fitting with custom equation

Hi I am havng troble in fitting the data with costoum eqaution: I/y = x/a + (1-x)/b. with and b are the fitting parametrs. x and y values are as given below.

y = [1.6E5 2.5E5 4.1E5 8E5 1E6 2E6 7E6 2E7]

x = [17 20 27 59 62 81 89 95]

Thank you in advance for your support!

Answer 1

[John D'Errico]

(I assume you intended to write 1/y, and not I/y, where the variable I is not defined.)

y = [1.6E5 2.5E5 4.1E5 8E5 1E6 2E6 7E6 2E7];
x = [17 20 27 59 62 81 89 95];

Often it is the case, when you cannot fit a model to your data, it means you model does not have the correct shape. This model would be a nice STRAIGHT line, when we plot it as as 1/y versus x.

If we plot 1/y versus x, we would expect to see that behavior in the data. And here, it looks like we have not at all a straight line.

plot(x,1./y,'o')

I might also point out that since your data is scaled to go from 0-100 in x, so the model you wanted to write was probably

I/y = x/a + (100-x)/b.

How can we estimate a and b in that model? Easily enough. We could even use polyfit. There is no need to use a custom model in fit. But you can. Next, note that I'll fit the model as

1/y = x*a + (100-x)*b

and then invert a and b. fit manages to make it converge better if I do so.

mdl = fittype('x*a + (100 - x)*b','indep','x')

mdl = 
     General model:
     mdl(a,b,x) = x*a + (100 - x)*b

fittedmdl = fit(x(:),1./y(:),mdl,'start',[0.001 0.001]) % note that fit wants COLUMN vectors of data.

fittedmdl = 
     General model:
     fittedmdl(x) = x*a + (100 - x)*b
     Coefficients (with 95% confidence bounds):
       a =  -7.796e-09  (-2.357e-08, 7.975e-09)
       b =   5.469e-08  (3.57e-08, 7.367e-08)

plot(fittedmdl,x,1./y)

a = 1./fittedmdl.a

a = -1.2826e+08

b = 1./fittedmdl.b

b = 1.8286e+07

Again though, I don't think your model fits the data very well.

Answer 2

[Star Strider]

This is probably as good as you can hope for —

%  I/y = x/a + (1-x)/b

y = [1.6E5 2.5E5 4.1E5 8E5 1E6 2E6 7E6 2E7];
x = [17 20 27 59 62 81 89 95];

figure
plot(x, y)
grid

figure
plot(x, 1./y)
grid

objfcn = @(b,x) x./b(1) + (1 - x)./b(2);
opts = optimset('MaxFunEvals',1E6);
[B,rn] = fminsearch(@(b) norm(1./y - objfcn(b,x)), randn(2,1)+1E10, opts)

B = 2×1

1.0e+05 * 1.8497 1.8286

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rn = 2.4935e-06

figure
plot(x, 1./y, 'pb',  'MarkerSize',12, 'MarkerFaceColor','b', 'DisplayName','Data')
hold on
plot(x, objfcn(B,x), '-r', 'LineWidth',2, 'DisplayName','Regression')
hold off
grid
xlabel('$x$', 'Interpreter','LaTeX', 'FontSize',12)
ylabel('$\frac{1}{y}$', 'Interpreter','LaTeX', 'FontSize',12)
legend('Location','best')
text(35, 5E-6, sprintf('$\\frac{1}{y} = \\frac{x}{%13.5E} + \\frac{1-x}{%13.5E}$',B), 'Interpreter','LaTeX', 'FontSize',14)

.

[Alex Sha]

First of all, the fitting function could be rewritten from "1/y = x*a + (100-x)*b" to "y = 1/(x*a + (100-x)*b)", the results will then be good enough.

Sum Squared Error (SSE): 3179655964395.29
Root of Mean Square Error (RMSE): 630441.904975717
Correlation Coef. (R): 0.99637702865579
R-Square: 0.992767183232941

Parameter	Best Estimate   
---------	-------------   
a        	519856.258368166
b        	514524.404512105