formula for nonlinear regression model

Dear MatLab Experts,

I have four column vectors with 14 elements representing respectively:

Area, Max.Diameter, Min.Diameter, Field Size Factor (FSF) of custom-made collimators.

I believe the FSF depends somehow on the other three quantities. The dependence is not linear.

I woud like to tryto model the FSF as a power law involving products of the other quantities.

For instance, FSF ~a*(Max.Diam*Min.Diam)^b + c*(Area/Max.Diam)^d + e*(Area/Min.Diam)^f + g*(Max.Diam/Min.Diam)^h + i

where a, b,c,d,e,f,g,h,i are the unknown model coefficients.

I am pretty sure most of the terms are useless. Perobably just one is necessary for the model. However, I do not know which one is the most important term. I expect the modeling function will figure that out.

My problem is that I do not know how to write the above formula in the proper format expected by the nonlinear regression functions.

I need some help to set up the above outlined particular model.

Thank you very much in advance.

Best regards,

Maura E. M.

Answer 1

[SYED IMTIAZ ALI SHAH]

Is this what you want?

x = MaxFDiam .* MinFDiam + Area ./ MaxFDiam + Area ./ MinFDiam + MaxFDiam ./ MinFDiam; % your formula

Linear model Poly7:

Your resultant equation with coefficients.

f(x) = p1*x^7 + p2*x^6 + p3*x^5 + p4*x^4 + p5*x^3 +

p6*x^2 + p7*x + p8

Coefficients (with 95% confidence bounds):

p1 = 2.961e-18 (-5.495e-18, 1.142e-17)

p2 = -6.914e-15 (-2.189e-14, 8.064e-15)

p3 = 6.708e-12 (-4.044e-12, 1.746e-11)

p4 = -3.496e-09 (-7.522e-09, 5.297e-10)

p5 = 1.054e-06 (2.132e-07, 1.895e-06)

p6 = -0.0001832 (-0.0002798, -8.656e-05)

p7 = 0.01691 (0.01138, 0.02244)

p8 = 0.3618 (0.2454, 0.4782)

Goodness of fit:

SSE: 5.048e-06

R-square: 0.9998

Adjusted R-square: 0.9995

RMSE: 0.0009173

[Maura E. Monville]

The result is amazing.

We can easily tell that the 1st observation is an outlier. This is found by my Monte Carlo simulation as well as my much simpler liner model which is a bad fit.

I had in mind to attemp a nonlinear model. That is the terms that you linearly added in the variable 'x' were originally (in my attempted formula) raised to powers whose exponents are coefficients of the model.

I see the message "Equation is badly conditioned. Remove repeated data point".

Why this messge and how can we go about it? Is it going to affect the model and why?

Did you use a MatLab functiion to obtain this model?

Did you use 'fitlm' or another one?

How did you get the idea of summing the terms I proposed rather than raising them to powers to be found by the model?

How can I try to fit terms tat are raised to powers whose coefficients are the model unknown?

Thank you

[SYED IMTIAZ ALI SHAH]

I use the Curve Fitting Toolbox which uses the Polyfit function. Means you can use both Polyfit and Curve Fitting Toolbox (It is more convenient).

I just used your equation without any coefficient and power, and let matlab find these automatically for me (without telling matlab).

As I am not a professional statistician, I don't fully understand the error, I used the equation derived from this toolbox in many practical experiments and it gave me best results.

Moreover some of the coefficients look somewhat odd to me also.

[Maura E. Monville]

Yes. The coefficients are very small. But the fit is close to perfection.

I have sdeveloped a Monte Carlo simulation that reproduces the trend of the experimental measurements but not as accurately as your model.

I tried to fit a much simpler linear model using 'fitlm'. The result was bad although the R^2 was more than 0.9. But some P_values were 'NaN'.

I am going to accept your very good solution. Although I really would like to learn how to play with nonlinear models.

Thank you.

Regards,

maura E. M.

[SYED IMTIAZ ALI SHAH]

You are welcome.