A multivariate approach for process variorums

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

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INTRODUCTIONForeword• 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 ?

• 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,

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INTRODUCTIONTheory of sampling

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INTRODUCTIONThe variographic approachTOS 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.

Relative heterogeneity:

(Semi-)Variogram:

Constitutional heterogeneity:

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.

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INTRODUCTIONThe variographic approach

Random effects (sampling, preparation, analysis)

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MULTIVARIATE VARIOGRAPHYPrevious 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.

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MULTIVARIATE VARIOGRAPHYApplication of multivariogram to process sampling• We now assume that the heterogeneity contribution is a multivariate measure:

Multivariate heterogeneity:

• Bourgault and Marcotte (1991) were the first to formalise the principle of a multivariate variogram which is define as:

Multivariogram:

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:

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MULTIVARIATE VARIOGRAPHYSampling and analysesThe 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

Analytical results

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Multivariate approach

CONTENTSComparing univariate and multivariate approaches

Univariate approach

PCA

PCA

Vario

gram

Multivariogram

Classical variograms

Multivariogram on raw data

Vario

gram

Variograms of PCA Multivariogram of PCA

Variogram Multivariogram

Analytical results

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Univariate approach

PCA

PCA

Vario

gram

Multivariogram



Vario

gram



CLASSIC APPROACHExperimental 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.

11Experimental variograms Vj of the 8

variables of interest. Experimental variograms Vj of 6 of the variables of

interest without LREE

CLASSIC APPROACHExperimental 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.

12 Std. deviation of sampling error for LREE grade

Analytical results

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Univariate approach

PCA

PCA

Vario

gram

Multivariogram



Vario

gram



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,

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MULTIVARIATE APPROACHVariograms 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 ?

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• PC3 is mostly loaded by LREE, as well as pulp density and RRslope.

How many PCs should be kept for variographic analysis ?

𝐹 𝑖=𝑠2𝑖

(𝑉 ¿¿1)𝑖=(𝐶𝐻𝐿)𝑖

(𝑉 ¿¿1)𝑖¿¿

• Explained variability• F-test (randomness)

PC #

Explained variability

(%)

Cumulated explained

variability (%)

Eigenvalues SpectrumF-test*

1 50.13 50.13

2.632 18.82 68.94 2.813 11.47 80.41 1.354 7.94 88.35 1.695 6.61 94.96 0.896 4.08 99.04 1.537 0.92 99.96 1.248 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,

• 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).

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Experimental variograms Vj , auxiliary functions wj and w’j of the 4 first PCs.

Analytical results

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Univariate approach

PCA

PCA

Vario

gram

Multivariogram



Vario

gram



• 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).

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MULTIVARIATE APPROACHMultivariogram

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.

• 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,

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MULTIVARIATE APPROACHMultivariogram

Global relative standard deviation of the sampling error

Need to reduce global sampling variance.

Analytical results

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Univariate approach

PCA

PCA

Vario

gram

Multivariogram



Vario

gram



• 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.

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MULTIVARIATE APPROACHMultivariogram 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.

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.

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CONCLUSIONS

References1. 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.

AcknowledgmentsThe 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.

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REFERENCES & AKNOWLEDGMENTS

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ANNEXComparing approaches

Univariate variogram Variograms of PCA MultivariogramNumber of variograms

p ≤ min(n-1, p) 1Sampling 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

Thank you for your attention

Quentin DEHAINEPhD Student at Université de Lorraine

Email: [email protected]

Science

A multivariate approach for process variorums