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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.
12
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.
13
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).
14
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”.
16
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
19
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
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