How can i find a parameter in a formula when i have three series od data and wanna to find best answer for fourth parameter?

I have three series od data for x,y and z I wanna find the fourth parameter m(like the best fit for it)in a Maxwell relaxation formula which that is:

z=2*m*x*y

and the series of other parameters are:

z=[0.0892 .158 0.890 1.26 2.47 3.64 6.71 11.6 18.1 27.7 42.0 65.4 97.5 141.0 196.0 283.0 440.0 661.0 863];
x=[0.05 0.628 0.0791 0.0995 0.125 0.158 0.199 0.25 0.315 0.396 0.5 0.628 0.79 0.995 1.25 1.58 1.99 2.5 3.15];
y=[1.42 1.78 2.25 2.83 3.56 4.49 5.65 7.08 8.79 11.01 13.87 17.46 21.8 27.16 33.75 42.34 53.53 67.03 84.11];

I will be grateful for helping me on this case.

Answer 1

[Star Strider]

This works:

z=[0.0892 .158 0.890 1.26 2.47 3.64 6.71 11.6 18.1 27.7 42.0 65.4 97.5 141.0 196.0 283.0 440.0 661.0 863];
x=[0.05 0.628 0.0791 0.0995 0.125 0.158 0.199 0.25 0.315 0.396 0.5 0.628 0.79 0.995 1.25 1.58 1.99 2.5 3.15];
y=[1.42 1.78 2.25 2.83 3.56 4.49 5.65 7.08 8.79 11.01 13.87 17.46 21.8 27.16 33.75 42.34 53.53 67.03 84.11];

xy = [x; y];
fz= @(m,xy) 2.*m.*xy(1,:).*xy(2,:);
CF = @(m) sum((z - fz(m,xy)).^2);

m = fminsearch(CF, 1);
ze = fz(m, xy);

figure(1)
plot3(x, y, z, '+b')
hold on
plot3(x, y, ze, '-r')
hold off
grid on
view([30 40])

It estimates: m = 1.794 and seems visually to give a good fit.

[amir]

tanx for this solution but I have one more question and that is: how do u find 'm' ? cause it just give the figurative of them but no answer

[Star Strider]

My pleasure!

If you run the code in my Answer to calculate ‘m’.

You find ‘m’ as the output of the fminsearch function in this line:

m = fminsearch(CF, 1);

When I ran your data, I got: m = 1.794.

I thought this could be a ‘prototype’ Question for a more complicated problem so I used fminsearch. Your problem as stated is actually a linear problem and can be solved for ‘m’ using the mldivide function:

m = [2*x'.*y']\z';

giving the same answer for ‘m’.

[amir]

sorry for asking so many questions but the reason I ask this question is that I wanna fit the answer on log-log scale figure in matlab...

[Star Strider]

No problem on the questions.

You are going to plot in three dimensions with plot3, so you will have to specify the log or linear scale you want on the separate x, y, and z axes. The 2-dimensional loglog plot will not work with your 3-dimensional data. I can help you with the axis scaling, but you will have to tell me what you want, and what version of MATLAB you are using (since some plotting commands have changed in R2014b).

I apologise for the delay in replying. Microsoft decided today that I absolutely had to upgrade to Windows 8.1, and that took nearly 4 hours to complete.

[amir]

u must sorry me to bothering u so much

I actually use matlab 2013a and I want to plot those data in log-log-log scale and 3D dimensional then find the best fit for that formula I had given before to estimate the value of n(calculating of m on log-log-log scale).

because on your advise when I change the lineat axes to log scales it doesn't affect on the value of m and it does estimate base on linear axes.

[Star Strider]

Questions are welcome.

If I solve your problem, I will appreciate it if you Accept my answer.

This code does what you want:

z=[0.0892 .158 0.890 1.26 2.47 3.64 6.71 11.6 18.1 27.7 42.0 65.4 97.5 141.0 196.0 283.0 440.0 661.0 863];
x=[0.05 0.628 0.0791 0.0995 0.125 0.158 0.199 0.25 0.315 0.396 0.5 0.628 0.79 0.995 1.25 1.58 1.99 2.5 3.15];
y=[1.42 1.78 2.25 2.83 3.56 4.49 5.65 7.08 8.79 11.01 13.87 17.46 21.8 27.16 33.75 42.34 53.53 67.03 84.11];

xy = [x; y];
fz= @(m,xy) 2.*m.*xy(1,:).*xy(2,:);
CF = @(m) sum((z - fz(m,xy)).^2);

m = fminsearch(CF, 1);
ze = fz(m, xy);

figure(1)
plot3(x, y, z, '+b')
hold on
plot3(x, y, ze, '-r')
hold off
grid on
view([30 40])
legend('Data', 'Fitted Regression', 'Location', 'SE')
text(0.2, 30, 800, sprintf('\\itz\\rm = %6.3f\\bullet\\itx\\rm\\bullet\\ity', m))
xlabel('X')
ylabel('Y')
zlabel('Z')
set(gca,'XScale','log')
set(gca,'YScale','log')
set(gca,'ZScale','log')

producing this plot:

<<www-mathworks-com-matlabcentral-answers-uploaded_files-19985-How-20can-20i-20find-20a-20parameter-20in-20a-20formula-20when-20i-20have-20three-20series-20od-20data-20and-20wanna-20to-20find-20best-20answer-20for-20fourth-20parameter-20--202014-2010-2025.png>>

We have already estimated your ‘m’ parameter, and the fit is quite good, so you need do nothing other than what is in my code.

[amir]

Thank you very much....

[Star Strider]

My pleasure!