Is it valid to transform ilrs using normal score, Box-Cox or other transformations in order to perform Sequential Gaussian Simulation (SGS)?
Is it true to use the combination of isometric logratios (ilrs) with Direct Sequential Simulation without any transformation?
Thanks in advance.
ilr(s) is a transformation. In my point of view it is true to do it without any further transformation.
ilr(s) is a transformation. In my point of view it is true to do it without any further transformation.
Thank you Matthias.
What do you suggest if we want to simulate ilr(s) in SGS where regarding to P-Values and Kolmogorov–Smirnov statistics they're not normally distributed?
The ilr-transform assigns (Cartesian) coordinates to a composition. These coordinates fully represent the composition and allow us to work with the coordinates as in the usual Euclidean space. Statistical methods designed for real variables can safely be applied to coordinates.
Related to your question, I would ask you why or for what you want to apply a Gaussian scores or Box-Cox to the ilr-coordinates. I assume that for conducting a conditional Gaussian simulation on a compositional space distributed problem. The ilr-coordinates may be approximately multivariate normal (MVN) distributed; in that case you can apply conditional Gaussian simulation directly (as suggested by M. Templ). Note that the raw composition cannot have a MVN distribution as the support is limited and most marginals are skewed or very skewed. This is not the case for the ilr-coordinates which span the whole real space and make the MVN a natural distribution for them.
However, there are cases in which you cannot assume that the ilr-coordinates are MVN. In this case you are free of transforming them as you would do with other real variables. In my opinion this case seldom occurs as it is very difficult to reject the normality hypothesis having only one realization of the field and probably doubts on the stationary and ergodicity of the field. For instance, taking all the data set as a sample, cheking or testing for normality is very naive; a rejection may be due to the shape of the spatial domain or its reduced dimension compared with the correlation length (range). Only very intensive samplings in large convex domains may support the reliability of such normality test.
I see J.J. Egozcue has answered in length your question. I would summarize his answer in saying: you can use a normal score transform, but do you really need it?
Thank you so much Professor Egozcue.
So I'll transform ilr(s) using normal score if it's needed. An alternative method to SGS is Direct Sequential Simulation as it doesn't need the normal distribution of input data. I may also try it.
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For clarification, I'm currently working on a dataset of coal proximate analysis and interested in comparing exploratory results, before and after compositional data analysis.
In order to model the spatial uncertainty in the deposit, I must simulate ilr(s) using SGS, like the recent work of Dr. Olea.
Thanks Professor Pawlowski-Glahn.
As mentioned above, I would like to model the spatial uncertainty in different geochemical environments (starting with a coal deposit with few variables). As I use SGS to reproduce multiple realizations, it's necessary to use normally distributed inputs (ilr(s)).
So I think when I want to simulate ilrs using SGS and they're skewed (and/or very skewed), I need to use a secondary transform to perform the simulation.
The Bigaus program in Deutsch and Journel (1998) can be used to check for bivariate normality, a necessary but not sufficient requirement for multivariate normality. It gets you a little closer to being confident that all is well. It's not often used.
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