SEMIPARAMETRIC IDENTIFICATION AND UNCERTAINTY QUANTIFICATION OF BENEFICIATION UNIT SEPARATION CHARACTERISTICS

Abstract

Identifying the separation characteristics of iron ore beneficiation units from process data provides the model and uncertainty information required for model predictive control. Parametric models, including the Rosin–Rammler and sigmoid models, have fixed analytical forms that may not describe the characteristics of individual units with sufficient accuracy. Point estimation without a dedicated statistical procedure does not quantify predictive uncertainty, while purely nonparametric estimates may be unstable when data are limited. This study developed a unified semiparametric identification method in which a physically motivated parametric prior mean function was augmented with a correction estimated by Gaussian process regression from empirical deviations of the observations. The same estimation procedure was used for model fitting, cross-validation, and sensitivity analysis. A semiparametric Nadaraya–Watson estimator constructed from deviations from the prior mean function served as a benchmark. Experimental recovery was calculated using a common mass-balance definition. The ratio between product mass flow rates was either obtained from measurements or estimated from the size-by-size balance. For the classifier, this ratio was well constrained by the data, with a 95% bootstrap interval of [1.95, 2.42]. The proposed semiparametric model reduces the within-sample root-mean-square error by approximately 82% relative to the Rosin–Rammler model and outperforms the stand-alone Gaussian process model in leave-one-out cross-validation, with coefficients of determination of 0.970 and 0.945, respectively. The mass-flow-rate ratio was re-estimated in every cross-validation iteration using only the training size classes. For the hydrocyclone, the ratio was estimated with a 95% bootstrap interval of [0.96, 2.56], and the proposed model retains the highest leave-one-out coefficient of determination, approximately 0.81, throughout this interval. Results for magnetic separators with measured mass flow rates confirm that the argument of a separation characteristic must reflect the physical separation mechanism. Negative leave-one-out coefficients of determination obtained when the separation characteristic was modelled as a function of particle size demonstrate that particle size is an unsuitable argument and justify representing the characteristic as a function of iron content, which more directly reflects the magnetic separation mechanism. Monte Carlo simulation under the specified noise model was used to quantify sensitivity; for the classifier, the mean relative width of the uncertainty band is approximately 14%.