Goodness of fit , residual plot for fminbnd fitting

I have the data fitting using fminbnd solver. I am wondering how to evaluate the goodness of the fit and obtain residual plots. I dont see any options to display the residuals for fminbnd solver like other solvers. Below is my code:

t1 = [0:300:28800]'; % input X data
% input Y data
y_obs = [         
   0
0.0350666
0.170773
0.298962
0.400482
0.481344
0.541061
0.588307
0.626498
0.657928
0.684406
0.705545
0.721963
0.738828
0.753222
0.765903
0.776001
0.786196
0.795698
0.804062
0.81206
0.820732
0.825598
0.832848
0.837778
0.8436
0.848495
0.852999
0.858091
0.863251
0.86657
0.870919
0.875362
0.879617
0.882049
0.884957
0.887106
0.889922
0.894813
0.896395
0.900105
0.903234
0.905787
0.907843
0.909099
0.913799
0.914104
0.916195
0.920424
0.922772
0.923837
0.922742
0.924935
0.927408
0.92851
0.930684
0.930988
0.933917
0.935012
0.938209
0.940926
0.942448
0.943642
0.942436
0.94564
0.946308
0.949709
0.950971
0.951911
0.954338
0.955225
0.958114
0.958801
0.962341
0.963808
0.965617
0.965214
0.966752
0.971954
0.971949
0.973827
0.977233
0.977157
0.980893
0.979747
0.981409
0.984914
0.986015
0.986951
0.990709
0.990882
0.991937
0.992701
0.996347
0.998733
0.999351
1
];


%************

% Guessing the initial assumption d0 by finding the minimum using min function
%Implementing fminbnd instead of lsqnonlin

pfun = @(d) norm( ypred(d, t1) - y_obs);



dsamps=linspace(0,15e-10,50);

[~,imin]=min(  arrayfun(pfun,dsamps) );

[best_d,fval,exitflag,output]=fminbnd(pfun,dsamps(imin-1), dsamps(imin+1),...
                              optimset('TolX',1e-14))

best_d = 
     6.545737727815822e-10

fval = 
     4.578375975388594e-01

exitflag = 
     1

output = struct with fields:
    iterations: 8
     funcCount: 9
     algorithm: 'golden section search, parabolic interpolation'
       message: 'Optimization terminated:↵ the current x satisfies the termination criteria using OPTIONS.TolX of 1.000000e-14 ↵'

exitflag,

exitflag = 
     1

best_d,

best_d = 
     6.545737727815822e-10

%****************





%************
t2 = [0:5/60:8]';
predicted_y = ypred(best_d, t1);
figure(2)
plot(t2, y_obs, 'ro', t2, predicted_y, 'b-');
grid on
set(gca,'XLim',[0 8])
set(gca,'XTick',(0:0.5:8))
ylim([ 0 1.4])
ylabel('At/Ainf') 
xlabel('Time in h')
legend({'observed', 'predicted'})
title('fitting upto full 8 hours; thickness = 2.63')

%***************

%Plotting the verification of minimum

figure(1)
fplot(pfun,[0,4e-10])

Warning: Function behaves unexpectedly on array inputs. To improve performance, properly vectorize your function to return an output with the same size and shape as the input arguments.

hold on; plot(best_d*[1,1],ylim,'--rx'); 

title('Verification of whether minimum was correctly found by fminbnd solver')
hold off

%***** Fitting model equation

function y_pred = ypred(d, t1)

    a=0.00263;
    gama = 0.01005;
    
    L2 = zeros(14,1);
    L3 = zeros(100,1);
    L4 = zeros(100,1);
    L5 = zeros(100,1);
    S= zeros(97,1);
    y_pred = zeros(97,1);
    
    % t = 0; 
    
    L1 = ((8*gama)/((pi*(1-exp(-2*gama*a)))));
    format longE
    
  
    for t = t1(:).'
        for n=0:1:100
            L2(n+1) = exp((((2*n + 1)^2)*-d*pi*pi*t)/(4*a*a));
            L3(n+1) = (((-1)^n)*2*gama)+(((2*n+1)*pi)*exp(-2*gama*a))/(2*a);
            L4(n+1)= ((2*n)+1)*((4*gama*gama)+((((2*n)+1)*pi)/(2*a))^2);
            L5(n+1) = ((L2(n+1)*L3(n+1))/L4(n+1));
        end
        S((t/300) +1) = sum(L5);
        y_pred((t/300)+1)= 1 -(L1*S((t/300) +1)); % predicted data 
    end
end

[Adam Danz]

I edited the question to run the code and produce plots.

[Adam Danz]

The prediction-observation plot does not look like a good fit to me.

Answer 1

[Adam Danz]

See matlab's documentation on goodness of fit. There are lots of options and you've got all the variables you need to move forward.

The residuals are just the difference between the predicted y-values and the actual y values. Here's how to plot them:

figure()
tiledlayout(2,1)
nexttile
stem(t2, predicted_y-y_obs)
xlabel('t2')
ylabel('residuals')

nexttile()
histogram(predicted_y-y_obs)
xlabel('residual')
ylabel('frequency')

[Anand Ra]

Thank you Adam!