Selected Topics for Further Research - Geomathematics: Theoretical Foundations, Applications and Future Developments - page 470

Obviously, application of ordinary statistical methods to data that are subject to
spurious correlations can lead to erroneous results. Methods of multivariate statis-
tical analysis often commence by calculating the variance-covariance matrix or the
correlation matrix if variances are replaced by ones and covariances by correlation
coefficients. The next step of multivariate analysis often consists of calculating the
principal components of the variance-covariance or correlation matrix (see, e.g.,
Agterberg 1974 ; or Davis 2002 ). The total variance (sum of all component vari-
ances) is decomposed into a number of principal components, which are linear
combinations of all variables, accounting for increasingly smaller proportions of
total variance. Factor analysis can result in improved results by considering random
variability of the variables (J¨reskog et al. 1976 ). The higher variance linear
combinations or component scores can be used in various ways; e.g., on maps
where they may provide more information than individual variables. A useful
multivariate technique along these lines is correspondence analysis (Greenacre
2009 ) which results in bi-plots for pairs of components that are shown as vectors
together with their scores on the original variables for easy comparison.
Aitchison ( 1986 ) has developed the powerful new methodology of composi-
tional data analysis for variables in closed-number systems. His methods avoid
misleading results based on spurious correlations. Multivariate analysis is applied
to log-ratios instead of to original data. Standard statistical techniques normally
apply to the log-ratios. If there are p variables in the closed number-system, the
log-ratios approach can be considered to be operational in ( p -1)-dimensional space
allowing unconstrained multivariate analysis. Theory of compositional data analy-
sis is clearly explained in Pawlowsky-Glahn and Buccianti ( 2011 ). This topic also
contains many recent examples of application. The approach also can be less
sensitive to outliers, particularly if robust estimators are used (Filzmoser and
Hron 2011 ). The topic of using geostatistics with compositional data is discussed
by Pawlowsky-Glahn and Olea ( 2004 ). Reyment et al. ( 2008 ) applied composi-
tional data analysis to study seasonal variation in radiolarian abundance.
There are three options for log-ratio transformation: (1) additive log-ratio ( alr ),
(2) centred log-ratio ( clr ), and (3) isometric log-ratio ( ilr ). The first two techniques
( alr and clr ) were developed by Aitchison ( 1986 ) and the third one ( ilr ) by Egozcue
et al. ( 2003 ). The alr method works with log-ratios in which the ratios are for
original variables divided by one of the variables. This might be problematic
because distances between points in the transformed space differ from divisor to
divisor. The clr method is not subject to this particular drawback because there is a
single divisor set equal to the geometric mean of all variables. However, some
standard multivariate techniques are difficult to apply under clr . The ilr method was
developed to overcome these problems but it also can be difficult to use for some
multivariate statistical techniques. On the whole, il r currently is the preferred
method because it often leads to a better understanding of the situation that is
being studied. The following example (after Aitchison 1986 ) is an application of the
clr method. The next section will contain an ilr example.
The ternary diagram for three rock components that form a closed system is a
commonly used tool in mathematical petrology and geochemistry. Aitchison's Data