Calculation of variance
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Calculation of variance

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ELSEVIER Int. J. Miner. Process. 55 (1998) 1–20

Calculation of variance and covariance of sampling errors in
complex mineral processing systems, using state–space

dynamic models

Azar Mirabedini, Daniel Hodouin *

Mining and Metallurgy Department, Laval University, Quebec City G1K 7P4, Canada
Received 6 December 1996; accepted 7 July 1998


This paper presents a method to calculate the variance and covariance matrix V of the
integration errors of the measured flow species in a complex mineral processing unit. The
integration error is due to the interaction between the sampling strategy and the dynamic
variations of the stream properties. First, a general formula is developed to relate V to the
autocovariance matrix Vyy of the flow species in the different streams of the unit. Then Vyy is
estimated using a dynamic model of the process. This dynamic model is obtained by assembling
summators, separators and transfer functions into a state–space formulation. Two examples of
the method are given: one for a flotation circuit and one for a leaching tank.  1998 Elsevier
Science B.V. All rights reserved.

Keywords: sampling; error variance; mineral processing circuits; flotation process; state-space

1. Introduction

Sampling errors have been extensively studied by Gy (1979, 1988) in the field
of particulate material processing. Gy’s theory of particulate material sampling (Pitard,
1992) contains a detailed analysis of the sources of measurement errors, and is extremely
useful for mineral processors willing either to evaluate the accuracy of their data or
to design improved sampling and measurement schemes. Errors can be classified into
three categories. First, there are sampling errors due to the constitution heterogeneity
of the particles (the fundamental error). Then, sampling errors due to the distribution
heterogeneity of the particles (the integration error). In the case of sampling of batch
� Corresponding author. Tel.: 1-418-656-5003; Fax: 1-418-656-5343; E-mail:

0301-7516/98/$19.00 c
 1998 Elsevier Science B.V. All rights reserved.
PII: S 0 3 0 1 - 7 5 1 6 ( 9 8 ) 0 0 0 2 4 - 6

2 A. Mirabedini, D. Hodouin / Int. J. Miner. Process. 55 (1998) 1–20

material, this distribution heterogeneity is related to the variation of the material
properties as a function of the sample location in the batch. For a flowing stream of
material, the heterogeneity is related to the time variation of the particle properties.
Finally, the last category includes all the other errors such as the sample increment
extraction error, the sample sub-sampling errors and the measurement device errors.

Since a mineral processing operation is never operating in perfect steady-state condi-
tions, the stream properties vary as a function of time, thus necessarily creating an inte-
gration error when the streams are sampled. Usually this is the dominant source of errors
in mineral processing data acquisition, and the only one which is studied in this paper.

It is assumed in this study that a property of a stream of material within a given time
interval is evaluated by averaging measured values of the property obtained at n different
times in the considered interval. The property is discretized using a discretization period
T , such that N T is the total time interval in which the property is evaluated, and kT the
sampling period corresponding to the n measurements. The stream property is evaluated
either by averaging n measurement values or by measuring the property of a composite
sample obtained by gathering n sample increments.

Gy developed an expression for the variance of the integration error of the average
material composition obtained by the analysis of a composite sample of the material.
The calculation makes use of the variogram of a property called heterogeneity (Gy,
1988) which is a combination of stream composition and flowrate. The same type of
expression for the integration error variance has been obtained by Hodouin and Ketata
(1994) using the stream property autocorrelogram. However, in their expression they do
not neglect the covariances between the sample increment errors, terms which in some
cases may be significant. In Gy’s approach, the variogram (David, 1977; Journel and
Huijbregts, 1991) is empirically modelled, while in the Hodouin and Ketata approach
(1994) the flow dynamics are modelled by an ARMA stochastic model (Autoregressive
and Moving Average Model, Box and Jenkins, 1994). In both cases a single stream, or
a set of independent streams are considered. Other papers deal with the same type of
approach (for instance Saunders et al., 1989).

In practice, samples are most of the time taken from the various streams of a
complex mineral processing unit. The time variations of the streams are correlated by
the process units themselves, thus necessarily leading to correlated integration errors. As
a consequence, in a complex flowsheet, the integration errors of the various connected
streams cannot be considered as independent of each other. This does not create
any problem when the properties of the various sampled streams are independently
processed, interpreted and used. However, this is usually not the case, since the various
stream measurements are simultaneously processed, in material balance reconciliation
techniques for instance (Hodouin and Everell, 1980; Hodouin and Flament, 1985; Makni
et al., 1995; Crowe, 1996), and simultaneously used and interpreted in the calculation of
process performance indicators such as metal recovery. The accuracy of these process
performance indicators are related to the variance of the errors made on the stream
properties as well as to their covariances, whatever raw or reconciled data are used to
calculate them (Hodouin et al., 1984, 1988). In addition, when data are reconciled it
may be important to account for the error correlation when designing the reconciliation
procedure (Hodouin et al., 1998).

A. Mirabedini, D. Hodouin / Int. J. Miner. Process. 55 (1998) 1–20 3

The objective of the present study is to develop a general expression of the integration
error variance–covariance for simultaneously sampled streams of interconnected process
units. This expression is derived in a first part as a function of the autocovariance matrix
of the stream properties. Then, in a second part, the autocovariance of the stream proper-
ties are evaluated through linear stochastic modelling of the plant using the state–space
representation. The model used is a multivariable extension of the ARIMA stochas-
tic model used to represent the autocovariance of the properties of a single stream.
Illustrations of the method will be considered for a leaching and a flotation plant.

2. Definition of process variables and integration error

The properties of a stream are usually described by:
– the overall material flowrate F.t/
– the mass fraction of the species ai .t/ (i D 1 to m)
– species flowrates xi .t/ D F.t/ai.t/
where t is an integer corresponding to the number of discretization periods T . To
characterize the average properties of the stream during an operating time interval N T ,
it is assumed that the process variables are measured with a sampling period kT . As a
consequence the stream properties are obtained from the following estimators:

Fn D 1


F[k0 C . j � 1/k] (1)

ain D


xi [k0 C . j � 1/k]


xin D 1


xi [k0 C . j � 1/k] (3)

where k0, the time of the first measurement in the interval [0; N T ], is taken as k=2 or
k C 1=2 for even and odd values of k, respectively.

The estimation errors of F , a and x can be decomposed into two parts. The first
one, the integration error (Gy, 1979), is related to the fact that the properties are only
measured at sampling instants. Three integration errors can be defined:

eF D .Fn � FN / (4)

eai D .ain � ai N / (5)

exi D .xin � xi N / (6)
where the subscript N indicates that the corresponding variables are defined by Eq. 1,
Eq. 2 or Eq. 3 where the