Using a regression model to find its coefficients

Hello,

I am testing different regression models and finding the a and b coefficents. I am having trouble puting the following model: V = a*(x/(x+b))

So far this is what I have but it gives me an error that 'a' is an unreconized variable. Please help, thank you!

w=  [0.3 3 4 15 35 80 600 1100]'; 
v = log([0.01 0.3 0.4 2 6 7 10 100]');
plot(w, v, 'b.-', 'LineWidth', 2, 'MarkerSize', 20)
grid on;

X = [ log(a) log(x) log(x-b)];

Unrecognized function or variable 'a'.

b = (X' * X) \ (X' * v)

[John D'Errico]

Why do you think you can use a bad formula to compute the results of a LINEAR regression to solve a nonlinear regression? In fact, what you have written will not even work crrectly for a LINEAR regression! (You got that formula wrong anyway.)

Why are you computing the logs of those terms? Yes, I know. You think that by computing the log, you have linearized the model. That does not work here.

This is a NONLINEAR regression. It cannot be linearized for a solution by taking the log.

My guess is you were told to linearize the problem, and your teacher only showed you how to linearize a model by taking the log, then threw you into the deep end of the pool for this assignment.

CAN you linearize it, by transforming the model in other ways? Well, yes. Technically, it can, though that screws around with the error structure in a very different way. But if you don't care about no steenkin error structure, then consider what happens if you multiply both sides of the expression

y = a*x/(x+b)

by the term (x+b). What happens now? Is this relation now linear in the unknowns, thus a and b?

Finally, learn how to correctly compute the unknowns. That is NOT by the expression (X' * X) \ (X' * v), which is not ony a bad way to solve it, it is not even correct at all. In fact, in the case of your problem, you can just use polyfit to estimate the unknowns at the end.

Answer 1

[Image Analyst]

That, a*(x/(x+b)), is the Michaelis-Menten rate equation. I have a demo for that. Plug in your data and try it.

%-----------------------------------------------------------------------------------------------------------------------------------
% 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.
% Initially, small increases in substrate concentration cause a substantial increase in reaction rate,
% but eventually, additional increases in concentration cause progressively smaller increases in reaction rate.
% Eventually reaction rate plateaus and does not respond to increases in concentration.
% The Michaelis-Menten function has this form:
% rate = Vmax*C / (K + C)
% where r is the growth rate (the dependent variable),
% C is the concentration (the independent variable),
% Vm is the maximum rate in the system, and
% K is the concentration at which the reaction rate is Vm/2.
% These last two parameters (Vm and K) are the Michaelis-Menten parameters to be fit.
% Rewritten in X and Y, you get:
% Y = b1 * X ./ (b2 + X)
% Ref: http://strata.uga.edu/8370/lecturenotes/nonlinearRegression.html
% https://www.graphpad.com/guides/prism/7/curve-fitting/index.htm?reg_classic_hyperbola.htm

[Image Analyst]

@Miriam Contreras Castillo Haven't heard from you. Did you have trouble adapting the code?

[Krista Blide]

Thanks for this code!