107
we present a case study in an iron ore deposit illustrating the practical application for the
work flow proposed.
2 PROJECTION PURSUIT MULTIVARIATE TRANSFORM
In mining projects very often it is required to characterize multiple correlated attributes. The
relationship between variables must be accounted for so that the final model is representative
of the geological phenomenon. The multivariate relationships imply the use of co-simulation
(Verly, 1983) techniques, which in turn demand the simultaneous modeling of the direct and
cross covariance functions for all variables considered. Also, traditional co-simulation methods are performed under the Gaussian formalism. Assuming that the attributes are multivariate Gaussian allows all relationships to be fully parameterized by the covariance matrix. Even
though, the traditional approaches allow obtaining realistic models, the covariance modeling is extremely tedious with more than two variables, and geological attributes rarely are
multivariate Gaussian in nature. So, prior to the co-simulation (Verly, 1983) process, nscore
(Journel and Huijbregts, 1976) transformation is often applied on the variables, so that the
Gaussian formalism may be used. However, the nscore transform only ensures that the variables are univariate Gaussian, even though this is a necessary condition, it is not sufficient.
Generally, the bivariate Gaussianity is verified and if valid, the multivariate Gaussianity is
assumed. This assumption, on the other hand, in some cases may not be met. An alternative
is to use Projection pursuit multivariate transform (Barnett et al., 2014). (PPMT), which
transforms any number of variables to be multivariate Gaussian and uncorrelated, enabling
the independent modeling of each attribute considered while reproducing the multivariate complexities on the final model (Manchuk et al., 2017). The PPMT approach involves
3 stages: a pre-processing stage where the properties of the data matrix are made sure to be
suitable for the projection pursuit algorithm; then the projection pursuit is performed on the
data matrix while Gaussianizing, as the process is called in the literature, the data; finally the
data may be back-transformed once the geostatistical model has been built (Barnett et al.,
2014). Each phase will be briefly explained, for further details on PPMT methodology refer
to (Barnett et al., 2014) and (Manchuk et al., 2017).
2.1 Pre-processing
Considering a random vector U of the N multiple attributes, a normal score transformation
is applied to remove all marginal complexities from the data (Barnett et al., 2014). Then, the
normal scored data Y are made orthogonal, ergo, uncorrelated through a sphering process,
according to equation 1:
X
Y
−
S
1
2 ,
(1)
where S
−
1
2 is the sphering matrix obtained through equation 2:
VD V
−
1
2 T
(2)
V and D are the eigenvector and eigenvalue matrices, respectively.
2.2 Projection pursuit
According to (Manchuk et al., 2017), considering a vector θ and the projection p of the
sphered data X upon θ, if X is multivariate Gaussian then the projection p shouldalso be
Gaussian. So, for this phase of the PPMT algorithm is defined a projection index I(θ) that
measures the non-Gaussianity of the projection (Manchuk et al., 2017). That is, if I(θ) = 0
then p is Gaussian. The aim of this step in the PPMT algorithm is to perform an optimized
we present a case study in an iron ore deposit illustrating the practical application for the
work flow proposed.
2 PROJECTION PURSUIT MULTIVARIATE TRANSFORM
In mining projects very often it is required to characterize multiple correlated attributes. The
relationship between variables must be accounted for so that the final model is representative
of the geological phenomenon. The multivariate relationships imply the use of co-simulation
(Verly, 1983) techniques, which in turn demand the simultaneous modeling of the direct and
cross covariance functions for all variables considered. Also, traditional co-simulation methods are performed under the Gaussian formalism. Assuming that the attributes are multivariate Gaussian allows all relationships to be fully parameterized by the covariance matrix. Even
though, the traditional approaches allow obtaining realistic models, the covariance modeling is extremely tedious with more than two variables, and geological attributes rarely are
multivariate Gaussian in nature. So, prior to the co-simulation (Verly, 1983) process, nscore
(Journel and Huijbregts, 1976) transformation is often applied on the variables, so that the
Gaussian formalism may be used. However, the nscore transform only ensures that the variables are univariate Gaussian, even though this is a necessary condition, it is not sufficient.
Generally, the bivariate Gaussianity is verified and if valid, the multivariate Gaussianity is
assumed. This assumption, on the other hand, in some cases may not be met. An alternative
is to use Projection pursuit multivariate transform (Barnett et al., 2014). (PPMT), which
transforms any number of variables to be multivariate Gaussian and uncorrelated, enabling
the independent modeling of each attribute considered while reproducing the multivariate complexities on the final model (Manchuk et al., 2017). The PPMT approach involves
3 stages: a pre-processing stage where the properties of the data matrix are made sure to be
suitable for the projection pursuit algorithm; then the projection pursuit is performed on the
data matrix while Gaussianizing, as the process is called in the literature, the data; finally the
data may be back-transformed once the geostatistical model has been built (Barnett et al.,
2014). Each phase will be briefly explained, for further details on PPMT methodology refer
to (Barnett et al., 2014) and (Manchuk et al., 2017).
2.1 Pre-processing
Considering a random vector U of the N multiple attributes, a normal score transformation
is applied to remove all marginal complexities from the data (Barnett et al., 2014). Then, the
normal scored data Y are made orthogonal, ergo, uncorrelated through a sphering process,
according to equation 1:
X
Y
−
S
1
2 ,
(1)
where S
−
1
2 is the sphering matrix obtained through equation 2:
VD V
−
1
2 T
(2)
V and D are the eigenvector and eigenvalue matrices, respectively.
2.2 Projection pursuit
According to (Manchuk et al., 2017), considering a vector θ and the projection p of the
sphered data X upon θ, if X is multivariate Gaussian then the projection p shouldalso be
Gaussian. So, for this phase of the PPMT algorithm is defined a projection index I(θ) that
measures the non-Gaussianity of the projection (Manchuk et al., 2017). That is, if I(θ) = 0
then p is Gaussian. The aim of this step in the PPMT algorithm is to perform an optimized