How to find uncertainties in Intercept and Slope of a fitted line

Hi everyone,

I am trying to calculate the uncertainties in the intercept and the slope of a fitted line seperately. Here is my code to fitting a linear line. Can anyone help me to find the uncertainties using polyfit function or any other function?

Thank you.

for bands = 1:7
    % Fit line to data using polyfit
    c(bands,:) = polyfit(Sun_Zenith_AA_ASD,ratio(:,bands),1);
    
    % Evaluate fit equation using polyval
    y_est(bands,:) = polyval(c(bands,:),Sun_Zenith_AA_ASD);
    
    % Add trend line to plot
    p_1= plot(Sun_Zenith_AA_ASD,y_est(bands,:),'b--','LineWidth',2)
    hold on
    
    % calculating r-sq and RMSE
    mdl{:,bands} = fitlm(Sun_Zenith_AA_ASD,ratio(:,bands));
    r_sq(bands,:) = round((mdl{:,bands}.Rsquared.Adjusted),4);
    RMSE(bands,:) = round((mdl{:,bands}.RMSE),4);
end

Answer 1

[William Rose]

@Dinithi Siriwardana Pathiranage,

The uncertainties in the fitted parameters are available in the model structure which is created when you call fitlm(). Use coefCI(mdl) to get the 95% confidence intervals for the estimated a0 and estimated a1. Here is what I mean.

Sun_Zenith_AA_ASD=[1:20]';  %x-values to demonstrate
%Next: generate 7 sets of y values: y=a0 + a1*x + noise
a0=70:-10:10;   %intercept values
a1=-3:3;        %slope values
ratio=a0.*ones(20,1)+a1.*Sun_Zenith_AA_ASD + 2*randn(20,7);  %y-values
%Next: Fit columns 1 and 2 of y-data.  
%Column 1: a0=70, a1=-3. Column 2: a0=60, a1=-2. Etc.
for i=1:7
   mdl=fitlm(Sun_Zenith_AA_ASD,ratio(:,i));       %fit the linear model
   a0est=mdl.Coefficients{1,1};
   a1est=mdl.Coefficients{2,1};
   confInt=coefCI(mdl);
   %Display results on console.
   fprintf('Model %d: Est.intercept=%.2f, 95%% C.I.=%.2f,%.2f\n',...
           i,a0est,confInt(1,:));
   fprintf('         Est. slope   =%.2f, 95%% C.I.=%.2f,%.2f\n',...
           a1est,confInt(2,:));
end

Model 1: Est.intercept=71.46, 95% C.I.=69.67,73.25

         Est. slope   =-3.14, 95% C.I.=-3.29,-2.99

Model 2: Est.intercept=58.84, 95% C.I.=56.54,61.14

         Est. slope   =-1.91, 95% C.I.=-2.10,-1.72

Model 3: Est.intercept=49.07, 95% C.I.=46.40,51.75

         Est. slope   =-0.93, 95% C.I.=-1.16,-0.71

Model 4: Est.intercept=38.90, 95% C.I.=37.13,40.66

         Est. slope   =0.05, 95% C.I.=-0.10,0.19

Model 5: Est.intercept=30.77, 95% C.I.=29.26,32.29

         Est. slope   =0.99, 95% C.I.=0.86,1.11

Model 6: Est.intercept=20.62, 95% C.I.=18.54,22.70

         Est. slope   =1.94, 95% C.I.=1.76,2.11

Model 7: Est.intercept=10.13, 95% C.I.=7.98,12.28

         Est. slope   =3.02, 95% C.I.=2.84,3.19

Try it. Good luck.