Can the Method of Least Squares be Used for Non-Linear Data?

[Prophet]

Many years ago I was given a problem involving the calibration of sets of weights. Each of two sets consisted of weights of 1.0 kg, 500.0 g, 300 g, 200 g, and 100 g. The two 1.0 kg weights were calibrated by the international agency in France and I was permitted to assume that their masses were known absolutely, i.e., 1.0 kg +/- a known correction.

The calibration of the remaining weights was carried out using a balance which would give the difference between the weights in the left pan and the weights in the right pan. This. was carried out in a vacuum to eliminate bouyancy.

Designating one set a A and the other as B the following comparisons were made:

A[1000] vs. B[1000]
A[500] + B[500] vs. A[1000]
A[500] + B[500] vs. B[1000]
A[500] + B[300] + B[200] vs. A[1000]
A[500] + B[300] + B[200] vs. B[1000]

Etc., etc.

Every possible combination of weights from either set which would nominally add to 1000 g was compared. It's been a long time but I think that every possible combination that would add to 500 (or 300 or 200) was also compared.

Now here my question: I was to use this data to solve for the most probable mass of each weight in the two sets with the exception of the 1.0kg weights. I was directed to use the method of least squares. Now I am familiar with using the method of least squares to find a straight line. For the life of me I could not, and still cannot, see any way of arranging this data so that the solution is a straight line. Was I using the wrong approach. Can the method of least squares be used to find something other than a straight line? Or a plane, etc.

The solution was found in a computer program written at the National Institute of Standards and Technology. I never figured out how the program worked but it compiled and my supervisors considered the problem solved. I was directed to move on the the next project. But to this day the whole thing bugs me.

 

[kuruman]

The method of least squares is a procedure that goes beyond linear regression. Several algorithms have been written for least-squares non-linear regression fits. The basic idea is that one calculates the sum of squares of the differences between "Experimental values" and "Theoretical values" and minimizes that difference with respect to appropriate parameters. In other words, you parametrize the problem and then find the optimum set of parameters that minimizes the sum of squares. In your case, the "Experimental values" would be the differences in the weights between the pans. You have five parameters, namely the weights that you want to calibrate, say B1000, B500, B300, B200 and B100. Your measured values are
Experimental (what you measured)
y_exp,1=Δ₁
y_exp,2=Δ₂
y_exp,3=Δ₃
...
The calculated theoretical values are obtained by moving everything to the left side in the equations that you showed above because you measured differences. Therefore you get (assuming that "A" are the known masses)
Theoretical (what it ought to be)
y_theo,1= 1000 - B1000
y_theo,2=-500 + B500
y_theo,3=500 + B500 - B1000
...
Then the program minimizes the difference $$\chi^2=\sum_{k=1}^N (y_{theo,k}-y_{exp,k})^2$$ with respect to the five unknown masses. Here N is the number of measurements and clearly you need many more measurements than you have parameters. That's why you measured all possible combinations.

Does this make sense?

 

[Prophet]

Does this make sense?

Thanks for your input. I have been quite busy lately, partly due to Christmas, and have not found time to do very much with the problem. But I will get to it ASAP. You've given me some new ideas.