CC BY
23
UDC 004.412:519.237.5
THE NON-LINEAR REGRESSION MODEL TO ESTIMATE THE SOFTWARE SIZE OF OPEN SOURCE JAVA-BASED SYSTEMS
Prykhodko N. V. - PhD, Associate Professor, Associate Professor of the Finance Department, Admiral Makarov National University of Shipbuilding, Mykolaiv, Ukraine.
Prykhodko S. B. - Dr. Sc., Professor, Head of the Department of Software of Automated Systems, Admiral Makarov National University of Shipbuilding, Mykolaiv, Ukraine.
ABSTRACT
Context. The problem of estimating the software size in the early stage of a software project is important, since the information obtained from estimating the software size is used for predicting the software development effort, including open-source Java-based information systems. The object of the study is the process of estimating the software size of open-source Java-based information systems. The subject of the study is the regression models for estimating the software size of open-source Java-based information systems.
Objective. The goal of the work is the creation of the non-linear regression model for estimating the software size of open-source Java-based information systems on the basis of the Johnson multivariate normalizing transformation.
Method. The model, confidence and prediction intervals of multiply non-linear regression for estimating the software size of open-source Java-based information systems are constructed on the basis of the Johnson multivariate normalizing transformation for non-Gaussian data with the help of appropriate techniques. The techniques to build the models, equations, confidence and prediction intervals of non-linear regressions are based on the multiple non-linear regression analysis using the multivariate normalizing transformations. The appropriate techniques are considered. The techniques allow to take into account the correlation between random variables in the case of normalization of multivariate non-Gaussian data. In general, this leads to a reduction of the mean magnitude of relative error, the widths of the confidence and prediction intervals in comparison with the linear models or nonlinear models constructed using univariate normalizing transformations.
Results. Comparison of the constructed model with the linear model and non-linear regression models based on the decimal logarithm and the Johnson univariate transformation has been performed.
Conclusions. The non-linear regression model to estimate the software size of open-source Java-based information systems is constructed on the basis of the Johnson multivariate transformation for SB family. This model, in comparison with other regression models (both linear and non-linear), has a larger multiple coefficient of determination, a larger value of percentage of prediction and a smaller value of the mean magnitude of relative error. The prospects for further research may include the application of other multivariate normalizing transformations and data sets to construct the non-linear regression model for estimating the software size of open-source Java-based information systems.
KEYWORDS: software size estimation, Java-based information system, non-linear regression model, univariate normalizing transformation, non-Gaussian data.
ABBREVIATIONS
HTML is hypertext markup language;
JSP is Java server pages;
KLOC is the thousand lines of code;
LB is lower bound;
MD is Mahalanobis distance;
MMR is a magnitude of relative error;
MMRE is a mean magnitude of relative error;
PHP is hypertext preprocessor;
PRED is percentage of prediction;
SQL is structured query language;
UB is upper bound;
VBA is visual Basic for application.
NOMENCLATURE
b is estimator for vector of linear regression equation
parameters, b = {,b2,...,bk} ;
bj is estimator for the i-th parameter of linear regression equation;
k is a number of independent variables (regressors); N is a number of data points;
N (0,1) is a Gaussian distribution with zero mathematical expectation and unit variance;
P is a non-Gaussian random vector,
P = {Y,Xi,X2,..„Xk}T ;
R2 is a multiple coefficient of determination; SN is a sample covariance matrix, SN = ] ; T is a Gaussian random vector, T = {Zy,Zi,Z2,...,Zk}T ;
taj2 v is a quantile of student's /-distribution with v
degrees of freedom and a/ 2 significance level; X1 is a total number of classes; X2 is a total number of relationships; X3 is an average number of attributes per class, Y is an actual software size in KLOC; ZX is a matrix of centered regressors that contains
the values Zj. - Z1, Z2. - Z2
Zk, - Zk '■.
zx is a vector with components Zt;
(zX ) is a transpose of zX ;
ZY is a sample mean of the values of the variable
Z7 ;
ZY is a prediction linear regression equation result; a is a significance level; P2 is a multivariate kurtosis;
Y is a vector of parameters of the Johnson
multivariate translation, y = (yy , Yi, Y2,—, Yk ) ;
e is a Gaussian random variable which defines residuals, e~N (0,l);
n is a vector of parameters of the Johnson multivariate translation, n = diag (, ni, • • •, nk);
X is a vector of parameters of the Johnson multivariate translation, X = diag ((, X1, —, X k); v is a number of degrees of freedom; £ is a covariance matrix, £ = ] ; 9 is a vector of parameters of the Johnson
multivariate translation, 9 = ((, 91,92, —, 9k ) ;
y is a vector of multivariate normalizing
transformation, y = (y
Y, y 2,•••, V k
} .
The subject of study is the non-linear regression models to estimate the software size of open-source Java-based information systems.
The known regression equation for estimating the software size of open-source Java-based information systems [2] is linear and generally have large widths of confidence and prediction intervals.
The purpose of the work is to construct the nonlinear regression model for estimating the software size of open-source Java-based information systems. The software size prediction results by constructed model should be better in comparison with other regression models, both linear and nonlinear, primarily on such standard evaluations as the multiple coefficient of determination and mean magnitude of relative error.
1 PROBLEM STATEMENT
Suppose given the original sample as the four-dimensional non-Gaussian data set: actual software size in the thousand lines of code (KLOC) Y, the total number of classes X1, the total number of relationships X2 and the average number of attributes per class X3 in conceptual data model from N information systems developed using the Java programming language with JSP, HTML and SQL. Suppose that there are bijective multivariate normalizing transformation of non-Gaussian random
INTRODUCTION
Java is a programming language and computing platform first released by Sun Microsystems in 1995 (https://www.java.com). Now Java is used practically everywhere from laptops to datacenters, game consoles to supercomputers, cell phones to the Internet, including information systems. Software size is one of the most important internal metrics of software including software of open-source Java-based information systems.
The information obtained from estimating the software size are useful for predicting the software development effort by such well-known model as COCOMO II. This leads to the need to develop appropriate models to estimate the software size [1-4].
The paper [2] proposed the linear regression equations for estimating the software size of some programming languages including Java. The proposed equation is constructed by multiple linear regression analysis on the basis of the metrics that can be measured from conceptual data model based a class diagram. However, there are four basic assumptions that justify the use of linear regression models, one of which is normality of the error distribution. But this assumption is valid only in particular cases. This leads to the need to use the non-linear regression models including for estimating the software size of Java-based open-source information systems.
The object of study is the process of estimating the software size of open-source Java-based information systems.
vector P = {Y, X1, X 2,-, Xk }T to Gaussian random vector T = (ZY, Z1, Z 2, •.., Zk }T is given by
T = y(P)
and the inverse transformation for (1)
P = y-1 (T).
(1)
(2)
It is required to build the non-linear regression model in the form Y = Y(X1,X2,X3,e) on the basis of the transformations (1) and (2).
2 REVIEW OF THE LITERATURE
In paper [2] the linear regression equation for estimating the software size of open-source Java-based information systems was proposed in the form
Y = b0 + b1X1 + b2 X 2 + b3 X3,
(3)
where b0 =-10.121, b1 = 1.201, b2 = 1.439 and b3 = 0.726 .
A normalizing transformation is often a good way to build the models, equations, confidence and prediction intervals of non-linear regressions [5-8]. According to [7] transformations are made for essentially four purposes, two of which are: firstly, to obtain approximate normality for the distribution of the error term (residuals) or the
dependent random variable, secondly, to transform the response and/or the predictor in such a way that the strength of the linear relationship between new variables (normalized variables) is better than the linear relationship between dependent and independent random variables.
Well-known techniques for building the equations, confidence and prediction intervals of multivariate nonlinear regressions are based on the univariate normalizing transformations (such as, the decimal logarithm, Box-Cox transformation), which do not take into account the correlation between random variables in the case of normalization of multivariate non-Gaussian data. Application of such univariate normalizing transformations for building the non-linear regression models does not always lead to good normality and linear relationship between normalized variables. This leads to the need to use the multivariate normalizing transformations.
In [9] the techniques to build the equations, confidence and prediction intervals of non-linear regressions for multivariate non-Gaussian data on the basis of the bijective multivariate normalizing transformations were proposed. The techniques consist of three steps. In the first step, a set of multivariate non-Gaussian data is normalized using a bijective multivariate normalizing transformation. In the second step, the equation, confidence and prediction intervals of linear regression for the normalized data are built. In the third step, the equations, confidence and prediction intervals of non-linear regressions for multivariate non-Gaussian data are constructed on the basis of the equation, confidence and prediction intervals of linear regression for the normalized data and the multivariate normalizing transformation. Note there is no the error term in nonlinear regression equation. The absence of the error term in non-linear regression equation does not allow modeling the random dependent variable for its prediction. This leads to the need to develop the non-linear regression model for estimating the software size of open-source Java-based information systems.
3 MATERIALS AND METHODS
After normalizing the non-Gaussian data by the transformation (1) the linear regression model is built for normalized data. The linear regression model for normalized data according to (1) will have the form
Zr = +6 = Zr + Z'
(zX) b
+6
(4)
After that the non-linear regression model is built on the basis of the linear regression model (4) for the normalized data and the transformations (1) and (2). The non-linear regression model will have the form
v = wJ
Zv +
(zx )b
+6
(5)
The technique to build a confidence interval of nonlinear regression is based on transformations (1) and (2), and a confidence interval of linear regression for normalized data
± ta/2,vSzr + (X )(Х )Z+X 1 1 (X )[
(6)
'X
where S^ = -Z(zY, - ZY¡) , v = N -k -1; (zX ) Z+ is the k x k matrix
(zX
(zX )
( Szlzl Szxz 2
Z X =
S Л
Z1Zk
Sz 7 S z z . . . Sz Z ¿^2^2 2 k
Sz z Sz z ••• Sz z
V Z1Zk z 2zk zkzk у
where SZqZr =Z [Zqi - Zq ] - Zr ], r = k .
,=1
The confidence interval for non-linear regression is built on the basis of the interval (6) and inverse transformation (2)
w1
± 'a/2,v szr IN+( X flte )zX ]1( x)
)V2 Л
The technique to build a prediction interval is based on multivariate transformation (1), the inverse transformation (2) and a prediction interval for normalized data
± 'a/2,v SZr J1 + N + (z X fi(zX )zX 1 1(zX)
1
(7)
The prediction interval for non-linear regression is built on the basis of the interval (7) and inverse transformation (2)
± ta/2,vSzr I1+N+(z x ГГ(Х )zX j 1(z x)
12 Л
For normalizing the multivariate non-Gaussian data, we use the Johnson translation system. In our case the Johnson normalizing translation is given by [10]
T = y + nh
к-1(P -ф)] ~ Nm (, Z),
(8)
where h[Y, yu-^ yk )]=K Oy \ hM V--, hk iyk)}r;
h (•) is one of the translation functions
h =
ln(y), for SL (log normal) family;
ln[y/(i - y)], forSB (bounded)family;
Arsh(y), for Sn (unbounded)family;
y for SN (normal)family.
(9)
Here y = (X - 9)) ; Arsh(y) = ln^y + ^y2 +1 j. In our
case X equals Y, X1, X2 or X3 respectively.
The equation, confidence and prediction intervals of non-linear regression to estimate the software size of open-source Java-based systems are constructed on the basis of the Johnson multivariate normalizing transformation for the four-dimensional non-Gaussian data set: actual software size in the thousand lines of code (KLOC) Y, the total number of classes X1 , the total number of relationships X2 and the average number of attributes per class X3 in conceptual data model from 30 information systems developed using the Java programming language with JSP, HTML and SQL. Table 1 contains the data from [2] on four metrics of software for 30 open-source Java-based systems.
Table 1 - The data set and squared MDs
No Y X1 X 2 X 3 Squared MD
univariate multivariate
1 11.717 8 6 4.25 0.99 0.99
2 47.52 23 19 9.565 1.65 1.50
3 84.01 26 40 11.462 10.13 6.55
4 26.999 15 14 8.933 1.33 1.89
5 41.72 20 15 5.9 0.25 0.50
6 13.015 5 6 12.4 5.89 6.64
7 30.402 18 7 6.611 1.59 3.10
8 29.159 23 10 6.957 2.26 3.13
9 53.443 28 25 4.179 3.53 4.02
10 18.694 13 9 6.615 0.39 0.89
11 26.384 16 6 5.125 2.23 4.06
12 38.721 19 16 6.579 0.19 0.19
13 75.643 26 30 6.154 1.87 2.74
14 46.72 21 24 6.048 1.31 1.66
15 6.413 7 5 4.143 2.41 6.07
16 79.534 20 37 4.85 7.06 8.20
17 36.343 18 17 5.333 0.35 0.47
18 59.684 22 31 6.182 2.49 2.62
19 50.454 15 20 11.6 2.51 3.35
20 3.055 4 1 7 10.83 7.10
21 63.257 34 17 3.971 9.16 8.29
22 91.28 35 28 13.571 17.73 11.22
23 32.707 11 17 7.545 0.98 1.54
24 11 5 5 3.6 6.15 6.36
25 5.543 6 4 3.833 2.54 5.16
26 22.686 12 11 6.667 0.11 0.22
27 3.911 3 2 6.667 7.26 5.52
28 20.841 14 7 3 8.17 7.21
29 9.269 6 5 3.5 3.23 2.82
30 7.732 7 2 11.143 5.42 5.98
For detecting the outliers in the data from Table 1 we use the technique based on multivariate normalizing transformations and the squared Mahalanobis distance (MD) [11]. There are no outliers in the data from Table 1 for 0.005 significance level and the Johnson multivariate transformation (8) for SB family. The same result was obtained in [12] for the transformation (8) for Sv family. In [2] it was also assumed that the data contains no outliers. The values of squared MD for normalized data by the Johnson univariate transformation (9) for SB family from Table 1 indicate the data of system 22 is multivariate outlier, since for this data row the squared MD equals to 17.73 is greater than the value of the quantile of the Chi-Square distribution, which equals to 14.86 for 0.005 significance level. Although note that without using normalization, the data of system 11 is multivariate outlier, since for this data row the squared MD equals to 15.44.
Parameters of the multivariate transformation (9) for SB family were estimated by the maximum likelihood method. Estimators for parameters of the transformation (9) are: Y Y = 9.63091, Y1 = 15.5355, Y2 = 25.4294,
Y3 = 0.72801, n Y = 1.05243, n = 1.58306,
n2 = 2.54714, n3 = 0.54312, 9Y =-1.4568, 92 = -6.9746,
cp1 =-1.8884, XY = 153102.605,
cp3 = 3.2925, L = 311229.5 and
X = 243051.0, x 2
X3 = 13.90. The sample covariance matrix SN of the T is used as the approximate moment-matching estimator of £
^1.0000 0.9514 0.9333 0.1574^
0.9514 1.0000 0.9006 0.1345
0.9333 0.9006 1.0000 0.0554
, 0.1574 0.1345 0.0554 1.0000
S N =
After normalizing the non-Gaussian data by the multivariate transformation (9) for SB family the linear regression model (3) is built for normalized data
ZY = ZY + e = b0 + ¿>1^1 + b2Z2 + b3Z3 + e .
'->2^2 ■
33
(10)
Parameters of the linear regression model (10) were estimated by the least square method. Estimators for
parameters of the equation (10) are such: b0 = 1.02 -10"5,
b1 = 0.56085 , b2 = 0.42491, b3 = 0.05846 .
After that the non-linear regression model (4) is built
Y = cp y + X y
1 + e~(
+ e-\ZY +e-YY УПY
-1
(11)
where Zj = y j + n j ln
j = 1,2,3.
Xj -P i
Pí Í -XÍ
Pj< Xj <pj+Xj
The model (11) is the non-linear regression model to estimate the software size of open-source Java-based information systems.
4 EXPERIMENTS
For comparison of the model (11) with other models two non-linear regression models are built on the basis of the data from Table 1 and two univariate normalizing transformations: the decimal logarithm transformation and the Johnson transformation.
The non-linear regression model is constructed on the basis of the linear regression model (4) for the normalized data and the decimal logarithm transformation
Y = 10s+b° xf1 Xb2 Xb3.
(12)
where the estimators for parameters of the model (12) are: b0 = -0.04536, b = 0.64235 , b2 = 0.56305 and
b3 = 0.18045 .
The non-linear regression model is constructed on the basis of the linear regression model (4) for the normalized data and the Johnson univariate transformation for SB family. In this case the estimators for parameters of the model (11) are: y Y = 0.46387, y1 = 0.38093,
y2 = 0.60545, Y3 = 0.65592, n Y = 0.50326,
in1 = 0.62689, f|2 = 0.62215, r)3 = 0.72789, cpY = 2.817, cp1 = 2.634, cp2 = 0.700, cp3 = 2.839,
iY = 89.930, ^ = 33.711, X2 = 41.428, X3 = 11.780,
b0 = 0, b1 = 0.46976, b2 = 0.53539 and b3 = 0.11397.
The computer program implementing the constructed models (11) and (12) was developed to conduct experiments. The program was written in the sci-language for the Scilab system. Scilab (http://www.scilab.org) is the free and open source software, the alternative to commercial packages for system modeling and simulation packages such as MATLAB and MATRIXx.
5 RESULTS
If the Gaussian random variable e equals zero the regression models (11) and (12) are the non-linear regression equations for which the prediction results for values of components of vector X = {x1, X2, X3} from Table 1 and values of MRE are shown in the Table 2. The prediction results by model (11) and values of MRE are shown in the Table 2 for two cases: the Johnson univariate and multivariate normalizing transformations. Table 2 also contains the prediction results by linear regression equation (3) from [2] for values of components of vector X from Table 1 and MRE values. Note, all prediction results by linear regression equation (3), nonlinear regression models (11) and (12) are positive.
Table 2 - The prediction results and confidence intervals of regressions for 30 open-source Java-based systems
No Linear regression Non-linear regression
univariate normalizing transformation the Johnson multivariate normalizing transformation
the decimal logarithm the Johnson transformation
Y RME LB UB Y RME LB UB Y RME LB UB Y RME LB UB
1 11.205 0.0437 8.301 14.109 12.197 0.0410 10.923 13.620 10.671 0.0893 8.989 12.756 11.978 0.0223 10.322 13.263
2 51.784 0.0897 48.396 55.171 53.248 0.1205 47.404 59.812 55.919 0.1768 49.992 61.568 52.316 0.1009 48.849 57.132
3 86.989 0.0355 79.427 94.551 90.515 0.0774 77.073 106.302 87.586 0.0426 83.866 89.805 87.262 0.0387 82.880 89.624
4 34.523 0.2787 31.823 37.223 33.653 0.2465 30.525 37.102 35.152 0.3020 30.502 40.115 33.719 0.2489 31.087 37.648
5 39.765 0.0469 36.726 42.804 39.052 0.0639 35.787 42.615 40.333 0.0333 35.838 44.986 38.969 0.0659 35.830 42.245
6 13.516 0.0385 6.843 20.188 10.941 0.1593 8.918 13.424 10.900 0.1625 7.968 15.255 11.602 0.1086 9.430 15.320
7 26.365 0.1328 21.326 31.404 24.255 0.2022 21.099 27.884 24.630 0.1898 20.414 29.485 23.956 0.2120 20.880 26.937
8 36.937 0.2668 30.570 43.305 35.027 0.2012 30.424 40.326 37.853 0.2982 31.806 44.302 35.358 0.2126 31.382 39.866
9 62.516 0.1698 57.570 67.462 60.729 0.1363 52.927 69.681 64.288 0.2029 57.204 70.537 62.131 0.1626 56.682 67.124
10 23.243 0.2433 20.890 25.595 22.674 0.2129 21.041 24.433 22.251 0.1903 19.396 25.446 22.451 0.2009 20.399 24.563
11 21.446 0.1872 16.885 26.006 19.692 0.2536 17.110 22.665 18.897 0.2838 15.560 22.881 19.239 0.2708 16.405 21.492
12 40.496 0.0459 38.185 42.808 39.963 0.0321 36.836 43.355 41.354 0.0680 37.021 45.812 39.831 0.0287 36.978 43.139
13 68.744 0.0912 64.647 72.842 68.808 0.0904 61.557 76.912 70.832 0.0636 65.599 75.335 68.708 0.0917 64.545 72.542
14 54.028 0.1564 50.795 57.260 52.739 0.1288 47.735 58.266 55.262 0.1828 49.875 60.436 53.278 0.1404 49.565 57.466
15 8.487 0.3234 5.441 11.534 10.056 0.5681 8.926 11.329 8.703 0.3571 7.338 10.443 9.863 0.5379 8.468 10.966
16 70.670 0.1115 61.411 79.928 62.671 0.2120 53.424 73.519 73.437 0.0767 64.730 79.996 70.101 0.1186 64.881 77.099
17 39.831 0.0960 37.362 42.300 38.454 0.0581 35.184 42.028 39.331 0.0822 34.828 44.018 38.677 0.0642 35.425 41.986
18 65.401 0.0958 59.939 70.864 63.010 0.0557 55.949 70.961 66.982 0.1223 60.723 72.447 64.714 0.0843 60.331 69.576
19 45.094 0.1062 39.943 50.244 43.124 0.1453 37.198 49.994 47.605 0.0565 39.805 55.417 44.662 0.1148 40.487 51.875
20 1.200 0.6071 -2.531 4.932 3.118 0.0206 2.525 3.850 3.430 0.1227 3.180 3.849 3.173 0.0388 2.947 3.480
21 58.055 0.0822 47.956 68.154 54.860 0.1328 45.880 65.597 70.088 0.1080 58.879 78.528 62.201 0.0167 56.926 73.899
22 82.053 0.1011 75.043 89.064 92.398 0.0123 77.034 110.827 88.089 0.0350 84.129 90.284 88.102 0.0348 84.265 91.745
23 33.031 0.0099 28.529 37.532 29.837 0.0878 26.096 34.113 31.165 0.0471 26.305 36.522 31.086 0.0496 27.987 35.574
24 5.692 0.4826 1.975 9.408 7.899 0.2819 6.666 9.359 6.502 0.4089 5.391 8.063 7.677 0.3021 6.421 8.778
25 5.622 0.0143 2.367 8.878 7.921 0.4290 6.917 9.071 6.795 0.2259 5.745 8.199 7.745 0.3972 6.621 8.678
26 24.958 0.1002 22.712 27.204 24.148 0.0644 22.338 26.104 23.874 0.0524 20.862 27.219 24.192 0.0664 22.075 26.610
27 1.197 0.6938 -2.722 5.117 3.796 0.0295 3.169 4.547 3.548 0.0929 3.253 4.038 3.763 0.0378 3.404 4.254
28 18.942 0.0911 14.883 23.001 17.897 0.1413 15.224 21.039 13.423 0.3559 9.477 19.245 15.903 0.2369 11.765 18.810
29 6.820 0.2642 3.356 10.284 8.835 0.0468 7.597 10.274 7.279 0.2147 5.991 9.053 8.532 0.0795 7.117 9.686
30 9.248 0.1960 3.644 14.851 7.176 0.0719 4.599 11.199 7.022 0.0918 5.480 9.390 7.094 0.0825 5.826 8.605
MMRE and PRED(0.25) are accepted as standard evaluations of prediction results by regression models and equations. The acceptable values of MMRE and PRED(0.25) are not more than 0.25 and not less than 0.75
respectively. The acceptable value of R2 is approximately the same as for PRED(0.25). The values of
R2 , MMRE and PRED(0.25) equal respectively 0.9621, 0.1734 and 0.7667 for linear regression equation (3), and equal respectively 0.9541, 0.1441 and 0.8667 for the model (12), and equal respectively 0.9574, 0.1579 and 0.8000 for the model (11) for the Johnson univariate transformation. The values of R 2 , MMRE and PRED(0.25) are better for the model (11) for the Johnson multivariate transformation in comparison with all previous equations, and are 0.9672, 0.1389 and 0.8667 respectively.
The confidence and prediction intervals of non-linear regression are defined for the data from Table 1. Table 2 contains the lower (LB) and upper (UB) bounds of the confidence intervals of linear and non-linear regressions on the basis of univariate and multivariate transformations respectively for 0.05 significance level. Note the lower bounds of the confidence interval of linear regression (3) from [2] are negative for the two rows of data: 20 and 27. All the lower bounds of the confidence interval of nonlinear regressions are positive. The widths of the confidence interval of non-linear regression on the basis of the Johnson multivariate transformation are less than for linear regression (3) from [2] for 21 rows of data: 1, 3, 6-8, 10, 11, 13, 15, 16, 18, 20-25, 27-30. Also the widths of the confidence interval of non-linear regression on the basis of the Johnson multivariate transformation are less for more data rows than for non-linear regressions following the univariate transformations, both decimal logarithm and the Johnson. The widths of the confidence interval of non-linear regression on the basis of the Johnson multivariate transformation are less than following the decimal logarithm univariate transformation for 24 rows of data: 2-5, 7-9, 11-14, 16-25, 27, 29 and 30. And ones are less than following the Johnson univariate transformation for 27 rows of data: 1, 2, 4-21, 23-26, 28-30. Approximately the same results are obtained for the prediction intervals of regressions.
Table 3 contains the lower (LB) and upper (UB) bounds of the prediction intervals of linear and non-linear regressions on the basis of univariate and multivariate transformations respectively for 0.05 significance level. Note the lower bounds of the prediction interval of linear regression (3) are negative for the eight rows of data: 1, 15, 20, 24, 25, 27, 29 and 30. All the lower bounds of the prediction interval of non-linear regressions are positive. The widths of the prediction interval of non-linear regression on the basis of the Johnson multivariate transformation are less than for linear regression (3) from [2] for 16 rows of data: 1, 3, 6, 7, 10, 11, 15, 20, 22, 2430. Also the widths of the prediction interval of non-linear regression on the basis of the Johnson multivariate transformation are less for more data rows than for non© Prykhodko N. V., Prykhodko S. B., 2018 DOI 10.15588/1607-3274-2018-3-17
linear regressions following the univariate transformations, both decimal logarithm and the Johnson. The widths of the prediction interval of non-linear regression on the basis of the Johnson multivariate transformation are less than following the decimal logarithm univariate transformation for 17 rows of data: 2-5, 8, 9, 12-14, 16-22 and 27. And ones are less than following the Johnson univariate transformation for 26 rows of data: 1, 2, 4-19, 21, 23-26, 28-30.
Table 3 - The bounds of the prediction intervals
No Bounds for linear regression Bounds for non-linear regression
Johnson univariate transformation Johnson multivariate transformation
LB UB LB UB LB UB
1 -0.004 22.413 5.679 22.412 7.536 19.277
2 40.441 63.127 32.573 75.467 36.600 68.504
3 73.784 100.194 77.736 91.114 73.863 96.264
4 23.365 45.680 17.436 58.472 21.814 48.847
5 28.521 51.009 20.726 63.360 25.763 54.772
6 0.799 26.232 5.559 24.105 7.061 19.398
7 14.424 38.305 11.701 46.283 14.924 36.873
8 24.378 49.497 18.875 61.455 22.764 51.145
9 50.614 74.418 40.607 80.655 45.398 77.523
10 12.164 34.321 10.693 42.556 14.102 34.502
11 9.699 33.192 9.052 37.791 11.900 30.337
12 29.427 51.566 21.431 64.236 26.458 55.673
13 57.169 80.319 49.028 83.854 52.398 82.637
14 42.730 65.325 32.094 74.958 37.508 69.345
15 -2.759 19.733 4.920 18.108 6.290 15.899
16 56.425 84.914 51.026 85.596 52.840 84.458
17 28.727 50.935 20.068 62.454 25.536 54.455
18 53.275 77.527 43.924 81.998 48.094 79.560
19 33.105 57.082 25.246 70.052 29.796 61.411
20 -10.250 12.651 3.009 4.748 2.516 4.443
21 43.250 72.859 45.506 84.384 44.137 78.697
22 69.156 94.951 78.586 91.334 74.127 97.180
23 21.307 44.755 15.088 54.334 19.782 45.949
24 -5.754 17.138 4.077 13.054 4.988 12.419
25 -5.682 16.927 4.203 13.643 5.062 12.421
26 13.902 36.014 11.471 44.858 15.245 36.852
27 -10.316 12.711 3.048 5.105 2.834 5.520
28 7.380 30.503 6.446 29.640 9.577 26.119
29 -4.546 18.186 4.360 14.963 5.483 13.826
30 -2.942 21.438 4.206 14.773 4.601 11.617
Following [13] multivariate kurtosis p2 is estimated for the data on metrics of software from Table I and the normalized data on the basis of the decimal logarithm transformation, the Johnson univariate and multivariate transformations for SB family. The estimator of multivariate kurtosis given by [13]
In our case, in the formula (13), the vectors Z and Z should be replaced by the vectors P and P or T and T , respectively, for the initial (non-Gaussian) or normalized data. It is known that p2 = m(m + 2) holds under multivariate normality. The given equality is a necessary
condition for multivariate normality. In our case p2 = 24. The estimators of multivariate kurtosis equal 27.17, 22.38, 32.05 and 24.02 for the data from Table 1, the normalized data on the basis of the decimal logarithm transformation, the Johnson univariate and multivariate transformations respectively. The values of these estimators indicate that the necessary condition for multivariate normality is practically performed for the normalized data on the basis of the decimal logarithm transformation and the Johnson multivariate transformation, it does not hold for other data.
6 DISCUSSION
As it evident from the Table 2 and Table 3, the values of lower bounds of the confidence and prediction intervals of linear regression (3) from [2] for estimating the software size of open-source Java-based information systems are negative for some data rows. In our opinion, the presence of negative values may be explained by two reasons. Firstly, for the initial data from Table 1, four basic assumptions that justify the use of linear regression model, one of which is normality of the error distribution, are not valid. Secondly, there is reason to reject the hypothesis that the sample of data from Table 1 comes from a multivariate normal distribution. Note all the lower bounds of the confidence and prediction intervals of nonlinear regressions are positive.
Also note that in our case for the data from Table 1, the poor normalization of multivariate non-Gaussian data using the Johnson univariate transformation leads to an increase in the widths of the confidence and prediction intervals of non-linear regression for a larger number of data rows compared to both the Johnson multivariate transformation and the decimal logarithm transformation.
The widths of the confidence and prediction intervals of non-linear regression on the basis of the Johnson multivariate transformation are less for more data rows than for linear regression and non-linear regressions following the univariate transformations, both decimal
logarithm and the Johnson. Also the values of R , MMRE and PRED(0.25) are better for the model (11) for the Johnson multivariate transformation in comparison with all previous equations and models, both linear and non-linear, based on univariate transformations. This may be explained best multivariate normalization and the fact that there is no reason to reject the hypothesis that the sample of data, which normalized by the Johnson multivariate transformation for SB family, comes from a multivariate normal distribution.
CONCLUSIONS
The important problem of increase of confidence of software size estimation for open-source Java-based information systems is solved.
The scientific novelty of obtained results is that the techniques to build the non-linear regression model for multivariate non-Gaussian data on the basis of the multivariate normalizing transformations is firstly
proposed. The non-linear regression model to estimate the software size of open-source Java-based information systems is constructed on the basis of the Johnson multivariate transformation for SB family. This model, in comparison with other regression models (both linear and non-linear), has a larger multiple coefficient of determination, a smaller value of the mean magnitude of relative error, a larger value of percentage of prediction and smaller widths of the confidence and prediction intervals of non-linear regression.
The practical significance of obtained results is that the software realizing the constructed model is developed in the sci-language for Scilab. The experimental results allow to recommend the constructed model for use in practice.
Prospects for further research may include the application of other multivariate normalizing transformations and data sets to construct the non-linear regression model for estimating the software size of open-source Java-based information systems.
REFERENCES
1. Kaczmarek J., Kucharski M. Size and effort estimation for applications written in Java, Information and Software Technology, 2004, Vol. 46, Issue 9, pp. 589-601. DOI: 10.1016/j.infsof.2003.11.001
2. Tan H. B. K., Zhao Y., Zhang H. Estimating LOC for information systems from their conceptual data models, Software Engineering : the 28th International Conference (ICSE '06), Shanghai. China, May 20-28, 2006 : proceedings, pp. 321-330. DOI: 10.1145/1134285.1134331
3. Tan H. B. K., Zhao Y., Zhang H. Conceptual data modelbased software size estimation for information systems, Transactions on Software Engineering and Methodology, 2009, Vol. 19, Issue 2, October 2009, Article No. 4. DOI: 10.1145/1571629.1571630
4. Kiewkanya M., Surak S. Constructing C++ software size estimation model from class diagram, Computer Science and Software Engineering : 13th International Joint Conference, Khon Kaen, Thailand, July 13-15, 2016 : proceedings, pp. 1-6. DOI: 10.1109/JCSSE.2016.7748880
5. Bates D. M., Watts D. G. Nonlinear Regression Analysis and Its Applications. New York, John Wiley & Sons, 1988, 384 p. DOI:10.1002/9780470316757
6. Seber G.A.F., Wild C. J. Nonlinear Regression. New York, John Wiley & Sons, 1989, 768 p. DOI: 10.1002/0471725315
7. Ryan T.P. Modern regression methods. New York, John Wiley & Sons, 1997, 529 p. DOI: 10.1002/9780470382806
8. Johnson R. A., Wichern D. W. Applied Multivariate Statistical Analysis. Pearson Prentice Hall, 2007, 800 p.
9. Prykhodko S. B. Developing the software defect prediction models using regression analysis based on normalizing transformations, Modern Problems in Testing of the Applied Software : the Research and Practice Seminar (PTTAS-2016), Poltava, Ukraine, May 25-26, 2016 : abstracts, pp. 6-7.
10. Stanfield P. M., Wilson J. R., Mirka G. A., Glasscock N. F., Psihogios J. P., Davis J. R. Multivariate input modeling with Johnson distributions, The 28th Winter simulation conference WSC'96, Coronado, CA, USA, December 8-11, 1996 : proceedings, ed. S. Andradyttir, K. J. Healy, D.H.Withers, and B. L. Nelson, IEEE Computer Society Washington, DC, USA, 1996, pp. 1457-1464.
11. Prykhodko S., Prykhodko N., Makarova L., Pugachenko K. Detecting Outliers in Multivariate Non-Gaussian Data on the basis of Normalizing Transformations, Electrical and Computer Engineering : the 2017 IEEE First Ukraine Conference (UKRCON) «Celebrating 25 Years of IEEE Ukraine Section», Kyiv, Ukraine, May 29 - June 2, 2017 : proceedings, pp. 846-849. DOI: 10.1109/UKRCON.2017.8100366
12. Prykhodko S., Prykhodko N., Makarova L., Pukhalevych A. Application of the Squared Mahalanobis Distance for Detecting Outliers in Multivariate Non-Gaussian Data,
УДК 004.412:519.237.5
Radioelectronics, Telecommunications and Computer Engineering : 14th International Conference on Advanced Trends (TCSET), Lviv-Slavske, Ukraine, February 20-24, 2018 : proceedings, pp. 962-965. DOI: 10.1109/TCSET.2018.8336353 13. Mardia K. V. Measures of multivariate skewness and kurtosis with applications, Biometrika, 1970, 57, pp. 519530. DOI: 10.1093/biomet/57.3.519
Received 18.05.2018.
Accepted 05.06.2018.
НЕЛШШНА РЕГРЕС1ЙНА МОДЕЛЬ ДЛЯ ОЦШЮВАННЯ РОЗМ1РУ ПРОГРАМНОГО ЗАБЕЗПЕЧЕННЯ
СИСТЕМ З В1ДКРИТИМ КОДОМ НА JAVA
Приходько Н. В. - канд. екон. наук, доцент, доцент кафедри фшанс1в Национального ун1верситету кораблебудування iMeHi адмiрала Макарова, Микола1в, Укра1на.
Приходько С. Б. - д-р техн. наук, професор, завщувач кафедри програмного забезпечення автоматизованих систем Национального ун1верситету кораблебудування iменi адмiрала Макарова, Миколагв, Украша.
АНОТАЦ1Я
Актуальнiсть. Проблема оцшювання розмiру програмного забезпечення на раннш стадп програмного проекту е важливою, оскiльки iнформацiя, отримана при оцшюванш розмiру програмного забезпечення, використовуеться для прогнозування трудомiсткостi по розробщ програмного забезпечення, включаючи iнформацiйнi системи на базi Java з вщкритим вихiдним кодом. Об'ектом дослiдження е процес оцшювання розмiру програмного забезпечення шформацшних систем з вiдкритим вихщним кодом на Java. Предметом дослщження е моделi регресп для оцшювання розмiру програмного забезпечення iнформацiйних систем з вщкритим вихiдним кодом на Java. Мета роботи - створення моделi нелшшно1 регресп для оцшювання розмiру програмного забезпечення iнформацiйних систем з вершим вимдним кодом на Java на основi багатовимiрного норматзуючого перетворення Джонсона.
Метод. Моделi, довiрчi штервали та штервали передбачення багатовимiрноi нелшшно1 регресп для оцшювання розмiру програмного забезпечення iнформацiйних систем з вершим вихiдним кодом на Java побудоваш на основi багатовимiрного норматзуючого перетворення Джонсона для негаусiвських даних за допомогою вiдповiдних методiв. Методи побудови моделей, р1внянь, довiрчих штерватв i штервал1в передбачення нелiнiйних регресш заснованi на багатовимiрному нелшшному регресшному аналiзi з використанням багатовимiрних нормалгзуючих перетворень. Розглянуто вiдповiднi методи. Ц методи дозволяють враховувати кореляцта мпж випадковими величинами в разi норматзацп багатовимiрних негаусiвських даних. Загалом, це призводить до зменшення середньо1 величини вiдносноi похибки, ширини довiрчих iнтервалiв i штервал1в передбачення в пор1внянш з лiнiйними моделями або нелшшними моделями, побудованими з використанням одновимiрних нормалiзуючих перетворень.
Результати. Здiйснено пор1вняння побудовано1 моделi з моделями лшшно! регресil та нелiнiйними регрес1ями на основ1 десяткового логарифму та одновимiрного перетворення Джонсона.
Висновки. Модель нелшшно! регресil для оцiнювання розмiру програмного забезпечення шформацшних систем з в^ригим вимдним кодом на Java побудована на основi багатовимiрного перетворення Джонсона для амейства SB . Ця модель в порiвняннi з шшими регресiйнiй моделi (як лiнiйними, так i нелiнiйними) мае б1льший множинний коефiцiенг детермiнацil i менше значення середньо1 величини вiдносноl похибки. Перспективи подальших дослiджень можуть включати застосування 1нших багатовимiрних нормалiзують перетворень i наборiв даних для побудови моделi нелiнiйноi регресil для оц1нювання розмiру програмного забезпечення 1нформац1йних систем з вщкритим вих^дним кодом на Java.
КЛЮЧОВ1 СЛОВА: оц1нювання розмiру програмного забезпечення, шформацшна система на основi Java, модель нел1И1йно1 регресii, одновимiрне нормалiзуюче перетворення, негаусiвськi данi.
УДК 004.412:519.237.5
НЕЛИНЕЙНАЯ РЕГРЕССИОННАЯ МОДЕЛЬ ДЛЯ ОЦЕНКИ РАЗМЕРА ПРОГРАММНОГО ОБЕСПЕЧЕНИЯ
СИСТЕМ С ОТКРЫТЫМ КОДОМ НА JAVA
Приходько Н. В. - канд. екон. наук, доцент, доцент кафедри фшанс1в Национального ушверситету кораблебудування iменi адмпрала Макарова, Микола1в, Укра1на.
Приходько С. Б. - д-р техн. наук, професор, завщувач кафедри програмного забезпечення автоматизованих систем Национального утверситету кораблебудування iменi адмiрала Макарова, Микола1в, Укра1на.
AННОТАЦИЯ
Актуальность. Проблема оценки размера программного обеспечения на ранней стадии программного проекта важна, поскольку информация, полученная при оценке размера программного обеспечения, используется для прогнозирования трудоемкости по разработке программного обеспечения, включая информационные системы на базе Java с открытым исходным кодом. Объект исследования - процесс оценки размера программного обеспечения информационных систем с © Prykhodko N. V., Prykhodko S. B., 2018 DOI 10.15588/1607-3274-2018-3-17
открытым исходным кодом на Java. Предмет исследования - модели регрессии для оценки размера программного обеспечения информационных систем с открытым исходным кодом на Java. Цель работы - создание модели нелинейной регрессии для оценки размера программного обеспечения информационных систем с открытым исходным кодом на Java на основе многомерного нормализирующего преобразования Джонсона.
Метод. Модели, доверительные интервалы и интервалы прогнозирования многомерной нелинейной регрессии для оценки размера программного обеспечения информационных систем с открытым исходным кодом на Java построены на основе многомерного нормализирующего преобразования Джонсона для негауссовских данных с помощью соответствующих методов. Методы построения моделей, уравнений, интервалов доверия и прогнозирования нелинейных регрессий основаны на многократном нелинейном регрессионном анализе с использованием многомерных нормализирующих преобразований. Рассмотрены соответствующие методы. Методы позволяют учитывать корреляцию между случайными величинами в случае нормализации многомерных негауссовских данных. В общем, это приводит к уменьшению средней величины относительной погрешности, ширины доверительных интервалов и интервалов предсказания по сравнению с линейными моделями или нелинейными моделями, построенными с использованием одномерных нормализирующих преобразований.
Результаты. Проведено сравнение построенной модели с линейной моделью и нелинейными регрессионными моделями на основе десятичного логарифма и одномерного преобразования Джонсона.
Выводы. Модель нелинейной регрессии для оценки размера программного обеспечения информационных систем с открытым исходным кодом на Java построена на основе многомерного преобразования Джонсона для семейства SB . Эта модель по сравнению с другими регрессионными моделями (как линейными, так и нелинейными) имеет больший множественный коэффициент детерминации и меньшее значение средней величины относительной погрешности. Перспективы дальнейших исследований могут включать применение других многомерных нормализующих преобразований и наборов данных для построения модели нелинейной регрессии для оценки размера программного обеспечения информационных систем с открытым исходным кодом на Java.
КЛЮЧЕВЫЕ СЛОВА: оценка размера программного обеспечения, информационная система на основе Java, модель нелинейной регрессии, одномерное нормализующее преобразование, негауссовские данные.
Л1ТЕРАТУРА / ЛИТЕРАТУРА
1. Kaczmarek J. Size and effort estimation for applications written in Java / J. Kaczmarek, M. Kucharski // Information and Software Technology. - 2004. - Vol. 46, Issue 9. -P. 589-601. DOI: 10.1016/j.infsof.2003.11.001
2. Tan H. B. K. Estimating LOC for information systems from their conceptual data models / H. B. K. Tan, Y. Zhao, H. Zhang // Software Engineering : the 28th International Conference (ICSE '06), Shanghai, China, May 20-28, 2006 : proceedings. - P. 321-330. DOI: 10.1145/1134285.1134331
3. Tan H. B. K. Conceptual data model-based software size estimation for information systems / H. B. K. Tan, Y. Zhao, H. Zhang // Transactions on Software Engineering and Methodology. - 2009. - Vol. 19, Issue 2. - October 2009. -Article No. 4. DOI: 10.1145/1571629.1571630
4. Kiewkanya M. Constructing C++ software size estimation model from class diagram / M. Kiewkanya, S. Surak // Computer Science and Software Engineering : 13th International Joint Conference, Khon Kaen, Thailand, July 13-15, 2016 : proceedings. - P. 1-6. DOI: 10.1109/JCSSE.2016.7748880
5. Bates D. M. Nonlinear Regression Analysis and Its Applications / D. M. Bates, D. G. Watts. - New York : John Wiley & Sons, 1988. - 384 p. DOI:10.1002/9780470316757
6. Seber G. A. F. Nonlinear Regression / G. A. F. Seber, C. J. Wild. - New York : John Wiley & Sons, 1989. - 768 p. DOI: 10.1002/0471725315
7. Ryan T. P. Modern regression methods / T. P. Ryan. - New York : John Wiley & Sons, 1997. - 529 p. DOI: 10.1002/9780470382806
8. Johnson R. A. Applied Multivariate Statistical Analysis / R. A. Johnson, D. W. Wichern. - Pearson Prentice Hall, 2007. - 800 p.
9. Prykhodko S.B. Developing the software defect prediction models using regression analysis based on normalizing transformations / S. B. Prykhodko // Modern Problems in Testing of the Applied Software : the Research and Practice Seminar (PTTAS-2016), Poltava, Ukraine, May 25-26, 2016 : abstracts. - P. 6-7.
10. Stanfield P. M. Multivariate input modeling with Johnson distributions / [P. M. Stanfield, J. R. Wilson, G. A. Mirka, et al.] // the 28th Winter simulation conference WSC'96, Coronado, CA, USA, December 8-11, 1996 : proceedings, ed. S. Andradyttir, K. J. Healy, D. H. Withers, and B. L. Nelson. - IEEE Computer Society Washington, DC, USA, 1996. - P. 1457-1464.
11. Prykhodko S. Detecting Outliers in Multivariate Non-Gaussian Data on the basis of Normalizing Transformations / S. Prykhodko, N. Prykhodko, L. Makarova, K. Pugachenko // Electrical and Computer Engineering : the 2017 IEEE First Ukraine Conference (UKRCON) «Celebrating 25 Years of IEEE Ukraine Section», Kyiv, Ukraine, May 29 -June 2, 2017 : proceedings. - P. 846-849. DOI: 10.1109/UKRC0N.2017.8100366
12. Application of the Squared Mahalanobis Distance for Detecting Outliers in Multivariate Non-Gaussian Data / [S.Prykhodko, N. Prykhodko, L. Makarova, A. Pukhalevych] // Radioelectronics, Telecommunications and Computer Engineering : 14th International Conference on Advanced Trends (TCSET), Lviv-Slavske, Ukraine, February 20-24, 2018 : proceedings. - P. 962-965. DOI: 10.1109/TCSET.2018.8336353
13. Mardia K. V. Measures of multivariate skewness and kurtosis with applications / K. V. Mardia // Biometrika. -1970. - 57. - P. 519-530. DOI: 10.1093/biomet/57.3.519
