How to fit a plot to a set of data points in a log-log scale?

Hi,

I have an equation of the form:

y = [1 + (1 - q) * x * m]^1/(1 - q)

I have 1000 values of y and x . I need to determine the values of 'q' and 'm' which best fits the equation in a log-log scale and also plot the same.

How to do it?

Answer 1

[Star Strider]

Perhaps something like this —

yfcn = @(x,m,q) (1 + (1 - q) .* x .* m).^1./(1 - q);                    % Objective Function

x = sort(rand(1,50));                                                   % Create Random Data
y = rand(1, 50);                                                        % Create Random Data

B0 = rand(2,1);                                                         % Initial Parameter Estimates
[B,fv] = fminsearch(@(b) norm(y - yfcn(x, b(1),b(2))), B0);             % Estimate Parameters

xv = linspace(min(x), max(x), 150);
yv = yfcn(xv,B(1),B(2));

figure
loglog(x, y, '.', 'DisplayName','Data')
hold on
plot(xv, yv, '-r', 'DisplayName','Regression')
hold off
grid
xlabel('X')
ylabel('Y')
legend('Location','best')
text(min(xlim)+0.01*diff(xlim), min(ylim)+0.01*diff(ylim), sprintf('$y = (1 + (1-%.2f) \\cdot x \\cdot %.2f)^\\frac{1}{1-%.2f}$',B(2),B(1),B(2)), 'Interpreter','latex', 'FontSize',12)

.

[Kashif Naukhez]

I am getting an error as "Exiting: Maximum number of function evaluations has been exceeded

- increase MaxFunEvals option.

Current function value: NaN "

I have attached the original data, could you please plot with this data?

[Star Strider]

The fit is not perfect, however the regression works for me —

T1 = readtable('https://www.mathworks.com/matlabcentral/answers/uploaded_files/1343429/data_points.xlsx')

T1 = 1023×2 table
        x           y   
    _________    _______

    0.0082825    0.67546
     0.023756    0.22581
    0.0086785    0.65298
    0.0029937    0.94721
     0.005617    0.78886
     0.017193    0.35288
     0.018741    0.31476
    0.0071413    0.72825
     0.020214    0.28837
      0.02242     0.2522
    0.0056988    0.78201
     0.004754    0.84164
     0.007176     0.7263
     0.032983    0.12512
    0.0045875    0.85239
     0.013214    0.47507

yfcn = @(x,m,q) (1 + (1 - q) .* x .* m).^1./(1 - q);                    % Objective Function

x = T1.x;
y = T1.y;

B0 = rand(2,1);                                                         % Initial Parameter Estimates
[B,fv] = fminsearch(@(b) norm(y - yfcn(x, b(1),b(2))), B0)              % Estimate Parameters

B = 2×1

-16.6911 -0.2806

fv = 4.8219

xv = linspace(min(x), max(x), 150);
yv = yfcn(xv,B(1),B(2));

figure
loglog(x, y, '.', 'DisplayName','Data')
hold on
plot(xv, yv, '-r', 'DisplayName','Regression')
hold off
grid
xlabel('X')
ylabel('Y')
legend('Location','best')
text(min(xlim)+0.01*diff(xlim), min(ylim)+0.01*diff(ylim), sprintf('$y = (1 + (1-%.2f) \\cdot x \\cdot %.2f)^\\frac{1}{1-%.2f}$',B(2),B(1),B(2)), 'Interpreter','latex', 'FontSize',12)

Warning: Negative data ignored

B0 = rand(2,1)*10;                                                         % Initial Parameter Estimates
yfcn2 = @(b,x) yfcn(x,b(1),b(2));
[B,fv] = lsqcurvefit(yfcn2, B0, x, y, zeros(1,2))                          % Estimate Parameters

Local minimum possible.

lsqcurvefit stopped because the final change in the sum of squares relative to 
its initial value is less than the value of the function tolerance.

B = 2×1

1.0e+06 * 0.0000 1.3517

fv = 293.7237

xv = linspace(min(x), max(x), 150);
yv = yfcn2(B,xv);

figure
loglog(x, y, '.', 'DisplayName','Data')
hold on
plot(xv, yv, '-r', 'DisplayName','Regression')
hold off
grid
xlabel('X')
ylabel('Y')
legend('Location','best')
text(min(xlim)+0.01*diff(xlim), min(ylim)+0.01*diff(ylim), sprintf('$y = (1 + (1-%.2f) \\cdot x \\cdot %.2f)^\\frac{1}{1-%.2f}$',B(2),B(1),B(2)), 'Interpreter','latex', 'FontSize',12)

The parameters are negative, so the display ed equation looks slightly strange, however there is no way in fminsearch to constrain them. If you have the Optimization Toolbox, use the lsqcurvefit function instead, and set the ‘lb’ srgument to zeros(1,2) to constrain them to be positive. I do not know if the regression will converge under those constraints, however.. (The negative values are ignored in the loglog plot because the logarithms of negative values are complex, and the logarithm of zero is negative infinity.)

Using lsqcurvefit with constrained parameters does not converge in any meaningful sense.

.

[Alex Sha]

Maybe the original fitting function: y = [1 + (1 - q) * x * m]^1/(1 - q)

should be: y = [1 + (1 - q) * x * m]^(1/(1 - q))

if so, the results will become a little better.