# Why, and How, Should Geologists Use Compositional Data Analysis/Print Version

WHY, AND HOW, SHOULD GEOLOGISTS USE COMPOSITIONAL DATA ANALYSIS
A Step-by-Step Guide for the Field Geologists
Special Edition for Wikibooks
By
Ricardo A. Valls, P. Geo., M. Sc Hector Nuñez Dr. Jorge Cruz Martin
January 1st, 2008

# Summary

Compositional data arise naturally in several branches of science, including geology. In geochemistry, for example, these constrained data seem to occur typically, when one normalizes raw data or when one obtains the output from a constrained estimation procedure, such as parts per one, percentages, ppm, ppb, molar concentrations, etc.
Compositional data have proved difficult to handle statistically because of the awkward constraint that the components of each vector must sum to unity. The special property of compositional data (the fact that the determinations on each specimen sum to a constant) means that the variables involved in the study occur in constrained space defined by the simplex, a restricted part of real space.
Pearson was the first to point out dangers that may befall the analyst who attempts to interpret correlations between Ratios whose numerators and denominators contain common parts. More recently, Aitchison, Pawlowsky-Glahn, S. Thió, and other statisticians have develop the concept of Compositional Data Analysis, pointing out the dangers of misinterpretation of closed data when treated with “normal” statistical methods.
It is important for geochemists and geologists in general to be aware that the usual multivariate statistical techniques are not applicable to constrained data. It is also important for us to have access to appropriate techniques as they become available. This is the principal aim of this book.
From a hypothetical model of a copper mineralization associated to a felsic intrusive, with specific relationships between certain elements, I will show how “normal” correlation methods fail to identify some of such embedded relationships and how we can obtain other spurious correlations. From there, I will test the same model after transforming the data using the CRL, ARL, and IRL transformations with the aid of the CoDaPack software.
Since I addressed this publication to geologists and geoscientists in general, I have kept to a minimum the mathematical formulae and did not include any theoretical demonstration. The “mathematical curios geologist”, if such category exists, can find all of those in a list of recommended sources in the reference section.
So let us start by introducing the model of mineralization that we will be testing.

# The Model

Figure 1 shows a very simplistic version of a copper mineralization associated to a zone of fractures within a granodiorite intrusive.
Table 1 shows the results of a random sampling using existing access to the top of the intrusive.
Table 1. Chemical composition of the sampling of the granodiorite intrusive.
| Sample | Easting | Northing | SiO2 | TiO2 | Al203 | Fe2O3 | MgO | CaO | K2O | Na2O | P2O5 |
| 259021 | 845962.7 | 833189 | 56.00 | 2.20 | 6.00 | 3.62 | 5.11 | 2.08 | 5.03 | 7.08 | 3.00 |
| 259022 | 831978.5 | 859092 | 57.00 | 1.25 | 4.00 | 2.54 | 5.40 | 2.19 | 5.08 | 7.19 | 3.74 |
| 259023 | 857644.3 | 846310 | 55.00 | 2.05 | 7.00 | 3.86 | 5.55 | 2.29 | 5.19 | 7.29 | 3.46 |
| 259024 | 840369.6 | 833920 | 62.00 | 2.19 | 0.00 | 2.47 | 5.63 | 2.29 | 5.29 | 7.29 | 3.80 |
| 259025 | 833367.4 | 857854 | 56.04 | 1.14 | 6.00 | 3.66 | 5.69 | 2.43 | 5.29 | 7.43 | 3.14 |
| 259026 | 841526.8 | 837288 | 58.36 | 1.48 | 2.00 | 2.75 | 5.89 | 2.45 | 5.43 | 7.45 | 3.70 |
| 259027 | 850152.3 | 854634 | 61.00 | 1.60 | 1.00 | 2.26 | 5.95 | 2.49 | 5.45 | 7.49 | 3.17 |
| 259028 | 836406.1 | 847121 | 57.34 | 1.58 | 4.02 | 2.94 | 6.25 | 2.54 | 5.49 | 7.54 | 3.26 |
| 259029 | 893494.7 | 891521 | 62.00 | 1.25 | 0.66 | 2.21 | 6.28 | 2.55 | 5.54 | 7.55 | 3.08 |
| 259030 | 898079.8 | 900849 | 61.00 | 1.60 | 1.85 | 3.19 | 6.44 | 2.60 | 5.55 | 7.60 | 3.96 |
| 259031 | 917124.2 | 893796 | 61.00 | 2.05 | 1.49 | 2.55 | 6.46 | 2.63 | 5.60 | 7.63 | 3.66 |
| 259032 | 904564.7 | 893086 | 59.02 | 1.15 | 1.04 | 2.70 | 6.47 | 2.65 | 5.63 | 7.65 | 3.85 |
| 259033 | 900474.9 | 898141 | 59.00 | 1.36 | 2.47 | 3.53 | 6.49 | 2.68 | 5.65 | 7.68 | 3.44 |
| 259034 | 894560.4 | 895311 | 59.00 | 1.61 | 2.34 | 3.56 | 6.82 | 2.71 | 5.68 | 7.71 | 3.36 |
| 259035 | 913232.2 | 903356 | 57.80 | 2.15 | 3.09 | 2.63 | 6.98 | 2.86 | 5.71 | 7.86 | 3.45 |
| 259036 | 893128.5 | 902967 | 56.44 | 1.15 | 6.00 | 3.48 | 6.98 | 2.91 | 5.86 | 7.91 | 3.40 |
| 259037 | 892202.8 | 894592 | 59.00 | 1.41 | 2.30 | 2.59 | 7.18 | 2.99 | 5.91 | 7.99 | 3.13 |
| 259038 | 913071.1 | 897870 | 57.30 | 1.06 | 4.00 | 3.62 | 7.22 | 3.02 | 5.99 | 8.02 | 3.82 |
| 259039 | 906249.3 | 907340 | 58.09 | 1.71 | 3.50 | 2.51 | 7.74 | 3.10 | 6.02 | 8.10 | 3.66 |
| 259040 | 916724.1 | 891607 | 57.00 | 1.16 | 4.00 | 3.27 | 7.83 | 3.15 | 6.10 | 8.15 | 3.83 |
| 259041 | 909848.3 | 915457 | 60.00 | 1.47 | 1.44 | 2.47 | 7.89 | 3.38 | 6.15 | 8.38 | 3.27 |
| 259042 | 915608.1 | 915306 | 57.47 | 1.83 | 3.00 | 3.57 | 7.95 | 3.56 | 6.38 | 8.56 | 3.28 |
| 259043 | 871460 | 863020 | 55.66 | 1.03 | 7.00 | 2.23 | 7.96 | 3.58 | 6.56 | 8.58 | 3.08 |
| 259044 | 877894.5 | 886973 | 59.00 | 1.13 | 2.00 | 3.33 | 8.07 | 3.68 | 6.58 | 8.68 | 3.99 |
| 259045 | 886538.3 | 888754 | 59.00 | 1.80 | 2.80 | 2.36 | 8.10 | 3.71 | 6.68 | 8.71 | 3.60 |
| 259046 | 860434.9 | 872223 | 53.00 | 1.36 | 5.00 | 3.16 | 8.28 | 3.82 | 6.71 | 8.82 | 3.90 |
| 259047 | 885897.4 | 864158 | 57.00 | 1.43 | 4.00 | 2.09 | 8.30 | 3.88 | 6.82 | 8.88 | 3.91 |
| 259048 | 867351 | 861364 | 54.30 | 2.16 | 4.00 | 3.98 | 8.57 | 3.90 | 6.88 | 8.90 | 3.83 |
| 259049 | 860971.4 | 864924 | 54.20 | 1.77 | 3.36 | 3.72 | 8.59 | 3.91 | 6.90 | 8.91 | 3.82 |
| 259050 | 866588.3 | 860513 | 58.50 | 1.00 | 2.00 | 3.10 | 8.83 | 1.00 | 6.91 | 2.00 | 3.65 |
| Sample | Easting | Northing | Cu | As | Pb | Co | Ni | Sc | L.O.I. | Other | |
| 259021 | 845962.7 | 833189 | 0.00191 | 0.00199 | 0.00002 | 0.00002 | 0.00002 | 0.01936 | 6.19 | 9.88 | |
| 259022 | 831978.5 | 859092 | 0.00190 | 0.00198 | 0.00003 | 0.00004 | 0.00005 | 0.01763 | 8.16 | 11.61 | |
| 259023 | 857644.3 | 846310 | 0.00188 | 0.00196 | 0.00003 | 0.00010 | 0.00012 | 0.02056 | 5.09 | 8.31 | |
| 259024 | 840369.6 | 833920 | 0.00182 | 0.00190 | 0.00006 | 0.00014 | 0.00017 | 0.00013 | 5.97 | 9.03 | |
| 259025 | 833367.4 | 857854 | 0.00171 | 0.00179 | 0.00006 | 0.00015 | 0.00018 | 0.01834 | 6.60 | 9.18 | |
| 259026 | 841526.8 | 837288 | 0.00168 | 0.00176 | 0.00004 | 0.00022 | 0.00026 | 0.01058 | 8.01 | 10.48 | |
| 259027 | 850152.3 | 854634 | 0.00158 | 0.00166 | 0.00005 | 0.00023 | 0.00027 | 0.00341 | 7.40 | 9.59 | |
| 259028 | 836406.1 | 847121 | 0.00156 | 0.00164 | 0.00003 | 0.00025 | 0.00030 | 0.01372 | 6.97 | 9.03 | |
| 259029 | 893494.7 | 891521 | 0.00138 | 0.00146 | 0.00005 | 0.00027 | 0.00032 | 0.00035 | 7.00 | 8.88 | |
| 259030 | 898079.8 | 900849 | 0.00115 | 0.00123 | 0.00002 | 0.00027 | 0.00033 | 0.00085 | 4.00 | 6.22 | |
| 259031 | 917124.2 | 893796 | 0.00110 | 0.00117 | 0.00005 | 0.00030 | 0.00036 | 0.00119 | 4.00 | 6.92 | |
| 259032 | 904564.7 | 893086 | 0.00102 | 0.00110 | 0.00002 | 0.00032 | 0.00038 | 0.00799 | 8.00 | 9.84 | |
| 259033 | 900474.9 | 898141 | 0.00099 | 0.00107 | 0.00002 | 0.00032 | 0.00039 | 0.00725 | 5.00 | 7.69 | |
| 259034 | 894560.4 | 895311 | 0.00091 | 0.00099 | 0.00004 | 0.00034 | 0.00041 | 0.00623 | 4.00 | 7.20 | |
| 259035 | 913232.2 | 903356 | 0.00086 | 0.00094 | 0.00003 | 0.00036 | 0.00043 | 0.01060 | 5.00 | 7.47 | |
| 259036 | 893128.5 | 902967 | 0.00071 | 0.00079 | 0.00001 | 0.00043 | 0.00052 | 0.01361 | 3.00 | 5.88 | |
| 259037 | 892202.8 | 894592 | 0.00068 | 0.00076 | 0.00001 | 0.00045 | 0.00054 | 0.00629 | 5.00 | 7.51 | |
| 259038 | 913071.1 | 897870 | 0.00065 | 0.00073 | 0.00005 | 0.00049 | 0.00059 | 0.01008 | 2.00 | 5.94 | |
| 259039 | 906249.3 | 907340 | 0.00063 | 0.00071 | 0.00005 | 0.00051 | 0.00061 | 0.00812 | 1.50 | 5.57 | |
| 259040 | 916724.1 | 891607 | 0.00060 | 0.00068 | 0.00004 | 0.00055 | 0.00066 | 0.01036 | 1.56 | 5.50 | |
| 259041 | 909848.3 | 915457 | 0.00055 | 0.00062 | 0.00004 | 0.00058 | 0.00069 | 0.00220 | 2.00 | 5.55 | |
| 259042 | 915608.1 | 915306 | 0.00054 | 0.00062 | 0.00004 | 0.00069 | 0.00083 | 0.00707 | 1.23 | 4.39 | |
| 259043 | 871460 | 863020 | 0.00049 | 0.00057 | 0.00001 | 0.00078 | 0.00094 | 0.01274 | 1.12 | 4.32 | |
| 259044 | 877894.5 | 886973 | 0.00045 | 0.00053 | 0.00004 | 0.00079 | 0.00095 | 0.00172 | 0.35 | 3.53 | |
| 259045 | 886538.3 | 888754 | 0.00043 | 0.00051 | 0.00004 | 0.00084 | 0.00101 | 0.00157 | 0.08 | 3.24 | |
| 259046 | 860434.9 | 872223 | 0.00029 | 0.00037 | 0.00003 | 0.00085 | 0.00103 | 0.00855 | 1.25 | 5.96 | |
| 259047 | 885897.4 | 864158 | 0.00029 | 0.00037 | 0.00004 | 0.00091 | 0.00109 | 0.00642 | 0.08 | 3.68 | |
| 259048 | 867351 | 861364 | 0.00019 | 0.00027 | 0.00001 | 0.00094 | 0.00113 | 0.00461 | 0.05 | 3.46 | |
| 259049 | 860971.4 | 864924 | 0.00008 | 0.00016 | 0.00003 | 0.00095 | 0.00114 | 0.00149 | 0.89 | 4.81 | |
| 259050 | 866588.3 | 860513 | 0.00003 | 0.00011 | 0.00005 | 0.00096 | 0.00115 | 0.00099 | 6.94 | 13.01 | |
The initial data respond to the following pre-established conditions:
* Is a close system, meaning that the sum of all the values is equal or very close to 100%
* There are no zero values present (I will deal with zero values in a separate example to avoid complicating the initial model).
* There are no statistical outliers (hurricane values) present.
* There is a big difference between the concentRations of the major oxides with respect to the trace elements.
* As shown in Figs. 2-4, there are embedded positive and significant correlation between Cu and As, Ni and Co, and MgO and K$_2$O.
* Finally, I introduced significant negative correlations between K$_2$O and CaO (Fig. 5), SiO$_2$ and Al$_2$O$_3$ (Fig 6), K$_2$O and Cu (Fig. 7), Na$_2$O and K$_2$O (Fig. 8), and Co with Cu (Fig 9).
According to these embedded conditions, any correlation analysis must give us two coefficients (Range Correlation Coefficient or RCC):

## Equation 1

    Range Correlation Coefficient type A for the initial data, according to the embedded correlations.

## Equation 2

    Range Correlation Coefficient type B for the initial data, according to the embedded correlations.

# Normal Processing of the Data
## Case One

For this case, I will process the initial dataset as a whole, without differentiating between major and trace elements. Since we established that there are no statistical outliers and no zero values within the data, the first step was to determine their distribution, using the kurtosis and skewness test as described by Kashdan et al, (1979). As one can see from Table 2, all elements responded to a Normal Distribution Law, except for CaO and Na2O, so the next step was to transform those values into Logarithms before testing their correlations.
Table 2. Results of the analysis of kurtosis and skewness.
| | SiO$_2$ | TiO$_2$ | Al$_2$O$_3$ | Fe$_2$O$_3$ | MgO | CaO | K$_2$O | Na$_2$O | P$_2$O$_5$ |
| Kurtosis | 0.20 | 0.83 | 0.94 | 0.13 | 0.08 | 0.78 | 0.72 | 7.42 | 0.52 |
| Skewness | 2.64 | 2.04 | 2.56 | 1.68 | 1.83 | 3.53 | 1.93 | 18.30 | 1.77 |
| Kurtosis | Cu | As | Pb | Co | Ni | Sc | L.O.I. | | |
| Skewness | 0.72 | 0.72 | 0.33 | 0.72 | 0.72 | 1.18 | 0.07 | | |
| | 1.93 | 1.93 | 2.06 | 1.93 | 1.93 | 2.45 | 1.64 | | |
Using Excel data analysis capabilities, I then determined the correlation analysis of the data (Table 3).
Table 3. Correlation analysis of the initial dataset.
| | SiO$_2$ | TiO$_2$ | Al$_2$O$_3$ | Fe$_2$O$_3$ | MgO | CaO | K$_2$O | Na$_2$O |
| SiO$_2$ | 1.00 | | | | | | | |
| TiO$_2$ | 0.11 | 1.00 | | | | | | |
| Al$_2$0$_3$ | -0.92 | -0.07 | 1.00 | | | | | |
| Fe$_2$O$_3$ | -0.46 | 0.10 | 0.41 | 1.00 | | | | |
| MgO | -0.13 | -0.21 | 0.01 | 0.03 | 1.00 | | | |
| CaO | 0.10 | 0.24 | 0.01 | -0.06 | -0.98 | 1.00 | | |
| K$_2$O | -0.16 | -0.19 | 0.04 | 0.03 | 0.98 | -0.94 | 1.00 | |
| Na$_2$O | 0.10 | 0.24 | 0.01 | -0.06 | -0.98 | 1.00 | -0.94 | 1.00 |
| P$_2$O$_5$ | 0.07 | -0.04 | -0.21 | 0.10 | 0.35 | -0.34 | 0.36 | -0.34 |
| Cu | 0.10 | 0.24 | 0.01 | -0.06 | -0.98 | 1.00 | -0.94 | 1.00 |
| As | 0.10 | 0.24 | 0.01 | -0.06 | -0.98 | 1.00 | -0.94 | 1.00 |
| Pb | 0.30 | 0.04 | -0.30 | -0.18 | -0.08 | 0.17 | -0.09 | 0.17 |
| Co | -0.16 | -0.19 | 0.04 | 0.03 | 0.98 | -0.94 | 1.00 | -0.94 |
| Ni | -0.16 | -0.19 | 0.04 | 0.03 | 0.98 | -0.94 | 1.00 | -0.94 |
| Sc | -0.80 | -0.05 | 0.80 | 0.31 | -0.47 | 0.48 | -0.46 | 0.48 |
| L.O.I. | 0.17 | -0.44 | -0.30 | -0.52 | -0.31 | 0.27 | -0.37 | 0.27 |
| | | | | | | | | (continue) |
| | P$_2$O$_5$ | Cu | As | Pb | Co | Ni | Sc | L.O.I. |
| P$_2$O$_5$ | 1.00 | | | | | | | |
| Cu | -0.34 | 1.00 | | | | | | |
| As | -0.34 | 1.00 | 1.00 | | | | | |
| Pb | 0.09 | 0.17 | 0.17 | 1.00 | | | | |
| Co | 0.36 | -0.94 | -0.94 | -0.09 | 1.00 | | | |
| Ni | 0.36 | -0.94 | -0.94 | -0.09 | 1.00 | 1.00 | | |
| Sc | -0.32 | 0.48 | 0.48 | -0.21 | -0.46 | -0.46 | 1.00 | |
| L.O.I. | -0.30 | 0.27 | 0.27 | 0.08 | -0.37 | -0.37 | 0.15 | 1.00 |
To determine the significance of the obtained correlations, I then proceeded to calculate the critical value of Student using equation 3 (Table 4).

### Equation 3

    Critical value of Student to determine the significance of the obtained correlations.
Where, tc- critical value of Student r- correlation n- amount of data
Table 4. Critical values of Student for the correlation analysis of the initial dataset.
| | SiO$_2$ | TiO$_2$ | Al$_2$O$_3$ | Fe$_2$O$_3$ | MgO | CaO | K$_2$O | Na$_2$O |
| SiO$_2$ | | | | | | | | |
| TiO$_2$ | 0.59 | | | | | | | |
| Al$_2$0$_3$ | 12.84 | 0.36 | | | | | | |
| Fe$_2$O$_3$ | 2.76 | 0.55 | 2.41 | | | | | |
| MgO | 0.68 | 1.12 | 0.04 | 0.17 | | | | |
| CaO | 0.53 | 1.30 | 0.05 | 0.34 | 24.01 | | | |
| K$_2$O | 0.83 | 1.02 | 0.24 | 0.18 | 23.31 | 14.85 | | |
| Na$_2$O | 0.53 | 1.30 | 0.05 | 0.34 | 24.01 | 100.00 | 14.85 | |
| P$_2$O$_5$ | 0.40 | 0.21 | 1.14 | 0.53 | 1.95 | 1.91 | 2.04 | 1.91 |
| Cu | 0.53 | 1.30 | 0.05 | 0.34 | 24.01 | 100.00 | 14.85 | 100.00 |
| As | 0.53 | 1.30 | 0.05 | 0.34 | 24.01 | 100.00 | 14.85 | 100.00 |
| Pb | 1.66 | 0.19 | 1.69 | 0.98 | 0.44 | 0.90 | 0.50 | 0.90 |
| Co | 0.83 | 1.02 | 0.24 | 0.18 | 23.31 | 14.85 | 100.00 | 14.85 |
| Ni | 0.83 | 1.02 | 0.24 | 0.18 | 23.31 | 14.85 | 100.00 | 14.85 |
| Sc | 7.01 | 0.27 | 6.95 | 1.73 | 2.80 | 2.86 | 2.73 | 2.86 |
| L.O.I. | 0.91 | 2.61 | 1.69 | 3.20 | 1.75 | 1.48 | 2.12 | 1.48 |
| | P$_2$O$_5$ | Cu | As | Pb | Co | Ni | Sc | |
| Cu | 1.91 | | | | | | | |
| As | 1.91 | 100.00 | | | | | | |
| Pb | 0.46 | 0.90 | 0.90 | | | | | |
| Co | 2.04 | 14.85 | 14.85 | 0.50 | | | | |
| Ni | 2.04 | 14.85 | 14.85 | 0.50 | 100.00 | | | |
| Sc | 1.81 | 2.86 | 2.86 | 1.12 | 2.73 | 2.73 | | |
| L.O.I. | 1.66 | 1.48 | 1.48 | 0.40 | 2.12 | 2.12 | 0.83 | |
It is a common practice in geology that for n > 30 and a probability of 0.05 (95%), if t$_c$ > 3, then the correlation is significant. Table 6 shows which correlations are significant from the initial dataset.
Table 5. Results of the significant correlation analysis of the initial dataset.
| | SiO$_2$ | TiO$_2$ | Al$_2$O$_3$ | Fe$_2$O$_3$ | MgO | CaO | K$_2$O | Na$_2$O |
| SiO$_2$ | | | | | | | | |
| TiO$_2$ | | | | | | | | |
| Al$_2$0$_3$ | -0.92 | | | | | | | |
| Fe$_2$O$_3$ | -0.46 | | 0.41 | | | | | |
| MgO | | | | | | | | |
| CaO | | | | | -0.98 | | | |
| K$_2$O | | | | | 0.98 | -0.94 | | |
| Na$_2$O | | | | | -0.98 | 1.00 | -0.94 | |
| P$_2$O$_5$ | | | | | | | | |
| Cu | | | | | -0.98 | 1.00 | -0.94 | 1.00 |
| As | | | | | -0.98 | 1.00 | -0.94 | 1.00 |
| Pb | | | | | | | | |
| Co | | | | | 0.98 | -0.94 | 1.00 | -0.94 |
| Ni | | | | | 0.98 | -0.94 | 1.00 | -0.94 |
| Sc | -0.80 | | 0.80 | | -0.47 | 0.48 | -0.46 | 0.48 |
| L.O.I. | | -0.44 | | -0.52 | | | -0.37 | |
| | P2O5 | Cu | As | Pb | Co | Ni | Sc | L.O.I. |
| As | | 1.00 | | | | | | |
| Pb | | | | | | | | |
| Co | | -0.94 | -0.94 | | | | | |
| Ni | | -0.94 | -0.94 | | 1.00 | | | |
| Sc | | 0.48 | 0.48 | | -0.46 | -0.46 | | |
| L.O.I. | | | | | -0.37 | -0.37 | | |

### Selecting the proper RCCs

Since the use of Range Correlation Coefficients (RCC) is not a common practice, I will explain in more detail the methodoLogy for their selection from the significant correlations. a. We start by ranging all correlations from the highest positive to the lowest negative correlation. b. Start at the bottom of the list and select all the negative significant correlations first to create the initial coefficient. c. After the first time you use a correlation pair, every time you get the same element, put a dot on top as shown in equation 4.

#### Equation 4

    Example of the calculation of the multiplicative factor on a RCC.
This example means that you had a significant positive correlation between Cu and As and two significant negative correlations between Co and Cu and Co and As.
d. Once you finish with the negative significant correlations, go back to the top of the list and repeat the same process for the positive correlations. e. If it is possible, do combine the obtained coefficients. f. If you get a contradictory result, e.g. an element that has “conflicting correlations” with previous elements, take those elements out from the coefficient and start a new one.
You can reduce the size of the obtained RCC by eliminating the less frequent elements and subtracting their influence from the overall coefficient. For example, let us assume that we obtained the RCC represented in equation 5 as follow:

#### Equation 5

    Hypothetical RCC to demonstrate the reduction process.
If we would like to eliminate the Sc and the L.O.I., we first eliminate the L.O.I. and subtract 2 from every element from equation 5.

#### Equation 6

    Hypothetical RCC without the L.O.I. component.
Then we would subtract the remaining Sc as shown in equation 7.

#### Equation 7

    Hypothetical RCC without the remaining Sc.

### RCC from the initial dataset

According to Table 5, and using the methodoLogy just described, I obtained the RCC shown in equation 8.

#### Equation 8

    RCC1 for the initial dataset.
In addition, because L.O.I. has a conflicting correlation with some elements from RCC1, I created a separated RCC for this case as shown in equation 9.

#### Equation 9

    RCC 2 for the initial dataset.
Before we use SURFER v. 8.0 to graphically plot these coefficients over our model of mineralization, let us graphically analyze these coefficients using Grapher v. 7.0, also from the Golden Software suite of programs (www.goldensoftware.com).

### Analysis of the RCCs from the initial dataset.

The objective of the processing of the data should be to obtain a RCC that will be as similar as possible to our theoretical ones represented by equations 1 and specially equation 2. We can see that we obtained all the embedded correlations, but “masked” by the presence of spurious (inexistent) ones. For example:
* There is no correlation whatsoever between Al2O3 and the other elements (Fig. 10)
* Same situation with the Fe2O3 (Fig. 11).
* Same situation with the Sc (Fig 12).
* Same situation with the TiO2 (Fig. 13).
* Same situation with the L.O.I. (Fig. 14).
Note however, that if we only choose the most strong correlations (r>0.92), then we get a RCC with just the embedded correlations.

## Case Two

A more common way to study this kind of data will be to separate the major oxides from the trace elements and treat them separately. It is often intuitively clear for geologists that when mixing percentages with ppm or ppb, the existing correlation between the trace elements is masked or eliminated by the relationships between the major oxides. So I proceeded to separate the initial dataset into major oxides and trace elements (Tables 7 and 8 in the file Initial data.xls [worksheet “Processing”], located in the attached CD).

### Major oxides

Table 6. Correlation for the major oxides from the initial dataset.
| | SiO$_2$ | TiO$_2$ | Al$_2$O$_3$ | Fe$_2$O$_3$ | MgO | CaO | K$_2$O | Na$_2$O | P$_2$O$_5$ | L.O.I. |
| SiO$_2$ | 1.00 | | | | | | | | | |
| TiO$_2$ | 0.11 | 1.00 | | | | | | | | |
| Al$_2$0$_3$ | -0.92 | -0.07 | 1.00 | | | | | | | |
| Fe$_2$O$_3$ | -0.46 | 0.10 | 0.41 | 1.00 | | | | | | |
| MgO | -0.13 | -0.21 | 0.01 | 0.03 | 1.00 | | | | | |
| CaO | 0.10 | 0.24 | 0.01 | -0.06 | -0.98 | 1.00 | | | | |
| K$_2$O | -0.16 | -0.19 | 0.04 | 0.03 | 0.98 | -0.94 | 1.00 | | | |
| Na$_2$O | 0.10 | 0.24 | 0.01 | -0.06 | -0.98 | 1.00 | -0.94 | 1.00 | | |
| P$_2$O$_5$ | 0.07 | -0.04 | -0.21 | 0.10 | 0.35 | -0.34 | 0.36 | -0.34 | 1.00 | |
| SiO$_2$ | 0.17 | -0.44 | -0.30 | -0.52 | -0.31 | 0.27 | -0.37 | 0.27 | -0.30 | 1.00 |
Table 7. Critical value of Student for the correlations of the major oxides of the original dataset.
| | SiO$_2$ | TiO$_2$ | Al$_2$O$_3$ | Fe$_2$O$_3$ | MgO | CaO | K$_2$O | Na$_2$O | P$_2$O$_5$ |
| SiO$_2$ | | | | | | | | | |
| TiO$_2$ | 0.59 | | | | | | | | |
| Al$_2$0$_3$ | 12.84 | 0.36 | | | | | | | |
| Fe$_2$O$_3$ | 2.76 | 0.55 | 2.41 | | | | | | |
| MgO | 0.68 | 1.12 | 0.04 | 0.17 | | | | | |
| CaO | 0.53 | 1.30 | 0.05 | 0.34 | 24.01 | | | | |
| K$_2$O | 0.83 | 1.02 | 0.24 | 0.18 | 23.31 | 14.85 | | | |
| Na$_2$O | 0.53 | 1.30 | 0.05 | 0.34 | 24.01 | 100.00 | 14.85 | | |
| P$_2$O$_5$ | 0.40 | 0.21 | 1.14 | 0.53 | 1.95 | 1.91 | 2.04 | 1.91 | |
| L.O.I. | 0.91 | 2.61 | 1.69 | 3.20 | 1.75 | 1.48 | 2.12 | 1.48 | 1.66 |
Table 8. Significant correlations for the major oxides from the initial dataset.
| | SiO$_2$ | TiO$_2$ | Al$_2$O$_3$ | Fe$_2$O$_3$ | MgO | CaO | K$_2$O |
| SiO$_2$ | | | | | | | |
| TiO$_2$ | | | | | | | |
| Al$_2$0$_3$ | -0.92 | | | | | | |
| Fe$_2$O$_3$ | -0.46 | | 0.41 | | | | |
| MgO | | | | | | | |
| CaO | | | | | -0.98 | | |
| K$_2$O | | | | | 0.98 | -0.94 | |
| Na$_2$O | | | | | -0.98 | 1.00 | -0.94 |
| P$_2$O$_5$ | | | | | | | |
| L.O.I. | | -0.44 | | -0.52 | | | -0.37 |

#### Analysis of the RCCs from the major oxides of the initial dataset.

As it was the case when we processed the whole dataset, here we obtain a RCC that contains the hypothetical one that we are looking to obtain (equation 1), but it is masked by the presence of other elements, many of them without real correlations between them (equation 10).
Equation 10. RCC3 for the major oxides of the initial dataset.

### Trace elements

Table 9. Correlation analysis of the trace elements of the initial dataset.
| | Cu | As | Pb | Co | Ni | Sc |
| Cu | 1.00 | | | | | |
| As | 1.00 | 1.00 | | | | |
| Pb | 0.17 | 0.17 | 1.00 | | | |
| Co | -0.94 | -0.94 | -0.09 | 1.00 | | |
| Ni | -0.94 | -0.94 | -0.09 | 1.00 | 1.00 | |
| Sc | 0.48 | 0.48 | -0.21 | -0.46 | -0.46 | 1.00 |
Table 10. Critical value of Student for the correlations of the trace elements of the original dataset.
| | Cu | As | Pb | Co | Ni |
| Cu | | | | | |
| As | 100.00 | | | | |
| Pb | 0.90 | 0.90 | | | |
| Co | 14.85 | 14.85 | 0.50 | | |
| Ni | 14.85 | 14.85 | 0.50 | 100.00 | |
| Sc | 2.86 | 2.86 | 1.12 | 2.73 | 2.73 |
Table 11. Significant correlations for the trace elements from the initial dataset.
| | Cu | As | Co | Ni |
| Cu | | | | |
| As | 1.00 | | | |
| Pb | | | | |
| Co | -0.94 | -0.94 | | |
| Ni | -0.94 | -0.94 | 1.00 | |
| Sc | 0.48 | 0.48 | -0.46 | -0.46 |

#### Analysis of the RCCs from the Trace Elements of the initial dataset.

As it was the case when we processed the whole dataset, here we obtain a RCC that contains the hypothetical one that we are looking to obtain (equation 2), but it is masked by the presence of other elements, many of them without real correlations between them (equation 11).
Equation 11. RCC 4 for the trace elements of the initial dataset.

## Graphical representation of the RCCs

Using SURFER v.8 from Golden Software Inc., (you can download the demos from www.goldensoftware.com), I obtained Figures 15 – 18.
RCC1 only maps the southwestern border of the mineralized target in a much-dispersed fan that makes it impossible to use as a targeting tool. This was our best RCC from the analysis of the whole dataset.
The only reason why RCC2 partially covers the ore body is the strong and real correlations between Ni, Co, and K$_2$O. None of these elements have however, a correlation with L.O.I:, therefore, this is a classic example of the formation of a spurious correlation because we applied correlation analysis to a “closed” dataset.
RCC3 contains one of the embedded correlations (SiO$_2$ vs. Al$_2$O$_3$), but it also contains several spurious correlations and, since it is a petrographic association, has little to do with the location of the ore body.
Finally, RRC4 is mostly our main embedded correlation, and although it also contains some spurious components (e.g. correlation with Sc), it is not surprising that it maps perfectly the ore body.

## Conclusions and recommendations from the processing of the initial dataset

Closed systems do provoke spurious correlations that mask the effectiveness of the established RCC. This is especially true when processing datasets that contain a combination of major oxides and trace elements. In those cases, I recommend to use only the extremely intense correlations.
A more useful solution is to separate major oxides from trace elements, and concentrate again only on the intense correlations. The disadvantage here is that we do not use the combine information of both groups of elements.
Will the transformation of the data be more efficient in the creation of RCCs that will help us target the mineralized zone within the granodiorite intrusive?

# Compositional Data Analysis

The CoDaPack software (which is included in the attached CD and the user guide is presented in Appendix 1) offers three type of transformation, the Centered Log-Ratio transformation (CRL), the Additive Log-Ratio transformation (ARL), and the Isometric Log-Ratio transformation (IRL). The last two require a column with the residual (100 minus the sum of all the other components).

## Centered Log-Ratio Transformation (CLR)

Appendix 1 contains the instructions on how to use the CoDaPack software. Since there are no zero values in our dataset, we can proceed directly to the CLR transformation (see table).
Table 12. Results of the CLR transformation of the initial dataset.
| SiO$_2$ | TiO$_2$ | Al$_2$O$_3$ | Fe$_2$O$_3$ | MgO | CaO | K$_2$O | Na$_2$O | P$_2$O$_5$ | Cu | As | Pb | Co | Ni | Sc | L.O.I. | Sum |
| 5.750 | 1.681 | 2.933 | 2.812 | 2.597 | 3.045 | 3.614 | 3.868 | 2.976 | -4.577 | -4.537 | -8.323 | -5.271 | -5.088 | -5.241 | 3.760 | 0 |
| 5.684 | 2.002 | 2.335 | 2.379 | 1.641 | 2.982 | 3.561 | 3.818 | 3.004 | -4.667 | -4.625 | -8.384 | -5.360 | -5.178 | -3.406 | 4.214 | 0 |
| 6.403 | 3.060 | 4.222 | 3.180 | 4.067 | 3.104 | 3.941 | 4.262 | 3.611 | -5.874 | -5.633 | -7.486 | -6.567 | -6.385 | -6.691 | 2.788 | 0 |
| 6.215 | 2.311 | 4.034 | 2.883 | 3.880 | 3.020 | 3.800 | 4.108 | 3.213 | -5.435 | -5.299 | -7.920 | -6.128 | -5.946 | -5.866 | 3.128 | 0 |
| 6.041 | 2.649 | 3.722 | 2.867 | 3.540 | 2.885 | 3.653 | 3.958 | 3.229 | -5.490 | -5.367 | -7.996 | -6.183 | -6.001 | -4.804 | 3.299 | 0 |
| 6.149 | 2.510 | 3.830 | 3.197 | 3.648 | 2.973 | 3.751 | 4.059 | 3.415 | -5.476 | -5.341 | -8.855 | -6.169 | -5.987 | -5.028 | 3.322 | 0 |
| 6.055 | 2.412 | 3.736 | 2.760 | 3.554 | 2.842 | 3.641 | 3.953 | 3.097 | -5.752 | -5.592 | -7.930 | -6.445 | -6.263 | -3.736 | 3.667 | 0 |
| 5.755 | 1.799 | 3.064 | 2.879 | 2.776 | 2.953 | 3.561 | 3.827 | 3.062 | -4.773 | -4.724 | -8.338 | -5.466 | -5.284 | -4.685 | 3.594 | 0 |
| 5.917 | 2.182 | 3.225 | 2.790 | 2.938 | 2.907 | 3.615 | 3.907 | 2.981 | -5.165 | -5.082 | -9.606 | -5.858 | -5.676 | -3.230 | 4.156 | 0 |
| 5.848 | 2.140 | 3.363 | 2.658 | 3.140 | 2.903 | 3.571 | 3.852 | 2.938 | -5.010 | -4.944 | -8.466 | -5.703 | -5.521 | -4.367 | 3.598 | 0 |
| 5.715 | 2.226 | 2.753 | 2.969 | 2.348 | 3.016 | 3.587 | 3.841 | 2.994 | -4.610 | -4.569 | -8.724 | -5.303 | -5.120 | -4.855 | 3.732 | 0 |
| 6.434 | 2.613 | 3.084 | 3.322 | 2.391 | 3.124 | 4.017 | 4.348 | 3.710 | -7.006 | -6.338 | -8.128 | -7.699 | -7.517 | -1.647 | 5.293 | 0 |
| 5.890 | 2.288 | 3.198 | 3.082 | 2.911 | 2.798 | 3.549 | 3.851 | 3.025 | -5.480 | -5.370 | -8.367 | -6.173 | -5.991 | -3.267 | 4.055 | 0 |
| 5.944 | 2.170 | 3.252 | 3.127 | 2.965 | 2.840 | 3.597 | 3.901 | 3.103 | -5.476 | -5.361 | -8.985 | -6.169 | -5.987 | -3.061 | 4.140 | 0 |
| 5.745 | 2.016 | 2.112 | 2.860 | 1.005 | 3.022 | 3.614 | 3.875 | 3.072 | -4.661 | -4.616 | -8.815 | -5.354 | -5.172 | -3.051 | 4.348 | 0 |
| 5.986 | 2.313 | 3.131 | 2.930 | 2.778 | 2.807 | 3.611 | 3.925 | 3.228 | -5.832 | -5.664 | -8.176 | -6.525 | -6.343 | -2.629 | 4.460 | 0 |
| 5.585 | 2.092 | 2.893 | 2.367 | 2.606 | 2.811 | 3.407 | 3.669 | 2.789 | -4.879 | -4.833 | -8.711 | -5.572 | -5.390 | -4.947 | 6.112 | 0 |
| 5.589 | 2.319 | 2.162 | 2.929 | 1.382 | 2.906 | 3.478 | 3.733 | 2.892 | -4.725 | -4.684 | -9.647 | -5.419 | -5.236 | -3.831 | 6.154 | 0 |
| 6.931 | 3.640 | -3.984 | 4.276 | -3.984 | 3.709 | 4.571 | 4.897 | 4.165 | -5.628 | -5.286 | -7.419 | -6.321 | -6.139 | -0.961 | 7.529 | 0 |
| 5.739 | 2.447 | 2.711 | 2.648 | 2.268 | 2.679 | 3.424 | 3.725 | 2.921 | -5.565 | -5.460 | -8.624 | -6.259 | -6.076 | -2.865 | 6.287 | 0 |
| 6.149 | 2.161 | 1.709 | 2.933 | -4.778 | 3.400 | 4.011 | 4.277 | 3.255 | -4.332 | -4.283 | -9.136 | -5.025 | -4.843 | -2.233 | 6.735 | 0 |
| 5.669 | 1.776 | 2.319 | 2.812 | 1.626 | 2.756 | 3.433 | 3.717 | 2.970 | -5.191 | -5.121 | -8.467 | -5.884 | -5.702 | -2.944 | 6.231 | 0 |
| 5.859 | 2.264 | 2.661 | 2.889 | 2.104 | 2.723 | 3.513 | 3.824 | 2.993 | -5.806 | -5.658 | -8.487 | -6.499 | -6.317 | -2.479 | 6.415 | 0 |
| 5.857 | 1.963 | 2.185 | 3.070 | 0.992 | 2.875 | 3.592 | 3.886 | 3.047 | -5.234 | -5.146 | -9.372 | -5.927 | -5.745 | -2.473 | 6.429 | 0 |
| 5.627 | 2.105 | 2.693 | 2.487 | 2.302 | 2.672 | 3.361 | 3.648 | 2.864 | -5.320 | -5.246 | -8.270 | -6.013 | -5.831 | -3.248 | 6.171 | 0 |
| 5.827 | 1.894 | 3.141 | 2.744 | 2.855 | 2.718 | 3.478 | 3.782 | 3.097 | -5.616 | -5.498 | -9.262 | -6.309 | -6.126 | -3.080 | 6.355 | 0 |
| 5.572 | 2.126 | 2.426 | 2.793 | 1.908 | 2.739 | 3.374 | 3.647 | 2.711 | -5.065 | -5.009 | -8.585 | -5.758 | -5.576 | -3.431 | 6.126 | 0 |
| 6.959 | 3.722 | 2.934 | 4.220 | -3.974 | 3.642 | 4.550 | 4.884 | 4.033 | -7.440 | -6.180 | -7.796 | -8.133 | -7.950 | -1.010 | 7.539 | 0 |
| 6.189 | 2.295 | 2.206 | 3.459 | -0.946 | 2.993 | 3.829 | 4.150 | 3.307 | -5.971 | -5.733 | -7.564 | -6.664 | -6.482 | -1.836 | 6.768 | 0 |
| 5.668 | 1.683 | 2.451 | 2.907 | 1.879 | 2.714 | 3.409 | 3.698 | 2.961 | -5.300 | -5.224 | -8.289 | -5.994 | -5.811 | -2.978 | 6.225 | 0 |
As Table 12 shows, the dataset is now “open”, since the sum of all the components is equal to zero, not 100%.
Once I achieved this transformation, I processed the data following the same steps as with the initial dataset. Tables 13 to 15 show the results of this process.
Table 13. Correlation analysis of the CLR transformed data.
| | SiO$_2$ | TiO$_2$ | Al$_2$O$_3$ | Fe$_2$O$_3$ | MgO | CaO | K$_2$O | Na$_2$O | P$_2$O$_5$ | Cu | As | Pb | Co | Ni | Sc | L.O.I. |
| SiO$_2$ | 1.00 | | | | | | | | | | | | | | | |
| TiO$_2$ | 0.87 | 1.00 | | | | | | | | | | | | | | |
| Al$_2$0$_3$ | -0.32 | -0.33 | 1.00 | | | | | | | | | | | | | |
| Fe$_2$O$_3$ | 0.87 | 0.78 | -0.49 | 1.00 | | | | | | | | | | | | |
| MgO | -0.51 | -0.43 | 0.68 | -0.61 | 1.00 | | | | | | | | | | | |
| CaO | 0.82 | 0.71 | -0.52 | 0.77 | -0.74 | 1.00 | | | | | | | | | | |
| K$_2$O | 0.96 | 0.83 | -0.45 | 0.88 | -0.67 | 0.94 | 1.00 | | | | | | | | | |
| Na$_2$O | 0.98 | 0.85 | -0.43 | 0.89 | -0.64 | 0.92 | 1.00 | 1.00 | | | | | | | | |
| P$_2$O$_5$ | 0.97 | 0.85 | -0.39 | 0.88 | -0.51 | 0.83 | 0.95 | 0.96 | 1.00 | | | | | | | |
| Cu | -0.69 | -0.65 | -0.09 | -0.61 | 0.12 | -0.26 | -0.53 | -0.58 | -0.65 | 1.00 | | | | | | |
| As | -0.57 | -0.54 | -0.16 | -0.48 | -0.05 | -0.05 | -0.36 | -0.42 | -0.53 | 0.96 | 1.00 | | | | | |
| Pb | 0.56 | 0.51 | -0.15 | 0.43 | -0.11 | 0.35 | 0.48 | 0.50 | 0.55 | -0.49 | -0.47 | 1.00 | | | | |
| Co | -0.69 | -0.65 | -0.09 | -0.61 | 0.12 | -0.26 | -0.53 | -0.58 | -0.65 | 1.00 | 0.96 | -0.49 | 1.00 | | | |
| Ni | -0.69 | -0.65 | -0.09 | -0.61 | 0.12 | -0.26 | -0.53 | -0.58 | -0.65 | 1.00 | 0.96 | -0.49 | 1.00 | 1.00 | | |
| Sc | 0.34 | 0.26 | -0.56 | 0.49 | -0.69 | 0.30 | 0.37 | 0.39 | 0.35 | -0.42 | -0.35 | 0.00 | -0.42 | -0.42 | 1.00 | |
| L.O.I | 0.14 | 0.18 | -0.58 | 0.33 | -0.71 | 0.20 | 0.21 | 0.22 | 0.16 | -0.23 | -0.16 | -0.07 | -0.23 | -0.23 | 0.76 | 1.00 |
Table 14. Critical value of Student of the CLR transformed data.
| | SiO$_2$ | TiO$_2$ | Al$_2$O$_3$ | Fe$_2$O$_3$ | MgO | CaO | K$_2$O | Na$_2$O | P$_2$O$_5$ | Cu | As | Pb | Co | Ni | Sc |
| SiO$_2$ | | | | | | | | | | | | | | | |
| TiO$_2$ | 9.32 | | | | | | | | | | | | | | |
| Al$_2$0$_3$ | 1.78 | 1.86 | | | | | | | | | | | | | |
| Fe$_2$O$_3$ | 9.27 | 6.60 | 2.99 | | | | | | | | | | | | |
| MgO | 3.12 | 2.52 | 4.93 | 4.12 | | | | | | | | | | | |
| CaO | 7.61 | 5.30 | 3.20 | 6.47 | 5.81 | | | | | | | | | | |
| K$_2$O | 18.35 | 7.97 | 2.67 | 9.62 | 4.72 | 15.22 | | | | | | | | | |
| Na$_2$O | 23.88 | 8.43 | 2.53 | 10.06 | 4.44 | 12.48 | 79.36 | | | | | | | | |
| P$_2$O$_5$ | 20.04 | 8.39 | 2.26 | 9.85 | 3.17 | 7.94 | 16.26 | 18.63 | | | | | | | |
| Cu | 5.03 | 4.51 | 0.47 | 4.02 | 0.66 | 1.42 | 3.31 | 3.74 | 4.58 | | | | | | |
| As | 3.67 | 3.35 | 0.83 | 2.86 | 0.28 | 0.28 | 2.06 | 2.45 | 3.31 | 19.40 | | | | | |
| Pb | 3.55 | 3.13 | 0.79 | 2.49 | 0.61 | 1.99 | 2.92 | 3.08 | 3.45 | 2.97 | 2.82 | | | | |
| Co | 5.03 | 4.51 | 0.47 | 4.02 | 0.66 | 1.42 | 3.31 | 3.74 | 4.58 | 100.00 | 19.40 | 2.97 | | | |
| Ni | 5.03 | 4.51 | 0.47 | 4.02 | 0.66 | 1.42 | 3.31 | 3.74 | 4.58 | 100.00 | 19.40 | 2.97 | 100.00 | | |
| Sc | 1.90 | 1.43 | 3.54 | 2.94 | 5.06 | 1.64 | 2.14 | 2.21 | 1.99 | 2.44 | 1.98 | 0.02 | 2.44 | 2.44 | |
| L.O.I | 0.77 | 0.95 | 3.74 | 1.87 | 5.36 | 1.06 | 1.16 | 1.17 | 0.88 | 1.26 | 0.87 | 0.39 | 1.26 | 1.26 | 6.27 |
Table 15. Significant correlations of the CLR transformed data.
| | SiO$_2$ | TiO$_2$ | Al$_2$O$_3$ | Fe$_2$O$_3$ | MgO | CaO | K$_2$O | Na$_2$O | P$_2$O$_5$ | Cu | As | Pb | Co | Ni | Sc |
| SiO$_2$ | | | | | | | | | | | | | | | |
| TiO$_2$ | 0.87 | | | | | | | | | | | | | | |
| Al$_2$0$_3$ | | | | | | | | | | | | | | | |
| Fe$_2$O$_3$ | 0.87 | 0.78 | -0.49 | | | | | | | | | | | | |
| MgO | -0.51 | -0.43 | 0.68 | -0.61 | | | | | | | | | | | |
| CaO | 0.82 | 0.71 | -0.52 | 0.77 | -0.74 | | | | | | | | | | |
| K$_2$O | 0.96 | 0.83 | -0.45 | 0.88 | -0.67 | 0.94 | | | | | | | | | |
| Na$_2$O | 0.98 | 0.85 | -0.43 | 0.89 | -0.64 | 0.92 | 1.00 | | | | | | | | |
| P$_2$O$_5$ | 0.97 | 0.85 | -0.39 | 0.88 | -0.51 | 0.83 | 0.95 | 0.96 | | | | | | | |
| Cu | -0.69 | -0.65 | | -0.61 | | | -0.53 | -0.58 | -0.65 | | | | | | |
| As | -0.57 | -0.54 | | -0.48 | | | -0.36 | -0.42 | -0.53 | 0.96 | | | | | |
| Pb | 0.56 | 0.51 | | 0.43 | | | 0.48 | 0.50 | 0.55 | -0.49 | -0.47 | | | | |
| Co | -0.69 | -0.65 | | -0.61 | | | -0.53 | -0.58 | -0.65 | 1.00 | 0.96 | -0.49 | | | |
| Ni | -0.69 | -0.65 | | -0.61 | | | -0.53 | -0.58 | -0.65 | 1.00 | 0.96 | -0.49 | 1.00 | | |
| Sc | | | -0.56 | 0.49 | -0.69 | | 0.37 | 0.39 | | -0.42 | | | -0.42 | -0.42 | |
| L.O.I | | | -0.58 | | -0.71 | | | | | | | | | | 0.76 |
Using SYSTAT SPSS 10.0 for Windows I constructed a matrix of scatter plots (Fig. 19) to confirm the results from table 15, as well as some individual graphics using Grapher 7.0 which clearly show that all the correlations now are real (Figs. 20 – 23).
Figure 19. Matrix of scatter plots for the CLR transformed data.
Equation 12 shows the RCC determined for the CLR transformed data and equation 13 shows the same RCC, but reduced by eliminating the SiO$_2$ and the L.O.I.
Equation 12. RCC5 for the CLR transformed data.
Equation 13. RCC5a for the CLR transformed data after reducing the SiO$_2$ and the L.O.I.
Fig. 24 shows that RCC5a can effectively target the copper mineralization within the granodiorite intrusive.
Figure 24. The RCC5a can effectively target the copper mineralization within the granodiorite intrusive.
Now, if we will use only the strongest correlations (r>0.95) then we will obtain RCC6 and RCC7 (equations 14 and 15).
Equation 14. RCC 6 for correlations stronger than ±0.95 for the CLR transformed data.
Equation 15. RCC 7 for correlations stronger than ±0.95 for the CLR transformed data.
As one can see from Fig. 25, RCC 6 is an almost perfect match with the location of the ore body. The RCC 7 (Fig 26) is similar to RCC 3 and represents a petroLogic association.
Figure 25. Almost perfect correspondence between the RCC 6 and the location of the ore body.
Figure 26. RCC 7 represents a petroLogic association of major oxides.

## Additive Log-Ratio Transformation (ARL)

Table 16 shows the results of the transformation of the original dataset. Tables 17 through 19 show the results of the correlation analysis.
Table 16. ALR transformed data.
| SiO$_2$ | TiO$_2$ | Al$_2$O$_3$ | Fe$_2$O$_3$ | MgO | CaO | K$_2$O | Na$_2$O | P$_2$O$_5$ | Cu | As | Pb | Co | Ni | Sc | L.O.I. |
| 2.916 | -0.322 | 0.682 | 0.176 | 0.521 | -0.235 | 0.506 | 0.892 | -0.011 | -7.368 | -7.327 | -11.839 | -12.176 | -11.994 | -5.054 | 0.713 |
| 3.048 | -0.772 | 0.392 | -0.063 | 0.691 | -0.058 | 0.631 | 1.027 | 0.324 | -7.259 | -7.218 | -11.513 | -11.084 | -10.902 | -5.033 | 1.105 |
| 3.087 | -0.204 | 1.026 | 0.432 | 0.794 | 0.047 | 0.727 | 1.112 | 0.321 | -7.194 | -7.153 | -11.263 | -10.165 | -9.983 | -4.805 | 0.708 |
| 3.307 | -0.036 | -7.728 | 0.084 | 0.908 | 0.168 | 0.846 | 1.219 | 0.515 | -7.129 | -7.086 | -10.582 | -9.663 | -9.481 | -9.787 | 0.967 |
| 3.533 | -0.360 | 1.299 | 0.803 | 1.245 | 0.568 | 1.173 | 1.572 | 0.651 | -6.865 | -6.820 | -10.220 | -9.320 | -9.137 | -4.492 | 1.394 |
| 3.679 | 0.006 | 0.306 | 0.623 | 1.386 | 0.694 | 1.304 | 1.685 | 0.921 | -6.774 | -6.728 | -10.484 | -8.832 | -8.650 | -4.936 | 1.693 |
| 4.529 | 0.886 | 0.418 | 1.234 | 2.201 | 1.597 | 2.115 | 2.528 | 1.571 | -6.032 | -5.983 | -9.456 | -7.971 | -7.789 | -5.262 | 2.420 |
| 4.637 | 1.042 | 1.979 | 1.667 | 2.420 | 1.777 | 2.291 | 2.702 | 1.771 | -5.874 | -5.825 | -9.709 | -7.722 | -7.539 | -3.701 | 2.529 |
| 6.582 | 2.678 | 2.034 | 3.250 | 4.293 | 3.691 | 4.167 | 4.588 | 3.580 | -4.131 | -4.076 | -7.553 | -5.761 | -5.579 | -5.499 | 4.401 |
| 4.813 | 1.174 | 1.316 | 1.861 | 2.565 | 1.944 | 2.416 | 2.838 | 2.079 | -6.061 | -5.995 | -10.191 | -7.505 | -7.322 | -6.364 | 2.089 |
| 3.890 | 0.498 | 0.181 | 0.716 | 1.645 | 1.022 | 1.502 | 1.916 | 1.078 | -7.037 | -6.968 | -10.146 | -8.334 | -8.152 | -6.955 | 1.166 |
| 6.110 | 2.176 | 2.076 | 3.027 | 3.899 | 3.284 | 3.761 | 4.172 | 3.379 | -4.853 | -4.779 | -8.979 | -6.026 | -5.844 | -2.797 | 4.112 |
| 4.957 | 1.183 | 1.785 | 2.141 | 2.750 | 2.218 | 2.611 | 3.056 | 2.116 | -6.041 | -5.964 | -9.972 | -7.156 | -6.974 | -4.048 | 2.489 |
| 4.479 | 0.878 | 1.253 | 1.672 | 2.321 | 1.782 | 2.139 | 2.596 | 1.614 | -6.602 | -6.519 | -9.778 | -7.583 | -7.401 | -4.677 | 1.788 |
| 5.576 | 2.285 | 2.647 | 2.485 | 3.462 | 2.901 | 3.262 | 3.714 | 2.759 | -5.538 | -5.451 | -8.786 | -6.421 | -6.239 | -3.027 | 3.129 |
| 5.188 | 1.294 | 2.946 | 2.400 | 3.098 | 2.586 | 2.923 | 3.372 | 2.377 | -6.092 | -5.987 | -10.041 | -6.596 | -6.414 | -3.142 | 2.253 |
| 7.545 | 3.811 | 4.299 | 4.418 | 5.440 | 4.908 | 5.244 | 5.689 | 4.610 | -3.824 | -3.715 | -7.978 | -4.229 | -4.047 | -1.601 | 5.077 |
| 4.747 | 0.762 | 2.085 | 1.987 | 2.676 | 2.255 | 2.489 | 2.975 | 2.040 | -6.643 | -6.528 | -9.209 | -6.914 | -6.732 | -3.898 | 1.392 |
| 4.589 | 1.067 | 1.780 | 1.449 | 2.573 | 2.102 | 2.323 | 2.813 | 1.826 | -6.838 | -6.720 | -9.308 | -7.051 | -6.869 | -4.287 | 0.933 |
| 4.856 | 0.963 | 2.199 | 1.999 | 2.871 | 2.401 | 2.620 | 3.105 | 2.157 | -6.607 | -6.484 | -9.280 | -6.697 | -6.515 | -3.757 | 1.258 |
| 5.031 | 1.323 | 1.298 | 1.841 | 3.003 | 2.537 | 2.754 | 3.235 | 2.121 | -6.578 | -6.443 | -9.283 | -6.521 | -6.338 | -5.184 | 1.630 |
| 5.085 | 1.639 | 2.132 | 2.306 | 3.107 | 2.634 | 2.887 | 3.332 | 2.223 | -6.489 | -6.353 | -9.072 | -6.246 | -6.063 | -3.918 | 1.241 |
| 5.567 | 1.580 | 3.494 | 2.351 | 3.622 | 3.171 | 3.429 | 3.857 | 2.673 | -6.068 | -5.920 | -9.718 | -5.607 | -5.425 | -2.815 | 1.661 |
| 5.155 | 1.199 | 1.771 | 2.279 | 3.165 | 2.708 | 2.962 | 3.391 | 2.463 | -6.618 | -6.458 | -8.938 | -6.066 | -5.883 | -5.285 | 0.028 |
| 8.201 | 4.709 | 5.153 | 4.983 | 6.216 | 5.787 | 6.023 | 6.453 | 5.405 | -3.628 | -3.460 | -6.095 | -2.956 | -2.773 | -2.330 | 1.598 |
| 3.412 | -0.253 | 1.051 | 0.591 | 1.555 | 1.109 | 1.345 | 1.774 | 0.803 | -8.691 | -8.453 | -11.084 | -7.623 | -7.441 | -5.319 | -0.335 |
| 5.583 | 1.900 | 2.926 | 2.278 | 3.656 | 3.258 | 3.460 | 3.899 | 2.903 | -6.610 | -6.368 | -8.485 | -5.462 | -5.279 | -3.508 | -0.986 |
| 7.715 | 4.491 | 5.107 | 5.102 | 5.870 | 5.443 | 5.650 | 6.081 | 5.064 | -4.831 | -4.489 | -7.474 | -3.246 | -3.064 | -1.658 | 0.725 |
| 6.454 | 3.032 | 3.673 | 3.775 | 4.612 | 4.224 | 4.393 | 4.843 | 3.800 | -6.935 | -6.267 | -7.917 | -4.496 | -4.314 | -4.049 | 2.345 |
| 2.267 | -1.802 | -1.109 | -0.671 | 0.376 | -1.802 | 0.132 | -1.109 | -0.507 | -12.176 | -10.916 | -11.806 | -8.753 | -8.571 | -8.724 | 0.135 |
Table 17. Correlation analysis for the ALR transformed data.
| | SiO$_2$ | TiO$_2$ | Al$_2$O$_3$ | Fe$_2$O$_3$ | MgO | CaO | K$_2$O | Na$_2$O | P$_2$O$_5$ | Cu | As | Pb | Co | Ni | Sc | L.O.I. |
| SiO$_2$ | 1.00 | | | | | | | | | | | | | | | |
| TiO$_2$ | 0.99 | 1.00 | | | | | | | | | | | | | | |
| Al$_2$0$_3$ | 0.71 | 0.69 | 1.00 | | | | | | | | | | | | | |
| Fe$_2$O$_3$ | 0.99 | 0.98 | 0.75 | 1.00 | | | | | | | | | | | | |
| MgO | 0.99 | 0.98 | 0.74 | 0.99 | 1.00 | | | | | | | | | | | |
| CaO | 0.99 | 0.98 | 0.74 | 0.98 | 0.99 | 1.00 | | | | | | | | | | |
| K$_2$O | 1.00 | 0.98 | 0.73 | 0.99 | 1.00 | 0.99 | 1.00 | | | | | | | | | |
| Na$_2$O | 0.99 | 0.98 | 0.72 | 0.98 | 0.98 | 1.00 | 0.98 | 1.00 | | | | | | | | |
| P$_2$O$_5$ | 1.00 | 0.98 | 0.72 | 0.99 | 1.00 | 0.99 | 1.00 | 0.99 | 1.00 | | | | | | | |
| Cu | 0.81 | 0.81 | 0.51 | 0.78 | 0.75 | 0.80 | 0.77 | 0.84 | 0.79 | 1.00 | | | | | | |
| As | 0.87 | 0.87 | 0.56 | 0.85 | 0.82 | 0.85 | 0.84 | 0.89 | 0.85 | 0.99 | 1.00 | | | | | |
| Pb | 0.94 | 0.93 | 0.62 | 0.92 | 0.94 | 0.94 | 0.94 | 0.93 | 0.94 | 0.75 | 0.81 | 1.00 | | | | |
| Co | 0.92 | 0.89 | 0.69 | 0.92 | 0.95 | 0.93 | 0.94 | 0.90 | 0.93 | 0.57 | 0.65 | 0.90 | 1.00 | | | |
| Ni | 0.92 | 0.89 | 0.69 | 0.92 | 0.95 | 0.93 | 0.94 | 0.90 | 0.93 | 0.57 | 0.65 | 0.90 | 1.00 | 1.00 | | |
| Sc | 0.70 | 0.69 | 0.88 | 0.74 | 0.71 | 0.74 | 0.71 | 0.74 | 0.71 | 0.66 | 0.68 | 0.59 | 0.61 | 0.61 | 1.00 | |
| L.O.I | 0.48 | 0.45 | 0.24 | 0.45 | 0.42 | 0.41 | 0.44 | 0.44 | 0.45 | 0.63 | 0.64 | 0.38 | 0.29 | 0.29 | 0.34 | 1.00 |
I used SYSTAT SPSS 10.0 for Windows to construct a matrix of scatter plots (Fig. 27) to confirm the results from table 17.
Figure 27. Matrix of scatter plots for the ARL transformed data.
Table 18. Critical values of Student for the ARL transformed data.
| | SiO$_2$ | TiO$_2$ | Al$_2$O$_3$ | Fe$_2$O$_3$ | MgO | CaO | K$_2$O | Na$_2$O | P$_2$O$_5$ | Cu | As | Pb | Co | Ni | Sc |
| SiO$_2$ | | | | | | | | | | | | | | | |
| TiO$_2$ | 31.32 | | | | | | | | | | | | | | |
| Al$_2$0$_3$ | 5.41 | 5.00 | | | | | | | | | | | | | |
| Fe$_2$O$_3$ | 35.09 | 25.48 | 6.03 | | | | | | | | | | | | |
| MgO | 51.30 | 24.81 | 5.76 | 34.14 | | | | | | | | | | | |
| CaO | 31.02 | 23.51 | 5.75 | 26.76 | 32.83 | | | | | | | | | | |
| K$_2$O | 72.03 | 27.59 | 5.70 | 37.43 | 140.28 | 33.75 | | | | | | | | | |
| Na$_2$O | 32.38 | 26.33 | 5.56 | 26.42 | 26.85 | 70.59 | 29.60 | | | | | | | | |
| P$_2$O$_5$ | 77.26 | 29.35 | 5.47 | 37.49 | 60.64 | 33.95 | 81.61 | 32.53 | | | | | | | |
| Cu | 7.19 | 7.36 | 3.14 | 6.55 | 5.97 | 7.01 | 6.32 | 8.10 | 6.72 | | | | | | |
| As | 9.36 | 9.48 | 3.59 | 8.36 | 7.58 | 8.65 | 8.08 | 10.19 | 8.63 | 38.77 | | | | | |
| Pb | 15.21 | 13.93 | 4.20 | 12.65 | 14.87 | 14.24 | 14.93 | 13.82 | 14.98 | 5.93 | 7.28 | | | | |
| Co | 12.63 | 10.27 | 4.99 | 12.12 | 16.35 | 13.16 | 14.79 | 11.07 | 13.57 | 3.65 | 4.53 | 11.04 | | | |
| Ni | 12.63 | 10.27 | 4.99 | 12.12 | 16.35 | 13.16 | 14.79 | 11.07 | 13.57 | 3.65 | 4.53 | 11.04 | 100.00 | | |
| Sc | 5.26 | 5.04 | 9.92 | 5.78 | 5.30 | 5.78 | 5.34 | 5.86 | 5.27 | 4.60 | 4.87 | 3.83 | 4.05 | 4.05 | |
| L.O.I | 2.91 | 2.66 | 1.33 | 2.69 | 2.47 | 2.35 | 2.56 | 2.61 | 2.63 | 4.28 | 4.45 | 2.17 | 1.60 | 1.60 | 1.90 |
Table 19. Significant correlations of the ALR transformed data.
| | SiO$_2$ | TiO$_2$ | Fe$_2$O$_3$ | MgO | CaO | K$_2$O | Na$_2$O | Cu | Co |
| SiO$_2$ | | | | | | | | | |
| TiO$_2$ | 0.99 | | | | | | | | |
| Al$_2$0$_3$ | | | | | | | | | |
| Fe$_2$O$_3$ | 0.99 | 0.98 | | | | | | | |
| MgO | 0.99 | 0.98 | 0.99 | | | | | | |
| CaO | 0.99 | 0.98 | 0.98 | 0.99 | | | | | |
| K$_2$O | 1.00 | 0.98 | 0.99 | 1.00 | 0.99 | | | | |
| Na$_2$O | 0.99 | 0.98 | 0.98 | 0.98 | 1.00 | 0.98 | | | |
| P$_2$O$_5$ | 1.00 | 0.98 | 0.99 | 1.00 | 0.99 | 1.00 | 0.99 | | |
| Cu | | | | | | | | | |
| As | | | | | | | | 0.99 | |
| Pb | | | | | | | | | |
| Co | | | | 0.95 | | | | | |
| Ni | | | | 0.95 | | | | | 1.00 |
| Sc | | | | | | | | | |
| L.O.I | | | | | | | | | |
Equations 16 and 17 shows the RCCs determined for the ALR transformed data. It is interesting to note that all the correlations here are positive.
Equation 16. RCC 8 of the ARL transformed data.
Equation 17. RCC 9 of the ARL transformed data.
Figures 28 and 29 show the spatial behavior of these RCCs with respect to the location of the ore body. If we combine both RCCs, we obtain equation 18 (Fig. 30).
Equation 18. Combination of RCC 9 and RCC 8 for the ARL transformed data.

## Isometric Log-Ratio Transformation (IRL)

Table 20 shows the results of the transformation of the original dataset. Tables 21 through 23 show the results of the correlation analysis.
Table 20. IRL transformed data.
| SiO$_2$ | TiO$_2$ | Al$_2$O$_3$ | Fe$_2$O$_3$ | MgO | CaO | K$_2$O | Na$_2$O | P$_2$O$_5$ | Cu | As | Pb | Co | Ni | Sc | L.O.I. |
| 2.289 | 0.502 | 0.793 | 0.306 | 0.940 | 0.108 | -0.267 | 0.615 | 7.530 | 6.773 | 10.502 | 9.984 | 9.068 | 1.737 | -3.959 | -3.027 |
| 2.701 | 0.610 | 0.825 | -0.036 | 0.655 | -0.085 | -0.444 | 0.271 | 7.437 | 6.688 | 10.218 | 8.987 | 8.144 | 1.912 | -4.155 | -2.831 |
| 2.327 | 0.339 | 0.755 | 0.260 | 0.894 | 0.126 | -0.251 | 0.525 | 7.599 | 6.834 | 10.174 | 8.304 | 7.512 | 1.991 | -3.475 | -2.578 |
| 2.364 | 7.645 | -1.359 | -1.790 | -0.786 | -1.292 | -1.468 | -0.631 | 6.687 | 6.009 | 8.832 | 7.241 | 6.528 | 6.374 | -4.451 | -3.242 |
| 2.753 | 0.235 | 0.595 | 0.066 | 0.672 | 0.007 | -0.367 | 0.545 | 7.618 | 6.848 | 9.506 | 7.879 | 7.119 | 2.140 | -3.697 | -2.121 |
| 2.598 | 1.255 | 0.613 | -0.208 | 0.462 | -0.175 | -0.508 | 0.273 | 7.544 | 6.780 | 9.785 | 7.414 | 6.689 | 2.639 | -3.950 | -2.068 |
| 2.576 | 1.869 | 0.615 | -0.389 | 0.234 | -0.281 | -0.631 | 0.346 | 7.523 | 6.758 | 9.494 | 7.307 | 6.589 | 3.693 | -3.983 | -1.394 |
| 2.542 | 0.702 | 0.767 | -0.080 | 0.522 | -0.035 | -0.414 | 0.513 | 7.711 | 6.928 | 10.043 | 7.329 | 6.609 | 2.445 | -3.746 | -1.065 |
| 2.760 | 2.120 | 0.446 | -0.588 | 0.070 | -0.382 | -0.725 | 0.311 | 7.594 | 6.816 | 9.552 | 7.065 | 6.365 | 5.849 | -4.115 | 0.404 |
| 2.573 | 1.370 | 0.496 | -0.245 | 0.368 | -0.126 | -0.504 | 0.271 | 7.965 | 7.141 | 10.536 | 7.111 | 6.408 | 5.040 | -3.471 | -1.234 |
| 2.399 | 1.644 | 0.699 | -0.289 | 0.332 | -0.164 | -0.528 | 0.324 | 7.989 | 7.160 | 9.579 | 7.070 | 6.370 | 4.774 | -3.397 | -2.060 |
| 2.782 | 1.688 | 0.370 | -0.493 | 0.159 | -0.308 | -0.651 | 0.173 | 7.965 | 7.134 | 10.534 | 6.852 | 6.168 | 2.799 | -4.071 | 0.165 |
| 2.668 | 1.050 | 0.434 | -0.209 | 0.315 | -0.097 | -0.501 | 0.445 | 8.136 | 7.286 | 10.489 | 6.943 | 6.252 | 2.994 | -3.529 | -0.900 |
| 2.546 | 1.164 | 0.460 | -0.224 | 0.309 | -0.069 | -0.488 | 0.496 | 8.238 | 7.372 | 9.850 | 6.952 | 6.261 | 3.197 | -3.269 | -1.336 |
| 2.327 | 1.048 | 0.881 | -0.191 | 0.356 | -0.033 | -0.452 | 0.502 | 8.321 | 7.443 | 9.988 | 6.915 | 6.226 | 2.694 | -3.441 | -0.197 |
| 2.753 | 0.240 | 0.643 | -0.126 | 0.364 | -0.004 | -0.424 | 0.564 | 8.540 | 7.625 | 10.841 | 6.663 | 5.993 | 2.418 | -2.962 | -0.596 |
| 2.641 | 1.126 | 0.693 | -0.377 | 0.178 | -0.161 | -0.556 | 0.528 | 8.473 | 7.560 | 10.982 | 6.501 | 5.843 | 3.077 | -3.588 | 1.555 |
| 2.818 | 0.546 | 0.472 | -0.251 | 0.180 | -0.065 | -0.511 | 0.431 | 8.623 | 7.690 | 9.587 | 6.614 | 5.947 | 2.799 | -2.504 | -1.001 |
| 2.491 | 0.856 | 0.892 | -0.315 | 0.173 | -0.058 | -0.509 | 0.482 | 8.650 | 7.712 | 9.518 | 6.587 | 5.923 | 3.019 | -2.230 | -1.190 |
| 2.753 | 0.580 | 0.584 | -0.328 | 0.161 | -0.067 | -0.511 | 0.443 | 8.711 | 7.761 | 9.762 | 6.498 | 5.841 | 2.773 | -2.261 | -0.904 |
| 2.622 | 1.534 | 0.614 | -0.563 | -0.034 | -0.230 | -0.649 | 0.478 | 8.679 | 7.722 | 9.769 | 6.332 | 5.687 | 4.179 | -2.689 | -0.944 |
| 2.437 | 1.004 | 0.559 | -0.283 | 0.200 | -0.064 | -0.472 | 0.629 | 8.828 | 7.855 | 9.774 | 6.275 | 5.634 | 3.173 | -2.027 | -0.701 |
| 2.820 | 0.065 | 1.036 | -0.335 | 0.139 | -0.122 | -0.506 | 0.670 | 8.892 | 7.901 | 10.849 | 6.031 | 5.408 | 2.513 | -1.984 | -0.251 |
| 2.797 | 1.148 | 0.372 | -0.505 | 0.006 | -0.230 | -0.601 | 0.345 | 8.924 | 7.919 | 9.603 | 6.074 | 5.448 | 4.494 | -0.941 | -0.857 |
| 2.470 | 1.063 | 0.898 | -0.407 | 0.060 | -0.169 | -0.548 | 0.505 | 9.021 | 7.999 | 9.825 | 6.022 | 5.400 | 4.599 | 0.498 | 2.018 |
| 2.592 | 0.431 | 0.704 | -0.317 | 0.148 | -0.093 | -0.482 | 0.490 | 9.446 | 8.317 | 10.112 | 5.976 | 5.357 | 2.938 | -2.079 | -2.277 |
| 2.604 | 0.666 | 1.033 | -0.433 | 0.010 | -0.179 | -0.565 | 0.441 | 9.418 | 8.289 | 9.593 | 5.920 | 5.305 | 3.227 | 0.577 | -0.414 |
| 2.280 | 0.813 | 0.580 | -0.238 | 0.196 | -0.027 | -0.426 | 0.583 | 9.909 | 8.636 | 10.743 | 5.819 | 5.212 | 3.494 | 0.961 | 1.606 |
| 2.420 | 0.874 | 0.529 | -0.339 | 0.078 | -0.091 | -0.500 | 0.543 | 10.670 | 9.014 | 9.809 | 5.736 | 5.135 | 4.524 | -1.959 | 0.435 |
| 2.877 | 1.095 | 0.395 | -0.630 | 1.474 | -0.545 | 0.689 | 0.040 | 11.105 | 8.844 | 8.926 | 5.278 | 4.710 | 4.533 | -4.338 | -3.943 |
Table 21. Critical value of Student of the ILR transformed data
| | SiO$_2$ | TiO$_2$ | Al$_2$O$_3$ | Fe$_2$O$_3$ | MgO | CaO | K$_2$O | Na$_2$O | P$_2$O$_5$ | Cu | As | Pb | Co | Ni | Sc | L.O.I. |
| SiO$_2$ | 1.00 | | | | | | | | | | | | | | | |
| TiO$_2$ | -0.22 | 1.00 | | | | | | | | | | | | | | |
| Al$_2$0$_3$ | 0.06 | -0.90 | 1.00 | | | | | | | | | | | | | |
| Fe$_2$O$_3$ | -0.09 | -0.86 | 0.78 | 1.00 | | | | | | | | | | | | |
| MgO | 0.11 | -0.57 | 0.45 | 0.66 | 1.00 | | | | | | | | | | | |
| CaO | -0.08 | -0.92 | 0.85 | 0.95 | 0.46 | 1.00 | | | | | | | | | | |
| K$_2$O | 0.22 | -0.64 | 0.51 | 0.53 | 0.90 | 0.43 | 1.00 | | | | | | | | | |
| Na$_2$O | -0.06 | -0.88 | 0.85 | 0.80 | 0.29 | 0.92 | 0.36 | 1.00 | | | | | | | | |
| P$_2$O$_5$ | 0.14 | -0.38 | 0.27 | -0.01 | 0.15 | 0.14 | 0.54 | 0.28 | 1.00 | | | | | | | |
| Cu | 0.11 | -0.44 | 0.35 | 0.04 | 0.06 | 0.23 | 0.46 | 0.39 | 0.98 | 1.00 | | | | | | |
| As | -0.03 | -0.47 | 0.43 | 0.50 | 0.11 | 0.54 | 0.05 | 0.54 | -0.03 | 0.06 | 1.00 | | | | | |
| Pb | -0.26 | 0.03 | 0.00 | 0.42 | 0.33 | 0.19 | -0.09 | -0.03 | -0.77 | -0.78 | 0.14 | 1.00 | | | | |
| Co | -0.26 | 0.03 | 0.00 | 0.42 | 0.33 | 0.19 | -0.09 | -0.03 | -0.77 | -0.78 | 0.14 | 1.00 | 1.00 | | | |
| Ni | -0.05 | 0.68 | -0.58 | -0.76 | -0.48 | -0.71 | -0.34 | -0.59 | 0.11 | 0.06 | -0.49 | -0.40 | -0.40 | 1.00 | | |
| Sc | -0.16 | -0.29 | 0.32 | 0.03 | -0.32 | 0.25 | -0.04 | 0.40 | 0.56 | 0.67 | 0.11 | -0.55 | -0.55 | 0.05 | 1.00 | |
| L.O.I | -0.03 | -0.20 | 0.29 | 0.01 | -0.43 | 0.23 | -0.24 | 0.43 | 0.27 | 0.38 | 0.43 | -0.46 | -0.46 | 0.11 | 0.58 | 1.00 |
Figure 31 shows the result of a matrix of scatter plots constructed with SYSTAT SPSS 10.0 for Windows to test the results from table 21.
Figure 31. Matrix of scatter plots for the IRL transformed data.
Table 22. Critical value of Student of the ILR transformed data.
| | SiO$_2$ | TiO$_2$ | Al$_2$O$_3$ | Fe$_2$O$_3$ | MgO | CaO | K$_2$O | Na$_2$O | P$_2$O$_5$ | Cu | As | Pb | Co | Ni | Sc |
| SiO$_2$ | | | | | | | | | | | | | | | |
| TiO$_2$ | 1.21 | | | | | | | | | | | | | | |
| Al$_2$0$_3$ | 0.33 | 11.08 | | | | | | | | | | | | | |
| Fe$_2$O$_3$ | 0.45 | 9.08 | 6.69 | | | | | | | | | | | | |
| MgO | 0.56 | 3.67 | 2.68 | 4.66 | | | | | | | | | | | |
| CaO | 0.42 | 12.02 | 8.40 | 15.47 | 2.74 | | | | | | | | | | |
| K$_2$O | 1.18 | 4.44 | 3.10 | 3.27 | 11.02 | 2.51 | | | | | | | | | |
| Na$_2$O | 0.30 | 9.57 | 8.40 | 7.02 | 1.59 | 12.18 | 2.02 | | | | | | | | |
| P$_2$O$_5$ | 0.73 | 2.15 | 1.49 | 0.07 | 0.79 | 0.72 | 3.37 | 1.55 | | | | | | | |
| Cu | 0.60 | 2.60 | 1.96 | 0.22 | 0.32 | 1.26 | 2.72 | 2.24 | 28.19 | | | | | | |
| As | 0.18 | 2.78 | 2.48 | 3.06 | 0.57 | 3.43 | 0.27 | 3.39 | 0.15 | 0.31 | | | | | |
| Pb | 1.45 | 0.14 | 0.01 | 2.44 | 1.83 | 1.03 | 0.50 | 0.17 | 6.35 | 6.69 | 0.77 | | | | |
| Co | 1.45 | 0.14 | 0.01 | 2.44 | 1.83 | 1.03 | 0.50 | 0.17 | 6.35 | 6.69 | 0.77 | 100.00 | | | |
| Ni | 0.27 | 4.85 | 3.75 | 6.19 | 2.87 | 5.40 | 1.93 | 3.91 | 0.60 | 0.31 | 2.99 | 2.32 | 2.32 | | |
| Sc | 0.84 | 1.57 | 1.78 | 0.14 | 1.81 | 1.34 | 0.23 | 2.29 | 3.61 | 4.72 | 0.56 | 3.51 | 3.51 | 0.27 | |
| L.O.I | 0.18 | 1.09 | 1.62 | 0.06 | 2.49 | 1.26 | 1.29 | 2.53 | 1.50 | 2.17 | 2.52 | 2.77 | 2.77 | 0.56 | 3.74 |
Table 23. Significant correlations of the ILR transformed data.
| | TiO$_2$ | Al$_2$O$_3$ | Fe$_2$O$_3$ | MgO | CaO | P$_2$O$_5$ | Pb |
| SiO$_2$ | | | | | | | |
| TiO$_2$ | | | | | | | |
| Al$_2$0$_3$ | -0.90 | | | | | | |
| Fe$_2$O$_3$ | -0.86 | | | | | | |
| MgO | | | | | | | |
| CaO | -0.92 | 0.85 | 0.95 | | | | |
| K$_2$O | | | | 0.90 | | | |
| Na$_2$O | -0.88 | 0.85 | 0.80 | | 0.92 | | |
| P$_2$O$_5$ | | | | | | | |
| Cu | | | | | | 0.98 | |
| As | | | | | | | |
| Pb | | | | | | | |
| Co | | | | | | | 1.00 |
| Ni | | | | | | | |
| Sc | | | | | | | |
| L.O.I | | | | | | | |
Equations 18 through 20 shows the RCCs determined for the ALR transformed data.
Equation 19. RCC 10 for the IRL transformed data.
Equation 20. RCC 11 for the IRL transformed data.
Equation 21. RCC 12 for the IRL transformed data.
Equation 22. RCC 13 for the IRL transformed data.
Figures 32 -35 show the result of the use of these RCCs as targeting tools.

### Conclusions and Recommendations from the Compositional Data Analysis

One very important effect of “opening” a dataset by using any of these transformations is that we get rid off all spurious correlations. The transformed data do contain unexpected correlations, but they are real.
Another important point is that we do not need to process the data separately (e.g. separating major oxides from trace elements), but can process the whole dataset taking advantage of the information contained in both groups.
From all the RCCs obtained so far, the RCC 8, RCC 9, and especially the RCC9/8 (ALR) were by far the most efficient one for targeting the copper mineralization.
The CRL transformed data did also provide for useful RCCs, especially if we concentrate in the higher correlations.
Finally, the IRL transformed data was effective for as long as the geochemist will “interpret” the coefficient and not plot them blindly. For example, a geochemist should know that elements like Pb and Co, usually concentrate bellow the ore body (inframinerals), and therefore while using RCC 13, the investigator should concentrate on the lower values as an indication of the location of the ore body. We have a similar situation with RCC 11. The investigator should know that a common effect of sodic metasomatism would be the lixiviation of MgO and K$_2$O; therefore, the geochemist should be looking for lower values of the RCC.
In general, I can state that transformed data are more effective for the location of the mineralized targets than the non-transformed dataset, and that the ARL method seems to be the most effective for processing this type of data. However, the geochemist should always use his background knowledge to help to decide the most efficient RCC for the studied area.
I recommend the use of the software CoDaPack for the processing of any type of “closed” dataset.

## References

* ↑ $^{Remembertoaddacolumnoftheresidualstotheoriginaldataset.}$
* ↑ $^{Remembertoaddacolumnoftheresidualstotheoriginaldataset.}$

# Factor Analysis

Factor analysis is a statistical data reduction technique used to explain variability among observed random variables in terms of fewer unobserved random variables called factors. It is useful to reduce the number of variables, by combining two or more variables into a single factor, thus “simplifying” the original dataset.
Factor analysis (FA) is especially useful in geochemistry when one has a known target or some other way to understand the meaning of the obtained associations. When failing this, the geologist is usually forced to “plot and see”, and then to select the FA that he believes is the most useful for the studied area.
I processed both the initial dataset and the three transformed versions using SYSTAT SSPS 10.0 for Windows, but you can use any other statistical program capable of factor analysis.

## Factor Analysis for the Initial Dataset

Figure 36 shows the plot for the initial dataset, while table 24 shows the principal components defined by the software.
Figure 36. Scree plot for the initial dataset.
Table 24. Principal component analysis (PCA) for the initial dataset.
Equations 23 – 25 show the three FA components for the initial dataset.
Equation 23. FA 1 for the initial dataset.
Equation 24. FA 2 for the initial dataset.
Equation 25. FA 3 for the initial dataset.
Figures 37 – 39 show the effectiveness of these FA as a targeting tool for our ore body.

### Conclusions and recommendations on the use of FA for the initial dataset

For as long as we have a known target to test the obtained FA, this method offers better results than the RCC. It also allows for the combined studied of all the elements together.
FA1 and FA2 do contain the embedded correlations I introduced in the initial dataset, thus their effectiveness, especially FA 1, in mapping the location of the ore body.
The next question will be: Will the transformed data be any more effective in helping us locate our target?

## CRL transformed data

Figure 40 shows the scree plot for the CLR transformed dataset, while table 25 shows the principal components defined by SYSTAT.
Figure 40. Scree plot for the CLR transformed dataset.
Table 25. Principal component analysis for the CLR transformed dataset.
Equations 26 – 28 show the three FA components for the CLR transformed dataset.
Equation 26. FA 4 for the CLR transformed dataset.
Equation 27. FA5 for the CLR transformed dataset.
Equation 28. FA6 for the CLR transformed dataset.
Figures 41 – 43 show the effectiveness of these FA as a targeting tool for our ore body.

## Factor Analysis for the ALR Transformed Dataset

Figure 44 shows the scree plot for the ALR transformed dataset, while table 26 shows the principal components defined by SYSTAT.
Figure 44. Scree plot for the ALR transformed dataset.
Table 26. Principal component analysis for the ALR transformed dataset.
Although table 26 shows two components, I will analyze only the second, which is a coefficient as shown in equation 29.
Equation 29. FA7 for the ALR transformed dataset.
This factor contains the embedded relationship from the initial dataset, but because of the presence of other elements, its usefulness as a targeting tool is more limited, as shown in Figure 45.
Figure 45. FA7 covers mostly the southeastern part of the ore body.

## Factor Analysis of the IRL Transformed Dataset

Figure 46 shows the scree plot for the IRL transformed dataset, while table 27 shows the principal components defined by SYSTAT.
Figure 46. Scree plot for the IRL transformed dataset.
Table 27. Principal component analysis for the IRL transformed dataset.
The fact that we have so many components as the result of the P.C.A., is an indication that we will not get good results this time. Equations 30 through 34 show the obtained factors.
Equation 30. FA8 for the IRL transformed dataset.
Equation 31. FA9 for the IRL transformed dataset.
Equation 32. FA10 for the IRL transformed dataset.
Equation 33. FA11 for the IRL transformed dataset.
Figures 47 through 50 shows the spatial distribution of these factors with respect to the location of our ore body.

### Conclusions and Recommendations on the Use of FA for the Transformed Datasets

As I mentioned earlier, for FA to be most useful, one needs to have a known target to calibrate it. The factor analysis applied to the CLR transformed data gave us three factors, but only one (FA5) was useful for targeting the ore body.
The factor analysis of the ALR transformed data (Factor 7) was good in general, but the best factors were obtained from the ILR transformed data, specially Factor 9 that not only gave the exact location of the ore body, but also its internal structure. Another efficient factor was FA11, but it definitively required calibration based on a known target.
So answering the question from page 41, yes, the factor analysis of the IRL transformed data will be more effective than the factor analysis of the raw data as a tool for locating the ore deposit.

# Dealing With Zero Values

/Dealing With Zero Values

# Conclusions and Recommendations

Conclusions and Recommendations
The treatment of “closed” dataset by normal statistical methods does create spurious correlations that lower the effectiveness of the obtained results. While there are ways to minimize this problem, like processing major oxides independently from trace elements, or using only the strongest correlations into the composition of the RCCs, I believe that the transformation of the initial dataset presents a better solution for the processing and interpretation of geological data.
For the estimation of the most efficient RCC, I propose the use of the ARL transformation, although the CLR is also effective.
When a target for the testing of the effectiveness of our coefficient is available, then we should use the factor analysis preferentially. I recommend the use of the IRL transformation to define the most effective combination of factors.
Finally, I introduced here a method for dealing with zero values. This method’s main advantage is that we do not obtain a fixed value for the “Rounded Zeros”, but one that depends on the real value of the other variable. The proposed method depends on the geological characteristics of the data, and therefore is less biased or random than other methods. It also presents a viable alternative to amalgamation and an effective way to deal with “Essential Zeros” in a population.
The sequence of the method is as follows:
* We transform the data using CoDaPack or other similar software.
* We select the lower quartile of real data for the element with the b.d.l. values.
* Within this dataset, we test the relationship between the elements with the b.d.l. values with one (or more) element without b.d.l. values. In most cases, these elements will correspond with well-established geological relationship like between Pb and Zn on polymetallic deposits, or between Au and Pb in hydrothermal deposits, or between Cu and Mo in porphyritic deposits, as in the case I presented here.
* We establish the regression equation.
* We then substitute the b.d.l. values by those estimated with the obtained equation of regression.
We can apply this method to any type of data, provided we establish first their correlation dependency. I would also like to see this method included as an option for dealing with zeros in the next version of CoDaPack.

# References
## Used in this text

* Aitchison, J, 2003 (2nd ed.). The statistical analysis of compositional data. The Blackburn Press, Caldwell, NJ (USA). 435 p. ISBN 1-930665-78-4.
* Aitchison, J. and J.W. Kay, 2003. Possible solutions of some essential zero problems in compositional data analysis. Proceedings of Compositional Data Analysis Workshop, 2003.
* Bacon-Shone, J., 2005. Modeling structural zeros in compositional data. Proceedings of Compositional Data Analysis Workshop, 2003.
* Kashan, A.B., Guskov, O.I. and A.A. Shimonsky, 1979. Mathematical modeling in prospection (original in Russian). Moscow: Nedra.
* Trusova, I. F and V. I. Chernov. Petrography of magmatic and metamorphic rocks (original in Russian). Moscow: Nedra.

## Suggested web links

3rd Compositional Data Analysis Workshop
http://ima.udg.edu/Activitats/CoDaWork08/
Bias arising from missing data in predictive models
http://www.scirus.com/srsapp/sciruslink?src=sd&url=http%3A%2F%2Fwww.sciencedirect.com%2Fscience%3F_ob%3DGatewayURL%26_origin%3DScienceSearch%26_method%3DcitationSearch%26_piikey%3DS0895435606002642%26_version%3D1%26_returnURL%3Dhttp%253A%252F%252Fwww.scirus.com%252Fsrsapp%252F%26md5%3D4868c0b11a79d4805f086c94316ef139
Compositional Data Analysis and Zeros in Micro Data.
http://www.scirus.com/srsapp/sciruslink?src=rpc&url=http%3A%2F%2Fideas.repec.org%2Fa%2Ftaf%2Fapplec%2Fv32y2000i8p953-59.html
Compositional Time Series: Past and Present
http://www.scirus.com/srsapp/sciruslink?src=rpc&url=http%3A%2F%2Fideas.repec.org%2Fp%2Fwpa%2Fwuwpem%2F0510002.html
Homepage of "compositions", an R package for compositional data analysis.
http://www.stat.boogaart.de/compositions/
Modeling Zeroes in Microdata.
http://www.scirus.com/srsapp/sciruslink?src=rpc&url=http%3A%2F%2Fideas.repec.org%2Fa%2Ftaf%2Fapplec%2Fv33y2001i3p383-92.html
New Features of CoDaPack. An User-friendly Compositional Data Package
http://ima.udg.edu/Activitats/CoDaWork05/CD/Session5/ThioHenestrosa-Tolosana-Gomez.pdf
Tectonic discrimination of basalts with classification trees.
http://www.scirus.com/srsapp/sciruslink?src=sd&url=http%3A%2F%2Fwww.sciencedirect.com%2Fscience%3F_ob%3DGatewayURL%26_origin%3DScienceSearch%26_method%3DcitationSearch%26_piikey%3DS0016703705009701%26_version%3D1%26_returnURL%3Dhttp%253A%252F%252Fwww.scirus.com%252Fsrsapp%252F%26md5%3De31d0170976cd137f78e7ce435aa9f32
Santiago Thió Fernández de Henestrosa.
http://ima.udg.edu/~thio/
Simcluster: clustering enumeRation gene expression data on the simplex space
http://www.scirus.com/srsapp/sciruslink?src=arx&url=http%3A%2F%2Farxiv.org%2Fabs%2Fq-bio%2F0703007
Some numerical considerations in the geochemical analysis of distal microtephra
http://www.scirus.com/srsapp/sciruslink?src=sd&url=http%3A%2F%2Fwww.sciencedirect.com%2Fscience%3F_ob%3DGatewayURL%26_origin%3DScienceSearch%26_method%3DcitationSearch%26_piikey%3DS0883292706001727%26_version%3D1%26_returnURL%3Dhttp%253A%252F%252Fwww.scirus.com%252Fsrsapp%252F%26md5%3Df9ab2e6094792c99aa80f30699cf5cb7
Statistical empirical index of chemical weathering in igneous rocks: A new tool for evaluating the degree of weathering
http://www.scirus.com/srsapp/sciruslink?src=sd&url=http%3A%2F%2Fwww.sciencedirect.com%2Fscience%3F_ob%3DGatewayURL%26_origin%3DScienceSearch%26_method%3DcitationSearch%26_piikey%3DS0009254107001052%26_version%3D1%26_returnURL%3Dhttp%253A%252F%252Fwww.scirus.com%252Fsrsapp%252F%26md5%3D8760181502f5d424d4ebb5fe21bf5ef6

# GNU Free Documentation License

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## 10. FUTURE REVISIONS OF THIS LICENSE

The Free Software Foundation may publish new, revised versions of the GNU Free Documentation License from time to time. Such new versions will be similar in spirit to the present version, but may differ in detail to address new problems or concerns. See http://www.gnu.org/copyleft/.
Each version of the License is given a distinguishing version number. If the Document specifies that a particular numbered version of this License "or any later version" applies to it, you have the option of following the terms and conditions either of that specified version or of any later version that has been published (not as a draft) by the Free Software Foundation. If the Document does not specify a version number of this License, you may choose any version ever published (not as a draft) by the Free Software Foundation. If the Document specifies that a proxy can decide which future versions of this License can be used, that proxy's public statement of acceptance of a version permanently authorizes you to choose that version for the Document.

## 11. RELICENSING

"Massive Multiauthor Collaboration Site" (or "MMC Site") means any World Wide Web server that publishes copyrightable works and also provides prominent facilities for anybody to edit those works. A public wiki that anybody can edit is an example of such a server. A "Massive Multiauthor Collaboration" (or "MMC") contained in the site means any set of copyrightable works thus published on the MMC site.
"CC-BY-SA" means the Creative Commons Attribution-Share Alike 3.0 license published by Creative Commons Corporation, a not-for-profit corporation with a principal place of business in San Francisco, California, as well as future copyleft versions of that license published by that same organization.
"Incorporate" means to publish or republish a Document, in whole or in part, as part of another Document.
An MMC is "eligible for relicensing" if it is licensed under this License, and if all works that were first published under this License somewhere other than this MMC, and subsequently incorporated in whole or in part into the MMC, (1) had no cover texts or invariant sections, and (2) were thus incorporated prior to November 1, 2008.
The operator of an MMC Site may republish an MMC contained in the site under CC-BY-SA on the same site at any time before August 1, 2009, provided the MMC is eligible for relicensing.

# How to use this License for your documents

To use this License in a document you have written, include a copy of the License in the document and put the following copyright and license notices just after the title page:
    Copyright (c) YEAR YOUR NAME.
    Permission is granted to copy, distribute and/or modify this document
    under the terms of the GNU Free Documentation License, Version 1.3
    or any later version published by the Free Software Foundation;
    with no Invariant Sections, no Front-Cover Texts, and no Back-Cover Texts.
    A copy of the license is included in the section entitled "GNU
    Free Documentation License".
If you have Invariant Sections, Front-Cover Texts and Back-Cover Texts, replace the "with...Texts." line with this:
    with the Invariant Sections being LIST THEIR TITLES, with the
    Front-Cover Texts being LIST, and with the Back-Cover Texts being LIST.
If you have Invariant Sections without Cover Texts, or some other combination of the three, merge those two alternatives to suit the situation.
If your document contains nontrivial examples of program code, we recommend releasing these examples in parallel under your choice of free software license, such as the GNU General Public License, to permit their use in free software.