Calculation of variance
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Calculation of variance

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summation is performed for the N discrete values in the
4 A. Mirabedini, D. Hodouin / Int. J. Miner. Process. 55 (1998) 1\u201320
time interval [0; N T ]. These errors are strictly related to the nature of the averaging
estimation technique. In addition to them, there are errors related to the particle
heterogeneity (the fundamental error), the sampling procedure, the sampling execution,
the sample handling, and finally the measuring devices.
This study is limited to the problem of evaluation of the integration error e, all the
other errors being lumped into a white additive error. As shown by Gy (1979), the
integration error of any stream property can be calculated from the knowledge of n, k
and the autocovariance of the corresponding stream property. For the flowrates F and x ,
the errors are simply obtained from a sum of the F or x properties taken at various times
in the interval [0; N T ] (see Eqs. 1, 3, 4 and 6). In this case the application of linear
statistics does not create any problem. However, the estimation error of the species mass
fractions (Eqs. 2 and 5) is not a linear function of F and a or F and x , thus leading to
more complex variance calculation problems.
In the first study by Gy (1979), two contributions to the integration error of a where
calculated, one related to the autocovariance of a and one related to the autocovariance
of F and the cross-covariance of F and a. In his most recent book (Gy, 1989), Gy
introduced a new variable (the heterogeneity h) which is a combination of a and F , thus
lumping the statistical properties of a and F in a single descriptor: the autocovariance
of the heterogeneity h. This new variable is defined by:
hi .t/ D ai.t/\ufffd ai N
ai N
and the relative integration error is expressed as:
ai N
D 1
hi [k0 C . j \ufffd 1/kT ] (8)
the variance of which depends only on the autocovariance of h, if ai N is assumed to
be a deterministic value. However, Eq. 8 is only an approximation, although this is not
explicitly discussed by Gy, which is valid if eF D 0, as it can be verified by expressing
the integration error of Eq. 5 in terms of hi defined by Eq. 7.
Finally, since the estimation of the integration error of the stream compositions
requires more complex calculations and also assumptions which limit the validity of
the calculated errors, this study is limited to the estimation of the integration error for
species flowrates (or overall flowrate which is just a special case of species flowrate).
Furthermore, extending the calculations of ea to multi-stream and multi-species would
have become cumbersome, and probably not bringing any additional understanding in
the area of estimation of integration errors for interconnected processes.
To illustrate the above problem, the flotation circuit of Fig. 1 is considered. The unit
processes 100 t=h of an ore, the critical components of which are zinc and copper. The
variables of interest are the flowrates of zinc and copper (and possibly ore if the latter is
processed using the same estimations scheme as zinc and copper) in the eight different
streams of the process. The variables are noted: xCu; j .t/ and xZn; j .t/ (and Fj.t/ if the
ore flowrate is considered), where j is the index of the stream. A discretization period
of 1 min is realistic to properly describe the process dynamics, taking into account
that the dominant time constants of the various process units of the plant are ranging
A. Mirabedini, D. Hodouin / Int. J. Miner. Process. 55 (1998) 1\u201320 5
Fig. 1. A flotation circuit.
from 5 to 10 min. Because of recycling streams the time constants of the overall circuit
(for instance the time constant between feed and reject) are larger, thus leading to
correlations between stream properties variations which can last for up to 1 h. If Cu
and Zn contents are measured by X-ray fluorescence analyzers, typical sampling periods
are 10 or 15 min. When the circuit behaviour has to be characterized every hour for
instance, typical sets of values of the sampling parameters are N D 60 and n D 4, for
k D 15, or n D 6, for k D 10. If composite samples are gathered during 8-h periods and
analyzed in the laboratory to characterize production shift performance indicators, then
a typical set of sampling parameter values is N D 480, k D 30 and n D 16 if increments
are collected every 30 min.
To illustrate the integration error expression, Eq. 6, applied to the copper in stream 5,
for k D 15, n D 4, N D 60, gives:
eCu;5 D 14
xCu;5.15 j \ufffd 7/\ufffd 160
xCu;5. j/ (9)
3. Integration error covariance matrix
Before calculating the variances of the integration errors, the following vector of
process variables .y/ is defined for a given plant:
yT D .x11; x12; : : :x1p; x21; : : :; xi j ; : : :; xmp/ (10)
where the stream index j varies from 1 to p and the species index i from 1 to m (T is
the matrix transposition operator).
In the following it will be assumed that the plant operates in a stationary state, i.e. that
6 A. Mirabedini, D. Hodouin / Int. J. Miner. Process. 55 (1998) 1\u201320
the first and second statistical moments of the random vector y.t/ are time independent.
Mathematically this is expressed by:
E[y.t/] D \ufffdI 8t (11)
y.t/\ufffd \ufffd\ufffd \ufffdy.t/\ufffd \ufffd\ufffdTo D Vyy.0/I 8t (12)
y.t C l/\ufffd \ufffd\ufffd \ufffdy.t/\ufffd \ufffd\ufffdTo D Vyy.l/I 8t (13)
where E is the mathematical expectation. Eq. 11 means that the mean value of the
statistical distribution of each species flowrate is the same at any time in the interval [0;
N T ]. This, however, does not mean that the average value of y in [0; N T ]
yN D 1N
y. j/ (14)
is equal to \ufffd, which is the limit value of yN when N tends to infinity. The matrix
Vyy.0/ contains in its diagonal the variances \ufffd 2i j of xi j.t/, which are assumed constant
in the interval [0; N T ]. This does not necessarily mean that the variance of xi j .t/
estimated from the data in [0; N T ] is equal to \ufffd 2i j , because these data are simply a
specific realization of a stochastic process characterized by (\ufffdi j , \ufffd 2i j ). The off-diagonal
terms of Vyy.0/ are the covariances between the species flowrates. One can distinguish
the covariances between the flowrates of different species in the same stream, the
covariances between the flowrates of the same species in different streams and, finally,
the covariances between the flowrates of different species in different streams. These
covariances are usually not zero for two reasons. First, the process units obviously
correlate the output stream properties to those of the input streams. Second, the amount
of different minerals in the same ore are usually correlated because of the inherent
correlated texture of the orebody, where the various minerals are usually associated.
Finally the matrix Vyy.l/ characterizes the autocovariance of y, i.e. the covariance
between values of y separated by l discretization periods T . More precisely, the diagonal
terms correspond to the autocovariances of the xi j .t/\u2019s, while the off-diagonal terms
correspond to the cross-covariances between flowrates of different species in different
streams. The remarks on the difference between the estimated values from the data in
[0; N T ] and the average values, made for the mean and variances of xi j , are also valid
for their covariances, autocovariances and cross-covariances.
Using these definitions of the statistical properties of the vector y.t/, it is now
possible to calculate the variance\u2013covariance matrix of the integration error i.e.:
V .n; k/ D E \ufffd.yn \ufffd yN /.yn \ufffd yN /T \ufffd (15)
where, as in the single stream case:
yn D 1
y.k0 C jk/ (16)
yN being defined by Eq. 14.
A. Mirabedini, D. Hodouin / Int. J. Miner. Process. 55 (1998) 1\u201320 7
To calculate the variance\u2013covariance matrix V of Eq. 15, it is decomposed into four
V D E \ufffdyn yTn \ufffdC E \ufffdyN yTN \ufffd\ufffd E \ufffdyn yTN \ufffd\ufffd E \ufffdyN yTn \ufffd (17)
For the first term one has:
yn yTn
\ufffd D 1
.n \ufffd i/Vyy.ik/C
.n \ufffd i/V Tyy.ik/
The second term is derived using