Isolate variables in nonlinear equation for regression

[edge333]

Hi all,

I have a nonlinear equation of the form:

$$\frac{TP_x}{TP_R} = c_0 + c_1 U_R^n + c_2 \frac{T_R^2}{\sqrt{U_R}}$$

This equation describes the relationship between tidal parameters and river discharge (velocity) in tidal rivers derived from the 1-D St. Venant equations. TP_x is some tidal property at station x along the river, TP_R is the same tidal property at a coastal reference station, U_R is the river flow (velocity), and T_R is the tidal range at the reference station.

What I am trying to do is calibrate this model using contemporary data where TP_x, TP_R, T_R, and U_R are known. Therefore, I am trying to determine the coefficients (c₀, c₁, c₂, and n) of this equation using Matlab's nlinfit function. The problem is that the last term in the equation complicates the regression because the T_R variable is on the right-hand side. Is there a way separating of the independent and dependent variables for regression? Consider any combination of TP_R, TP_x, and T_R to be the independent variable. I should also mention that 0.5 < n < 1.5.

thanks

 

[Stephen Tashi]

I don't use Matlab, but after some web browsing, I suggest you consider doing total least squares regression (http://www.mathworks.com/matlabcentral/fileexchange/31109-total-least-squares-method) instead of ordinary regression.

In a least square regression fit for Y = F(x1,x2,..) to data, it is assumed that x1,x2,... are measured without "error" and that the object is to find a curve that fits noisy data for Y. So if you one of your variables $TP_X,TP_R,T_R,U_R$ is measured very precisely and free of any conceptual random variations, you could us it for Y. However, if all your variables are on an equal footing as far as measurement errors or random fluctuations go, then total least squares regression would be a better choice.

 

[chiro]

Hey edge333.

Building on the post from Stephen Tashi - if the dependent variable spans the real line then linear regression can be used. If the variables contributing to the dependent variable are independent then it makes the whole process easier.

If not you will need to use correlation/covariance information to get accurate estimates (basically the standard errors will be under-estimated if things are correlated in some manner).