Paper presented at the 7th World Conference on Sampling and Blending (WCSB7) in June 2015.

1. Q. DEHAINE*, L. O. FILIPPOV
*PhD student at Université de Lorraine,
A MULTIVARIATE APPROACH FOR PROCESS
VARIOGRAMS
WCSB7 – Bordeaux, 12th June 2015
Work performed under FP7, STOICISM project

2. 1
INTRODUCTION
Foreword
• In every mining project, economic improvements pass through metallurgical
assessment based on series of tests aiming to improve the process,
• The effectiveness of these improvements will depend on the representativeness
of the samples initially collected for the tests (Abzalov, 2013),
• Process samples are typically collected to obtain representative samples
regarding one property, i.e. grade, mineralogy or physical characteristics,
• But response to one test usually depends not only on one property but on a
certain range of p properties of interest.
How to assess the global representativeness of the samples for all the
properties of interest ?

3. • Process streams can be seen as elongated objects:1D model,
• The preferred method for sampling 1D lots is the increment sampling,
• The choice of the sampling mode is very important as it changes the variance
of lots mean,
2
INTRODUCTION
Theory of sampling

4. 3
INTRODUCTION
The variographic approach
TOS introduced the variogram as a tool which provide critical information on (Gy,
2004; Petersen and Esbensen, 2005):
 the process variability over time,
 the lot mean and the uncertainty of a single measurement,
 the optimal design and scheme for the sampling protocol.
ℎ𝑖 =
𝑎𝑖 − 𝑎 𝐿
𝑎 𝐿
𝑀𝑖
𝑀𝑖
, 𝑖 = 1, ⋯ , 𝑁
𝑣𝑗 =
1
2(𝑁 − 𝑗)
𝑖
𝑁−𝑗
ℎ𝑖 − ℎ𝑖+𝑗
2
, 𝑗 = 1, ⋯ ,
𝑁
2
𝐶𝐻𝐿 = 𝑠2
ℎ𝑖 =
1
𝑁
𝑖
𝑁
ℎ𝑖
2
Relative heterogeneity:
(Semi-)Variogram:
Constitutional heterogeneity:

5. TOS introduced the variogram as a tool which provide critical information on (Gy,
2004; Petersen and Esbensen, 2005):
 the process variability over time,
 the lot mean and the uncertainty of a single measurement,
 the optimal design and scheme for the sampling protocol.
4
INTRODUCTION
The variographic approach
Random effects (sampling, preparation, analysis)

6. 5
MULTIVARIATE VARIOGRAPHY
Previous work
• The process samples need to be representative not only for one property but for a certain
range of properties.
• A solution was proposed in spatial-data analysis by Oliver and Webster (1989)4 who
suggested to study the variogram of Principal Component Analysis (PCA) scores of the
first few principal components.
• Minkkinen and Esbensen (2014) have illustrated the advantages of ‘variograms of PCA’
as a way to perform a combined multivariate chemometric-variogram,
• It is also worth mentioning a reverse approach, introduced by Kardanpour et al. (2014),
which consists in applying a PCA analysis on the individual variograms.

7. 6
MULTIVARIATE VARIOGRAPHY
Application of multivariogram to process sampling
• We now assume that the heterogeneity contribution is a multivariate measure:
Multivariate heterogeneity: 𝐻𝑖 = ℎ1, ⋯ , ℎ 𝑘 , ⋯ , ℎ 𝑝 𝑖
𝑡
, 𝑖 = 1, ⋯ , 𝑁
• Bourgault and Marcotte (1991) were the first to formalise the principle of a multivariate
variogram which is define as:
Multivariogram: 𝑉𝑗 =
1
2(𝑁−𝑗) 𝑖
𝑁−𝑗
𝐻𝑖 − 𝐻𝑖+𝑗 𝑀 𝐻𝑖 − 𝐻𝑖+𝑗
𝑡
, 𝑗 = 1, ⋯ , 𝑁/2
where M is a positive definite p x p matrix which defines the metric for calculating the "distances"
between the units.
Examples: - Euclidean metric: M=Ipxp or
- Mahalanobis metric: M=[Cov(H)]-1
Multivariate constitutional heterogeneity: 𝐶𝐻𝐿 = 𝑠2
𝐻𝑖 =
1
𝑁 𝑖
𝑁
𝐻𝑖 𝑀𝐻𝑖
𝑡

8. 7
MULTIVARIATE VARIOGRAPHY
Sampling and analyses
The studied material is a residue stream from a kaolin mining plant (St Austell, UK), considered
as a potential source of metals (LREE, Sn, Nb) as by-products (Dehaine and Filippov, 2015).
This stream must be sampled feasibility studies of pre-concentration by gravity concentration,
• Sampling: 50 increments extracted over 2 h (1 shift), with a 2 min frequency, for variographic
study.
• Analyses:
 Chemical analysis: Sn, Nb and LREE (La, Ce, Nd),
 Moisture content: Pulp density
 Particle size analysis: D10, D50, D90 and Rosin-Rammler slope (RRslope)
8 important properties

9. Analytical
results
8
Multivariate
approach
CONTENTS
Comparing univariate and multivariate approaches
Univariate
approach
PCA
Multivariogram
Classical variograms
Multivariogram on
raw data
Variograms of PCA Multivariogram of
PCA

10. Analytical
results
9
Multivariate
approach
CONTENTS
Comparing univariate and multivariate approaches
Univariate
approach
PCA
Multivariogram
Classical variograms
Multivariogram on
raw data
Variograms of PCA Multivariogram of
PCA

11. CLASSIC APPROACH
Experimental individual variograms
• The individual variograms allow distinguishing two main groups:
– A high-sill variables group including LREE (+D90 and Sn),
– a low-sill variables group (Pulp density, D10, D50, RRSlope and Nb).
• At first, all variograms look flat except LREE which display a local minimum,
• But the other high-sill variograms also display local minimums, indicating possible periodic
phenomenon.
10
Experimental variograms Vj of the 8
variables of interest.
Experimental variograms Vj of 6 of the variables of
interest without LREE

12. CLASSIC APPROACH
Experimental individual variograms
• The classical conclusion will be to focus on the highest sill variogram, i.e. LREE grade,
• But the sampling protocol will not consider the different periodic phenomena,
This approach doesn’t account for the multivariate nature of heterogeneity,
which can lead to an underestimation of global sampling variance.
11 Std. deviation of sampling error for LREE grade

13. Analytical
results
12
Multivariate
approach
CONTENTS
Comparing univariate and multivariate approaches
Univariate
approach
PCA
Multivariogram
Classical variograms
Multivariogram on
raw data
Variograms of PCA Multivariogram of
PCA

14. PCA is a variable reduction procedure which simplify the data in a smaller number of more
relevant components.
• PC1 is mostly loaded by size distribution properties (D50, D90 and RRslope),
• PC2 is mostly loaded by pulp density, LREE, and to a lesser extent by D10,
13
MULTIVARIATE APPROACH
Variograms of PCA
PCA 3D biplot
• PC3 is mostly loaded by LREE, as
well as pulp density and RRslope.
How many PCs should be
kept for variographic
analysis ?

15. 14
MULTIVARIATE APPROACH
Variograms of PCA
• PC3 is mostly loaded by LREE, as
well as pulp density and RRslope.
How many PCs should be
kept for variographic
analysis ?
𝐹𝑖 =
𝑠2
𝑖
(𝑉1)𝑖
=
(𝐶𝐻𝐿)𝑖
(𝑉1)𝑖
• Explained variability
• F-test (randomness)
PC #
Explained
variability
(%)
Cumulated
explained
variability (%)
Eigenvalues Spectrum
F-test*
1 50.13 50.13 2.63
2 18.82 68.94 2.81
3 11.47 80.41 1.35
4 7.94 88.35 1.69
5 6.61 94.96 0.89
6 4.08 99.04 1.53
7 0.92 99.96 1.24
8 0.05 100.00 1.42
*Critical values for the F-test are: P(F=0.90)=1.22, P(F=0.95)=1.30 and
P(F=0.99)=1.47 (Minkkinen and Esbensen, 2014).
PCA is a variable reduction procedure which simplify the data in a smaller number of more
relevant components.
• PC1 is mostly loaded by size distribution properties (D50, D90 and RRslope),
• PC2 is mostly loaded by pulp density, LREE, and to a lesser extent by D10,

16. • V(PC1) reflect particle size variability within time, reaching a sill for the last lags,
reflecting a long range variation.
• V(PC2) display a short range with a sill reached quickly, in-line with the short range
variation displayed by the variograms of pulp density and LREE both loading PC2.
• V(PC3) is related to a cyclic variation with a short period (approx. 5 lags).
15
MULTIVARIATE APPROACH
Variograms of PCA
Experimental variograms Vj , auxiliary functions wj and w’j of the 4 first PCs.

17. Analytical
results
16
Multivariate
approach
CONTENTS
Comparing univariate and multivariate approaches
Univariate
approach
PCA
Multivariogram
Classical variograms
Multivariogram on
raw data
Variograms of PCA Multivariogram of
PCA

18. • Size distribution and metal grade multivariograms both display high sills,
• 3 distinct ranges: long (size distribution), medium (metal grade) and short (pulp density),
• Each class-multivariogram is best modelled by a bounded linear model (blinear).
17
MULTIVARIATE APPROACH
Multivariogram
Multivariograms for size distribution properties (D10, D50, D90 and RRslope),
metal grades (Nb, Sn and LREE) and pulp density and the global multivariogram.
• Global multivariogram is
best fitted by pentaspherical
model,
Consequence of small, to
long range displayed by the
class-multivariograms,
Size Distribution has the
highest sill with a possible
periodic phenomenon.

19. • The global multivariogram displays a relatively high sill, as a consequence of the metric
used for computation,
• Hence, the sampling variance is much more important comparing to the univariate case,
• The std. deviation stay very high even for high number of increments collected to make the
final sample,
18
MULTIVARIATE APPROACH
Multivariogram
Global relative standard deviation of the sampling error
Need to reduce global sampling
variance.

20. Analytical
results
19
Multivariate
approach
CONTENTS
Comparing univariate and multivariate approaches
Univariate
approach
PCA
Multivariogram
Classical variograms
Multivariogram on
raw data
Variograms of PCA Multivariogram of
PCA

21. • The std. deviation of sampling error increase
with the number of PCs retained,
• Even with the first fours PCs retained, the
associated std. deviation, is lower than for the
global multivariogram of the raw data.
• This is due to the fact that only the significant
PCs were retained to compute this last
multivariogram,
• PCA act as a noise-filter on the raw data to
keep only relevant information on global
variability of the dataset.
20
MULTIVARIATE APPROACH
Multivariogram of PCA
Influence of the number of principal components on the
multivariogram shape and std. deviation of sampling error.
Global shape of the variogram is already
reached with the two first PCs.

22.  Univariate variogram can lead to a misinterpretation and underestimation of
global sampling error if interpretation is based on the highest variance property only,
 Variograms of PCA scores highlight distinct spatial patterns thorough variable
grouping in a reduced number of variograms,
 Multivariogram summarise time variation of multiple properties and highlights the
multivariate time auto-correlation aspects of these variables which however results
in a high sampling variance,
 An alternative approach is proposed, multivariogram of PCA scores, which filter
noise from the data, and keep only the relevant data information,
 Most of the variability of the St Austell residue stream is due to LREE according to the
univariate approach, whereas multivariate approach point size distribution as the most
important contribution.
21
CONCLUSIONS

23. References
1. Abzalov, M., 2013. Representativiness of bulk samples, in: Beniscelli, J., Felipe Costa, J., Domínguez, O., Duggan, S., Esbensen,
K., Lyman, G., Sanfurgo, B. (Eds.), Proceedings of the 6th World Conference on Sampling and Blending. Lima, pp. 257–264.
2. Gy, P., 2004. Sampling of discrete materials III. Quantitative approach—sampling of one-dimensional objects. Chemometrics and
Intelligent Laboratory Systems 74, 39–47.
3. Petersen, L., Esbensen, K.H., 2005. Representative process sampling for reliable data analysis—a tutorial. Journal of
Chemometrics 19, 625–647.
4. Oliver, M.A., Webster, R., 1989. A geostatistical basis for spatial weighting in multivariate classification. Mathematical Geology 21,
15–35.
5. Minkkinen, P., Esbensen, K.H., 2014. Multivariate variographic versus bilinear data modeling. Journal of Chemometrics 28, 395–
410.
6. Kardanpour, Z., Jacobsen, O.S., Esbensen, K.H., 2014. Soil heterogeneity characterization using PCA (Xvariogram) - Multivariate
analysis of spatial signatures for optimal sampling purposes. Chemometrics and Intelligent Laboratory Systems 136, 24–35.
7. Bourgault, G., Marcotte, D., 1991. Multivariable variogram and its application to the linear model of coregionalization.
Mathematical Geology 23, 899–928.
8. Dehaine, Q., Filippov, L.O., 2015. Rare earth (La, Ce, Nd) and rare metals (Sn, Nb, W) as by-product of kaolin production,
Cornwall: Part1: Selection and characterisation of the valuable stream. Minerals Engineering 76, 141–153.
Acknowledgments
The authors wish to thank Imerys Ltd., UK, and especially S. Moradi, P. Chauhan and A. Coe for their help during the sampling
exercise. We are also grateful to S. Lightfoot and P. Budge for their advices and technical support. We thank C. Gauthier for its help in
the sample preparation process. This work has been financially supported by the European FP7 project “Sustainable Technologies for
Calcined Industrial Minerals in Europe” (STOICISM), grant NMP2-LA-2012-310645.
22
REFERENCES & AKNOWLEDGMENTS

24. 23
ANNEX
Comparing approaches
Univariate variogram Variograms of PCA Multivariogram
Number of variograms
p ≤ min(n-1, p) 1
Sampling error(s) estimation
 p sampling errors
 Good estimation of the singular sampling
errors
± As many sampling errors as significant PC
selected
 One sampling error only
 Estimation of the overall sampling error
sampling error depends on the metric used
for calculation
Representativeness
 Representative of the singular variability
contribution
 Underestimation of the global variability
 Representative of (part of) the overall
multivariate variability
– Depends on the cumulative variability explained
by the selected principal components
 Representative of the overall multivariate
variability
Sampling protocol design
– Choosing the variable with highest sampling
error lead to underestimation of the real
sampling variance
+ Allow to design the optimum sampling protocol
regarding all the variables
 The number of increments required can be very
large
 Allow to design the optimum sampling
protocol regarding all the variables
 The number of increments required can be
very large
Objectives
To estimate the singular sampling errors
To perform variable grouping (PCA) and
explain distinct spatial patterns
To summarize the overall variability and
representativeness of a sample

25. Thank you for your attention
Quentin DEHAINE
PhD Student at Université de Lorraine
Email: quentin.dehaine@gmail.com