How can add confidence intervals in the plot generated by the Curve Fitting Toolbox?

I have used the Curve Fitting Toolbox to fit a custom equation (modified Ratkowsky sqaure root model) to my data and generated using the Nonlinear least square method with the trust region algorithm. The app generates the 95%confidence limit of each of the parameters, but how can i use this to generate th confidence limits of the fit in the plot?

I have generated the code of the fitting from the app to edit the plot and the code is given below:

(aplogies if this is not enough informatio to asnwer the question. I am relatively new to coding, so if anyone needs more info, I can provide it).

Thanks,

Dipon

%% Fit: 'untitled fit 1'.
[xData, yData] = prepareCurveData( mid35x, mid35y );

% Set up fittype and options.
ft = fittype( 'a*(x-c)*(1-exp(b*(x-d)))', 'independent', 'x', 'dependent', 'y' );
opts = fitoptions( 'Method', 'NonlinearLeastSquares' );
opts.Display = 'Off';
opts.StartPoint = [0.260107478852342 0.290941052369803 0.995531610700984 0.800330575352401];

% 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 );
xlim([0,45]);
legend('observed growth rate', 'fitted model', 'Location', 'NorthWest' )
% Label axes
xlabel 'Temperature (in °C)'
ylabel 'Sq GR'

grid on
print(gcf,'foo.png','-dpng','-r300')

Answer 1

[John D'Errico]

I lack your data. But it is simple enough to make some up.

x = rand(50,1);
y = 1 + 2*exp(0.75*x) + randn(size(x))/10;
plot(x,y,'o')

●

So some arbitrary crap data. The model is exponential.

mdl = fittype('a + b*exp(c*x)','indep','x');

fittedmdl = fit(x,y,mdl,'start',[1 1 1])

fittedmdl = 
     General model:
     fittedmdl(x) = a + b*exp(c*x)
     Coefficients (with 95% confidence bounds):
       a =       1.166  (-0.2608, 2.593)
       b =       1.844  (0.4799, 3.208)
       c =       0.783  (0.3754, 1.191)

Pretty noisy data, so the parameters are not that close to the underlying model. fittedmdl is an object from the curve fitting TB.

whos fittedmdl

  Name           Size            Bytes  Class    Attributes

  fittedmdl      1x1               837  cfit               

If you don't know what methods apply there, then use methods!!!!!!!

methods(fittedmdl)

Methods for class cfit:

argnames       coeffnames     dependnames    fitoptions     integrate      numcoeffs      probnames      type           
category       coeffvalues    differentiate  formula        islinear       plot           probvalues     
cfit           confint        feval          indepnames     numargs        predint        setoptions     

Scan through the list. There, we see confint is one of the methods. Any bets what confint does? How about predint? Be careful, because you have both confint and predint available. You NEED to know the difference. confint just gives you intervals on the parameters.

xint = linspace(min(x),max(x),100);
CIF = predint(fittedmdl,xint,0.95,'Functional');
CIO = predint(fittedmdl,xint,0.95,'obs');
plot(fittedmdl)
hold on
plot(x,y,'o')
plot(xint,CIF,':b')
plot(xint,CIO,':g')

●

The wider set of bands allow you to predict where a new observation would fall. The narrow bands are around the estimated function itself. So the observational bands essentially have the estimated process noise added back in.

[Dipon Sarkar]

Dear John,

Thanks for the very detailed answer. I took the example that you have shown and used in my data and generated both the confidence intervals.

Im again sorry for my ignorance, but my supervisors asked me 'to provide confidence interval around the fitted model'. In this case should I be using the narrow bands for the function estimate or the observational bands?

Just to provide some background: these are growth rate data of a bacteria generated at different temperatures and the model is being fitted to create a predictive model that can be used to predict the growth rate in this temperature range.

x= 4,4,10,10,15,15,20,20,25,25,30,30,35,35,40,40

y=0.1157, 0.1233, 0.1865, 0.2520, 0.2519, 0.2814, 0.3680, 0.3186, 0.4549, 0.4559, 0.4241, 0.4697, 0.4953, 0.5767, 0.4163, 0.4279

Also here is my code and results

mdl= fittype( 'a*(x-c)*(1-exp(b*(x-d)))', 'independent', 'x', 'dependent', 'y' );
opts = fitoptions( 'Method', 'NonlinearLeastSquares' );
opts.Display = 'Off';
opts.StartPoint = [0.260107478852342 0.290941052369803 0.995531610700984 0.800330575352401];

[fitresult, gof] = fit( xData, yData, ft, opts )

xint = linspace(min(x),max(x),100);
CIO = predint(fitresult,xint,0.95,'obs');
CIF = predint(fittedmdl,xint,0.95,'Functional');
plot(fitresult)
hold on
plot(x,y,'o')
plot(xint,CIF,':b')
plot(xint,CIO,':g')

Thanks,

Dipon

[Salman Salamn]

You can do that by adding 'predobs' to the line where you the plot function:

h = plot( fitresult, xData, yData, 'predobs' );

This works with 2018a version. I am not sure if it is still valid for other versions.

Then you can find the lower/upper limits by confint(Fitresult, 0.95), where Fitresult is the output from the curve fitting tool, and the second argument specifies the confidence interval.

Hope this helps.

[Editor]

Hi @John D'Errico. Thank you for your practical example. It has really helped me a lot. However, I was wondering whether you could provide a link to a documentation so that I learn more about it?

Thank you in anticipation.

[John D'Errico]

I have no idea what links you are looking for. You can find the help for all of it in MATLAB. Just use the help. Or use doc.

help cfit/predint

 PREDINT  Prediction intervals for a fit result object or new observations.
    CI = PREDINT(FITRESULT,X,LEVEL) returns prediction intervals
    for a new Y value at the specified X value.  LEVEL is the
    confidence level and has a default value of 0.95.
 
    CI = PREDINT(FITRESULT,X,LEVEL,'INTOPT','SIMOPT') specifies the
    type of interval to compute.  'INTOPT' can be either 'observation'
    (the default) to compute bounds for a new observation, or
    'functional' to compute bounds for the curve evaluated at X.
    'SIMOPT' can be 'on' to compute simultaneous confidence bounds
    or 'off' to compute non-simultaneous bounds.
 
    If 'INTOPT' is 'functional', the bounds measure the uncertainty
    in estimating the curve.  If 'INTOPT' is 'observation', the bounds
    are wider to represent the addition uncertainty in predicting a
    new Y value (the curve plus random noise).
 
    Suppose the confidence level is 95% and 'INTOPT' is 'functional'.
    If 'SIMOPT' is 'off' (the default), then given a single pre-determined
    X value you have 95% confidence that the true curve lies between
    the confidence bounds.  If 'SIMOPT' is 'on', then you have 95%
    confidence that the entire curve (at all X values) lies between
    the bounds.

    Documentation for cfit/predint
       doc cfit/predint

[Editor]

@John D'Errico Thank you very much

[Dani A]

@Dipon Sarkar Please mark this answer as correct, especially given the time that John put in to writing this extremely detailed answer.