How can i get prediction Bounds from curve fitting?

I know how to fit the data to a custom equation using the fitting tool as yous see in the picture, and get the prediction bounds as well. I want to get the equation or matrix output for these prediction bounds. How can I do that?

Answer 1

[William Rose]

@David Nielsen-Franco,

I generated some data:

t=0:20; c=1; k=0.4; s=0.1; CAs3=c*exp(-k*t)+s*randn(size(t));
plot(t,CAs3,'rx'); grid on; xlabel('t'); ylabel('CAs3'); hold on

I fitted the data with cftool, using the Exponential model: y=a*exp(b*x). I saved the fit results to the workspace, by clicking Fit > Save to Workspace. I checked all three boxes, to save the fit, goodness of fit, and fit output, respectively. I accepted the default names for these items: 'fittedmodel', 'goodness', and 'output'. I also selected Tools > Predictioin Bounds > 95% to get 95% lines on the plot in the tool. Screenshot below:

I compute the best fit curve, using the fit results:

yhat=feval(fittedmodel,t);

The lower 95% boundary curve is constructed by multiplying the root mean square error of the fit by 2.5% point on the T distribution with df degrees of freedom, where df=number of fitted points-number of fitted parameters. In this case, df=21-2=19.

The upper 95% boundary curve is constructed as above, but we use the 97.5% point on the T distribution. The T distribution is symmetric about zero, therefore we can just compute the 97.5% point, and use its negative when computing the lower curve.

df=goodness.dfe;  %degrees of freedom
w=icdf('T',0.975,df);  %half-width multiplier for the 95% confidence interval
shat=goodness.rmse;  %estimate of sigma for this fit (pardon me)
ylow=yhat-shat*w*ones(size(t));
yhigh=yhat+shat*w*ones(size(t));

Add the bounds to the plot:

plot(t,yhat,'-k',t,ylow,'--k',t,yhigh,'--k');
legend('Data','Fit','lower 95% CI','upper 95% CI');

This produces the plot below:

We expect 5% of the fitted points to be outside the lines. In this case, one out of 21 points is outside the lines - as expected.

[William Rose]

@David Nielsen-Franco,

The answer I gave above produces confidence bounds that seem to match the plotted bounds in cftool. The formula I gave produces bounds with equal vertical height at all x-values. You have probably seen confidence bounds for linear regression that have an hourglass shape: skinny in the middle and wide at the ends. Such bounds are more realistic than the bounds I gave. You can generate those kinds of bounds if you wish. To do it, read pages 7-52 to 7-56 (also known as pages 260-264) of this document.

[David Nielsen-Franco]

Thank you so much!

[William Rose]

@David Nielsen-Franco, You're welcome. Good luck with your chemistry.