2009 Oxford Business & Economics Conference Program
ISBN : 978-0-9742114-1-1
THE USE OF PRINCIPAL COMPONENT ANALYSIS TO INTEGRATE
BLASTING INTO THE MINING PROCESS
David P. Lilly, P.E., MBA
Senior Consulting Engineer
DynoConsult, a Division of Dyno Nobel Inc
Pawleys Island, South Carolina
John Cory
Blasting Specialist North America
Lafarge North America
Ottawa, Ontario
Bill Hissem
Senior Mining/Application Engineer
Sandvik Mining and Construction
Appleton, Wisconsin
INTRODUCTION
In blasting many variables are highly correlated (powder factor, pattern area, explosive
weight and density, hole size, etc.) and principal component analysis can combine these
variables into a more concise form to solve problems relating to specific goals such as
mining productivity, environmental concerns and highlight the effects of major changes.
Principal component analysis was developed by Pearson in 1901 and has been
successfully adapted to many industries to create summary information. The summary
information concisely expresses the status of a process from the diverse data that best
expresses the result or a specific outcome. This paper is the 3rd in a series that utilizes
well-known statistical processes to integrate blasting into the mining process (“A
Statistical Approach to Integrating Blasting Into the Mining Process”, Oxford 2007; and
“The Enterprise Solution to Integrating Blasting Into the Mining Process”, Oxford 2008).
The principal component analysis method creates an overall measure, which is a refined
dashboard type of performance measurement using statistical data compression
techniques. The variable reduction technique creates artificial variables and reduces
redundancy in the correlations. This paper will develop the method of the approach and
use a simplified example of work the authors are using to optimize productivity at the
Lafarge plant on Texada Island near Vancouver, British Columbia.
June 24-26, 2009
St. Hugh’s College, Oxford University, Oxford, UK
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2009 Oxford Business & Economics Conference Program
ISBN : 978-0-9742114-1-1
THE PRINCIPAL COMPONENT METHOD
Basically Principal Component Analysis (PCA) takes a set of data in matrix form,
preconditions the data, extracts eigenvalues and eigenvectors, rotates the data and
condenses many correlated variables into a lesser number of uncorrelated principle
components. A principle component is a linear-weighted combination of optimally
weighted observed variables. There are several methods to conduct PCA including the
variance mode, and the correlation mode. The variance mode uses the covariance matrix.
and the correlation method uses a matrix of Pearson’s R values. The correlation method
must be used if the data has different units of measure which is the case with most
blasting data. We favor this method also because of the easily recognized R values. While
PCA can be daunting, the XL-Stat program for Excel can simplify the calculations and is
used in these examples.
The procedure to conduct a PCA using the correlation matrix is as follows:
1) Center the data variables on their mean and standard deviation.
2) Calculate a correlation matrix. Also a scatter plot can be made to help
visualize the correlations between the variables.
3) From the correlation matrix calculate the eigenvalues and eigenvectors. Eigen
refers to “characteristic”. The eigenvalues and eigenvectors describe the cloud
of data as an ellipse characterized by the eigenvalues with direction
characterized by the eigenvectors. A good website to calculate eigenvalues
and eigenvectors is:
http://www.arndt-bruenner.de/mathe/scripts/engl_eigenwert.htm
4) Determine the number of principal components by using the eigenvalues or a
scree plot. If using the eigenvalues, a good rule of thumb is to eliminate any
principle components in which the eigenvalue is less than one. The scree
value plot gives the cumulative % of variance explained as principle
components are added.
5) Form a feature vector factor loading which creates regression weights of
original variables into the principle component
6) Use varimax rotation for optimal orthogonal rotation. Orthogonal refers to
independent or uncorrelated representation.
7) Make biplot graphs to show the data in relationship to the variables selected.
In a biplot graph variables are symbolized by arrows and transformed data by
the points.
A SIMPLE EXAMPLE – PRODUCTIVITY AT THE LAFARGE TEXADA PLANT
The Lafarge Texada Plant is located on Texada Island near Vancouver, British Columbia.
The plant has produced close to 12 million tonnes of limestone aggregate, cement stone,
and high purity chemical stone. This analysis uses raw data from 2006, 2007, and 2008
and is aimed at determining the main drivers of productivity in tonnes per manhour and
tonnes per operating hour. A preliminary PCA was conducted using many input variables
including bench height, bench elevation, truck and loader productivity, etc. The main
June 24-26, 2009
St. Hugh’s College, Oxford University, Oxford, UK
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2009 Oxford Business & Economics Conference Program
ISBN : 978-0-9742114-1-1
drivers remaining after the preliminary analysis were powder factor (kg/m3) and
operating hours.
Summary statistics:
Variable
Tonnes/Manhour
KG/M3
Op Hours
Observations
33
33
33
Minimum Maximum
21.015
42.600
0.434
0.782
215.750 326.000
Mean Std. deviation
32.172
4.997
0.648
0.092
273.051
27.830
A correlation matrix was produced to determine the eigenvalues and eigenvectors. In
addition a scatter plot was made to visually inspect the relationship between the variables.
Correlation matrix (Pearson (n)):
Variables
Tonnes/Manhour KG/M3 Op Hours
Tonnes/Manhour
1
0.441
0.629
KG/M3
0.441
1
0.100
Op Hours
0.629
0.100
1
Values in bold are different from 0 with a significance level alpha=0.05
45
45
40
40
Tonnes/Manhour
Tonnes/Manhour
Scatter Plots
35
30
25
35
30
25
20
20
0.4
0.5
0.6
0.7
0.8
200
220
240
KG/M3
260
280
300
320
340
360
Op Hours
0.9
0.8
KG/M3
0.7
0.6
0.5
0.4
210
230
250
270
290
310
330
Op Hours
June 24-26, 2009
St. Hugh’s College, Oxford University, Oxford, UK
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2009 Oxford Business & Economics Conference Program
ISBN : 978-0-9742114-1-1
Eigenvalues:
F1
1.817
60.575
60.575
Eigenvalue
Variability (%)
Cumulative %
F2
0.906
30.192
90.767
F3
0.277
9.233
100.000
Eigenvectors:
F1
0.685
0.441
0.581
Tonnes/Manhour
KG/M3
Op Hours
F2
-0.046
0.821
-0.569
F3
0.728
-0.363
-0.582
Scree plot
2
100
1.6
80
Eigenvalue
1.4
1.2
60
1
0.8
40
0.6
0.4
20
Cumulative variability (%)
1.8
0.2
0
0
F1
F2
F3
axis
Factor loadings:
Tonnes/Manhour
KG/M3
Op Hours
F1
0.923
0.594
0.783
F2
-0.044
0.781
-0.542
F3
0.383
-0.191
-0.306
The factor loadings indicate that factor F1 is dependent upon all of the variables
but mainly operating tonnes/hour and operating hours. Factor F2 is mainly
dependent upon powder factor (kg/m3). F3 adds little to the analysis and will be
removed.
June 24-26, 2009
St. Hugh’s College, Oxford University, Oxford, UK
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2009 Oxford Business & Economics Conference Program
ISBN : 978-0-9742114-1-1
The biplot below is the final result of the analysis. The results indicate that tonnes per
manhour are increased with increasing powder factor (kg/m3) and with increasing
number of operating hours. This biplot could be used as a dashboard to indicate monthly
status and possible corrective action. It is worth noting that powder factor and operating
hours are almost perpendicular to each other indicating they are independent variables.
Principal component 1 is highly loaded upon and dependent upon operating hours.
Principal component 2 is highly loaded upon and dependent upon powder factor.
Together 90.77% of the variability is explained by these 2 principal components. Because
of the closer proximity to tonnes per hour operating hours has the higher correlation with
tonnes per hour and is the major influence.
Biplot (axes D1 and D2: 90.77 %)
after Varimax rotation
4
KG/M 3
TONNES PER MANHOUR
3
35+ TPMH
D2 (38.98 %)
2
To nnes/M anho ur
30 TO 35 TPMH
LESS THAN 30 TPMH
1
0
Op Ho urs
-1
-2
-3
-4
-3
-2
-1
0
1
2
3
4
5
D1 (51.78 %)
In like manner when a similar analysis is conducted of operating tonnes per hour, it
appears that powder factor is the major driving variable as shown below.
Biplot (axes D1 and D2: 81.51 %)
after Varimax rotation
4
Op Ho urs
3
TONNES PER OPERATINGHOUR
D2 (36.78 %)
2
To nnes/Hr
1500+ TPMH
1450 TO 1500 TPH
LESS THAN 1450 TPH
1
0
KG/M 3
-1
-2
-3
-3
-2
-1
0
1
2
3
4
D1 (44.73 %)
June 24-26, 2009
St. Hugh’s College, Oxford University, Oxford, UK
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2009 Oxford Business & Economics Conference Program
ISBN : 978-0-9742114-1-1
A couple of cautions and rules of thumb: PCA is a large sample procedure and works best
with 300 or more rows of data, and it is desirable to have at least 3 variables that load on
a component. The simple example above does not meet this criteria it but is still
informative. PCA in it’s basic form is linear. Since many blasting variables are not linear,
kernal PCA must be performed in which the variables are transformed into a linear form
before completing the analysis. It is also important that the principal component and the
variables that load on it share the same conceptual meaning and that the variables that
load on different components seem to measure different constructs.
Below is a more complex example from the blasting data from 814 blasts in 2006 and
2007. The purpose is to determine what factors greatly influence powder factor (kg/m3)
which was earlier determined to be a major driver of productivity.
Biplot (axes D1 and D2: 82.48 %)
after Varimax rotation
10
Kg/M 3
D2 (29.09 %)
POWDER FACTOR KG/M3
0.9 KG/M3 +
0.5 TO .0.9 KG/M3
LESS THAN 0.4 KG/M3
5
bench
0
Ho le Diameter
B xS A reaSqFt
-5
-9
-4
1
6
D1 (53.39 %)
The analysis indicates that bench elevation, hole diameter, and pattern area are significant
influences to the final powder factor. Higher benches have higher powder factors and
larger hole diameters and pattern area tend to decrease powder factor. While most
blasters calculate powder factors before blasting, subtle effects can change the desired
value and this is a method to make fine adjustments when trying to adjust powder factors
to obtain maximum productivity.
THE FUTURE
The authors have developed a database of drill penetration rates at the Texada Island
plant, which they believe may be a proxy for rock hardness, and affect productivity
analysis. They plan to incorporate this into the PCA model also. In addition, air blast and
vibration analysis generates a substantial amount of correlated data that could utilize the
PCA method to minimize public complaints.
June 24-26, 2009
St. Hugh’s College, Oxford University, Oxford, UK
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2009 Oxford Business & Economics Conference Program
ISBN : 978-0-9742114-1-1
REFERENCES
Harris, Jeffery; Meredith, Donald. L.; and Stanley, Donald A. (1991), Principal
Components Factor Analysis in Mineral Processing, U.S. Bureau of Mines
Koch, George. S., and Link, Richard F. (1971), Statistical Analysis of Geological Data.
ISBN 0-486-64040-X
Lilly, David P. (2007), A Statistical Approach to Integrating Blasting Into the Mining
Process, Oxford Business and Economic Conference.
Lilly, David P. (2007), The Enterprise Solution to Integrating Blasting Into the Mining
Process, Oxford Business and Economic Conference.
Mutmansky, Jan M., and Singh, Madan M. (1965) A Statistical Study of Relationships
Between Rock Properties.
Smith, Lindsay I. (2007), A Tutorial on Principle Component Analysis (Cornell
University)
Yavuz, M. (2007) An Equipment Selection Application Using the AHP Method. SME
Annual Meeting Feb 25-28, Denver, CO
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