How to obtain the standard error for each of the fitted parameters

I am using the curvit toolbox of Matlab and I was wondering how can I get the standard error for each of the fitted parameters:a, b, c, and z. Let's say the x and y data points are:

x = [10, 20, 30, 40]
y = [0.1, 0.02, 0.01, 0.001]

you can visuzlize the data points by the code below

plot(x,y,'*')

The x, y data should fit to function below

function [fitresult, gof] = createFit(x, y)
[xData, yData] = prepareCurveData( x, y );

% Set up fittype and options.
ft = fittype( 'a*exp(b/(x-c)^z)', 'independent', 'x', 'dependent', 'y' );
opts = fitoptions( 'Method', 'NonlinearLeastSquares' );
opts.Display = 'Off';
opts.Lower = [0 0 0 0];
opts.StartPoint = [0.001 0.952083907850712 0.7812 2];
opts.Upper = [1 Inf Inf 3];

% Fit model to data.
[fitresult, gof] = fit( xData, yData, ft, opts );

% Plot fit with data.
figure( 'Name', 'untitled fit 1' );
h = plot( fitresult, xData, yData );
legend( h, 'y vs. x', 'untitled fit 1', 'Location', 'NorthEast', 'Interpreter', 'none' );
% Label axes
xlabel( 'x', 'Interpreter', 'none' );
ylabel( 'y', 'Interpreter', 'none' );
grid on
end

Answer 1

[Matt J]

If you're prepared to assume the parameter estimates have Gaussian errors, perhaps you can find the 95% confidence interval width using confint and divide that result by 3.92.

[Boy]

Thanks @Matt J I added the confint function to the fitting function as follows

function [fitresult, gof,ci] = createFit(x, y)
load census
[xData, yData] = prepareCurveData( x, y);

% Set up fittype and options.
ft = fittype( 'a*exp(b/(x-c)^z)', 'independent', 'x', 'dependent', 'y' );
opts = fitoptions( 'Method', 'NonlinearLeastSquares' );
opts.Display = 'Off';
opts.Lower = [0 0 0 0];
opts.StartPoint = [0.001 0.952083907850712 0.7812 2];
opts.Upper = [1 Inf Inf 3];

% Fit model to data.
[fitresult, gof] = fit( xData, yData, ft, opts );
ci=confint(fitresult,0.95)
% Plot fit with data.
figure( 'Name', 'untitled fit 1' );
h = plot( fitresult, xData, yData );
legend( h, 'tow vs. T', 'untitled fit 1', 'Location', 'NorthEast', 'Interpreter', 'none' );
% Label axes
xlabel( 'T', 'Interpreter', 'none' );
ylabel( 'tow', 'Interpreter', 'none' );
grid on
end

but I get the following errors:

"Error using cfit/confint (line 36)

Cannot compute confidence intervals if #observations<=#coefficients."

and

"Error in creatFit (line 17)

confint(fitresult,0.95)"

I don't understand what's wrong with the code.

[Matt J]

The problem is that you have 4 unknown variables and only only 4 data points to estimate them with.

[Boy]

thanks @Matt J

Answer 2

[Jeff Miller]

The notion of a standard error assumes some kind of random sampling. For example, the standard error of your 'b' parameter reflects the variation in b estimates that would be found across from many random samples like the one you have. It isn't really clear from your dataset what is varying randomly, or by how much it varies. You would need that kind of info to get standard error estimates.

[Boy]

Thanks @Jeff Miller

Can we creat a sample data with random samples to test the fit function and to obtain the standard error?

[Jeff Miller]

Yes, in principle. Generate (say) 1000 random samples. Estimate b for each sample. Compute the standard deviation of the 1000 estimated b values. That standard deviation is your estimated "standard error of b". Similarly you get estimates of the other parameters at the same time, and you can look at the correlation (across 1000 samples) between parameter estimates to get a feel for the tradeoffs between parameters. Of course, 10,000 or more is better if you are patient.

[Boy]

@Jeff Miller Ok, thanks