Is least squares minimisation appropriate for Mass Balancing?

Introduction

Least squares minimisation is the most common form of setting the objective function for the mineral processing mass balancing problem.

This article is targeted toward readers who are already familiar with the mass balance problem and least squares minimisation.

It also presumes the reader is familiar with Lagrange multipliers.

The article questions whether the conventional approach of using least squares for the mass balance problem in mineral processing is valid.

The suggestion is that the conventional approach of using least squares for the mass balance problem in mineral processing is not valid; and that better validity is achieved by using mass-flow weighting.

The basis of least squares minimisation

Least squares minimisation is accredited to Legendre and Gauss.

Classic texts generally say that least squares was initially developed and commonly applied in regression.

Weighted least squares minimisation is a variation of least square minimisation where the variance (or reliability) of a measurement is known, and this is incorporated into the least square objective.

Least square minimisation is often linked to maximum likelihood, which is a probabilistic framework to determine the most likely probability.

Another probabilistic approach is called information theory, or maximum entropy.

It is very difficult to come to terms with the issue of when maximum likelihood should be used and when maximum entropy should be used.

My reading is that maximum likelihood is used to adjust model parameters.

Although maximum likelihood is applied to the mass balance problem to ‘correct’ the data; i.e. make the flows mass balance-consistent this may well be out of scope with the purpose of maximum likelihood.

Here I do not specifically ‘solve’ the problem I am raising.?I simply question the least squares approach.

For a start I am not specifically concerned with the use of least squares for the solid flow problem.?This does not mean I agree it is valid, but rather if it is not valid, it is not obvious.

I am focusing primarily on the 1 dimensional (1D) problem, e.g. assays or size distribution.

Problem example

Suppose there is a unit with known 100 tph feed, 90 tph tails and 10tph con.

The measured size component in a particular size-class is measured at 10%, 16% and 16% respectively.

It is understood that this is not meant to be a realistic example.

What would be the new size components after mass balancing.?Clearly:

13%, 13% and 13%

would be mass-balance consistent; and this is what one would get (or at least very close to) if one were to use an information theory approach.

But what do we get if we use least square minimisation; and to keep the problem simple we take the variance as the same so that the problem is a least squares minimisation, rather than the more complex weighted least squares.

Let us denote the streams with the subscripts 1, 2 and 3 respectively, and use f for flow and s for size.

And let us use the superscripts e and * for experimental and calculated (or adjusted).

The numeric solution is as given in Table 1.?Notice there is hardly any adjustment to the con.

Alternative strategy

It is not easy to argue that the solution is incorrect – but it is arguable that the solution is concerning – particularly when the information theory approach gives results that are more uniform.

Diehards will of course say that the least square approach is indeed correct and the solution as given is also correct.

Yet here I give an alternative argument.

Suppose our objective was not to minimise the above (the deviations relative to the variances) – but that we wanted the overall discrepancy for particles to be minimised.

Summary

The fact that the alternative approach gives results which are aligned with my expectation neither proves nor disproves that the alternative strategy is superior.

What it does show is the conventional approach has limited justification, and the method yield results that are concerning.

My personal view is that there are many problems with least squares – even beyond the focal problem discussed here; and consideration should be given to either:

1.???????Modifying weighted least squares minimisation so that the weights yield results that are more plausible, or

2.??????Abandoning weighted least squares entirely, or

3.??????Seeking input from statistics/probability specialists with a view to identifying a valid approach.

4.??????Abandoning least squares entirely in favour of an information theory approach.

The author favours strategy 4 but would certainly welcome strategy 3 – although recognising that there appears to be an absence of such specialists who would either be interested in the problem, or there may be insufficient funding to pursue such an approach.

As a consequence the author will mainly focus on strategy 4, for which there does not seem to be any reason against this approach.

Action

The author is currently developing a mass balance system based on information theory rather than least squares. This mass balance system is to be released via online courses by the end of 2023.

If you wish to keep informed please contact Stephen or Janet Rayward.