',15);
plot(ys, xs);
plot(yss, xs,'g');
legend('raw data','peak point','smoothed raw data','streched smoothed data')

[Malak]

Thank you very much for your reply. I tried your suggestion and I have the following concern:

Does this method garantee that the final fitted curve will have its peak exactly at the indicated point or is it close to it? because when I tried it, I can see that the final curve is not intersecting the point, it is just close to it.

[Mathieu NOE]

hello again

this is an improvement over the previous code

now we are 100 % sure to be going through the peak point

load data.mat
[myb,ib] = max(yb);
mxb = xb(ib);

% spline smoothing or simply with smoothdata ? 
xs = xb;
ys = smoothdata(yb,'sgolay',17);

% make sure that the ys(ib) value cannot be lower than max(ys)
if max(ys) > ys(ib)
    ys(ib) = max(ys);
end

% non linear y  shift ? 
% trying to create a y shift based on x distance from peak point (exp
% distribution)
dx = abs(xs-mxb);
dy = abs(myb-ys(ib));
y_shift = dy*exp(-dx.^2/3000);
yss = ys + y_shift;

figure(1);
plot(yb,xb,'og');
hold on 
plot(myb,mxb,'*b','Markersize',15);
plot(ys, xs);
plot(yss, xs,'g');
legend('raw data','peak point','smoothed raw data','streched smoothed data')

[Mathieu NOE]

I somehow tried to find a simpler code and at the same time match your requirement

I wonder if the shape of the curve is exactly what you want of if this "corner" must be avoided

we can simply remove that "disturbing" neighbour if we want to

[Mathieu NOE]

hello

problem solved ?

[Malak]

Yes. Thank you for your efforts, I really appreciate it.

[Mathieu NOE]

as always, my pleasure !

Answer 2

[Bruno Luong]

GIGO, using my FEX submission BSFK

load('data.mat')
[yp,imax] = max(yb);
xp = xb(imax);
xb(imax) = [];
yb(imax) = [];
% force fit pass through (xp,yp) with horizontal ta,gential (peak)
pntcon = struct('p',{0 1},'x',{xp xp},'v',{yp 0});
options = struct('pntcon', pntcon);
% https://www.mathworks.com/matlabcentral/fileexchange/25872-free-knot-spline-approximation
pp = BSFK(xb,yb,4,20,[],options);

close all

xi = linspace(min(xb),max(xb),600);
yi = ppval(pp,xi);
plot(xi,yi,'r',xb,yb,'ob',xp,yp,'r+');
grid on

Zoom yo make sure that peak is at the (xp,yp) coordinates

[Bruno Luong]

If I increase the spline order to pentic instead of cubic, the weird shape becomes apparent

...
pp = BSFK(xb,yb,6,20,[],options)
...

[Bruno Luong]

or quadratic spline

[Alex Sha]

try the fitting function below:

Sum Squared Error (SSE): 1.1865504761041E23
Root of Mean Square Error (RMSE): 62890128427.9206
Correlation Coef. (R): 0.97834912925758
R-Square: 0.957167018719065

Parameter	Best Estimate    
---------	-------------    
y0       	372158039488.346 
a        	591488381059.073 
xc       	339.045269108107 
w        	-41.8638567923805
p1       	-462109113.796853