How to fit a multivariate nonlinear function to experimental data?

I've developed a function that should fit my experimental data. I'm trying to use the nonlinear regression model, but I'm having some difficulties executing the command with the function and associated data.

The function (shown below), in its full expression, is comprised of 8 variables (i.e., b(1), b(2), b(3), b(4), b(5), b(6), b(7), and x). x is the independent variable. b(4) is fixed at 0 or +/-Pi/6 in these examples.

Three representative data sets are found in the attachment that vary in complexity. I provide starting coefficient guesses for each set based on beta0 = [b(1) b(2) b(3) b(4) b(5) b(6) b(7)].

Data 1:

beta0 = [0.89 0.81 0 0 0 0 0]

Data 2:

beta0 = [0.38 0.97 0 0 0 0.54 0]

Data 3:

beta0 = [0.71 0.835 0 -0.523599 -0.15 -0.045 0]

The full functional form is illustrated here:

modelfun = @(b,x)b(1)^4 * cos(x)^2 * cos(b(4) - b(6) * sin(x))^4 + b(1)^4 * cos(b(4) - b(5) * sin(x))^4 * sin(x)^2 + b(3)^2 * (cos(x)^2 * (3 * b(2)^2 * cos(b(4) - b(6) * sin(x))^2 - 2 * b(1) * b(2) * cos(b(4) - b(6) * sin(x)) * cos(b(4) - b(5) * sin(x)) + b(1)^2 * cos(b(4) - b(5) * sin(x))^2) + (b(2)^2 * cos(b(4) - b(6) * sin(x))^2 - 2 * b(1) * b(2) * cos(b(4) - b(6) * sin(x)) * cos(b(4) - b(5) * sin(x)) + 3 * b(1)^2 * cos(b(4) - b(5) * sin(x))^2) * sin(x)^2) * sin(b(4) - b(7) * cos(x))^2 + b(3)^4 * sin(b(4) - b(7) * cos(x))^4

[Star Strider]

... b(5) depends on b(1) ...

If they are functionally related, express ‘b(5)’ as a function of ‘b(1)’, and use that functional relationship for ‘b(5)’ instead of a separate parameter.

You have three columns of dependent variables. Do you want to fit all at once, or one column at a time?

Also, ‘modelfun’ will fail because of the upper-case initial letters in the trigonometric functions.

And it would be best to use Vectorization (link) in ‘modelfun’.

[Carbon]

I'd rather keep the variables as they are for now. I'd like to simply fit one column at a time. The trig functions are now lower-case.

I'm not sure what you mean by using vectorization in relation to the function (I did look at the link). Can you explain this?

[Greg Heath]

Do not use the title word COMPLEX when dealing with real-valued functions and numbers.

My first guess at a more suitable word is "COMPLICATED"

Greg

[Star Strider]

By ‘vectorization’ I mean using element-wise operators instead of operators that could be interpreted as array operations.

Further reading: Array vs. Matrix Operations (link)

[Carbon]

Thank you. Vectorization became a more significant contributor to my issue than the degrees/radians usage.

Answer 1

[Walter Roberson]

Your x values appear to be in degrees, but your Cos and Sin are going to be interpreted as requests to calculate with radians. You would need cosd and sind to calculate with degrees. However, if you are calculating with degrees, then I would tend to expect that b(4) would potentially be much larger than negative 1/2 degree.

[Carbon]

Yes! That's correct. The experimental data is in degrees, but the function is in radians. I would need to run it from 0 to 2Pi.

However, I'm still not sure how to set up the command with my data.

[Walter Roberson]

You had written

Cos(b(4) - b(5) * Sin(x))

we could tell your x was in degrees, which would lead to the suggestion you use sind(x) instead of Sin(x). However, we cannot tell whether you expect that your phase b(4) would be in degrees or in radians? When you wrote beta0 = [0.71 0.835 0 -0.523599 -0.15 -0.045 0] was the -0.523599 intended to convey degrees or radians ?

[Carbon]

This value for for b(4) is in radians.

[John D'Errico]

I would strongly suggest that you work ENTIRELY in degrees or radians. Otherwise, you will constantly be fighting bugs. Convert your data or your parameters to be consistent.