Non Linear Regression Problem

I have tried to attempt this question however I don't know if I'm on the right track. I have already done (i), my difficulty is in (ii).

clear all
clc
syms x y a b 
 
X=[1:1:10];
Y=[0.5379,0.827,1.2654,1.5324,1.717,1.3983,1.6844,1.6892,1.506,1.707];
 
 
a = [1;0;0;0;0;0;0;0;0;0;0];
b = [1;0;0;0;0;0;0;0;0;0;0];
c = [0;0;0;0;0;0;0;0;0;0;0];
 
u = [((a(1)*x)/(b(1)+x)) , ((a(1)*2*x)/(b(1)+2*x)), ((a(1)*3*x)/(b(1)+3*x)), ((a(1)*4*x)/(b(1)+4*x)), ((a(1)*5*x)/(b(1)+5*x)), ((a(1)*6*x)/(b(1)+6*x)), ((a(1)*7*x)/(b(1)+7*x)), ((a(1)*8*x)/(b(1)+8*x)), ((a(1)*9*x)/(b(1)+9*x)), ((a(1)*10*x)/(b(1)+10*x))]';
 
B0=[a(1);b(1)];
U = [0;0;0;0;0;0;0;0;0;0;0];
 
 
for i=2:11
    
    U(i-1) = [((a(i-1)*1)/(b(i-1)+1)) , ((a(i-1)*2)/(b(i-1)+2)), ((a(i-1)*3)/(b(i-1)+3)), ((a(i-1)*4)/(b(i-1)+4)), ((a(i-1)*5)/(b(i-1)+5)), ((a(i-1)*6)/(b(i-1)+6)), ((a(i-1)*7)/(b(i-1)+7*x)), ((a(i-1)*8)/(b(i-1)+8)), ((a(i-1)*9)/(b(i-1)+9)), ((a(i-1)*10)/(b(i-1)+10))]';
 
    Jac = jacobian([((a*1)/(b+1)) , ((a*2)/(b+2)), ((a*3)/(b+3)), ((a*4)/(b+4)), ((a*5)/(b+5)), ((a*6)/(b+6)), ((a*7)/(b+7)), ((a*8)/(b+8)), ((a*9)/(b+9)), ((a*10)/(b+10))],  [a,b] );
 
    r = u - U(i-1);
    
    c = [a(i-1);b(i-1)] - inv(J(i-1)'.* J(i-1)).*(J(i-1)'.*r(i-1));
    
    c
    
    a(i) = c(1);
    b(i) = c(2);
end

Answer 1

[Image Analyst]

That equation is "the rate equation" and I have attached a demo for solving it using fitnlm().

% Fit a vertically shifted version of the the Fluorescence Recovery Curve (Michaelis-Menten function).
% One common non-linear equation is the Michaelis-Menten function,
% which is used to describe the reaction rate as a function of substrate concentration.
% The Michaelis-Menten function is characterized by a steep rise that gradually flattens into a plateau.

For your assignment you would want the Equation 1 version where no vertical offset is allowed (shown by the red equation above, not the green one).

[J.Cam]

Thank you for your help!

[J.Cam]

However, I need to use the equations as in the question to find the parameters. I cannot use ( modelFunction = @(b, tbl) b(1) * tbl(:, 1) ./ (b(2) + tbl(:, 1)); )

[Image Analyst]

J, why do you think that the model function is not of the form Y = aX / (b + X)??? It is exactly of that form.

b(1) is a, tbl(:, 1) is X, and b(2) is b. You don't need to do all that stuff with J and B - fitnlm() handles all that internally and just gives you back the coefficients, which is really all you care about.

[J.Cam]

So what is left in my assignment is to do a for loop to obtain estimates of b(1) and b(2) for 10 times right?

[Image Analyst]

Well it looks like your homework wants you to do it a certain way instead of using the built in function - they want you to do it manually according to their recipe. Sorry, but I don't have the time to devote to it.