Dear Saeid Esmaeiloghli

Your question is a topic of debate in the recent geochemical studies. My PhD thesis is aimed to investigate "what is the difference of using CoDa in mineral reserve evaluations, in contrast with the ordinary geostatistical methods?".

The result is: in "some elements" the difference is huge! But for some other, it is negligible. Phosphorous, copper and molybdenum are examples of the first type and iron and carbon are examples for the latter.

Back in Girona and as a collaboration with Vera Pawlowsky-Glahn and Juan Jose Egozcue , we studied 5 real case studies:

1. Parvadeh IV coking coal deposit

2. Qarechar gold deposit

3. Sungun porphyry copper - molybdenum deposit

4. Choghart iron ore deposit

5. Zaghia iron ore and vanadium deposit.

We focused on the statistical/geostatistical modelling of the above deposits and compared CoDa with ordinary tonnage and mean-grade curves, grade distribution maps and more important, we compared the uncertainty. We found that, although CoDa does not improve the quality of estimation in some cases (such as iron grade in an iron ore deposit) but it is drastically reduce the uncertainty and prevent the results to be mixed with inconsistencies such as negative and/or impossible grades (two example diagrams are attached).

Again, I came back to the main question: "is it necessary to perform CoDa for mineral reserve modelling?" The answer is: sometimes yes, sometimes no (if we only look at the results, not the mathematical necessity).

The problem is that no one knows how much of the error/uncertainty is due to spurious correlation/covariance. Therefore, it is wiser to choose the safe side and always perform the compositional data analysis.

Dear Hossein Molayemat

Thank you for your comprehensive, enlightening response as well as the interesting results you shared.

The motivation for asking the current question was justifications that have recently been published in some papers, e.g., section 4.1 of Xiang et al. (2019) (https://doi.org/10.1016/j.gexplo.2018.12.004), pointing out that CoDA is not required when using only trace elements.

However, in my recently published paper (https://doi.org/10.1007/s11053-021-09933-2), I discovered improved results for the factor model using CoDA, specifically for using a combination of CLR and ILR transformations to tune a typical covariance matrix as:

original data -> ilr-transformed data -> covariance matrix in the ilr space -> back-transforming ilr covariance to the clr covariance -> implement factor analysis model -> interpretation in the clr space;

where it made inter-elemental relationships more accurately and interpretably identified.

Dear Saeid Esmaeiloghli

I do not agree with "CoDA is not required when using only trace elements". At least, I personally studied a counter-example: Poster Comparison of compositional and noncompositional approaches ...

In the above case study we worked on the geostatistical modelling of an 8-part regionalized compositional data, consists of only trace elements: Ag, As, Au, Cu, Mo, Pb, Sb, Zn. It is obvious that compositional results are closer to reality, at the all defined grade cut-offs (Please see the attached diagram(s)). Anyway, again I recommend to perform compositional data analysis for any statistical and especially, any geostatistical study.

p.s.: your way of using coda in your paper is interesting!

Dear Hossein Molayemat

Thank you for providing a clear response to this critical issue.

I think the documents you attached / linked are quite enlightening.

Compositional data is a controversial topic right now, and I would like say you can acknowledge or avoid it in the research, unless you are professional in mathematics and geochemistry. For the compositional data analysis in mineral exploration, you can read the paper published by Prof. Renguang Zuo https://www.researchgate.net/profile/Renguang-Zuo-2.

Dear Haofan Xu

Thank you very much for the answer and the link you provided.

Before I could answer the question by Saeid Esmaeiloghli, Hossein Molayemat did it in a way I fully support. However, the answer added by Haofan Xu pushes me to add a new comment. Haofan Xu seems to be eclectic with respect to compositional techniques or traditional ones. He thinks the choice depends on the user's training in mathematics.

Each analytical option includes a choice between models that built on basic assumptions and hypotheses. This is so, even when these assumptions are not explicit for the analyst. For instance, when a geologist describes a data set using the average of each element concentration in mg/kg, he/she is implicitly assuming that the main operation is the sum and the division by the sample size is a multiplication: the geologist is assuming that the concentrations in mg/kg are real variables and that differences between them are in an absolute scale. An this is highly adventurous, irrespective of the mathematical background of the geologist. Another geologist might decide to use logarithms of the concentrations, thus accounting for a relative (ratio) scale of concentrations in mg/kg. Now, the change in concentration would be measured by differences of logarithms (log-ratio), which points out that the scale of logarithms is assumed absolute and the basic operation is the multiplication. However, there are still some underlying assumptions. For instance, that changes in the concentration of Na, K, Mg, Fe and some other major elements do not affect to concentration of Li, La, Th, although all of them are a part of the same reference kg.

Compositional data (CoDa) analysis assumes that the scale of concentrations is relative, that the basic operation is the multiplication (e.g. decreasing 3% means multiplication by 0.97), that expressing concentrations in mg/kg or in g/kg provides the same information. And, importantly, that the relationship between some elements does not depend on which elements were measured in the same sample.

Understanding these kind of principles does not require more than maths taught in the elementary school. The demanding point is to have a critical vision of Geology or Science in general.

To finish this comment, models and methods should be selected according to the consistency and adequateness of their assumptions, and not to the mathematical background required to use them.

The background recommended for using CoDa methods may be the concepts of Euclidean vector spaces and elementary statistics (first year of most graduate degrees), complemented with a dose of critical view on the traditional methods. In fact, historically, traditional methods provided important results, but the progress of science requires new methods to go ahead.

Some compositional readings:

P. Filmoser, K. Hron, M. Templ (2018): Applied Compositional Data Analysis. Springer

V. Pawlowsky-Glahn, J.J. Egozcue, R. Tolosana-Delgado (2015): Modeling and analysis of compositional analysis. Wiley

K.G. van den Boogart, R. Tolosana-Delgado (2013): Analyzing dompositional with R, Springer

A. Buccianti, G. Mateu-Figueras, V. Pawlowsky-Glahn (Eds.) (2006): Compositional data analysis in the geosciences. From theory to practice. Geological Society of London

Thanks to all participants in this interesting discussion!

I appreciate the intellectual, enlightening explanation by Professor Juan Jose Egozcue .

Hi Saeid,

I'd like to bring the discussion down to the practical and empirical level. I have worked in many different mineral systems and deposit styles, with nice clean data and not so nice legacy (meaning I wasn't involved in the design and collection of it!) inherited data. In the work that I do the trace elements drive our pathfinder and vectoring workflows...CoDa principals make the difference between learning something (maybe something that you don't fully understand or appreciate and the time of discovery) and nonsensical or useless interpretations. That is to say I have yet to be led astray by applying CoDa principals in exploration.

To add to Juan Jose's comment:

"To finish this comment, models and methods should be selected according to the consistency and adequateness of their assumptions, and not to the mathematical background required to use them."

The models and methods should be selected according to the data you have to work with and the question you are trying to answer or hypothesis you are trying to test!

Natalie Caciagli Warman

Thank you Natalie for adding this to the page.

Considering the totality of the discussions, it seems that we have reached a good consensus on the use of CoDa and its effect on improving interpretations related to geochemical mineral exploration.

Dear all,

Sometimes CoDa is not applied due to lack of support for it in some commercial applications. I am working on an open-source machine learning solution (https://github.com/italo-goncalves/geoML) which supports CoDa in spatial modeling. I hope you find it useful.

Dear Ítalo Gomes Gonçalves ,

I have already visited your Python projects in GITHUB repositories, mainly because of my interest in geospatial pattern recognition and geochemical signal processing. They are excellent tools capable of solving complex problems in the spatial analysis of geoscience data.

I sincerely appreciate your work.

Hi,

probably a bit late in the discussion, but I'll provide my answer anyhow. There has been and is still lots of debate on this issue. Some people advocate very strongly for log-ratio in all situations; some other don't. My answer is that .... it all depends.

It depends on the purpose of the analysis and on the data at hand. For example, when the purpose of the analysis is to understand the relationships between a small number of parts (eg votes at an election, expenses in a constrained budget, ...), it really makes sense to acknowledge i) the relative nature of the variables and ii) the strong negative correlations induced by the fact that the variables sum to a fixed quantity (eg 100%).

On the other hand, when the purpose of the analysis is prediction or estimation of a total amount (or a mean) and when the variable under consideration is only one of many parts (such as trace elements, or ore grade), I would not recommend CoDA. The reason is quite simple: if the goal is prediction of an ore grade, for example, what matters is the prediction error in absolute value, because absolute this error translates immediately in loss (expressed in $). If one uses a log-ratio transform such as ILR, and then perform optimal prediction in the transformed (relative) space, optimality is obtained in that relative space. Then the prediction needs to back-transformed to the original space, entailing loss of optimality. The mining engineer should therefore use the prediction method with the smallest prediction score (such as MSE) in the original, absolute space.

More details about this issue in two recent publications of mine:

Article Means and Covariance Functions for Geostatistical Compositio...

Working Paper Means and covariance functions for spatial compositional dat...

Preprint A new class of $\alpha$-transformations for the spatial anal...

The second one is under (positive) revision a Spatial Statistics

Dear Denis Allard

Thank you very much for the clear discussion you provided and the links you shared.

I believe these can be a beacon for those who work in geostatistics and spatial statistics with compositional variables.