How to smooth the matlab plot to get the desired plot shape?

Is there a way to change figure one to figure 2 (like the lines I draw in red and black color) without changing the values of y1 and y2? Please help.

Figure 1:

Figure 2:

Below is my code

clear all; close all; clc;
x= [0,0.1,0.2,0.3,0.4,0.5,0.6,0.7,0.8,0.9,1];
y1 = [0    0.0833    0.1583    0.2167    0.1500    0.3250    0.3750    0.3000    0.5917    0.3750    0.5000];
y2=  [ 0         0    0.0167    0.0750    0.1000    0.0917    0.1167    0.1583    0.1083    0.2000    0.1833];
figure
plot (x,y1,'o')
hold on
plot (x,y2,'o')
Xi = 0:0.005:1;
Yi = pchip(x,y1,Xi);
Yi_spline = spline(x,y1,Xi);
h(1) = plot(Xi,Yi,'-','color',lines(1));
h(2) = plot(Xi, Yi_spline, '--', 'color', lines(1));
Yj = pchip(x,y2,Xi);
Yj_spline = spline(x, y2, Xi);
h(3) = plot(Xi,Yj,'-','color',[0.85, 0.325, 0.098]);
h(4) = plot(Xi,Yj_spline,'--','color',[0.85, 0.325, 0.098]);
legend(h, "Yi pchip", "Yi spline", "Yj pchip", "Yj spline", "location", "NW")

Answer 1

[Angelo Yeo]

I'm not sure about your intention. But the easiest way to smooth signals is moving average. See the doc below for more information about moving average.

https://www.mathworks.com/help/releases/R2023a/matlab/ref/movmean.html

x= [0,0.1,0.2,0.3,0.4,0.5,0.6,0.7,0.8,0.9,1];
y1 = [0    0.0833    0.1583    0.2167    0.1500    0.3250    0.3750    0.3000    0.5917    0.3750    0.5000];
y2=  [ 0         0    0.0167    0.0750    0.1000    0.0917    0.1167    0.1583    0.1083    0.2000    0.1833];
figure
plot (x,y1,'o')
hold on
plot (x,y2,'o')
dt = 0.005;
Xi = 0:dt:1;
Yi = pchip(x,y1,Xi);
Yi_spline = spline(x,y1,Xi);
h(1) = plot(Xi,Yi,'-','color',lines(1));
h(2) = plot(Xi, Yi_spline, '--', 'color', lines(1));
Yj = pchip(x,y2,Xi);
Yj_spline = spline(x, y2, Xi);
h(3) = plot(Xi,Yj,'-','color',[0.85, 0.325, 0.098]);
h(4) = plot(Xi,Yj_spline,'--','color',[0.85, 0.325, 0.098]);

%% Smoothing
Yi_smooth = movmean(Yi_spline, 100);
Yj_smooth = movmean(Yj_spline, 100);

h(5) = plot(Xi, Yi_smooth, 'r','linewidth',2);
h(6) = plot(Xi, Yj_smooth, 'k','linewidth',2);
legend(h, "Yi pchip", "Yi spline", "Yj pchip", "Yj spline", "Yi smoothed", "Yj smoothed", "location", "NW")

[Haya Ali]

Is there a way that the value of Yi smoothed remains zero at zero?

[Image Analyst]

You could do

Yi_smooth(1) = 0;

but why would you want that? Can you give a theoretical/physical reason why that would be best other than the original smoothed data? It doesn't seem like it would describe any real world situation.

[Angelo Yeo]

I agree with @Image Analyst. What's your intention, @Haya Ali?

Anyways, if you insist that the resultant curve should pass (0, 0), you can think of something like curve fitting for a quadratic polynomial without a bias term.

clear; close all; clc;
x= [0,0.1,0.2,0.3,0.4,0.5,0.6,0.7,0.8,0.9,1];
y1 = [0    0.0833    0.1583    0.2167    0.1500    0.3250    0.3750    0.3000    0.5917    0.3750    0.5000];
y2=  [ 0         0    0.0167    0.0750    0.1000    0.0917    0.1167    0.1583    0.1083    0.2000    0.1833];
figure
plot (x,y1,'o')
hold on
plot (x,y2,'o')
dt = 0.005;
Xi = 0:dt:1;
Yi = pchip(x,y1,Xi);
Yi_spline = spline(x,y1,Xi);
h(1) = plot(Xi,Yi,'-','color',lines(1));
h(2) = plot(Xi, Yi_spline, '--', 'color', lines(1));
Yj = pchip(x,y2,Xi);
Yj_spline = spline(x, y2, Xi);
h(3) = plot(Xi,Yj,'-','color',[0.85, 0.325, 0.098]);
h(4) = plot(Xi,Yj_spline,'--','color',[0.85, 0.325, 0.098]);
legend(h, "Yi pchip", "Yi spline", "Yj pchip", "Yj spline", "location", "NW")

%% Fitting a quadratic curve
% Set up fittype and options.
[xData, yData] = prepareCurveData( Xi, Yi_spline );
ft = fittype( 'p1*x^2+p2*x', 'independent', 'x');
opts = fitoptions( 'Method', 'NonlinearLeastSquares' );
opts.StartPoint = [0, 0];
Yi_fit = fit( xData, yData, ft, opts );

[xData, yData] = prepareCurveData( Xi, Yj_spline );
ft = fittype( 'p1*x^2+p2*x', 'independent', 'x');
opts = fitoptions( 'Method', 'NonlinearLeastSquares' );
opts.StartPoint = [0, 0];
Yj_fit = fit( xData, yData, ft, opts );

h(5) = plot(Xi, Yi_fit.p1*Xi.^2 + Yi_fit.p2*Xi,'r','linewidth', 2);
h(6) = plot(Xi, Yj_fit.p1*Xi.^2 + Yj_fit.p2*Xi,'k','linewidth', 2);
legend(h, "Yi pchip", "Yi spline", "Yj pchip", "Yj spline", "Yi smoothed", "Yj smoothed", "location", "NW")

[Haya Ali]

Actually I was recommended to take avergae of my results 100 times ti smooth my original plot. I didnt want to spend many days on one graph that is why I am trying to get a plot that is smoothest. Thank you so much for your help.