Inference of phase properties from sorting experiments and MLA data

In the last 20 years the development of new analytical methods and devices provided the possibility of high-resolution data in almost every field of science. Information is much easier to retrieve and in a depth never known before. But often these methods are expensive and a lot of time is needed for proper data acquisition and analysis. For example, in geosciences the Mineral Liberation Analyser (MLA) provides quantitative mineralogical microstructural information. This is a scanning electron microscope with automated software for high resolution images of rock specimen and sample compounds from mineral processing. The information can be used for evaluating the effect of mineral processing on a given ore sample in order to find the optimal processing parameters of each step and predict the overall recovery and grade the requested value minerals.

For example, the magnetic susceptibility of a mineral phase determines its behaviour during magnetic separation. It can be modelled as a linear combination of the susceptibilities of each occurring mineral phase with respect to its mass fraction:
\begin{equation} \label{equ} \overline{\chi_s} = \sum_{i=1}^{n} \frac{m_i}{m_s}\chi_i. \end{equation} (chi_s: susceptibility of the whole sample, chi_i: susceptibility of the i-th mineral phase, m_s: mass of the whole sample, m_i mass of the i-th mineral phase)

Unfortunately, quite often only the susceptibility of the composition can be measured in an experiment due to several reasons, e.g. if the composition consists of too many distinct components and the contained mineral particles consist of several mineral phases. During the separation the sample is split into several classes. The susceptibility can only be measured for such a class.
But we would like to infer the susceptibility for every single mineral phase. The common approach is a linear model, which fails if we have more mineral phases than susceptibility classes found in the experiment.

Our approach uses bootstrapping for constructing new subsamples out of the measured ones. Since every particle has the given mean property, taking such subsamples is like repeating the experiment. This provides a broader base with subsamples having a much higher variability of phase compositions. We repeat this procedure for every susceptibility class.

Furthermore we often do not only have one single value for each class found in the experiment, but a set of them within a certain bounded range. Instead of using the average we arbitrarily assign one of them to each new sample. This additionally prevents us from too many linear dependent equations using (\ref{equ}). We end up in an over-determined system of linear equations. For the solution we use the Moore-Penrose inverse, giving us the possibility to compute an estimation error for every mineral phase relying on the corresponding eigenvalue.

We will discuss simulation results and apply the method to actual experimental data.