ESTIMATING PROJECT DEVELOPMENT EFFORT USING CLUSTERED REGRESSION APPROACH

Jan Zizka (Eds) : CCSIT, SIPP, AISC, PDCTA - 2013
pp. 493–507, 2013. © CS & IT-CSCP 2013 DOI : 10.5121/csit.2013.3654
ESTIMATING PROJECT DEVELOPMENT
EFFORT USING CLUSTERED REGRESSION
APPROACH
Geeta Nagpal1
, Moin Uddin2
and Arvinder Kaur3
1
Dept. of Computer Science and Engg,
National Institute of Technology, Jalandhar
sikkag@gmail.com
2
Pro Vice Chancellor, Delhi Technological University, Delhi
prof_moin@yahoo.com
3
University School of Information Technology, GGSIPU, Delhi
arvinderkaurtakkar@yahoo.com
ABSTRACT
Due to the intangible nature of “software”, accurate and reliable software effort estimation is a
challenge in the software Industry. It is unlikely to expect very accurate estimates of software
development effort because of the inherent uncertainty in software development projects and the
complex and dynamic interaction of factors that impact software development. Heterogeneity
exists in the software engineering datasets because data is made available from diverse sources.
This can be reduced by defining certain relationship between the data values by classifying
them into different clusters. This study focuses on how the combination of clustering and
regression techniques can reduce the potential problems in effectiveness of predictive efficiency
due to heterogeneity of the data. Using a clustered approach creates the subsets of data having
a degree of homogeneity that enhances prediction accuracy. It was also observed in this study
that ridge regression performs better than other regression techniques used in the analysis.
KEYWORDS
Software estimation, Clustering, Grey relational analysis, Feature weighted Grey relational
based clustering
1. INTRODUCTION
Development of software is a creative process where each person’s efficiency is different. It is
initially difficult to plan and estimate as most software projects have deficient information and
vague associations amongst effort drivers and the required effort. Software developers and
researchers are using different techniques and are more concerned about accurately predicting the
effort of the software product being developed.
Most commonly used algorithmic models for software cost estimation include Boehm’s
COCOMO [1], Albrecht’s Function Point Analysis [2], and Putnam’s SLIM [3]. These models
require some cost drivers to estimate the effort of the project. In recent years, a number of soft
computing and computationally intelligent techniques have been proposed in order to handle the
unpredictability and inherent uncertainty contributed by cost drivers’ judgment and environment

494 Computer Science & Information Technology (CS & IT)
complexity of the projects. These techniques include artificial neural network, genetic algorithm,
support vector regression, genetic programming, neuro-fuzzy inference system and case base
reasoning. They use the historical datasets of completed projects as training data and predict the
values for new project’s effort based on previous training. Though, a significant improvement has
been achieved using soft computing techniques in software cost estimation, yet there exist some
limitations due to the heterogeneity in the datasets.
Soft computing techniques estimate accurately if there is some relationship between the tuples of
the dataset. Due to this heterogeneity that exists amongst software projects, these techniques are
not able to estimate optimally. This heterogeneity of data can be reduced by clustering the data
into some similar groups. The goal of clustering is to create the groups of data that have similar
characteristics. The clustering divides the data set X into k disjoint subsets that have some
dissimilarity between them.
A clustered regression approach has been proposed in this study in order to generate more
efficient estimation sub models. A feature weighted grey relational based clustering method has
been integrated with regression techniques. The feature weighted grey relational clustering
algorithm uses grey relational analysis for weighting features and also for clustering. The results
obtained showed that clustering could decrease the effect of irrelevant projects on accuracy of
estimations. Cluster specific regression models for the four publicly available data sets are
generated. Empirical results have shown that regression when applied on clustered data provides
some outstanding results, thus indicating that the methodology has great potential and can be used
for software effort estimation. The results are subjected to statistical testing using Wilcoxon
signed rank test and Mann_Whitney U Test.
The rest of the paper is organized as follows. Section 2 reviews some related works on clustering
algorithm and GRG as a similarity measure for feature weighting. Section 3, introduces the
modeling techniques. Further, in Section 4 we present the proposed methodology. Section 5,
gives description of the data sets used in the study and the experimental results that demonstrate
the use of proposed clustered regression approach in software effort estimation. In the end the
conclusion is made in Section 6.
2. REVIEW OF LITERATURE
A number of data clustering techniques have been developed to find the optimal subsets of data
from the existing datasets [4],[5],[6]. The main aim of clustering is to partition an unlabeled data
set into subsets according to some similarity measure, called unsupervised classification.
Clustering algorithms can be categorized into two main families: input clustering and input-
output clustering [7]. In input clustering algorithms all the attributes are considered as
independent. Hard c-means [8] and fuzzy c-means [9] algorithms fall into this category. In the
case of input-output clustering each multi-attribute data point is considered as a vector of
independent attribute with some corresponding dependent value. Let S = {(x1,y1), (x2,y2) ,… (xn,
yn)} be a set of unlabelled input-output data pairs. Each independent input vector xi = [x1i, x2i,…
xki] has a corresponding dependent value yi. Research work has been done to motivate this
category of classification [10],[11].
Kung and Su [12] developed an effective approach to establish affine Takagi-Sugeno (T-S) fuzzy
model for a nonlinear system from its input – output data. Chunheng, Cui and Wang [13]
proposed FCM-SLNMM clustering algorithm, consisting of two stages. The FCM algorithm was
applied in the first stage and supervised learning normal mixture model was applied in the second
stage. The clustering results of the first stage were used as training data. The experiments on the
real world data from the UCI repository showed that the supervised learning normal mixture

Computer Science & Information Technology (CS & IT) 495
model can improve the performance of the FCM algorithm sharply. Lin and Tsai [14] proposed a
hierarchical grey clustering approach in which the similarity measure was a globalized modified
grey relational grade instead of traditional distances. Chang and Yeh [15] generalized the concept
of grey relational analysis in order to develop a technique for analyzing the similarity between
given patterns. They also proposed a clustering algorithm to find cluster centers of a given
dataset.
In this study, GRA a technique of GST which utilizes the concept of absolute point-to-point
distance between features [16],[29] has been applied. GST a recently developed system
engineering theory, was first established by Deng [18],[19],[20]. It draws out valuable
information by generating and developing the partially known information. So far, GST has been
applied in different areas of image processing [23], mobile communication [24], machine vision
inspection [25], decision making [26], stock price prediction [27] and system control [28]. The
success of GST motivated us to investigate its application in software effort estimation.
3. MODELING TECHNIQUES
The data available for software cost estimation is inherently non linear and hence accurate
estimation of effort is difficult. Efficient estimation can be achieved if this non linearity that
exists can be treated by tracing relationships among data values. In this study, we try to reduce the
heterogeneity by applying feature weighted grey relational clustering methodology.
3.1 Regression
As discussed, a large number of techniques have been applied to the field of software effort
estimation. The aim of this study is to assess which regression techniques perform best to
estimate software effort. The following techniques are considered.
3.1.1 Ordinary Least Square Regression
It is the most popular and widely applied technique to build software cost estimation models.
According to principle of least squares the `best fitting' line is the line which minimizes the
deviations of the observed data away from the line. The regression parameters for the least square
line, are estimates of the unknown regression parameters in the model. This is referred to as
multiple linear regression and is given by:
ikikii xxy εβββ +++= ,1,10 ..... (1)
where, yi is a dependent variable where as x1,x2,........xk are k independent variables. βo is the y
intercept, β1 , β2 are the slope of y, εi is the error term.
3.1.2 Ridge Regression (RR)
RR is an alternative regression technique that tries to address the potential problems with OLS
that arise due to highly correlated attributes. In regression, the objective is to “explain” the
variation in one or more “response variables”, by associating this variation with proportional
variation in one or more “explanatory variables”, but the problem arises when the explanatory
variables vary in similar ways reducing , their collective power of explanation. The phenomenon
is known as near collinearity. As the different variables are correlated the covariance matrix X X
will be nearly singular and as a result the estimates will be unstable. A small variation in error
will have large impact on βˆ
. Ridge regression reduces the sensitivity by adding a number δ to

the elements on the diagonal of the matrix to be inverted. δ is called the ridge parameter and it
yields the following estimator of β.
( ) ( )eXIXX n ''ˆ 1−
+= δβδ
(2)
where, In represents the identity matrix of rank n.
3.1.3 Forward Stepwise Regression
The purpose of stepwise regression is to generate regression model in which the detection of most
predictive variables is carried out. It is carried out by a series of F tests. The method evaluates the
independent variables at each step, adding or deleting them from the model based on user-
specified criteria. In the first step, each of the independent variables are evaluated individually
and the variable that has the largest F value greater than or equal to the F to enter value, is entered
into the regression equation. In the subsequent steps, when a variable is added to the model based
on their F value, the method also examines variables included in the model based on F to remove
criteria, and if any variables are found they are removed.
3.1.4 Backward Stepwise Regression
The backward stepwise elimination procedure is basically a series of tests for significance of
independent variables. The process starts with the maximum model, it eliminates the variable
with the highest p -value for the test of significance of the variable, conditioned on the p -value
being bigger than some pre-determined level (say, 0.05). In the next step, it fits the reduced
model after having removed the variable from the maximum model, and also removes from
the reduced model the variable with the highest p-value for the test of significance of that variable
(if p>=0.05 ) and so on. The process ends when no more variables can be removed from the
model at significance level 5%.
3.1.5 Multiple Adaptive Regression Splines(MARS)
MAR splines focuses on the development and deployment of accurate and easy-to-understand
regression models. The MAR splines model is designed to predict continuous numeric and high
quality probability models. MAR spline model is a regression model which automatically
generates non-linearities and interactions between variables and is thus a promising technique to
be used for software effort estimation[21]. MAR splines fits the data to the following equation.
( )( )jxhbbe ii
K
k
L
iki ∑=
=Π+=
1
10
(3)
In this bo and bk are the intercept and slope. Parameters hi (xi (j)) are the hinge functions. They take
the form max (0, xi (j)-b) where, b is the knot. MAR splines is a multiple piece wise linear
regression by adding multiple hinge functions.
3.2 Grey Relational Analysis
GRA is comparatively a novel technique in software estimations. It is used for analyzing the
relationships that exists between two series. The magnetism of GRA to software effort estimation
shoots from its flexibility to model complex nonlinear relationship between effort and cost drivers
[16].

Grey Relational Grade by Deng’s Method [18],[19],[20]
GRA is used to quantify all the influences of various factors and the relationship among data
series that is a collection of measurements [16].The three main steps involved in the process are:
Data Processing: The first step is the standardization of the various attributes. Every attribute has
the same amount of influence as the data is made dimensionless by using various techniques like
upper bound effectiveness, lower bound effectiveness or moderate effectiveness.
Upper-bound effectiveness (i.e., larger-the-better) is given by:
( ) ( ) ( )
( ) ( )kxkx
kxkx
kx
iiii
iii
i
minmax
min*
−
−
=
(4)
where i=1,2,…,m and k=1,2,…,n.
Difference Series: GRA uses the grey relational coefficient to describe the trend relationship
between an objective series and a reference series at a given point in a system.
( ) ( )( )
( ) max,
maxmin
0 ,
∆+∆
∆+∆
=
ζ
ζ
γ
k
kxkx
io
i
(5)
where;
∆0,i(k) = |x0(k) − xi(k)| is the difference of the absolute value between x0(k) and xi(k);
∆min = minjmink |x0(k) − xj(k)| is the smallest value of ∆0,j ∀j ∈ {1, 2, . . . , n};
∆max = maxjmaxk |x0(k) −xj(k)| is the largest value of ∆0,j∀j ∈ {1, 2, . . . , n}; and
ζ is the distinguishing coefficient, ζ ∈ (0, 1].
The ζ value will change the magnitude of γ(x0(k), xi(k). In this study the value of ζ has been taken
as 0.5 [17].
Grey Relational Grade: GRG is used to find overall similarity degree between reference tuple xo
and comparative tuple xi. When the value of GRG approaches 1, the two tuples are “more closely
similar”. When GRG approaches a value 0, the two tuples are “more dissimilar”. The GRG Γ(x0,
xi) between an objective series xi and the reference series x0 was defined by Deng as follows:
( ) ( ) ( )( )∑=
=Γ
n
k
ii kxkx
n
xx
1
0,0 ,
1
γ
(6)
4. PROPOSED METHODOLOGIES
4.1 Clustered regression using grey relational analysis
In this methodology, in order to reduce the heterogeneity that exists in the datasets, the initial
focus is to partition datasets into subsets according to some similarity measure, called
unsupervised classification. The proposed clustering algorithm uses grey relational analysis for
feature selection as well as for clustering. In this the maximum mean grey relational grade

between data points acts as an objective function instead of the minimum distance used by k-
means. The structural flow chart is shown in figure 1.
The three main steps involved in the algorithm are:
Step 1: Using Grey relational analysis for finding feature weights.
Step 2: Using Grey relational analysis for clustering the datasets based on these feature-weights.
Step 3: Applying regression techniques on the clustered datasets.
Step 4: Effort Prediction by Regression Techniques
The detailed algorithm is described as follows:
Using Grey Relational Analysis for finding feature weights.
Feature Selection by GRA [16]
a. Construction of data: The columns in each cluster dataset are treated as series . The effort
series xe ={e1,e2,e3.......en} is taken as the reference series and the attribute columns are
regarded as objective series.
b. Normalization: Each data series is normalized according to equation 4, so that they have
same degree of influence on the dependent variable “effort”.
c. Generation of Grey Relational Grade: Grey relational grade(GRG) is calculated for each
series wrt reference series according to equation 6.
The GRG’s are generated, normalized and used as the corresponding features weight wk .
Using Grey Relational Analysis for clustering the datasets based on these feature-weights.
After finding the weights of the features from the first step it applies the clustered approach based
on grey relational analysis. The detailed algorithm is described as follows:
a. Weight of each feature is generated as described earlier.
b. Normalize the data with larger the better as per equation 4,
c. Calculate distance between data points based on weighted GRG
• Consider the ith
data point as reference series xo
• All the other data points as the objective series, { x1,x2,x3, ……xn-1 }
• Calculate the grey relational coefficient with ζ=0.5 [17].Calculate weighted GRG
of the reference series and feature weight calculated as in step 1 for all objective
series.
d. Randomly select the number of desired of clusters center ck.
e. GRG distance from data points to cluster centers is used as a basis to select cluster
members, for which it has maximum GRG value.
f. Update the cluster centers by selecting centers based on the maximum mean GRG, then
repeat step e.
g. Repeat steps e and f, until there is no change in the cluster head updating or the difference
between the mean is less than some predefined threshold value.
5. EXPERIMENTAL RESULTS
5.1 Dataset
In order to evaluate the models based upon the proposed methodology, four well established
datasets from the Promise repository [22] have been used for validating our model. These datasets
are Desharnais, Finnish, Albrecht and Maxwell. The descriptive statistics of the datasets are
shown in Table 1 given below. All the datasets have a varied range of effort values. These

datasets have been treated individually as they have distinct features. Also the clusters from each
dataset have been treated separately. The prediction accuracy for all the models with and without
clustering are then compared. In order to measure the accuracy of the software estimation, we
have used three most popularly used evaluation criteria in software engineering i.e MMRE,
MdMRE and Pred(n).
Figure1. Structural flowchart for feature weighted Grey Relational Clustering
Table1. Descriptive Statistics of the data sets
Cases Features Effort Mean Minimum
(effort
value)
Maximum
(effort
value)
Effort Std.
Dev.
Albrecht 24 8 21.875000 0.50 105.20 28.417895
Desharnais 77 11 4833.9090 546 23940 4188.1851
Finnish 38 8 7678.2894 460 26670 7135.2799
Maxwell 62 23 8223 583 63694 10499.903
5.1.1 Comparison over Desharnais data set
The results obtained suggest that applying regression technique on clustered data produces more
accurate estimation models than applying regression on the entire datasets. This is evident from
the results obtained shown in Table 2. The Pred(25) accuracy has improved from 35.06 % to 50
% using OLS regression whereas, the MMRE and MdMRE has fallen from 0.5 to 0.32 and from
0.31 to 0.25 respectively. Similar observations can be notified from the table below for all other
regression models also. Best results have been obtained on using the proposed feature weighted
grey relational clustering.
Select Continuous Attributes
Data Processing
Grey Relational Coefficient
Grey Relational Grade
Historical data
sets
Cluster of each data set
Ordinary Least
Square Regression
Ridge Regression
Forward Stepwise
Backward Stepwise
Estimated Effort by Regression
techniques
MAR Splines
Regression Techniques
Data Processing
Grey Relational Coefficient
GRA for
Clustering
using the
Feature
weights
GRA for
finding
Feature
weights
Grey Relational Grade

Table 2. Prediction accuracy results (Desharnais data set)
Figure 2. Boxplot of Absolute Residuals for Desharnais
The Boxplot of absolute residuals provide good indication of the distribution of residuals and can
help better understand the mean magnitude of relative error and Pred (25).The results obtained
were subjected to statistical tests using Wilcoxon Signed rank test and Mann Whitney U test.
The box plots of absolute residuals shown in figure 2 suggest that:
The medians for all regression techniques applied on Desharnais _Cluster are more close to zero,
as it is clear from the values on the Y-axis, indicating that the residuals were closer to the
minimum value. The Outliers are few and less extreme in cases of Desharnais_Clusters as
compared to Desharnais data set. As the p-value in all the cases shown in Table 3 is greater 0.05
where in we conclude that the residuals obtained in all approaches are not significantly different
from the test value zero. As a result, the proposed methods can be used for software effort
estimation. The statistical tests were performed using SPSS 19 for windows.
Table 3. Wilcoxon signed rank test Test
The results of Mann_Whitney U Test are provided in Table 4. Predictions obtained using the
clustered approach presented statistically significant estimations.
OLS Ridge
Regression
Forward
Stepwise
Backward
Stepwise
MAR Splines
Desharnais
MMRE 0.5 0.47 0.5 0.5 0.51
MdMRE 0.31 0.3 0.31 0.31 0.32
Pred(25) 35.06 41.56 37.66 37.66 35.06
Desharnais(Cluster_1) using Grey Relational Clustering
MMRE 0.32 0.35 0.33 0.39 0.39
MdMRE 0.25 0.23 0.24 0.21 0.20
Pred(25) 50 55.56 50 52.78 58.33
Desharnais Desharnais_Cluster_1
using Grey Relational
Z Asymp. Sig. (2-tailed) Z Asymp. Sig. (2-
tailed)
OLS-Actual -.419a .675 -.236a .814
Ridge – Actual -.551a .582 -.299a .765
Forward - Actual -.449a .653 -.189a .850
Backward- Actual -.449a .653 -.314a .753
MARS– Actual -.566a .571 -.236a .814

Table 4. Results Mann-Whitney U Test
5.1.2 Comparison over Finnish data set:
For the Finnish dataset, some significant results (as shown in Table 5.) were obtained on the
clustered data. The Pred(25)accuracy improved from 36.84 % to 100 % using OLS regression
whereas, the MMRE and MdMRE has fallen from 0.75 to 0.02 and from 0.36 to 0.02 respectively.
Similar observations can be notified from the table below for all other regression models also.
The boxplot of absolute residuals for Finnish dataset and Finnish_cluster is shown in Figure 3.
They suggest that:
The medians for all regression techniques applied on Finnish _Cluster are very close to zero, as it
is clear from the values on the Y-axis, indicating that the estimates were closer to the minimum
value. Outliers are less extreme in case of Finnish_Cluster. One sample Wilcoxon signed rank test
has been applied in order to investigate the significance of the results by setting level of
confidence to 0.05. From the results obtained as shown in Table 6, we can conclude that no
significant difference exists between the residual median and hypothetical median.
Table 5. Prediction accuracy results (Finnish data set)
Unsurprisingly, predictions based on clustered regression model presented statistically significant
accurate estimations, measured using absolute residuals, confirmed by the results of boxplot of
absolute residuals and also verified using Mann-Whitney U test (Table 7.)
Figure 3. Boxplot of Absolute Residuals for Finnish
Desharnais vs. Desharnais_Cluster_1
using Grey Relational Z
OLS Regression -4.018
Ridge Regression -4.079
Forward Stepwise -4.252
Backward Stepwise -4.227
MAR Splines -4.240
OLS Ridge
Regression
Forward
Stepwise
Backward
Stepwise
MAR Splines
Finnish
MMRE 0.75 0.71 1.01 0.76 0.08
MdMRE 0.36 0.32 0.43 0.42 0.07
Pred(25) 36.84 36.84 36.84 36.84 97.37
Finnish(Cluster_1) using Grey Relational Clustering
MMRE 0.02 0.025 0.23 0.022 0.022
MdMRE 0.02 0.024 0.022 0.023 0.023
Pred(25) 100 100 100 100 100

Table 6. Wilcoxon signed rank test
5.1.3 Comparison over Albrecht data set:
The results obtained using the proposed clustered regression approach produced more accurate
models. This is evident from the Pred(25) accuracy that improved from 37.5 % to 85.71 % using
OLS whereas, the MMRE and MdMRE has fallen from 0.9 to 0.09 and from 0.43 to 0.05
respectively. Similar observations can be notified for all other regression models also (Table 8).
Table 8. Prediction accuracy results (Albrecht data set)
The box plots of absolute residuals suggest that:
The medians for all regression techniques applied on Albrecht_Clusters are very close to zero, as
it is clear from the values on the Y-axis, indicating that the residuals were closer to the minimum
value. Outliers are less extreme in case of Finnish_Cluster.
Finnish Finnish_Cluster_1
Z AsympSig (2-
tailed)
Z Asymp.Sig.
(2tailed)
OLS- Actual -.268a .788 -.408a .683
Ridge – Actual -.355a .722 -.220b .826
Forward- Actual -.268a .788 -.157b .875
Backward- Actual -.152a .879 -.345a .730
MARS– Actual -.558a .577 -.282a .778
Finnish vs.
Finnish_Cluster_1
MAR Splines -1.939
OLS Ridge
Regression
Forward
Stepwise
Backward
Stepwise
MAR Splines
Albrecht
MMRE 0.9 0.91 0.86 1 1.23
MdMRE 0.43 0.52 0.5 0.49 0.6
Pred(25) 37.5 37.5 41.67 37.5 29.17
Albrecht(Cluster_1) using Grey Relational clustering
MMRE 0.092 0.21 0.19 0.08 0.225
MdMRE 0.05 0.24 0.22 0.025 0.175
Pred(25) 85.71 50 57.14 85.71 57.14

Figure 4. Boxplot of Absolute Residuals for Albrecht
The results of Wilcoxon signed rank conclude that no significant difference exists between the
residual median and hypothetical median, thus indicating good predictions (Table 9.)
The results obtained using Mann-Whitney U test shown in Table 10, however didn’t prove
significant difference between the proposed approaches. This is because of the small size of the
dataset. The data set comprises of 24 projects. It was divided into two clusters one with 20
projects and other with 4 projects. Clustered regression approach was applied on 20 projects.
5.1.4 Comparison over Maxwell Dataset
The results obtained in Table 11 using the proposed clustered regression approach produced more
accurate models for Maxwell dataset also. This is evident from the Pred(25) accuracy that
improved from 38.71 % to 51.51 % using OLS regression whereas, the MMRE and MdMRE has
fallen from 0.59 to 0. 51 and from 0.38 to 0.24 respectively. For Ridge regression also, the
Pred(25) accuracy has increased from 43.55% to 60.60% which is significant improvement, the
MMRE and MdMRE has gone low from 0.54 to 0.33 and 0.3 to 0.18 respectively.
Albrecht Albrecht_Cluster_1
Z Asymp.Sig.
(2-tailed)
Z Asymp.Sig (2-
tailed)
OLS- Actual -.029a .977 -.031a .975
Ridge – Actual -.057a .954 -.031a .975
Forward - Actual -.057a .954 -.031a .975
Backward Actual -.086b .932 -.031a .975
MARS– Actual -.029b .977 -.031a .975
Albrecht vs.
Albrecht_Cluster_1
Forward Stepwise -.943
MAR Splines -1.108

Table 11. Prediction accuracy results(Maxwell data set)
The box plots of absolute residuals suggest that:
The medians for all regression techniques applied on Maxwell _Cluster are more close to zero, as
it is clear from the values on the Y-axis, indicating that the estimates were closer to the minimum
value. The medians are more skewed to the minimum value indicating that the predictions are
good. Outliers are few and less extreme in case of Maxwell_Cluster as compared to entire dataset.
Figure 5. Boxplot of Absolute Residuals for Maxwell
Results of Wilcoxon signed rank test suggest that no significant difference exists between the
residual median and hypothetical median. The results of Wilcoxon signed rank test are given in
Table 12.
Concerning the statistical test based on Mann-Whitney U(Table 13), we found no significant
difference between clustered regression approach and regression approach.
OLS Ridge
Regression
Forward
Stepwise
Backward
Stepwise
MAR
Splines
Maxwell
MMRE 0.59 0.54 0.53 0.59 0.7
MdMRE 0.38 0.3 0.32 0.33 0.46
Pred(25) 38.71 43.55 38.71 37.1 32.26
Maxwell(Cluster_1) using Grey Relational Clustering
MMRE 0.51 0.33 0.46 0.60 0.75
MdMRE 0.24 0.18 0.25 0.25 0.52
Pred(25) 51.51 60.60 48.48 48.48 24.24
Maxwell Maxwell_Cluster_1
Z Asymp. Sig
(2-tailed)
Z Asymp. Sig (2-
tailed)
OLS-Actual -249a .803 -.068a .946
Ridge – Actual -.508a .611 -.616b .538
Forward – Actual -.691a .490 -.068b .946
Backward-Actual -.831a .406 -.023b .982
MARS – Actual -.109a .913 -.205a .838

6. CONCLUSION
This work resolves the heterogeneity problems that exist in the datasets. In order to confirm the
effectiveness of proposed work, four different data sets have been used for software estimation.
Simulation results obtained provide a comparison of clustered regression approach over only
regression. The results confirm that the proposed feature weighted grey relational clustering
algorithm performed appreciably for software effort estimation. The statistical test based on
Mann-Whitney U, further confirmed that statistical significant difference exists between the
proposed clustered-regression models and regression models.
Further, this work can be extended by using clustered approach with different soft computing
techniques with different similarity measures for feature selection. GRA can also be analyzed for
feature subset selection Also, for enhanced efficiency in software estimation the techniques
should be applied on large data sets with different clustering algorithms.
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Authors
Geeta Nagpal
Geeta Nagpal, Ph D in Computer Science & Engineering from National Institute of
Technology, Jalandhar, INDIA. She completed her Master’s degree in Computer
Science from Punjab Agricultural University, Ludhiana. She is presently working as
Associate Professor in the Department of Computer Science and Engineering at
National Institute of Technology, Jalandhar. Her research interests are Software
Engineering, Databases and Data mining.
Prof. Moin Uddin
Prof. Moin Uddin, Pro Vice Chancellor, Delhi Technological University, Delhi
,INDIA. He obtained his B.Sc. Engineering and M.Sc. Engineering (Electrical)
from AMU, Aligarh in 1972 and 1978 respectively. He obtained hid Ph. D degree
from University of Roorkee, Roorkee in 1994. Before Joining as the Pro Vice
Chancellor of Delhi Technological University, he was the Director of NIT,
Jalandhar for five years. He has worked as Head Electrical Engineering Department
and Dean Faculty of Engineering and Technology at Jamia Millia Islamia (Central
University) New Delhi. He supervised 25 Ph. D thesis and more than 30 M.Tech
dissertations. He has published more than 40 research papers in reputed journals
and conferences. Prof. Moin Uddin holds membership of many professional bodies. He is a Senior Member
of IEEE.
Dr. Arvinder Kaur
Dr. Arvinder Kaur, Professor, University School of IT, Guru Gobind Singh
Indraprastha University, Delhi, India. She completed her masters degree in
Computer Science from Thapar Institute of Engineering and Technology and Ph D
from Guru Gobind Singh Indraprastha University, Delhi. Prior to joining the school,
she worked with Dr. B. R. Ambedkar Regional Engineering College, Jalandhar and
Thapar Institute of Engineering and Technology. Her research interests include
Software Engineering, Object-Oriented Software Engineering, Software Metrics,
Microprocessors, Operating Systems, Artificial Intelligence, and Computer networks. She is a lifetime
member of ISTE and CSI. She is also a member of ACM. She has published 45 research papers in national
and international journals and conferences. Her paper titled, “Analysis of object oriented Metrics”was
published as a chapter in the book Innovations in Software Measurement (Shaker-Verlag, Aachen 2005).

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