How to find a variable in a function that mostly match a set of data points

I have a one-variable function and a set of data points. The variable of the function is x.

If I mannuly change the variable from 1.0 to 3.0, the function's result will be closer to the data points I have as shown in the pictures below.

I was wondering what kind of MATLAB function I can use to automatically find the best value of x to match the function to the data points the most?

[Wilmer Alexander - Conferencista en Automatización y Robótica más influyente en Iberoamérica]

Thanks for asking. If possible. I remain at your disposal for any questions.

Gracias por preguntar. Si es posible. Quedo a su disposición para cualquier consulta.

Answer 1

[Star Strider]

First, I would experiment with fminsearch, since it uses a derivative-free approach. With one parameter, this should not be difficult.

For more difficult problems, I tend to favor the ga function. I have no idea how difficult your problem is.

With either of those, the function you would use would be something like this:

fcn = @(x) norm(ydata(:) - yourFunction(x,xdata(:));

This would be easier to respond to with the function and the data.

.

[KEDI YAN]

Hi Star,

I was, at the begining, trying to use fminsearch, but didn't really understand how to use it. Below is my MATLAB script. The variable x I mentioned in my question is actually Rp2, the function I mentioned is S21, and the data points I mentioned is S21_real.

You can mannuly try different value of Rp2 to match the function results to the data points. How do I use the fminsearch to automatically find the best Rp2 then?

clear
clc
wo = 2*pi*13.56*10^6;
Rsource = 50; Rload = 50;
Rp1 = 0.2; Rp4 = 0.2
L1 = 106e-9; L4 = 106e-9;
C1 = 1/(wo^2*L1); C4 = 1/(wo^2*L4);
Rp2 = 1.6; % The only variable is Rp2
Rp3 = Rp2; 
L2 = 1.84e-6; L3 = 1.84e-6;
C2 = 1/(wo^2*L2); C3 = 1/(wo^2*L3);
d23_array = []; S21_array = []; i = 0;
S21_real = [0.028 0.066 0.126 0.237 0.339 0.468 0.589 0.631 0.596 0.519 0.447 ...
    0.367 0.305 0.249 0.209]; % The data points
for d23=10e-3:10e-3:150e-3
    i = i+1;
    Z11 = Rp1+Rsource+1i*wo*L1-1i/(wo*C1);
    Z22 = Rp2+1i*wo*L2-1i/(wo*C2);
    Z33 = Rp3+1i*wo*L3-1i/(wo*C3);
    Z44 = Rp4+Rload+1i*wo*L4-1i/(wo*C4);
    M12 = MI_loop_spiral_50mm_Ben();
    M34 = MI_loop_spiral_50mm_Ben();
    M23 = MI_Tx_Rx_Ben(d23);
    Z12 = 1i*wo*M12;
    Z34 = 1i*wo*M34;
    Z23 = 1i*wo*M23;
    % VL/VS
    Numerator = Z12*Z23*Z34*Rload;
    Denominator = -Z11*Z22*Z33*Z44-Z12^2*Z34^2+Z11*Z22*Z34^2+Z11*Z44*Z23^2+Z33*Z44*Z12^2;
    VL_over_Vs = Numerator./Denominator;
    S21 = 2*VL_over_Vs*(Rsource/Rload)^0.5; % The function
    d23_array(i) = d23;
    S21_array(i) = abs(S21);
end
plot(d23_array,S21_array,'r-o');
hold on
plot(d23_array,S21_real,'bd');
hold off
legend('Function','Data Points')

[Star Strider]

I cannot run your code.

I need either:

MI_loop_spiral_50mm_Ben()

and:

MI_Tx_Rx_Ben(d23)

or vector or scalar values to substitute for them that have the appropriate magnitudes and signs.

What function do you want to fit to ‘S21_real’? Please describe what you want to do.

I would like to help, however until I have those missing bits of information, I can do nothing.

[KEDI YAN]

MI_loop_spiral_50mm_Ben()

function [M] = MI_loop_spiral_50mm_Ben()
    N_loop = 1;
    N_spiral = 7;
    R_loop = 19e-3;
    R_spiral = [12.4 14.5 16.6 18.7 20.8 22.9 25]*10^-3; % m
    uo = 1.2566e-6; 
    d= 15e-3;
    M = 0;
    for i=1:N_loop
        for j=1:N_spiral
            func = @(theta) cos(theta)./sqrt(R_loop(i)^2+R_spiral(j)^2+d^2-2*R_loop(i)*R_spiral(j)*cos(theta));
            M = M + uo*R_loop(i)*R_spiral(j)*integral(func,0,pi);
        end
    end  

MI_Tx_Rx_Ben(d23)

function [M23] = MI_Tx_Rx_Ben(d23)
    N_spiral = 7;
    R_spiral = [12.4 14.5 16.6 18.7 20.8 22.9 25]*10^-3; % m
    uo = 1.2566e-6; 
    M23 = 0;
    for i=1:N_spiral
        for j=1:N_spiral
            func = @(theta) cos(theta)./sqrt(R_spiral(i)^2+R_spiral(j)^2+d23^2-2*R_spiral(i)*R_spiral(j)*cos(theta));
            M23 = M23 + uo*R_spiral(i)*R_spiral(j)*integral(func,0,pi);
        end
    end

[Star Strider]

I cannot get your code to run correctly.

The ‘d23_array’ is (1x7), and that will not work in the plot call, since ‘S21_real’ is (1x15).

I cannot figure out what the problem is with it.

Stopping here.

.

[KEDI YAN]

I am so sorry for the inconvinence~~ I compressed all the necessary files into the ZIP file here. Could you try to run the FC_system.m and see if it works?

[KEDI YAN]

In the FC_system.m script, you will see two array: S21_array and S21_real

S21_array is a list of data points directly derived from the function S21.

S21_real is a list of data points I measured from my experiment.

Eventually, I am looking forward to matching the S21_array to S21_real.

The only variable in the function S21 is the Rp2, and the data in S21_array will be varied by adjusting the Rp2.

I am looking for a way to automatically find the best Rp2 so that the S21_array will mostly match the S21_real.

Let me know if that makes sense to you

[Star Strider]

I did my best to optimise that fit, and could not. I could not manage to optimise ‘Rp2’ to create a good fit to your data.

I will delete my Answer in a few hours.

[KEDI YAN]

Thanks so much for your help anyway! I really appreciate your patience. In this case, I guess the fminsearch and ga function do not work, is that correct?

[Star Strider]

My pleasure!

The problem is that the code is so complicated that it is difficult for me to understand how best to code a function that I can present to an optimisation function to fit it. I isolated the loop (and the supporting code that defined the constants) in a specific additional function, and did my best to optimise it by minimising:

norm(S21_array - S21_real)

and calculating ‘Rp2’ from that, however I was not able to get a value for ‘Rp2’ that would minimise it effectively. The fminsearch or ga functions are not the problem. The complexity of the code (and my apparent inability to understand it effectively) is.

[KEDI YAN]

Could you elaborate more about the ruction you first replied? fcn = @(x) norm(ydata(:) - yourFunction(x,xdata(:)); I guess in my case, it will be fcn = @(Rp2) norm(S21_real(:) - S21(Rp2)). How do I solve the fcn function and why would this give me the best Rp2 to match my function results to S21_real data?

[Star Strider]

What I posted:

b = norm(S21_array - S21_real);

was essentially what I tried (returning ‘b’ as the function output). It ‘worked’ in the sense that ‘Rp2’ changed from the initial estimate that I gave to fminsearch, however it did not work at all to fit the curve.

This idea:

fcn = @(x) norm(ydata(:) - yourFunction(x,xdata(:));

was simply not appropriate for your code, because your code does not work that way.

[KEDI YAN]

Sorry about the late response.

Could you sort of exlain how you feed Rp2 to fminsearch and how you used the output from the fminsearch?

[Star Strider]

I did something like this:

function b = Rp2_fcn(Rp2)
wo = 2*pi*13.56*10^6;
Rsource = 50; Rload = 50;
Rp1 = 0.2; Rp4 = 0.2
L1 = 106e-9; L4 = 106e-9;
C1 = 1/(wo^2*L1); C4 = 1/(wo^2*L4);
% Rp2 = 1.6; % The only variable is Rp2
Rp3 = Rp2; 
L2 = 1.84e-6; L3 = 1.84e-6;
C2 = 1/(wo^2*L2); C3 = 1/(wo^2*L3);
d23_array = []; S21_array = []; i = 0;
S21_real = [0.028 0.066 0.126 0.237 0.339 0.468 0.589 0.631 0.596 0.519 0.447 ...
    0.367 0.305 0.249 0.209]; % The data points
for d23=10e-3:10e-3:150e-3
    i = i+1;
    Z11 = Rp1+Rsource+1i*wo*L1-1i/(wo*C1);
    Z22 = Rp2+1i*wo*L2-1i/(wo*C2);
    Z33 = Rp3+1i*wo*L3-1i/(wo*C3);
    Z44 = Rp4+Rload+1i*wo*L4-1i/(wo*C4);
    M12 = MI_loop_spiral_50mm_Ben();
    M34 = MI_loop_spiral_50mm_Ben();
    M23 = MI_Tx_Rx_Ben(d23);
    Z12 = 1i*wo*M12;
    Z34 = 1i*wo*M34;
    Z23 = 1i*wo*M23;
    % VL/VS
    Numerator = Z12*Z23*Z34*Rload;
    Denominator = -Z11*Z22*Z33*Z44-Z12^2*Z34^2+Z11*Z22*Z34^2+Z11*Z44*Z23^2+Z33*Z44*Z12^2;
    VL_over_Vs = Numerator./Denominator;
    S21 = 2*VL_over_Vs*(Rsource/Rload)^0.5; % The function
    d23_array(i) = d23;
    S21_array(i) = abs(S21);
end
b = norm(S21_array - S21_real);
end

I remember making other changes in the code to make it more efficient, however I do not remember what they were.

the fminsearch call was essentially:

Rp2 = fminsearch(Rp2_fcn, rand);

EDIT — (3 Aug 2020 at 19:08)

Corrected typographical errors.

.