470 XXXI International Mineral Processing Congress 2024 Proceedings/Washington, DC/Sep 29–Oct 3
Simple Linear Regression (SLR)
The successful implementation of sorting relies on the
capability to establish a connection between the ore com-
position and the sensor responses. Typically, this correla-
tion between the actual calcite grade and sensor responses
is achieved through models. A widely used model for pre-
dicting a property based on various types of information
is simple linear regression (Li, Klein, Sun, &Kou, 2020).
In this study, 75 specimens were randomly selected, and
their information was used to correlate the actual calcite
grade to the responses from different sensors, as illustrated
in Equation 1, where y represents the assay grade of calcite,
x is the sensor response, and a and b are the slope and inter-
cept coefficients determined through model fitting using
the least square approach. The sorting performance of the
assessed sensors was evaluated by calculating the correlation
coefficient (R2).
y ax b =+(1)
To compare SRL with ROC, the remaining 25 rocks were
sorted based on the model derived from the analysis of the
initial 75 rocks. The calcite grade was calculated for the
accepted and rejected fractions.
Receiver Operating Characteristic (ROC)
A ROC graph is a tool for visualizing and assessing the per-
formance of classifiers (Fawcett, 2006). In the context of
a sensor and a target calcite grade, there are four potential
scenarios. If the actual calcite grade of the rock is higher
than the desired grade, and the classifier places it in the
accepted fraction, it is considered a True Positive (TP). If
the sensor classifies it as negative, it is counted as a False
Negative (FN). If the actual calcite grade of the rock is
below the desired grade and it is classified as rejected, it is a
True Negative (TN) if it is classified in the accepted frac-
tion, it is a False Positive (FP).
Sensitivity and specificity were calculated to evaluate the
sorting performance of the four tested sensors. Sensitivity
assesses the fraction of actual concentrate properly con-
sidered as such, calculated as per Equation 2. Specificity
evaluates the proportion of actual waste accurately sorted as
such, determined by Equation 3.
TP FN
TP sensitivity =+(2)
FP TN
TN specificity =+(3)
A graphical representation known as ‘ROC space’ can be
constructed, as depicted in Figure 2. In this representation,
“1 – specificity” and “sensitivity” are the x and y axes, respec-
tively. In an ideal scenario where the sensor does not make
any errors, the point (0,1) would point out a True Positive
(TP) rate of 1 and a False Positive (FP) rate of 0. Informally,
one point in ROC space is considered superior to another
if it is positioned closer to the point representing perfect
classification. The diagonal line y=x depicts the random
classification.
With different cut-off thresholds of the sensor response,
for a specified target grade, sensitivity and specificity would
Figure 1. Sample preparation and selection of specimens
Simple Linear Regression (SLR)
The successful implementation of sorting relies on the
capability to establish a connection between the ore com-
position and the sensor responses. Typically, this correla-
tion between the actual calcite grade and sensor responses
is achieved through models. A widely used model for pre-
dicting a property based on various types of information
is simple linear regression (Li, Klein, Sun, &Kou, 2020).
In this study, 75 specimens were randomly selected, and
their information was used to correlate the actual calcite
grade to the responses from different sensors, as illustrated
in Equation 1, where y represents the assay grade of calcite,
x is the sensor response, and a and b are the slope and inter-
cept coefficients determined through model fitting using
the least square approach. The sorting performance of the
assessed sensors was evaluated by calculating the correlation
coefficient (R2).
y ax b =+(1)
To compare SRL with ROC, the remaining 25 rocks were
sorted based on the model derived from the analysis of the
initial 75 rocks. The calcite grade was calculated for the
accepted and rejected fractions.
Receiver Operating Characteristic (ROC)
A ROC graph is a tool for visualizing and assessing the per-
formance of classifiers (Fawcett, 2006). In the context of
a sensor and a target calcite grade, there are four potential
scenarios. If the actual calcite grade of the rock is higher
than the desired grade, and the classifier places it in the
accepted fraction, it is considered a True Positive (TP). If
the sensor classifies it as negative, it is counted as a False
Negative (FN). If the actual calcite grade of the rock is
below the desired grade and it is classified as rejected, it is a
True Negative (TN) if it is classified in the accepted frac-
tion, it is a False Positive (FP).
Sensitivity and specificity were calculated to evaluate the
sorting performance of the four tested sensors. Sensitivity
assesses the fraction of actual concentrate properly con-
sidered as such, calculated as per Equation 2. Specificity
evaluates the proportion of actual waste accurately sorted as
such, determined by Equation 3.
TP FN
TP sensitivity =+(2)
FP TN
TN specificity =+(3)
A graphical representation known as ‘ROC space’ can be
constructed, as depicted in Figure 2. In this representation,
“1 – specificity” and “sensitivity” are the x and y axes, respec-
tively. In an ideal scenario where the sensor does not make
any errors, the point (0,1) would point out a True Positive
(TP) rate of 1 and a False Positive (FP) rate of 0. Informally,
one point in ROC space is considered superior to another
if it is positioned closer to the point representing perfect
classification. The diagonal line y=x depicts the random
classification.
With different cut-off thresholds of the sensor response,
for a specified target grade, sensitivity and specificity would
Figure 1. Sample preparation and selection of specimens