Mathematical Modeling of Effort of Mobile Application Development in a Planning Phase

Abstract

Mathematical modeling of effort of development of mobile applications (apps) by non-linear regression model using multivariate normalizing transformation is performed. A three-factor non-linear regression model to estimate the effort (in man-hours) of developing the mobile apps in a planning phase is constructed on the basis of the Johnson four-variate transformation for SB family. This model is constructed around the Requirement Analysis Document (RAD) variables: number of screens, number of functions, and number of files. Comparison of the constructed model with the linear regression model and non-linear regression models based on the univariate normalizing transformations is performed. This model, in comparison with other regression models, 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 regression. Such a good result for the constructed model may be explained best multivariate normalization of the non-Gaussian data set, which used to build the three-factor non-linear regression model based on the Johnson four-variate transformation for SB family.

Keywords

• Mathematical Modeling
• Effort Estimation
• Mobile Application
• Non-linear Regression Model
• Prediction Interval

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The problem of estimating software development effort is one of the important ones in the planning phase, which is the first of the five phases of the software development lifecycle [1]. Today, the solution of this problem is carried out, including using mathematical modeling. One of the more well-known mathematical models for estimating software development effort is COCOMO II. But its use for mobile apps has some difficulties. First, the main factor for this model is the size of the software, which is still unknown in the planning phase. Second, COCOMO II is a non-linear regression equation built on a univariate transformation in the form of a decimal logarithm, which does not always allow for proper normalization of the data. In addition, the regression equation does not include random variables [2-4] as and a effort estimation model based on Function Points Analysis method [5]. And, as you know, the Copyright © 2019 for this paper by its authors.

Use permitted under Creative Commons License Attribution 4.0 International (CC BY 4.0). effort is a random variable. Third, while mobile app development is similar to web app development and has its roots in more traditional software development, however, one significant difference is that mobile apps are often written specifically to take advantage of the unique features that a particular mobile device offers [6].

Therefore, over the last decade, the various models for forecasting the effort of developing the mobile apps in a planning phase, including regression ones [7, 8], were constructed. It is the regression models that describe an effort as a random variable. And since the effort distribution is not Gaussian, it is necessary to use non-linear regression models, and their construction should be based on multivariate normalizing transformations [9]. 2

Model construction

At first, the three-factor linear regression model to estimate the effort Y (in manhours) of developing the mobile apps in a planning phase is constructed for the fourdimensional data set from Table 1. This model is constructed around the Requirement Analysis Document (RAD) variables: number of screens X1, number of functions X2, and number of files X3. The data set from Table 1 was obtained by combining two data sets for 17 mobile apps from [5] and for 21 mobile apps (rows 18 to 38). Also, Table 1 contains the values of squared Mahalanobis distance (MD2). We use the technique based on the squared Mahalanobis distance [10] for detecting the outliers in the data from Table 1. There are no outliers in the data from Table 1 for 0.005 significance level, since for all data rows, the MD2 values are smaller than the value of the quantile of the ChiSquare distribution, which equals to 14.86.

Following [2-4] the three-factor linear regression model has the form

Y  bˆ0  bˆ1X1  bˆ2 X 2  bˆ3 X 3   x , ( 1 ) where  x is a Gaussian random variable which defines residuals,  x  N 0,  x  ; the estimators for parameters of the

model ( 1 ) are: bˆ0  0.26513 , bˆ1  0.23116 , bˆ2  -0.00082 , bˆ3  0.08374 . Parameters of the model ( 1 ) were estimated by the least square method.

To judge the prediction accuracy of linear regression model ( 1 ) we first used the well-known standard metrics of prediction accuracy, i.e., a multiple coefficient of determination R2, a mean magnitude of relative error MMRE and percentage of prediction at the level of magnitude of relative error (MRE), which equals 0.25, PRED(0.25) [11, 12]. The values of R2, MMRE, and PRED(0.25) equal respectively 0.5449, 0.5713, and 0.5789 for the linear regression model ( 1 ). These values show us bad prediction results of the regression model ( 1 ).

Besides, the null hypothesis that the observed frequency distribution of residuals for the linear regression model ( 1 ) is the same as the normal distribution was tested by Pearson's chi-squared test. There is a reason to reject the null hypothesis that the distribution of residuals for the model ( 1 ) is the same as the normal distribution, since the chi-squared test statistic value equals to 13.33 is higher than the critical value of the chi-square, which equals to 7.81 for 3 degrees of freedom and 0.05 significance level. Also, for the distribution of residuals in linear regression model ( 1 ), estimators of skewness and kurtosis equal to 0.78 and 5.69, respectively. Although for the Gaussian distribution, the values of skewness and kurtosis equal to 0 and 3, respectively.

It is known [2], one of the underlying assumptions that justify the use of linear regression models is the normality of the distribution of residuals. But this assumption is not valid for the linear regression model ( 1 ). What leads to the need to construct a multiple non-linear regression model to estimate the effort of developing the mobile apps in a planning phase.

The three-factor non-linear regression model to estimate the effort of developing the mobile apps in a planning phase was constructed based on the Johnson fourvariate transformation for SB family according [9]. The three-factor non-linear regression model has the form [9] 1 Y  ˆY  ˆ Y 1  eZˆY ˆY  ˆY  ,  ( 2 ) where  is a Gaussian random variable which defines residuals,   N 0,1 ; ZˆY is a prediction result by linear regression equation for normalized data, which were transformed using the Johnson four-variate transformation for SB family, ZˆY  bˆ0  bˆ1Z1  bˆ2Z2  bˆ3Z3 ;

Z j   j   j ln

X j   j  j   j  X j ,  j  X j   j   j , j  1,2,3 ; the estimators for parameters of the Johnson four-variate transformation for SB family are: ˆY  5.69898 , ˆ1  0.524119 , ˆ 2  0.776179 , ˆ 3  0.540973 , ˆY  2.40219 , ˆ1  0.743879 , ˆ2  0.79545 , ˆ3  0.534447 , ˆY  -114.5452 , ˆ1  1.7242 , ˆ2  1.6885 , ˆ3  0.90 , ˆ Y  3328.564 , ˆ 1  12.3743 , ˆ 2  12.09 1 , ˆ 3  8.30648 ; the estimators for parameters of the linear regression equation for normalized data are: bˆ0  0 , bˆ1  0.808152 , bˆ2  -0.928296 , bˆ3  0.854262 . Parameters of the linear regression equation for normalized data were estimated by the least square method.

The values of R2, MMRE, and PRED(0.25) equal respectively 0.5789, 0.4933 and 0.5263 for non-linear regression model ( 2 ). These values show us bad prediction results of the non-linear regression model ( 2 ) approximately also as for the linear regression model ( 1 ).

Because of this, the method [13] for improving non-linear regression models was further used to construct a non-linear regression model to estimate the effort of developing the mobile apps in a planning phase. The method [13] consists of four stages. In the first stage, a set of multivariate non-Gaussian data is normalized using a multivariate normalizing transformation. After that, normalized data are checked for outliers, and, if ones are detected, outliers are cut off. The method based on the squared Mahalanobis distance [14] is used for outlier detection. In the second stage, the nonlinear regression model is constructed based on the multivariate normalizing transformation [9]. In the third stage, the prediction intervals of non-linear regression is built according [9]. And finally, at the fourth stage, it is checked whether among the data for which the non-linear regression model was built, those that go beyond the found boundaries of the prediction interval. And, if the outliers are detected, they are cut off, and we repeat all the stages, starting with the first, for new data.

For the non-linear regression model ( 2 ) with the parameter estimators obtained from the data in Table 1 of the 38 mobile apps, it turned out that Y values for the three apps (5, 6, and 11) go beyond the prediction interval. In Table 2, the lower bound of the prediction interval obtained in the first iteration is denoted as LB1, and the upper bound is denoted as UB1. In the second iteration, data from three mobile apps (5, 6, and 11) were cut off, and data from the remaining 35 apps were used for model construction. For the model ( 2 ) with the parameter estimators obtained from the data in Table 1 of the 35 mobile apps, it turned out that the value of Y for app 17 goes beyond the prediction interval. There were four such iterations, after which 30 mobile apps remained ( 1, 3, 4, 7, 9, 10, 12-14, 18-38 ). At the fifth iteration, there were no outliers; the repeat of the stages was completed, the nonlinear regression model ( 2 ) was constructed using data from 30 apps. In Table 2, the lower bound of the prediction interval obtained in the fifth iteration is denoted as LB5, and the upper bound is denoted as UB5. The row numbers (i.e., mobile apps) with the outliers in data are highlighted in bold. A dash (-) depicts the exclusion of the corresponding numbers of data in the relevant iteration (i.e., iteration 5). ˆ3  0.73913 , ˆ Y  500.266 , ˆ1  11.3796 , ˆ 2  13.2488 , ˆ 3  8.52637 ; the estimators for parameters of the linear regression equation for normalized data are: bˆ0  0 , bˆ1  1.1190 , bˆ2  -1.3765 , bˆ3  1.2027 .

The values of R2, MMRE, and PRED(0.25) equal respectively 0.965, 0.117 and 0.867 for non-linear regression model ( 2 ). These values show us good prediction results of the non-linear regression model ( 2 ) with parameter estimators obtained from the data in Table 1 of the 30 mobile apps.

Following [9], appropriate equations were constructed to determine the lower and upper bounds of the non-linear regression prediction intervals 

 YPI  Y1 ZˆY  t 2,SZY 1  ˆ

N1  zX T ZX T ZX 1zX 1 2  , ( 3 ) where Y

is a first component of a vector of normalizing transformation, ψ  Y , 1, 2 ,, k T ; k is a number of factors (regressors or independent variables); t 2, is a quantile of student's t-distribution with  2 significance level and  degrees of freedom; ZX is a matrix of centered regressors that contains the values of normalized data Z1i  Z1 , Z 2i  Z 2 , , Z ki  Zk ; zX is a vector with components Z1i  Z1 , Z 2i  Z 2 , , Z ki  Zk for i-row; SZ2Y    T ZX is k  k matrix Z X 1 N 2  i1 ZYi  ZˆYi  ,   N  k 1 ;  SZ1Z1   T ZX  

 SZ1Z2 Z X    S  Z1Zk

SZ1Z2 SZ2Z2

 SZ2Zk 

SZ1Zk   SZ Z 

2 k  ,     SZk Zk 

N where SZqZr  i1 Zqi  Zq Zri  Zr , q, r  1,2,, k . In our case, k=3.

In the fifth iteration, for the data which normalized by the Johnson four-variate transformation for SB family from 30 mobile apps, 3 3 matrix  29.8    T ZX   25.5 Z X Also, for comparison of the model ( 2 ) with other models, a linear regression model and nonlinear regression models on the basis of the univariate decimal logarithm transformation (Log10) and the Johnson univariate transformation for the SB family were constructed for data from Table 1 of the 30 mobile apps. The three-factor linear regression model for data from Table 1 of the 30 apps has the form

Yˆ  40,250  28,973 X1 - 41,798 X 2  50,665 X 3   x . ( 4 ) The three-factor non-linear regression model is constructed based on the decimal logarithm transformation for data from Table 1 of the 30 apps

ˆ ˆ ˆ Y  10x bˆ0 X1b1 X 2b2 X 3b3 , ( 5 ) where the estimators for parameters are: bˆ0  1.73898 , bˆ1  1.6687 , bˆ2  -2.1116 , bˆ3  1.30125 .

The three-factor non-linear regression model based on the Johnson univariate transformation for the SB family has the form ( 2 ) with only the following parameter estimators: bˆ3  1.1148 ˆY  0.25204 , ˆ1  0.10255 , ˆ 2  0.49345 , ˆ 3  0.61963 , ˆY  0.58192 , ˆ1  0.51359 , ˆ2  0.63352 , ˆ3  0.58967 , ˆY  19.9286 , Johnson transformation for S B family were estimated by the maximum likelihood method.

The values of R2, MMRE and PRED(0.25) equal respectively 0.838, 0.237 and 0.733 for linear regression model ( 4 ), and equal respectively 0.789, 0.206 and 0.733 the model ( 5 ), and equal respectively 0.878, 0.190 and 0.767 for the model ( 2 ) with estimators of parameters for the Johnson univariate transformation. The values of R2, MMRE, and PRED(0.25), which equal respectively 0.965, 0.117, and 0.867, is better for the model ( 2 ) with estimators of parameters for the Johnson four-variate transformation in comparison with all previous models.

The null hypothesis that the distribution of residuals for the linear regression model ( 4 ) is the same as the normal distribution was tested by Pearson's chi-squared test. There is a reason to reject the null hypothesis that the distribution of residuals for the linear regression model ( 4 ) is the same as the normal distribution, since the chisquared test statistic value equals to 10.78 is higher than the critical value of the chisquare, which equals to 7.81 for 3 degrees of freedom and 0.05 significance level. Also, for the distribution of residuals in linear regression model ( 4 ), estimators of skewness and kurtosis equal respectively to 1.52 and 7.73. There is no reason to reject the null hypothesis that the distribution of residuals for nonlinear regression models ( 2 ) and ( 5 ) is the same as the normal distribution, since the chi-squared test statistic values are less than the critical value of the chi-square, which equals to 7.81. The chisquared test statistic values equal to 4.78, 2.91, and 2.30 for the distribution of residuals in nonlinear regression models ( 5 ), ( 2 ) with estimators of parameters for the Johnson univariate transformation and ( 2 ) with estimators of parameters for the Johnson four-variate transformation respectively. For the distribution of residuals in nonlinear regression models ( 2 ) and ( 5 ), estimators of skewness and kurtosis are close to 0 and 3, respectively. Only the estimator of kurtosis equals to 5.39 for the distribution of residuals in the nonlinear regression model ( 2 ) with estimators of parameters for the Johnson univariate transformation for the SB family.

The lower (LB) and upper (UB) bounds of the linear regression and non-linear regression prediction intervals were also determined by ( 3 ) based on the decimal logarithm transformation, Johnson's univariate and four-variate transformations for a significance level of 0.05. These bounds are shown in Table 3. Note that the width of the non-linear regression prediction interval based on the Johnson four-variate transformation is less than after the Johnson univariate transformation for 29 from 30 data rows (except one with number 25), smaller than after decimal log transformation and less compared with the linear regression prediction interval width for all 30 data rows. Approximately the same results were obtained for the confidence intervals of regressions. Herewith a confidence interval of non-linear regression is defined as ( 3 ) with the only difference that in the sum in curly brackets, there will not be 1.

Such good prediction results for the constructed model may be explained best multivariate normalization of the non-Gaussian data set, which used to build the threefactor non-linear regression model based on the Johnson four-variate transformation for SB family. The measures of multivariate skewness 1 and kurtosis 2 [15] allow one to test two hypotheses that are compatible with the assumption of multivariate normality. In our case for 30 apps 1  4 and 2  24 . The estimators of multivariate skewness and kurtosis equal 8.42, 5.44, 12.86, 6.82, and 26.78, 23.08, 33.57, 25.71 for the data for 30 apps from Table 1, the normalized data on the basis of the decimal logarithm transformation, the Johnson univariate and four-variate transformations respectively. The values of these estimators indicate that the necessary condition for multivariate normality is approximately performed for the normalized data on the basis of the decimal logarithm and the Johnson four-variate transformation. Also, multivariate normality was tested by MD2 [16]. A multivariate normality condition is only performed for the normalized data on the basis of the decimal logarithm and the Johnson four-variate transformation, since for all 30 rows of the normalized data, the MD2 values are smaller than the value of the quantile of the Chi-Square distribution, which equals to 14.86 for 0.005 significance level. 4

Conclusions

Mathematical modeling of effort of development of mobile apps by non-linear regression model using multivariate normalizing transformation is performed. A threefactor non-linear regression model to estimate the effort of developing the mobile apps in a planning phase is firstly constructed on the basis of the Johnson four-variate transformation for SB family. This model, in comparison with other regression models (both linear and non-linear), has a more significant multiple coefficient of determination, a smaller value of the mean magnitude of relative error, a more significant value of percentage of prediction, and smaller widths of the confidence and prediction intervals of regression. An example of the construction of the three-factor non-linear regression model confirms the efficiency of the method for improving non-linear regression models on the basis of multivariate normalizing transformations, the squared Mahalanobis distance, and prediction intervals. Prospects for further research may include the application of other data sets to construct the multiple non-linear regression models for estimating the effort of developing the mobile apps in a planning phase.

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