Indian Journal of Engineering & Materials Sciences
Vol. 17, June 2010, pp. 179-185
Modeling slump of concrete using the group method data handling algorithm
Li Chen* & Tai-Sheng Wang
Department of Civil Engineering and Engineering Informatics, Chung Hua University, Hsinchu, Taiwan 30067, R.O.C.
Received 1 December 2008; accepted 15 April 2010
This paper proposes the group method data handling (GMDH) algorithm and applies it to estimate the slump of highperformance concrete (HPC). It is known that HPC is a highly complex material whose behaviour is difficult to model,
especially for slump. To estimate the slump, it is a nonlinear function of the content of all concrete ingredients, including
cement, fly ash, blast furnace slag, water, superplasticizer, and coarse and fine aggregate. Therefore, slump estimation is set
as a function of the content of these seven concrete ingredients and additional four important ratios. The GMDH algorithm
presented in this paper has the advantage of a heuristic self-organized and gradually complicated model for the complicated
multi-variable HPC slump estimation. The model establishes the input-output relationship of a complex system using a
multilayered perception-type structure that is similar to a feed-forward multilayer artificial neural network (ANN), but it
expresses relationships using more explicit functions than ANN. Moreover, the GMDH has the ability to select significant
variables and combine them properly and automatically. The results show that GMDH obtains a more accurate mathematical
equation through learning procedures which outperforms the traditional multiple linear regression analysis (RA) and ANN,
with lower estimating errors for predicting the HPC slump.
Keywords: Group method data handling, Slump, High-performance concrete, Regression analysis
Workability in concrete technology is one of the key
properties that must be satisfied1 which consists of at
least two main components consistency and
cohesiveness. To measure the consistency or flow
characteristic of a concrete mixture, the slump test is a
fairly good method. The slump can be deduced by
measuring the drop from the top of the slumped fresh
concrete. However, these tests, if carried out on site
by site workers, may give inadequate results due to
lack of professional knowledge and proper training2.
The essence of high-performance concrete (HPC) is
emphasized on such characteristics as high strength,
high workability with good consistency, dimensional
stability and durability3. Nowadays, HPC can be made
with about four to ten different components as a
highly complex material that modeling its behaviour
is a difficult task, especially for the slump. In addition
to the three basic ingredients in conventional
concrete, i.e., Portland cement, fine and coarse
aggregates, and water, the making of HPC needs to
incorporate supplementary cementitious materials,
such as fly ash and blast furnace slag, and chemical
admixture, such as superplasticizer4. In other words,
the number of properties to be adjusted has also
——————
*Corresponding author (E-mail: [email protected]).
increased results in the waste of materials, laborers
and time. Furthermore, in laboratory, to obtain desired
concrete strength with suitable workability, technical
personnel must try several mix proportions5.
Modeling slumps from laboratory are not adequate
to include many factors that need to be considered
when designing HPC mixes. Therefore, it becomes
more difficult to estimate the slump of concrete with
these complex materials described above. The
traditional approach used in modeling the effects of
these parameters on the slump of concrete starts with
an assumed form of analytical equation and is
followed by a regression analysis using experimental
data to determine unknown coefficients in the
equation6. Unfortunately, rational and easy-to-use
equations are not yet available in design codes to
accurately predict the slump. In recent years, artificial
neural networks (ANNs) have shown exceptional
performance as regression tools7. They are highly
nonlinear, and can capture complex interactions
among input/output variables in a system without any
prior knowledge about the nature of these
interactions. The main advantage of ANNs is that one
does not have to explicitly assume a model form,
which is a prerequisite in the parametric approach8.
There are a lot of recent applications of neural
networks in civil engineering materials3,5,7-16.
180
INDIAN J. ENG. MATER. SCI., JUNE 2010
Group method of data handing (GMDH) algorithm
is another useful data process for identifying complex
systems, which was presented by Ivakhnenko17-19. The
main idea is that the gradually complicated models
are generated based on the evaluation of their
performances on a set of multi-input-single-output
data pairs20. It has an advantage over traditional
statistical methods because it is distribution free, i.e.,
no prior knowledge is needed about the statistical
distribution of the data like the ANN. In other words,
we need not to know the properties of system; GMDH
can generate and compare all possible combinations
between input and output variables automatically.
However, this method has received very little
attention in the concrete mixture literature despite
successful use in broad areas such as education,
economic systems, weather modeling, manufacturing,
pattern recognition, physiological experiments19,21-26.
In this study, the GMDH algorithm therefore was
represented and used to estimate the slump of
concrete. The results were compared with those
obtained from the regression analysis and ANN.
By means of GMDH algorithm a model can be
represented as set of neurons in which different pairs
of them in each layer are connected through a
quadratic polynomial and thus produce new neurons
in the next layer. Such representation can be used in
to map inputs to outputs. The formal definition of the
^
identification problem is to find a function f so that
can be approximately used instead of actual one, f in
^
order to predict output Y for a given input
vector x = ( x 1 , x 2 , . . . . . , x n ) as close
as possible to its actual output Y . Therefore, assume
the output variable Y is a function of the input
variables ( x 1 , x 2 , . . . . . , x n ) , as in the
following equation
(Y
1
, x
2
, ....., x
n
)
… (1)
The Kolmogorov-Gabor polynomial28-33:
Yˆ = a 0 +
n
n
∑ax
i
i
i =1
n
GMDH Algorithms
GMDH is a heuristic self-organization method that
models the input-output relationship of a complex
system using a multilayered Rosenblatt's perceptiontype network structure, which is similar to a feedforward multilayer neural network. Each element in
the network implements a non-linear equation of two
inputs and its coefficients are determined by a
regression analysis. Self-selection thresholds are
given at each layer in the network to delete those
useless elements which cannot estimate the correct
output. Only those elements whose performance
indices exceed the threshold are allowed to pass to
succeeding layers, where more complex combinations
are formed. These steps are repeated until the
convergence criterion is satisfied or a predetermined
number of layers are reached. In general, the
advantageous characteristics of the GMDH algorithm
for modeling or problem solving can be summarized
as follows: (i) a small training set of data is required;
(ii) the computational burden is reduced; (iii) the
procedure automatically filters out input properties
that provide little information about the location
and shape of the decision hypersurface and
(iv) a multilayers structure is a computationally
feasible way to implement multinomials of
very high degree27.
f ( x
=
+∑
i =1
n
+
n
∑∑a
i =1
ij
xi x j
j =1
n
∑∑ a
ijk
… (2)
xi x j xk + ....
j =1 k =1
can simulate the input-output relationship perfectly
and has been widely used as a complete description of
the system model. By combining the so-called partial
polynomial of two variables in multilayers, GMDH
algorithm can easily solve these problems. The main
process is summarized in the following sequence.
Step 1. Select input variables
N useful input variables are chosen. In the case of
the estimation of slump, the components of concrete
may be chosen. The model is set as Eq. (1):
Y
=
f ( x 1 , x 2 ,.....,
x n ) ,
where Y and xi represent vectors of the output and ith
input, respectively.
Step 2. Divide the original data into a training and a
testing set.
Step 3. Construct new intermediate variables
In this step, all of the independent variables are
taken two at a time to construct the partial polynomial
equation
Yˆi = f ( x j , x k ) = a 0 i + a1i x j + a 2 i x k
+ a3i x 2j + a4i xk2 + a5i x j xk
i = 1,..., q; j = 1,..., n; k = 1,..., n − 1; q =
n(n − 1)
2
… (3)
CHEN & WANG : GROUP METHOD DATA HANDLING ALGORITHM
The method of least squares is used to estimate the
coefficients so that the equation will best fit the
observed slump of concrete, Y.
The coefficients matrix A = [a0i,…a5i] can be
calculated as:
XA = Y→(XT X ) A =XT Y
A = (XT X )-1 XT Y,
where XT is the transpose of X and (XTX)-1XT is the
pseudo-inverse of X if XTX is non-singular.
Step 4. Select the new variables.
Evaluate the total RMSE in the preceding step. We
keep only n best nodes in each layer which are
allowed to pass to the succeeding layer. These new
variables can be interpreted as new improved
variables that have better predictability powers than
the previous generation.
Step 5. Truncate the multilayered iterative
computation.
Compare the best result of the present layer with
the preceding layer; if the improvement does not
exceed the defined threshold or reach the maximum
layer, the stop criterion is satisfied; otherwise go to
step 3.
Step 6. Compute the predicted value.
The prediction model can be obtained as the
intermediate variables remaining in the final layer.
Modeling the Slump of Concrete
System models
The properties of concrete are mainly influenced
by the mix proportion. This system identification
problem may be viewed as a search for a proper
model, which maps input values of ingredients onto
an output value of slump of HPC by using GMDH
described in this paper. There are seven ingredients
used to produce the HPC: (i) cement (C, kg/m3);
(ii) fly ash (FL, kg/m3); (iii) blast furnace slag (SL,
kg/m3); (iv) water (W, kg/m3); superplasticizer (SP,
kg/m3); coarse aggregate (CA, kg/m3); and fine
aggregate (FA, kg/m3). Table 1 presents the general
properties of the concrete evaluated in this study. In
addition to the seven components, four ratios were
included as input features defined as follows.
Water-to-cement ratio: W/C = (W+SP) / (C)
Water-to-binder ratio: W/B = (W+SP)/ (C+FL+SL)
Water-to-solid
ratio:
W/S
=
(W+SP)
/(C+FL+SL+CA+FA)
Total aggregate-to-binder ratio: TA/B = (CA+FA) /
(C+FL+SL)
181
Therefore, in this approach, slump of concrete is a
nonlinear function of these eleven input variables
described above.
Data set
Experimental data from Chen34,35 and Lien36 was
used to construct of the slump model. The fresh
concrete was assessed by the slump test. To collect
training and testing data systematically, mix
proportions were performed using the design of
mixture experiment. In this study, the experiments
were designed according to a simplex-centroid design
(SCD)3. In all 100 concrete samples from the above
investigations were evaluated, each containing seven
components and four ratios, total eleven of the input
vector and one output value, slump (from 0 to 30 cm).
Modeling procedures
All data were grouped in two sets, called the
training (calibration) set and the testing (validation)
set. When the training process had been completed,
the constructed model was used to predict the output
values for the data in the testing set (which the
process had never seen during the training stage).
Therefore, using these HPC data for learning by
GMDH depends on randomly splitting 100 records
into two groups: (i) The first group is used for training
the model called the training set, which consists of
seventy five records and (ii) The second group is used
to measure the performance of the model called the
testing set, which consists of twenty five records.
Results and Discussion
First, all the eleven input variables are standardized
from 0.1 to 0.9, then the GMDH algorithms are
applied to the slump estimation. The number of input
variables of the first layer is set to be eleven, while
Table 1—Statistical properties of components
Components
Minimum Maximum
3
Cement (kg/m )
Fly ash (kg/m3)
Blast furnace slag (kg/m3)
Water (kg/m3)
Superplasticizer (kg/m3)
Coarse aggregate (kg/m3)
Fine aggregate (kg/m3)
Water-to-cement ratio (W/C)
Water-to-binder ratio (W/B)
Water-to-solid ratio (W/S)
Total aggregate-to-binder ratio (TA/B)
Slump (cm)
137.0
0.0
0.0
160.0
4.4
708.0
640.6
0.5
0.3
0.075
2.363
0.0
374.0
193.0
260.0
240.0
19.0
1049.9
902.0
1.736
0.678
0.125
5.562
29.0
INDIAN J. ENG. MATER. SCI., JUNE 2010
182
the number of input variables of the rest of the layers
is also limited to eleven. During the training stage, the
GMDH model builds up through twenty layers with
eleven nodes in each layer. The convergence diagram
is shown in Fig. 1. At the final (20th) layer, the root
mean square error (RMSE) equals 3.07 (cm). To
realize the mechanism in detail, the fittest function
between eleven input features and the slump of
concrete generated from GMDH with only two layers
was shown as Fig. 2 and Eq. (4).
Y = −0.153 + 0.504 f ( xFL , xW ) + 1.657 f ( xW , xW
2
−2.82 f ( xFL , xW ) − 4.312 f ( xW , xW
+5.884 f ( xFL , xW ) f ( xW , xW
B
B
)
B
)
2
)
f ( xFL , xW ) = 0.411 + 0.013xFL + 1.199 xW
2
−0.909 xFL
− 1.009 xW2 + 0.857 xFL xW
f ( xW , xW
B
) = 0.019 + 0.896 x
+ 1.723 xW
W
2
W
2
W B
−0.726 x − 1.919 x
B
+ 0.424 xW xW
B
… (4)
where Y is the slump of HPC.
The nodes in grey represent the optimal solution of
this problem, which consists of three nodes in the
input, two nodes in the first layer, and only one node
in the second layer. This shows that only three input
variables were available, including fly ash (FL), water
(W) and water-to-binder ratio (W/B). The RMSE at
this stage (two layers) equals 4.55 (cm) also shown in
Fig. 1. These two components and one ratio are very
significant variables to model the slump of HPC.
Comparison with multiple linear regressive analysis (RA)
In the conventional material modeling process,
regression analysis (RA) is an important tool for
building a model. Because we don’t know the proper
form of these functions, only the simplest linear type
was considered. The established form including
eleven variables was given by:
Fig. 1—Convergence diagram of slump estimation by GMDH
Y = 466.41- 0.4536C - 0.459 FL - 0.4663SL
+3.4929W + 3.1385SP - 0.1761CA
-0.1791FA + 2.6744(W / C ) -199.8158(W / B )
-5625.2125(W / S ) - 4.6645(TA / B)
… (5)
Comparison with artificial neural network (ANN)
Fig. 2—Structure of GMDH with two layers
The artificial neural network with backpropagation algorithm, called back-propagation
network (BPN), might be one of the most widely used
models for estimation. The same data were selected
for use in the training and testing stages to compare
the performance of GMDH with that of BPN. In the
BPN with the gradient descent algorithms, there are
some combinations of neural parameters that are set
by trials. It uses two hidden layers with eight nodes at
each layer and is terminated after 1000 iterations for
training procedure.
The criteria of root mean square error (RMSE) and
coefficient of determination (R2) were used for
evaluating the performance of these three models,
which are summarized in Table 2. Obviously, the
results of GMDH (RMSE = 3.07 cm for the training
set; 4.54 cm for the testing set) are better than those of
RA (RMSE = 4.96 cm for the training set; 8.82 cm for
CHEN & WANG : GROUP METHOD DATA HANDLING ALGORITHM
the testing set). The RMSE of BPN equals 3.20 cm
for the training set, which is slightly worse than that
of GMDH, but the RMSE of BPN equals 7.46 cm for
the testing set, which is much worse than that of the
GMDH.
Table 2—The results of three models at two stages
Criteria
R2 for the training set
RMSE for the training set (cm)
R2 for the testing set
RMSE for the testing set (cm)
GMDH
0.8582*
3.0687*
0.7396*
4.5357*
RA
0.6214
4.9619
0.2002
8.8242
*represents the best result of these three models.
BPN
0.8380
3.2018
0.5366
7.4636
183
According to R2, it indicates a significant enough
correlation by using GMDH (R2 = 0.93 for the training
set; 0.86 for the testing set). On the contrary, the
coefficient of determination R2 is 0.2002 by RA and
0.5366 by BPN for the testing set, both indicate low
correlations. Figures 3- 5 show the scatter diagrams of
predicted slump values versus values observed in the
laboratory for these three models at the training stage.
Figures 6-8 show the scatter diagrams of predicted
slump values versus values observed in the laboratory
for these three models at the testing stage. One can
tell that the predicted values of GMDH are much
[
Fig. 3—Scatter plots of GMDH for the training set
Fig. 5—Scatter plots of BPN for the training set
Fig. 4—Scatter plots of RA for the training set
Fig. 6—Scatter plots of GMDH for the testing set
184
INDIAN J. ENG. MATER. SCI., JUNE 2010
Fig. 7—Scatter plots of RA for the testing set
Fig. 8—Scatter plots of BPN for the testing set
closer to the ideal line than the other two methods. It
is also indicated that the model obtained by GMDH
more accurately predicts the experimental results for
both the training and testing data in the range of
concrete slump in this study. In contrast with GMDH,
it verifies that when the testing set is used instead of
the training set as the basis for evaluating the slump
model derived with RA or BPN, the predictions
become much more inaccurate for the model used in
this study.
traditional multiple regression analysis (RA) and
back-propagation network (BPN), the performances
of GMDH with twenty layers are much better than
those of the RA and slightly better than those of BPN.
Conclusions
The main contribution of this paper is to provide a
self-organization method called group method data
handling (GMDH) algorithm, which creates potentials
to predict the slump of concrete. This model can deal
easily with nonlinear problems through multilayer
network among seven components including
(i) cement (C, kg/m3); (ii) fly ash (FL, kg/m3); (iii)
blast furnace slag (SL, kg/m3); (iv) water (W, kg/m3);
(v) superplasticizer (SP, kg/m3); (vi) coarse aggregate
(CA, kg/m3); and (vii) fine aggregate (FA, kg/m3) and
four ratios, versus the slump of high-performance
concrete (HPC). The highly nonlinear equation
obtained using GMDH with two layers helps us
realize the mixture mechanisms in a transparent way,
containing only three significant input variables
including fly ash (FL), water (W), and water-to-binder
ratio (W/B). The results also show that the GMDH
presented in this paper is a very efficient and robust
system identified model. Compared with the
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