INTERNATIONAL JOURNAL OF GEOMATICS AND GEOSCIENCES
Volume 5, No 1, 2014
© Copyright by the authors - Licensee IPA- Under Creative Commons license 3.0
Research article
ISSN 0976 – 4380
Computing pit excavation volume using Multiple Regression Analysis
Ragab Khalil
Civil Engineering Department, Faculty of Engineering, Assiut University, Assiut, Egypt
and Landscape Architecture Department, Faculty of Environmental Design, KAU, Saudi
Arabia
khalilragab@yahoo.com
ABSTRACT
Volume estimation of borrow pits is common application in civil Engineering. Several
methods for volume estimation have been presented in literature. In general, they rely on a
specific polynomial to fit the surface heights. Practically, each site’s topography is unique
and may not follow that polynomial. In this paper, regression analysis to find the most
suitable equation that fit each site is presented. Using numerical examples, results from the
proposed approach are presented and accuracy compared with existing methods.
Keywords: Estimate, Excavation, Regression, Pit, Volume.
1. Introduction
Volume estimation of borrow pits is common application in civil Engineering. Several
methods have been developed for computing pit volume. Easa (1988) proposed a seconddegree polynomial based on Simpson method in each direction of equal interval grid.
Chambers (1989) applied the equation of Easa (1988) on a grid of unequal intervals. Chen
and Lin (1991) developed the cubic spline method, which provides smooth connections
between the approximating third-degree polynomials. Easa (1998) developed a method based
on the cubic Hermite polynomial. Yanalak (2005) studied the use of natural neighbor
gridding technique to transform scattered field data to uniform grid and computed the
volumes of rectangular prisms. Yilmaz (2010) applied photogrammetry to obtain
measurements and use Surfer software to compute the volume. Davis (1994) and Mukherji
(2012) proposed using finite elements technique for computing pit excavation volume.
Data points obtained from classical field survey are often irregular and scattered. Each site
has its unique topography and may not coincide with any of the mentioned polynomials. This
paper introduces a technique to deal with each site as a unique and find the suitable equation
that represents its surface using regression analysis. Regression analysis has been used to
predict earthquake effects (Youd et al 2002), trip generation model (Sekhar et al 1997),
thermal conductivity (Sanjaya et al 2011), Concrete compressive strength (Chou ans Tsai
2012), overbreak in underground mining (Jang and Topal 2013), housing demand (Ng et al
2008), selecting retaining wall systems (Choi and Lee 2010) and other engineering subjects.
The aim of this paper is to compare the existing methods of volume estimation with volume
estimated using regression technique on example 2 of Chen and Lin (1991), which was also
used by others. The results are compared with those in Mukherji (2012). The applied
technique is introduced, then applications and conclusions are given.
2. Regression analysis
Submitted on June 2014 published on August 2014
43
Computing pit excavation volume using Multiple Regression Analysis
Ragab Khalil
Statistical models of the relationship between dependent variable (response variable) and one
or more independent variables (explanatory variables) can be developed using linear
regression. The general formula for multiple regression models is:
(1)
Where “y” is a dependent variable, β0 is a constant, βi is a regression coefficient and “xi” is
independent variable (i=1, 2,…, n).
The regression is still linear in coefficients even the independent variables have been raised to
a power of any order as in equation 2.
(2)
In case of volume estimation, the dependent variable is the excavation depth (z) and the
independent variables are the point plane coordinates (x, y).
(3)
The volume could be estimated using double integral over the pit area
(4)
Regression is usually carried out by specialist programs such as Microcal Origin, Sigma Plot
or Graphpad Prism; however these programs tend to be expensive (Brown 2001). Popular
spreadsheet programs, such as Quattro Pro, Microsoft Excel, and Lotus 1-2-3 provide
comprehensive statistical program packages, which include a regression tool among many
others (Orlov 1996). Microsoft Excel is probably included in the computer package as part of
Microsoft Office, and thus no additional expense is required. Spreadsheet programs are
among the most commonly used software, and most Engineers have experience with them
even if at an elementary level. Excel offers a friendly user interface, flexible data
manipulation, built-in mathematical functions and instantaneous graphing of data (Brown
2001). Excel contains two functions for data analysis, REGRESSION for linear regression
model and SOLVER for non-linear model. Orlov (1996) explain how to use REGRESSION
function for linear model. Using SOLVER function was explained by (Brown 2001).
McCormick (2010) explained using REGRESSION function for non-linear model. ElGebeily and Yushau (2007) explained using Excel to perform numerical Integration.
3. Select the best equation
A regression equation can be used for several purposes. The set of variables that may be best
for one purpose may not be best for another. The purpose for which a regression equation is
constructed should be kept in mind in the variable selection process. Some of the purposes
may be broadly summarizes as Prediction, Description and Control. Prediction means that the
regression model provides best prediction of dependent variable. When a regression equation
International Journal of Geomatics and Geosciences
Volume 5 Issue 1, 2014
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Computing pit excavation volume using Multiple Regression Analysis
Ragab Khalil
is used for prediction, the variables are selected with an eye toward minimizing the mean
square error of prediction (max R2). The goal description model is to quantify the relationship
between one or more independent variables of interest and dependent variable, controlling for
the other variables. In situations where description the prime goal, the smallest number of
independent variables that explains the most substantial part of the variation in the dependent
variable are chosen. When a regression model is used for control, the purpose for
constructing the equation may be to determine the magnitude by which the value of an
independent variable must be altered to obtain a specified value of dependent variable (Karim
2004). For volume estimation, description is the suitable model. To select the effective
variables, a technique called Backward Elimination Procedure was used. The process of
selecting the equation variables was as follows:
1. Determine the fitted regression equation containing all possible independent variables,
considering
.
2. Remove variable that its coefficient close to zero or its P value is greater than 0.05
(Dallal, 2012).
3. Re-compute the regression equation for the remaining variables.
4. Repeat steps 2 and 3 to reach the smallest number of independent variables.
The intercept coefficient is included if it increases the values of R2 of the equation.
4. Application
The data in Example 2 of Chen and Lin (1991), which was also used by Easa (1998), Yanalak
(2005) and Mukherji (2012), were used for this application, because the purpose of this paper
was to compare volumes determined by the existing methods with volumes calculated by
integration of regression equation. The example in Chen and Lin (1991) involved a pit whose
ground surface is expressed with the function:
, where
and
(values are in meters) with exact volume = 118800 m3. The
following three cases shown in figure 1 for constructing the data grid were considered:
1. A 6 x 5 grid with equal intervals in the ( x ) directions (20 m) but with unequal
intervals in the ( y ) direction (25, 10, 30, 15, 10).
2. A 6 x 5 grid with equal intervals in the ( y ) directions (18 m) but with unequal
intervals in the ( x ) direction (15, 30, 10, 35, 10, 20).
3. A 6 x 5 grid with unequal intervals in both the ( x ) and ( y ) directions, ( x ) intervals
as described in cases 2 and ( y ) intervals as in case 1.
By applying the regression on case 1, 2 and 3 the surface can be expressed as
for case 1
for case 2
for case 3
Because the ground surface is expressed mathematically, the volume can be determined using
integration as 120570, 134726 and 123704 m3 respectively. True rational errors of the
volumes were calculated as ratio of the true error, which is the difference between estimated
and exact volume, to the exact volume. The new calculated volumes and rational true errors
International Journal of Geomatics and Geosciences
Volume 5 Issue 1, 2014
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Computing pit excavation volume using Multiple Regression Analysis
Ragab Khalil
are shown in Table 1 with those of the existing methods in Mukherji (2012) so that the results
can be compared easily.
Figure 1: Grids for application example (cases 1, 2, and 3)
By considering Table 1, which provides a summary of earlier investigations, including the
proposed method the following results can be outlined for the volume calculations handled in
this study:
1. Volume calculation with regression analysis is better than that of the trapezoidal
formula with the original data for the three cases. Regression method has rational true
errors of 1.5, 13.4, and 4.1%, but the trapezoidal formula with the original data has
values of 25.4, 19.2, and 19.2% for Cases 1–3, respectively.
2. Volume calculation with regression analysis is the best, with a rational true error of
1.5% for Case 1. The worst is Chambers (1989), with a value of 29.3%.
3. Volume calculation with regression analysis, which has the rational true error of
13.4%, is better than the trapezoidal method with 19.2% and the method of Yanalak
(2005) with 16.2% but worse than the methods of Chambers (1989) with 3.8%, Chen
and Lin (1991) with 2.7%, Easa (1998) with 3.7%, and Mukherji (2012) with 1.6%
for Case 2.
International Journal of Geomatics and Geosciences
Volume 5 Issue 1, 2014
46
Computing pit excavation volume using Multiple Regression Analysis
Ragab Khalil
4. With 4.1% rational true error, volume calculation with regression analysis is better
than the trapezoidal method, with 19.2%, and the methods of Chambers (1989),
Yanalak (2005) and Mukherji (2012) having the values of 15.5%, 8.3% and 13.6%
respectively and worse than the methods of Chen and Lin (1991) and Easa (1998),
which have the values of 2.6 and 3.7%, respectively, for Case 3.
Table 1: Comparative Application Results
Case 1
Case 2
Case 3
Method
Volume
Error %
Volume
Error %
Exact volume
118,800
—
118,800
—
Trapezoidal
149,009
25.4
141,615
19.2
Chambers (1989)
153,551
29.3
122,820
3.8
Chen and Lin (1991)
139,568
17.5
122,009
2.7
Easa (1998)
138,280
16.4
123,207
3.7
Yanalak (2005)
(best of 3)
136,117
14.6
138,085
16.2
Mukherji (2012)
137,096
15.4
120,649
1.6
Proposed regression
120,570
1.5
134,726
13.4
Volume
118,80
0
141,61
4
137,15
4
121,86
0
123,19
9
128,63
2
134,97
7
123,70
4
Error %
—
19.2
15.5
2.6
3.7
8.3
13.6
4.1
5. Conclusion
Volume calculation with regression analysis method can be used for both regular and
scattered data. It is very simple and fast method for volume estimation. It does not need
specific software; regression analysis tool is available with Microsoft Excel in almost all
computer machines. It just needs arranging data in Excel spreadsheet and apply the regression
analysis to get the coefficients of the equation that represents the surface .volume can be
estimated by integration of the gotten equation. Applying the regression analysis method on
the example used by the earlier investigators shown that it gave the best results for case 1, it
deviates from the true volume only by 1.5%. For case 3, the proposed regression analysis
deviates from the true volume only by 4.1% which is close to the best results gotten by earlier
authors.
6. References
1. Brown A.M., (2001), A step-by-step guide to non-linear regression analysis of
experimental data using a Microsoft Excel spreadsheet, Computer Methods and
Programs in Biomedicine, 65, pp 191–200.
2. Chambers, D. W., (1989), Estimating pit excavation volume using unequal intervals,
Journal of Surveying Engineering, 115(4), pp 390–401.
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Computing pit excavation volume using Multiple Regression Analysis
Ragab Khalil
3. Chen, C. S. and Lin, H. C., (1991), Estimating pit excavation volume using cubic
spline volume formula, Journal of Surveying Engineering, 117(2), pp 51–66.
4. Choi M. and Lee G., (2010), Decision tree for selecting retaining wall systems based
on logistic regression analysis, Automation in Construction, 19, pp 917–928.
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combined classification and regression technique, Automation in Construction, Vol.
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6. Dallal, G. E., (2012), The Little Handbook of Statistical Practice, available at
http://www.jerrydallal.com/LHSP/LHSP.HTM, accessed on 25 May 2014.
7. Easa, S. M., (1988), Estimating pit excavation volume using nonlinear ground profile,
Journal of Surveying Engineering, 114(2), pp 71–83.
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