84
5 CONCLUSIONS
Principal component analysis is an orthogonal transformation that can be used to convert
a set of correlated variables into a set of almost linearly uncorrelated components. In this
paper, we show a detailed methodology to apply principal component analysis to a set of geochemical variables from an exploration campaign of a Nickel laterite deposit. The method
requires dealing with the compositional nature of the data, thus requiring a transformation
of the grades into log-ratios. These log-ratios are then decorrelated using PCA. The decorrelation is checked by computing cross variograms between the principal components, which
confirms that almost all linear correlation is removed by the transformation into principal
components. These principal components are then independently simulated using sequential
Gaussian simulation, which in turn requires a normal score transformation of the data.
The methodology is therefore presented as a sequence of three transformations: a logratio transformation using the additive approach, a decorrelation using principal component analysis, and a normal score transformation to use Gaussian simulation. The simulated
results must be back-transformed to bring them back from Gaussian simulated deviates, into
simulated principal components, then into log-ratios and finally, into simulated grades.
Results are checked to ensure the correlation statistics are preserved, which is confirmed
by the scatter plots of the simulated variables, where correlation coefficients are well preserved, and the general correlation structure is reproduced. The method cannot capture some
non-linear features of the relationships, which is expected due to its linear nature. Overall,
results are satisfactory, confirming that PCA is a suitable approach to model spatially correlated variables.
ACKNOWLEDGEMENTS
We acknowledge the support of the Natural Sciences and Engineering Research Council of
Canada (NSERC), funding reference number RGPIN-2017-04200 and RGPAS-2017-507956.
REFERENCES
Aitchison, J. 1982. The statistical analysis of compositional data (with discussion). Journal of the Royal
Statistical Society, Series B (Statistical Methodology) 44 (2): 139–177.
Aitchison, J. 1986. The statistical analysis of compositional data. Monographs on Statistics and Applied
Probability, London, Chapman & Hall Ltd., 416 p.
Almeida, A.S. & Journel, A.G. 1994. Joint simulation of multiple variables with a Markov-type coregionalization model. Mathematical geology 26(5): 565–588.
Chiles, J.-P. & Delfiner, P. 2012. Geostatistics – Modeling Spatial Uncertainty, Second Edition.
Willey, 699 p.
Davis, B.M. & Greenes, K.A. 1983. Estimating using spatially distributed multivariate data: An example
with coal quality. Mathematical Geology, 15(2): 287–300. https://doi.org/10.1007/BF01036071.
Davis, J.C. 1986. Statistics and Data Analysis in Geology, 2nd Edition, John Wiley & Sons, New York, 646 p.
Desbarats, A.J. & Dimitrakopoulos, R. 2000. Geostatistical simulation of regionalized pore-size distributions using min/max autocorrelation factors. Mathematical Geology 32(8): 919–942.
Deutsch, C.V. & Journel, A.G. 1998. GSLIB: Geostatistical Software Library and User’s Guide, Oxford
University Press, New York, 2nd edition.
Egozcue, J.J., Pawlowsky-Glahn, V., Mateu-Figueras, G. Barcelo-Vidal C (2003) Isometric Logratio
Transformations for Compositional Data Analysis, Mathematical Geology 35 (3): 279–300 https://
doi.org/10.1023/A:1023818214614.
Goovaerts, P. 1997. Geostatistics for natural resources evaluation. New York, N.Y.: Oxford University
Press.
Hotelling, H. 1933. Analysis of a complex of statistical variables into principal components. Journal of
Educational Psychology, 24: 417–441, 498–520.
Howarth, R.J. 2017. Dictionary of Mathematical Geosciences – With Historical Notes, Springer, 893
pages. DOI 10.1007/978-3-319-57315-1.
5 CONCLUSIONS
Principal component analysis is an orthogonal transformation that can be used to convert
a set of correlated variables into a set of almost linearly uncorrelated components. In this
paper, we show a detailed methodology to apply principal component analysis to a set of geochemical variables from an exploration campaign of a Nickel laterite deposit. The method
requires dealing with the compositional nature of the data, thus requiring a transformation
of the grades into log-ratios. These log-ratios are then decorrelated using PCA. The decorrelation is checked by computing cross variograms between the principal components, which
confirms that almost all linear correlation is removed by the transformation into principal
components. These principal components are then independently simulated using sequential
Gaussian simulation, which in turn requires a normal score transformation of the data.
The methodology is therefore presented as a sequence of three transformations: a logratio transformation using the additive approach, a decorrelation using principal component analysis, and a normal score transformation to use Gaussian simulation. The simulated
results must be back-transformed to bring them back from Gaussian simulated deviates, into
simulated principal components, then into log-ratios and finally, into simulated grades.
Results are checked to ensure the correlation statistics are preserved, which is confirmed
by the scatter plots of the simulated variables, where correlation coefficients are well preserved, and the general correlation structure is reproduced. The method cannot capture some
non-linear features of the relationships, which is expected due to its linear nature. Overall,
results are satisfactory, confirming that PCA is a suitable approach to model spatially correlated variables.
ACKNOWLEDGEMENTS
We acknowledge the support of the Natural Sciences and Engineering Research Council of
Canada (NSERC), funding reference number RGPIN-2017-04200 and RGPAS-2017-507956.
REFERENCES
Aitchison, J. 1982. The statistical analysis of compositional data (with discussion). Journal of the Royal
Statistical Society, Series B (Statistical Methodology) 44 (2): 139–177.
Aitchison, J. 1986. The statistical analysis of compositional data. Monographs on Statistics and Applied
Probability, London, Chapman & Hall Ltd., 416 p.
Almeida, A.S. & Journel, A.G. 1994. Joint simulation of multiple variables with a Markov-type coregionalization model. Mathematical geology 26(5): 565–588.
Chiles, J.-P. & Delfiner, P. 2012. Geostatistics – Modeling Spatial Uncertainty, Second Edition.
Willey, 699 p.
Davis, B.M. & Greenes, K.A. 1983. Estimating using spatially distributed multivariate data: An example
with coal quality. Mathematical Geology, 15(2): 287–300. https://doi.org/10.1007/BF01036071.
Davis, J.C. 1986. Statistics and Data Analysis in Geology, 2nd Edition, John Wiley & Sons, New York, 646 p.
Desbarats, A.J. & Dimitrakopoulos, R. 2000. Geostatistical simulation of regionalized pore-size distributions using min/max autocorrelation factors. Mathematical Geology 32(8): 919–942.
Deutsch, C.V. & Journel, A.G. 1998. GSLIB: Geostatistical Software Library and User’s Guide, Oxford
University Press, New York, 2nd edition.
Egozcue, J.J., Pawlowsky-Glahn, V., Mateu-Figueras, G. Barcelo-Vidal C (2003) Isometric Logratio
Transformations for Compositional Data Analysis, Mathematical Geology 35 (3): 279–300 https://
doi.org/10.1023/A:1023818214614.
Goovaerts, P. 1997. Geostatistics for natural resources evaluation. New York, N.Y.: Oxford University
Press.
Hotelling, H. 1933. Analysis of a complex of statistical variables into principal components. Journal of
Educational Psychology, 24: 417–441, 498–520.
Howarth, R.J. 2017. Dictionary of Mathematical Geosciences – With Historical Notes, Springer, 893
pages. DOI 10.1007/978-3-319-57315-1.