Nonlinear Equation Fitting: Exploring Solutions

[patric44]

Homework Statement

why nonlinear least square method will not work well with the function
y=kx/(5+cx)

Relevant Equations

y=kx/(5+cx)

I was trying to fit a set of data to the nonlinear equation
$$
y=\frac{kx}{5+cx}
$$
and find the parameters k,c that will result in a best fit, but (I was told without explanation) that the parameters change as we increase x, so regular fitting techniques such as nonlinear least square will not work?
can any one explain this to me, if the parameters vary as a function of the independent variable what is the best way for the fitting, and is that even possible?

 

[BvU]

The idea of fitting is based on the assumption that the parameters do NOT change with x.
Start with a plot of the data.

 

[FactChecker]

If there are no limits on the behavior of the parameters $k$ and $c$, it is not only possible to fit the data, it is too easy. In that case, you could always just set $c=1$ and $k_i = y_i \frac {5+x_i} {x_i}$. If there are no repeated $x_i$ values associated with non-equal $y_i$ values, this would give you a perfect fit. If there are some repeated values in $x_i$s, you can just use the mean of the associated $y_i$ values. In some cases, this might give you a reasonable model, but I would not count on it.
You do not say if there is any random behavior in the data, or what the nature of the random behavior might be.

 

[BvU]

I was told without explanation

Yeah...

What's the range of your $x$ ? If it's in the several thousands $y$ is a constant !

$\$

 

[patric44]

there is a limit on the values of k,c I suppose, they are related to some constants about the system.
the values of x is discrete like 2,4,6,... and no values in between, the values of y ranges from 100 to say 3000 and so on, what I am confused about is how its possible for a parameter to vary as a function of the independent variable like i was told?!, I mean if they really change then no fit will be accurate? not only the non linear least squares

 

[FactChecker]

what I am confused about is how its possible for a parameter to vary as a function of the independent variable like i was told?!, I mean if they really change then no fit will be accurate? not only the non linear least squares

If the parameters $k$ and $c$ change and you perform a non-linear least squares fit to determine the best CONSTANT $k$ and $c$, then the fit will not be accurate. However, if you allow a different model, then you might get a good enough fit. Or you might try something like a cubic spline, which would fit the data in pieces with a continuous derivative. It all depends on what you want to use the model for.

 

[patric44]

If the parameters $k$ and $c$ change and you perform a non-linear least squares fit to determine the best CONSTANT $k$ and $c$, then the fit will not be accurate. However, if you allow a different model, then you might get a good enough fit. Or you might try something like a cubic spline, which would fit the data in pieces with a continuous derivative. It all depends on what you want to use the model for.

the idea is that I want to determine the values of k,c for the specific model given by $y=kx/(5+cx)$, because I will use the formula of the model in further calculations. from what I unstrood from you guys now I believe that the person who told me that the parameters vary could be mistaken or misinterpreted something, I really don't know