Q. Dehaine1, L.O. Filippov2 and H.J. Glass3

1. Postdoctoral Research Associate, University of Exeter, Camborne School of Mines

2. Professor, Université de Lorraine, GeoRessources laboratory,

3. Rio Tinto Professor of Mining and Minerals Engineering, University of Exeter, Camborne School of Mines

Optimising multivariate variographic analysis with information

from multivariate process data modelling (PLS-R)

• Predicting process performance variability over time is crucial for many industrial processes and

particularly for the mining industry,

• Mineral processing operations are susceptible to process variations, operating parameters

changes or ore variability, which could generate significant losses of performance for the whole

process.

• Process performance depends not only on one but on a certain range of p properties/variables.

1

How Theory Of Sampling (TOS) could help to reduce risks linked to process

performance variability?

INTRODUCTION Forewords

TOS introduced the variogram as a tool which provide critical information on (Gy, 2004; Petersen

and Esbensen, 2005)1,2:

the process variability over time,

the lot mean and the uncertainty of a single measurement,

the optimal design and scheme for the sampling protocol.

INTRODUCTION The variographic approach

Random effects (sampling, preparation, analysis)

2

3

INTRODUCTION Forewords

• Many practical situations within science and industry deals with multivariate data,

• However, variographic analysis, as described in TOS, has often be limited to univariate

applications, looking at one property/variable at a time,

• It is often believed that “All one needs to consider for appropriate sampling is the single

property/variable with the most heterogeneous distribution”,

• This may be true in many cases but there are situations in which multivariate approaches are of

significant value,

• While this situation is well known in geostatistics, only a limited number of studies have combined

TOS and multivariate data analysis (Minkkinen and Esbensen, 2014; Kardanpour et al., 2014) 3,4.

4

INTRODUCTION Multivariate variography

Bourgault and Marcotte (1991)5 were the first to formalise the principle of a multivariate variogram for spatial

data analysis,

But it has only recently been applied to TOS (Dehaine & Filippov, 2015)6.

Univariate Multivariate

Relative heterogeneity ℎ𝑖 =𝑎𝑖 − 𝑎𝐿𝑎𝐿

𝑀𝑖

𝑀 𝑖 𝐻𝑖 = ℎ1, ⋯ , ℎ𝑘 , ⋯ , ℎ𝑝 𝑖

𝑡

(Semi-) Variogram 𝑣𝑗 =1

2(𝑁 − 𝑗) ℎ𝑖 − ℎ𝑖+𝑗

2𝑁−𝑗

𝑖

𝑉𝑗 =1

2(𝑁 − 𝑗) 𝐻𝑖 − 𝐻𝑖+𝑗 𝑀 𝐻𝑖 −𝐻𝑖+𝑗

𝑡𝑁−𝑗

𝑖

Constitutional heterogeneity 𝑐ℎ𝐿 = 𝑠2 ℎ𝑖 =

1

𝑁 ℎ𝑖

2

𝑁

𝑖

𝐶𝐻𝐿 = 𝑠2 𝐻𝑖 =

1

𝑁 𝐻𝑖 𝑀𝐻𝑖

𝑡

𝑁

𝑖

Mahalanobis metric: M=[Cov(H)]-1

5

INTRODUCTION Previous work After Dehaine et al. (2016)7

0.5

0.14

How to weight the contribution of each variable according to its importance for the process?

How to characterize, and if possible predict, process performance variability over time?

Some variables contributing extensively to the global (multivariate) sampling variance could be less important

for the process compared to the other variables,

6

COMBINING TOS AND MULTIVARIATE DATA MODELLING Theory

Lets consider the case where the q process performance Y-responses indexes could be linked to

the p process X-variables/properties by linear models such as:

𝑌 = 𝑋 ∙ 𝐵 + 𝑟

B could be obtained using various process modelling techniques/multivariate regression methods:

• Design Of Experiments (DOE),

• Multiple Linear Regression (MLR),

• preferentially Partial Least Squares (PLS) regression.

An estimate of 𝐻𝑌 could therefore be expressed as:

𝐻 𝑌 = 𝐻𝑋 ∙ 𝐵

Weights (p x q)

Residuals (n x q)

Process variables

heterogeneity (n x p)

𝐻𝑌 𝐻𝑋 Process responses

heterogeneity (n x p)

heterogeneities

heterogeneities

7

COMBINING TOS AND MULTIVARIATE DATA MODELLING Theory

The multivariogram of Y, noted 𝑉𝑌𝑗, is expressed as:

𝑉𝑌𝑗 =1

2(𝑁 − 𝑗) 𝐻𝑌𝑖 − 𝐻𝑌𝑖+𝑗 𝑀 𝐻𝑌𝑖 − 𝐻𝑌𝑖+𝑗

t, 𝑗 = 1,⋯ ,𝑁/2

𝑁−𝑗

𝑖=1

Using the relationship between 𝐻𝑌 and 𝐻𝑋 : 𝐻 𝑌 = 𝐻𝑋 ∙ 𝐵

𝑉𝑌𝑗 =

1

2(𝑁 − 𝑗) 𝐻𝑋𝑖 − 𝐻𝑋𝑖+𝑗 𝐵𝑀

𝐵𝑡 𝐻𝑋𝑖 − 𝐻𝑋𝑖+𝑗t

𝑁−𝑗

𝑖=1

𝑀 = 𝐶𝑜𝑣(𝐻 𝑌)−1= 𝐶𝑜𝑣(𝐻𝑋 ∙ 𝐵)

−1 = 𝐵𝑡𝐶𝑜𝑣 𝐻𝑋 𝐵−1

𝑉𝑌𝑗 =

1

2(𝑁 − 𝑗) 𝐻𝑋𝑖 − 𝐻𝑋𝑖+𝑗 𝐵 𝐵

𝑡𝐶𝑜𝑣 𝐻 𝐵 −1𝐵𝑡 𝐻𝑋𝑖 − 𝐻𝑋𝑖+𝑗t

𝑁−𝑗

𝑖=1

Leading to:

Where:

Change in metric: 𝑉𝑌 = 𝑉𝑋 𝑤𝑖𝑡ℎ 𝑀′ = 𝐵 𝐵𝑡𝐶𝑜𝑣 𝐻 𝐵 −1𝐵𝑡

𝑀’

8

COMBINING TOS AND MULTIVARIATE DATA MODELLING PLS-R

• PLS-R allows to model correlations between the multivariate X-data (predictors), and the

dependent Y-data (responses), by regression: Y=XB

• PLS models can be viewed as interrelated PCA scores of the predictors, t, and the responses,

u, maximizing cov(t,u) (Høskuldsson 1996)8,

Why PLS-R?

X Y T U

W

P

Q X=TPt+E

Y=UQt+F T=XW* B=W (PtW)-1Qt

• PLS-R defines which of the X variables have the highest

weight in predicting the Y responses for future data,

• PLS only extracts the systematic features of the process

variations and descriptor variables,

• Two main outputs will be used:

Matrix of loading-weights (W)

Matrix of regression coefficients (B)

9

APPLICATION Case Study

Industrial mineral processing plant producing a clay concentrate for the paper, paints and plastics industries.

15 variables, recorded every 5 mins by sensors (flowmeters, pressure gauges, weightometers)

2 responses: clay recovery & product density, obtained by metallurgical balance or further lab analyses,

Laborious, costly and can only provide a posteriori information on the process performance → not suitable

for a continuous process.

10

APPLICATION PLS-R Results

• PLS-R was applied to a set of raw X-Y data points representing 1 week production,

• 7-component PLS model

• PLS-R applied to the corresponding X-Y heterogeneity data display similar results.

7

→ predicts both clay recovery and final product density with

satisfactory validation results,

11

APPLICATION Process performance prediction

PLS-R models are applied to a set of 57 consecutive X data to be used for a variographic analysis:

Acceptable correlations for direct on-site

process performance prediction based

on real-time process data.

PLS-R models can accurately

characterise process performance.

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APPLICATION Application to multivariate variography

Comparison between actual Y-multivariogram and the non-weighted/ weighted X-multivariogram:

PLS-Model (B) weighted X-multivariogram displays the same characteristics than the Y-multivariogram,

Global Standard Deviation of the Sampling Error (SDSE) is best approximated using PLS-Model (B) weigths,

Using PLS regression coefficients to weight the X-multivariogram can help to optimize the sampling

procedure according to the actual global process performance by using real-time process data.

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APPLICATION Predicting process variability

• Other benefit → use PLS regression coefficients to decompose the variogram of each process

performance response,

• Considering the ith process performance response, an estimate of this response could be assessed using:

𝑦𝑖 = 𝑋. 𝐵𝑖 ith process performance

response estimate

ith PLS regression coefficient vector Predictors

Change in the metric of X-Multivariogram

Predicted and experimental variograms display the same characteristics (nugget effect, range and sill).

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CONCLUSIONS & FUTURE TRENDS

Summary

• Some of the variables accounting for a high proportion of the global (multivariate) sampling variance may not

be significant for the process performance,

• Using PLS to weight the variables dampens the effect of those variables that are less important in predicting

the future responses,

• Strictly speaking the method does not reduce the sampling variance but helps to optimize the sampling

procedure according to the actual global process performance by using real-time process data.

• Using the PLS regression coefficients even allows accurate prediction of the overall and individual process

performance variability,

Loading-weights (W) Regression coefficients matrix (B) Regression coefficient vector (Bi)

PLS on raw X-Y Weight the raw-X-

Multivariogram Weight the raw-X-Multivariogram

Predict process responses, ie

individual yi

PLS on X-Y

heterogeneity

Weight the H(X)-

Multivariogram

Predict global process performance

variability

Help to optimize the sampling

procedure

Predict individual process

response (yi) variability

15

CONCLUSIONS & FUTURE TRENDS

Application to geometallurgy

• Geometallurgy aims to combine geological and metallurgical information to create spatial predictive model

for mineral processing plants, to be used in production management (Lamberg, 2011)8,

• Geometallurgy documents variability within an orebody and quantifies the impact of geology, mineralogy,

geotechnical properties on metallurgical responses (Williams and Richardson, 2004)9,

• Optimised-multivariate variography could potentially help in predicting process performance variability as a

function of ore properties variability within the ore deposit.

Application to geometallurgical modelling

X Y

B

References

1. Gy, P., 2004. Sampling of discrete materials III. Quantitative approach—sampling of one-dimensional objects. Chemometrics and Intelligent Laboratory Systems 74,

39–47.

2. Petersen, L., Esbensen, K.H., 2005. Representative process sampling for reliable data analysis—a tutorial. Journal of Chemometrics 19, 625–647.

3. Minkkinen, P., Esbensen, K.H., 2014. Multivariate variographic versus bilinear data modeling. Journal of Chemometrics 28, 395–410.

4. 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.

5. Bourgault, G., Marcotte, D., 1991. Multivariable variogram and its application to the linear model of coregionalization. Mathematical Geology 23, 899–928.

6. Dehaine Q., Filippov L., 2015. A multivariate approach for process variogram. TOS Forum - Proceedings of the 7th World Conference on Sampling and Blending.

Bordeaux: IM Publishers, Chichester, pp. 169–174. doi: 10.1255/tosf.76.

7. Dehaine, Q., Filippov, L.O., Royer, J.J., 2016. Comparing univariate and multivariate approaches for process variograms: A case study. Chemom. Intell. Lab. Syst.

152, 107–117. doi:10.1016/j.chemolab.2016.01.016

8. Lamberg, P., 2011. Particles – the bridge between geology and metallurgy, in: Proceedings of the Conference in Mineral Engineering. Luleå, pp. 1–16.

9. Williams, S.R., Richardson, J.M., 2004. Geometallurgical mapping: a new approach that reduces technical risks, in: Proceedings of 36th Annual Meeting of the

Canadian Mineral Processors Conference. CIM, Ottawa, ON, Canada, pp. 241–268.

Acknowledgments

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 as well as by the NERC Project "CoG3: Investigating the recovery of Cobalt”.

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THANK YOU FOR YOUR ATTENTION!

17

INTRODUCTION Previous work

After Dehaine et al. (2016)7

0.5

0.14

18

APPLICATION Case Study

Case study: Industrial mineral processing plant producing a clay minerals concentrate for the

paper, ceramics, paints, plastics and rubber industries.

15 variables (see table),

2 responses: Clay Recovery & Product density.

N° Variables Code Units Location

1 Calculated Matrix Feed Flow CM-F Tons/h 1

2 Stone Belt Weigher STB-W Tons/h 2

3 Gravel Belt Weigher GB-W Tons/h 3

4 Sand Belt Weigher SB-W Tons/h 4

5 Secondary Cyclone Product

Density

2CP-D kg/m3 5

6 Secondary Product Flow 2P-F m3/h 5

7 Secondary Residue Flow 2R-F m3/h 6

8 Stone Belt Weigher - 15 min

average

STB-

W15A

Tons/h 2

9 Gravel Belt Weigher - 15 min

average

GB-

W15A

Tons/h 3

10 Sand Belt Weigher - 15 min

average

SB-

W15A

Tons/h 4

11 Cavex Cyclones Pressure CC-P Bar 7

12 LP Water Flow Rate LPW-F m3/h 8

13 HP Water Flow HPW-F m3/h 9

14 Primary Cyclones Feed Pressure PCF-P Bar 10

15 Secondary Cyclones Feed

Pressure

SCF-P Bar 11

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INTRODUCTION Multivariate variography

Data/Method PCA Variograms Multivariogram

Raw data

PCA on raw data Variograms on raw data Multivariogram on raw data

Perform variable reduction,

Filter noise from the data.

Study the spatial characteristics of all

individual parameters,

Design the optimal sampling protocol

for one property.

Summarize the overall variability in

one variogram,

Assess the global sampling’s

representativeness.

Principal

Components

Analysis

(PCA) scores

Variograms on PCA scores Multivariogram on PCA scores

Highlight distinct spatial patterns

through variable grouping in a reduced

number of variograms (Minkkinen and

Esbensen, 2014),

Design the optimal sampling protocol.

Summarize the (filtered) overall

variability in one variogram,

Assess the (filtered) global sampling’s

representativeness.

Variograms

PCA on variograms

Study and then summarize the spatial

characteristics of all individual analytes

(Kardanpour et al., 2014)

After Dehaine et al. (2016)7

• 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,

20

INTRODUCTION Theory of sampling