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Q. DEHAINE*, L. O. FILIPPOV*PhD student at Universit de Lorraine, A MULTIVARIATE APPROACH FOR PROCESS VARIOGRAMS

WCSB7 Bordeaux, 12th June 2015Work performed under FP7, STOICISM project

1INTRODUCTIONForewordIn 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 ?

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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,2INTRODUCTIONTheory of sampling

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3INTRODUCTIONThe variographic approachRelative heterogeneity:(Semi-)Variogram:Constitutional heterogeneity:

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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.

4INTRODUCTIONThe variographic approach

Random effects (sampling, preparation, analysis)

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5MULTIVARIATE 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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6MULTIVARIATE VARIOGRAPHYApplication of multivariogram to process sampling

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7MULTIVARIATE 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 densityParticle size analysis: D10, D50, D90 and Rosin-Rammler slope (RRslope)

8 important properties

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Analytical results8Multivariate approach

CONTENTSComparing univariate and multivariate approaches

Univariate approach

PCAPCAVariogramMultivariogramClassical variogramsMultivariogram on raw data

VariogramVariograms of PCAMultivariogram of PCA

VariogramMultivariogram

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Analytical results9Multivariate approach

CONTENTSComparing univariate and multivariate approaches

Univariate approach

PCAPCAVariogramMultivariogramClassical variogramsMultivariogram on raw data

VariogramVariograms of PCAMultivariogram of PCA

VariogramMultivariogram

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Classic approachExperimental individual variogramsThe 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.

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Experimental variograms Vj of the 8 variables of interest. Experimental variograms Vj of 6 of the variables of interest without LREE

10From these individual heterogeneity contributions, the individual variograms are calculated as well as the auxiliary functions noted wj and wj and the nugget effect V0 is estimated by backward extrapolation (Figure 3A). The individual variograms allow distinguishing two main groups, a high-sill variables group (LREE, D90 and Sn) and a low-sill variables group (Pulp density, D10, D50, RRSlope and Nb). The overall range is difficult to estimate using directly the variograms, but the auxiliary functions suggest a range varying between 5-7 lags. The variograms of pulp density and Nb display a classic increasing variogram shape. A minimum can be observed in the variogram of LREE and Sn at j=15 and j=23 respectively, indicating the existence of a possible cyclic fluctuations with a too long period (i.e. j=15=30 min and j=23=46 min respectively) to see another minimum in the variogram plot. Similar observations can be made for the variograms of properties reflecting the size distribution (D10, D50, D90, and RRslope). Indeed, for all these properties, a local minimum is observed at j=7-9 and a tentative repetition at j=20. This suggests correlations between these properties and the existence of some periodic phenomenon affecting the size distribution with a rather short period of approximately 9 lags (i.e. j=9=18 min).

Classic approachExperimental individual variogramsThe 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 doesnt account for the multivariate nature of heterogeneity, which can lead to an underestimation of global sampling variance.

11Std. deviation of sampling error for LREE grade

11The classical conclusion at this point will be to focus the sampling protocol on the property with the highest sill, i.e. LREE grade. Figure 6 show the standard deviation of the sampling error for the LREE grade according to the sampling scheme. It can be seen that the 3 sampling schemes are quite close but the systematic sampling stay the sampling scheme with the lowest variance.Even with this simplification, the recommended sampling protocol is hard to define because of the different periodic phenomena. It could be recommended to use a stratified random sampling or systematic sampling with at least 5 or 10 increments with a sampling frequency higher than two per period of 18 min and 30 min (Petersen and Esbensen, 2005). Since the average shift duration is around 3h this would imply to sample not 5 or 10 but at least 20 increments.

Analytical results12Multivariate approach

CONTENTSComparing univariate and multivariate approaches

Univariate approach

PCAPCAVariogramMultivariogramClassical variogramsMultivariogram on raw data

VariogramVariograms of PCAMultivariogram of PCA

VariogramMultivariogram

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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 D10,13MULTIVARIATE 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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14MULTIVARIATE APPROACHVariograms of PCA

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

How many PCs should be kept for variographic analysis ?

Explained variabilityF-test (randomness)PC #Explained variability (%)Cumulated explained variability (%)Eigenvalues SpectrumF-test*150.1350.132.63218.8268.942.81311.4780.411.3547.9488.351.6956.6194.960.8964.0899.041.5370.9299.961.2480.05100.001.42

*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,

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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).

15MULTIVARIATE APPROACHVariograms of PCA

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

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Analytical results16Multivariate approach

CONTENTSComparing univariate and multivariate approaches

Univariate approach

PCAPCAVariogramMultivariogramClassical variogramsMultivariogram on raw data

VariogramVariograms of PCAMultivariogram of PCA

VariogramMultivariogram

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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).

17MULTIVARIATE 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