What is estimated data?

I have tow data sets, one of them are 69 geochemical stream sediment samples that each sample analysed for 33 elements. the next data set is lithological samples for each element.

I realized that you have a real distribution and two empirical distributions for comparison. The sample-based benchmarks such as root mean square error (RMSE) and mean absolute error (MAE) are fundamentally the most frequently used paradigms. Depending on the numerical conditions, Pearson moment correlation coefficient (Gaussian distributions) and Spearman's rank correlation coefficient (non-Gaussian distributions) may also be useful.

About your issue, however, the “geochemical background distribution” seems to be of greater importance than the numerical difference of elemental magnitudes, because in catchment basin mapping of geochemical anomalies, we are interested not in elemental magnetite values, but in the distribution of synergetic-lithological components that should be avoided to reduce spurious anomalies. For this, statistics such as the t-student test and the Z-test can work effectively. But the problem is that they are not quantitative criteria but statistics for comparing critical values (from standard look-up tables). The methods derived from “Rényi Divergence” may help tackle the problem. In a more detailed discussion, the Bhattacharyya distance can be used to determine the relative closeness of the two distributions being considered. In its simplest formulation, the Bhattacharyya distance (BD) between two classes p and q, can be calculated as

BD(p,q)=0.25*ln(0.25*(sqrt(sigma_p)/sqrt(sigma_q)+ sqrt(sigma_q)/sqrt(sigma_p)+2)+0.25(sqrt(mu_p-mu_q)/sqrt(sigma_p)+ sqrt(sigma_q)).

In a geochemical sense, as a comprehensive appraisal, you may consider the performance of two empirical background distributions in delineating pre-identified anomalies and improving probability-area curves. This is my advice if you are considering a journal paper.

Dear Saeid,

thanks for taking the time to answer my question. Your answer is very comprehensive. I will follow your suggested methods.

Dear Taher,

when you are comparing two estimations, the "distance from reality" is a simple but informative way of comparison. We can work on distances, sample by sample or we can sum them up and report a single value for the whole estimation. I recommend you to test these graphs/maps:

- distance - distance plots (e.g. Aitchison (for compositional data), Euclidean, ...)

- distance maps

some examples are attached.

p.s.: please note the distributions before setting the color scale for map(s) and also, before choosing linear or logarithmic scales for distance - distance plot(s).

Dear Hossein, thanks for your attention. Can you introduce me the software that does this for me?

Taher Najafian

Dear Taher,

I have used R programming environment to calculate the Aitchison distances between two sets of samples. The above diagram is drawn using Grapher software and the map is plotted using ArcMap software.

the note is that you can calculate the distances in excel and then go after presenting the results.

Aitchison distance is nothing but euclidean distance of clr-transformed multivariate data.

some notes about the Aitchison distance:

The Aitchison distance (A.D) is a measure of difference between two vectors of multivariate data (i.e. two compositions). The Aitchison distance is scale invariant, permutation invariant, perturbation invariant and subcompositionally dominant. Thus, it is a good criterion to calculate the dissimilarity of two compositions.

for more details you may see Article The impact of the compositional nature of data on coal reser...