fitting to a power function with an additional constant

Hello Matlab! I have this dataponits t and y. I want to fit those data points to a function y=a(t+c)^b, and get the parameters a,c and b.I have tried below, but not converging. Can anyone suggest me Thanks GH

close all
clear all
clc

t=[0.097
0.477
0.934
4.835
15.590
46.645
69.371
100.709
165.249
277.846
382.978
1000.64
1519.83
1991.60
2544.96
3023.40
3589.94
4159.93
4473.43
5186.36
6316.71
7134.15
9125.71
10571.90
12565.82
14916.64
16859.92
];

y=[20.96
21.33
20.77
20.96
21.14
21.33
20.24
19.71
17.90
15.98
14.51
10.05
8.37
7.40
6.49
6.05
5.54
5.17
4.82
4.57
4.26
3.87
3.52
3.25
3.08
2.80
2.61
];

loglog(t,y,'o')

F = @(x,xdata)x(1)*(x(2)+xdata).^x(3);
x0 =  [80 -0.1 130];
[x,resnorm,~,exitflag,output] = lsqcurvefit(F,x0,t,y)
hold on
plot(t,F(x,t))
hold off

Answer 1

[Walter Roberson]

You used an unconstrained curve fitting, which did a trial run with some values that were invalid for your function.

Your x0 is not especially close to the best parameters, which are approximately 851.271574244121 -0.604937228501513 436.783547500979

[friet]

Hello Walter! Thanks for you help. by try and error I can find the values of a,b,c. as coded below.

But still using those values I cant get it converged.

 close all
  clear all
  clc
  t=[0.097
  0.477
  0.934
  4.835
  15.590
  46.645
  69.371
  100.709
  165.249
  277.846
  382.978
  1000.64
  1519.83
  1991.60
  2544.96
  3023.40
  3589.94
  4159.93
  4473.43
  5186.36
  6316.71
  7134.15
  9125.71
  10571.90
  12565.82
  14916.64
  16859.92
  ];
  y=[20.96
  21.33
  20.77
  20.96
  21.14
  21.33
  20.24
  19.71
  17.90
  15.98
  14.51
  10.05
  8.37
  7.40
  6.49
  6.05
  5.54
  5.17
  4.82
  4.57
  4.26
  3.87
  3.52
  3.25
  3.08
  2.80
  2.61
  ];

a=850;
b=-0.58;
tt=0.1:10:10000;
c=700;
y_fitted=a.*((tt+c).^b);

figure(4)
loglog(t,y)
hold on
loglog(tt,y_fitted,'x')

[Walter Roberson]

a=850;
b=-0.58;
tt=t;
c=700;
y_fitted=a.*((tt+c).^b);
sum(y_fitted - y).^2

this will give you a residue of 56.8047627311654 . The coordinates I indicated, 851.271574244121 -0.604937228501513 436.783547500979, has a residue of 3.94159980868792

[friet]

Thanks for your kind help.

even with your suggested intial values, I cant get it work. Can you check the code.

 close all
  clear all
  clc

t=[0.097
0.477
0.934
4.835
15.590
46.645
69.371
100.709
165.249
277.846
382.978
1000.64
1519.83
1991.60
2544.96
3023.40
3589.94
4159.93
4473.43
5186.36
6316.71
7134.15
9125.71
10571.90
12565.82
14916.64
16859.92
];

y=[20.96
21.33
20.77
20.96
21.14
21.33
20.24
19.71
17.90
15.98
14.51
10.05
8.37
7.40
6.49
6.05
5.54
5.17
4.82
4.57
4.26
3.87
3.52
3.25
3.08
2.80
2.61
];

loglog(t,y,'o')

F = @(x,xdata)x(1)*(x(2)+xdata).^x(3);
x0 =  [851.271574244121 -0.604937228501513 436.783547500979];
[x,resnorm,~,exitflag,output] = lsqcurvefit(F,x0,t,y)
hold on
plot(t,F(x,t))
hold off

[Walter Roberson]

You have x(2)+xdata, where xdata is t. min(t) is 0.097 so if your trial x(2) is less than -0.097 then you would get a negative value for x(2)-min(t), and you would then try to raise that negative to a floating point power. You thus need a constraint that x(2) >-min(t) to avoid that negative raised to a floating point.

I will need to investigate further as to why I did not encounter this problem in my own tests.