How to do a nonlinear fit using least squares

Question

Andrew 2019-9-14

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评论： Star Strider 2019-9-15

I have a set of data points giving me the values for the second virial coefficient,

for various values of T, of the virial expansion which is an equation that corrects the ideal gas law for empirical deviations:

I'm trying to do a least squares fit to determine how well the van der Waals equation predicts

using MATLAB.

The equation you derive for B using van der Waals' equation ends up being:

where a and b are unknown constants I need to determine and R is just the ideal gas constant.

Now my data points for true

values are here:

B = [-160 -35 -4.2 9 16.9 21.3]*10^-6 ;

T = [100 200 300 400 500 600]; (kelvin)

How can I write some code which will use the least squares method to generate estimates for a and b using the given data points?

My problem is that this is essentially a

equation which I'm not sure how to represent in MATLAB.

I can only find options for quadratic, cubic, polynomial fit etc.

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Answer 1

Star Strider 2019-9-15

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编辑：Star Strider 2019-9-15

在 MATLAB Online 中打开

This is actually a linear problem, so a linear approximation will estimate the parameters correctly:

R = 8.314462;                                                     % J K^−1 mol
B = [-160 -35 -4.2 9 16.9 21.3];
T = [100 200 300 400 500 600];
ab = [ones(size(B(:))) - 1./(R*T(:))] \ B(:);                     % Linear Approximation Parameter Estimation
Bfit = [ones(size(B(:))) - 1./(R*T(:))] * ab;                     % Linear Fit
figure
plot(T, B, 'pg')
hold on
plot(T, Bfit,'-r')
hold off
grid
xlabel('T')
ylabel('B')
legend('Data','Linear Least-Squares Fit', 'Location','E')

The parameters are:

a = 64.2001320479734
b = 182307.574287957

EDIT —

Added plot figure:

How to do a nonlinear fit using least squares - 2019 09 14.png

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Star Strider 2019-9-15

在 MATLAB Online 中打开

All you need to do is scale ‘B’, and the parameters will scale accordingly.

The parameters are the result of configuring the design matrix corectly, then using the mldivide,\ operator (or function) to do the linear fit. (If you want confidence intervals on the parameters as well, you can use essentially the same code with the regress function or its friends.)

Essentially the parameters are the x solution of

where A is the design matrix:

[ones(size(B(:))) - 1./(R*T(:))]

and b here, is appropriately, ‘B’. Both ‘T’ and ‘B’ must be column vectors for this to work correctly. (Capital letters in general use denote matrices, and lower-case letters denote vectors.)

The solution x corresponds to the constant (or intercept) term (the first column of the design matrix) and the slope term, corresponding to the second column of the design matrix. Here, x is the column vector ‘ab’, so the first row is ‘a’, corresponding to the first column of the design matrix, and the second row is ’b’, corresponding to the second column of the desigh matrix. The calculation of ‘ab’ is essentially:

without the instabilities associates with calculating

. (The mldivide function does not explicitly calculate the inverse, so the operation is more efficient and more mathematically stable.) Other than that, it is essentially straightforward linear algebra. (I also could have used polyfit, however this is more efficient.)

So

ab(1) = a
ab(2) = b

Explore the documentation on mldivide to understand in detail how it works. (The documentation explains it much better than I ever could.)

Andrew 2019-9-15

got it. Thanks for the clarification!

Star Strider 2019-9-15

My pleasure!

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Answer 2

Jackson Burns 2019-9-15

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createFit.m

Attached is code generated by cftool using a custom equation. Hope this helps!

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Jackson Burns 2019-9-15

If you're making this call in a function, it will tell you that result is unused if you don't later on reference result or return it from the function.

Andrew 2019-9-15

got it. Thanks!

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