Calculation of variance
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Calculation of variance

Disciplina:Processamento de Minerais I206 materiais2.072 seguidores
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mineral processing flowsheet. This is presented in an other publication
(Hodouin et al., 1998).

7. Nomenclature

ai .t/ flowrate of species i at time t
ain average of n equally spaced values of ai .t/ in a window of width N T
A, B, C coefficients of the state–space model of the species flowrates (matrices)
eF material flowrate integration error
eai ith species content integration error
exi ith species flowrate integration error
F.t/ or Fj.t/ material flowrate at time t in the jth stream
Fn or Fjn average of n equally spaced values of F.t/ or Fj.t/ in a window of width

g; g0 model coefficients
G first-order discrete transfer function
h Gy’s heterogeneity
k number of discretization periods separating two sampling instants
k0 number of discretization periods corresponding to the first sample in the

window [0; N T ]
l number of discretization periods for the estimation of an autocovariance
m number of species in the processed material
n number of samples in a window [0; N T ]
N number of discretization periods in a window [0; N T ]
p number of streams in the flowsheet
s marginal separation coefficient of a mineral separation device
t time index (integer corresponding to the number of discretization periods)
T discretization period

A. Mirabedini, D. Hodouin / Int. J. Miner. Process. 55 (1998) 1–20 19

V .n; k/ variance–covariance of the integration error of y.t/ for n samples sepa-
rated by k discretization periods

Vyy.0/ variance–covariance of y.t/
Vyy.l/ autocovariance of y.t/ at lag l
Vzz.l/ autocovariance of z.t/ at lag l
V� variance–covariance of �
xi .t/ ith species flowrate
xi j .t/ ith species flowrate in stream j
xin average value of n samples of xi.t/ in the window [0; N T ]
y.t/ vector of species flowrates xi j .t/
yn average value of n samples of y.t/ in [0; N T ]
z�1 backshift operator
z.t/ state variable of the flowrate dynamic model
�, �, 
 model coefficients
� mean value of y.t/
�i j mean value of xi j .t/
� 2i j variance of xi j.t/
�.t/ white noise


This study is part of a generic project on the application of automatic control to
the mineral processing and extractive metallurgy industries. It is supported by the
Natural Resources Departments of the Quebec and Canada Governments as well as by a
consortium of fourteen Canadian mining companies.


A¨ stro¨m, K.J., Wittenmark, B., 1990. Computer-Controlled Systems: Theory and Design. Prentice Hall,
Englewood Cliffs, NJ.

Box, G.E.P., Jenkins, G.M., 1994. Time Series Analysis: Forecasting and Control. Holden Day, San
Francisco, CA.

Crowe, C.M., 1996. Data reconciliation progress and challenges. J. Proc. Contrib. 6 (213), 89–98.
David, M., 1977. Geostatistical Ore Reserve Estimation. Elsevier, Amsterdam.
Gy, P.M., 1979. Sampling of Particulate Materials — Theory and Practice. Elsevier, Amsterdam.
Gy, P.M., 1988. E´ chantillonnage, Homoge´ne´isation. Masson, Paris, 607 pp.
Gy, P.M., 1989. Heterogeneity — Sampling — Homogenization. Elsevier, Amsterdam.
Hodouin, D., Everell, M.D., 1980. A hierarchical procedure for adjustment and material balancing of

mineral processing data. Int. J. Miner. Process. 7, 91–116.
Hodouin, D., Flament, F., 1985. Material balance computation for process evaluation and modelling:

BILMAT computer program, SPOC manual, Chapter 3.1. Rep. SP85-1=3.1, CANMET, Dep. of Natural
Resources, Ottawa.

Hodouin, D., Ketata, C., 1994. Variance of average stream composition obtained by automatic incremental
sampling. Int. J. Miner. Process. 40, 199–223.

Hodouin, D., Bazin, C., Trusiak, A., 1984. Reliability of calculation of mineral process efficiencies and rate
parameters from balanced data. In: Herbst, J.A. (Ed.), Control’84. AIME-SME=TMS, New York.

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

Hodouin, D., Flament, F., Bazin, C., 1988. Reliability of material balance calculations — a sensitivity
approach. Miner. Eng. 2 (2), 157–170.

Hodouin, D., Mirabedini, A., Makni, S., Bazin, C., 1998. Reconciliation of mineral processing data
containing correlated measurement errors. Int. J. Miner. Process. 54, 201–215.

Journel, A.G., Huijbregts, Ch.J., 1991. Mining Geostatistics. Academic Press, New York.
Makni, S., 1996. Re´conciliation des donne´es des syste`mes dynamiques. Application aux proce´de´s de

flottation. Ph.D. Dissertation, Laval University, Que´bec.
Makni, S., Hodouin, D., Bazin, C., 1995. On-line reconciliation by minimization of a weighted sum of

squared residuals and node imbalance. Proc. XIXth IMPC, San Francisco, SME=AIME, pp. 233–238.
Pitard, F., 1992. Pierre Gy’s Sampling Theory and Sampling Practice. CRC, Baton Rouge, LA, Vols. I and

Saunders, I.W., Robinson, G.K., Lwin, T., Holmes, R.J., 1989. A simplified variogram method for deter-

mining the estimation error variance in sampling from a continuous stream. Int. J. Miner. Process. 25,