This paper introduces the usage of regression models and equations for Data Science and Machine Learning Java applications size estimation. Size estimation of applications plays one of the key planning tasks at the early stages of project planning for the successful implementation of software development projects. Application size estimation is used to predict software development effort estimation using parametric models such as COCOMO, COCOMO II, etc. The aim of the study is to increase the reliability and accuracy of size estimation of Data Science and Machine Learning Java applications at the early stage of software project planning using class diagram metrics by building a nonlinear regression model. The object of research is the process of size estimation for open-source Data Science and Machine Learning Java applications. The subject of the study is the regression equations and nonlinear regression models to estimate the software size. To achieve this goal, we analyzed and compared the existing mathematical regression models and equations for Java applications size estimating on the sample of code metrics information from open-source Java applications of Data Science and Machine Learning. Proven the necessity of building the the three-factor nonlinear regression model for estimating the software size of Data Science and Machine Learning Java applications on the basis of the decimal logarithm normalizing transformation using the software code metrics such as the total quantity of classes, the total visible methods quantity, and the average fields quantity per class. The obtained nonlinear regression model is compared with the existing models by the regression models quality criteria such as the determination coefficient, mean magnitude of relative error and the percentage of prediction of the relative error level 0.25. The comparison confirms increasing the accuracy of software size estimation using the given sample by the obtained nonlinear regression model.
Software development effort estimation is one of the significant indicators of budget,
resources and duration planning of any project for software development business.
Reliable estimates of software development effort provide valuable information at the
early stages of project planning, and it helps to take into account risks, to recognize
planning gaps and to increase the efficiency of the development process. If the estimates
are close to the targets, then the plans can assume less risks. It is impossible to create
modern software products without integrating Data Science and Machine Learning
technologies, because the information technologies market requires them to integrate
these technologies to obtain a competitive advantage and offer better software solutions
and services to their users. The capitalization and active growth of the Machine Learning
market indicates the prospects for the development of this area of the IT industry. Thus,
the expected capitalization growth for 2030 is 528.1B USD, compared to 72.17B in 2022.
The appearance of large language models in 2022 and 2023 has caused a real revolution in
the direction of Data Science and Machine Learning in the IT and other industries [
The information about the software size for Data Science and Machine Learning Java
applications allows us to predict the software development effort estimation at the early
stages of project planning using well-known parametric models COCOMO, COCOMO II,
SLIM, SEER-SEM [
Both the linear [
Nowadays, both the linear and nonlinear regression equations and models have been developed to estimate the number code lines of Java applications and information systems, depending on the metrics of the conceptual data model based on a class diagram.
Thus, in [
The paper [
In the paper [
In the paper [
In the paper [
The analysis of the literature has shown that the linear equations and nonlinear regression models exist for estimating Java-application code size. For the existing models, there is a necessity to confirm the possibility of using to estimate the size of Data Science and Machine Learning Java applications and, otherwise, there is a necessity to build a regression model based on the metrics of Data Science and Machine Learning Java applications for the size estimation.
Let's make the assumption that the models [
In case of refutation of the assumption, it is necessary to build an appropriate
multivariate nonlinear regression model using proper normalization transformations and
compare it with the existing models [
The aim of the study is to increase the reliability and accuracy of size estimation of Data Science and Machine Learning Java applications at the early stages of software development project planning using the metrics from a conceptual data model by building a nonlinear regression model.
The object of the study is the process of software size estimation of open-source Data Science and Machine Learning Java applications.
The subject of the study is regression equations and nonlinear regression models for estimating the size of open-source Data Science and Machine Learning Java applications.
To achieve the aim of the paper, it is necessary to analyze and compare the existing
mathematical models and equations for estimating the size of Java applications on the
sample of code metrics of open-source Data Science and Machine Learning Java
applications; to justify the necessity of a nonlinear regression model building for
estimating the size of Data Science and Machine Learning Java applications; to build a
three-factor nonlinear regression model for estimating the size of Data Science and
Machine Learning Java applications using quantitative code metrics, such as the total
quantity of classes, the total quantity of visible methods, the average fields quantity per
class; compare the obtained nonlinear regression model with the existing models
[
There are different approaches for estimating the quantity of lines of code variable using both linear and non-linear regression models. Typically, the software code metrics have non-Gaussian distribution. It limits the ability to use linear models to estimate the variable of lines of code. One of the theoretical conditions for the usage of linear regression models says that the regression residuals ε should have Gaussian distribution.
The following approaches can be used to normalize data: decimal logarithm, square root,
Box-Cox transformation,
transformation, etc.
The normalizing transformations allow the construction of linear regression models based on normalized data with their further inverse transformation into nonlinear regression models.
The following quality criteria are used for forecasting quality assessment of regression models: the coefficient of determination 2, a mean magnitude of relative error MMRE and percentage of prediction for magnitude of relative error (MRE) level 0.25 (0.25).
The MMRE criterion is defined as where N - sample size and
is the value of the magnitude of relative error for the i-th line of data of a random variable.
= 1 ∑
=1 , = |( − ̂ )|.
The calculation of prediction percentage (PRED) for the magnitude of relative error
level 0.25
(0.25) = 1 ∑
=1{0
1
The
acceptable
values of
≤ 0.25 and
(0.25) ≥ 0.75
for the
measurement of the regression models accuracy of prediction results. The coefficient of
determination (R2) value is acceptable if it is more or equals to 0.75 [
The nonlinear regression models construction techniques are proposed in the papers
[
There is bijective normalizing transformation of a non-Gaussian random vector of the the form (1) (2) (3) (4) (5) ℎ
≤ 0.25 . = ( ), = −1( ), where k is number of factors (regressors or independent variables) and the inverse where
is a vector of bijective normalizing transformation functions, = The linear regression model for multivariate normalized data according to (4) will have = ̂ + = ̂ 0 + ̂1 1 + ̂ 2 2+. . . + ̂ + .
where ε is Gaussian random variable, ε ∼ (0, σε2 ); ̂ 0, ̂1, ̂ 2, . . . , ̂ - estimators for parameters of the linear regression model (6). The estimators are calculated by the least square method. To build a nonlinear regression model, the inverse transformation (5) is applied to the linear regression model (6). The nonlinear regression model will have the form
= −1( ̂ + ) = −1( ̂ 0 + ̂1 1 + ̂ 2 2+. . . + ̂ + ).
The choice of factors in regression models should take into account the level of
multicollinearity, because a high level of correlation between regression factors increases
the sensitivity of the model to random changes in the data. The multicollinearity level is
determined by the variance inflation factors (VIFs) of the independent variables. For a
multiple linear regression model with k factors Xi , i=1 2,..., k, the VIFs are represented by
the diagonal elements of the inverse correlation k×k matrix. If the value of the VIF
coefficient exceeds 10 (the threshold value), a high level of
multicollinearity exists
between the independent variables [
normalization transformation is chosen to normalize multivariate data =
10( ), where Х takes the values , 1, 2, . . . , respectively.
Normality distribution of multivariate data is checked with Mardia test [
The test statistics for 1, has the form 6 1, ≤ 2 , (6) (7) (8) (9) (10) (11)
(12) (13) 1, = 1 ∑
2 2, = 1 ∑
=1 ∑ =1[( − ) −1( − )] ,
3 =1[( − ) −1( − )] ,
2 = 1 ∑ =1( − )( − ) , where X is k-dimensional vector of random variable, = ( 1, 2, . . . , ) and SN is a biased sample variance matrix of multivariate random variable X. it is given by where is a means vector of independent variable of the sample, = ( 1, 2, . . . , ) . Accordingly to Mardia test, the following conditions should be verified to confirm if the where 2 is approximated Chi-Square distribution with ( + 1)( + 2)/6 degrees of freedom and α is a significant level (Accordingly to the proposed technique, α=0.005).
For 2, , the test statistics is 1-α quantile of the normal distribution with the mathematical expectation = ( + 2) and variance 2 = 8 ( + 2)/
2, ≤ 1− ( , 2) .
Outliers of the sample are detected using the techniques proposed in [
where SN is a biased sample variance matrix (11), Z is a normally distributed random variable. According to the technique the elements of the main diagonal of 2, = 1,2, . . . matrix are detected as outliers if the values are exceeded the threshold value of the ChiSquare 2 distribution quantile for the significant level - α. These outliers should be excluded from the sample.
The prediction interval outlier detection technique is based on prediction intervals of the nonlinear regression model. If the outliers are detected, ones are discarded from the sample. For the nonlinear regression model (7), the prediction interval is given by ̂ where ̂ is the upper and lower bounds of the prediction interval (PI) of the linear regression (6), −1 is the inverse normalizing transformation function of the linear regression estimators. The linear regression prediction interval is given by ̂ 1/2 , where /2, is a quantile of T-Student distribution with = − − 1 degrees of freedom and α/2 significant level; 2 = 1 ∑ =1( − ̂ )2; + is a vector of central moments of independent variables of the sample which is given by { 1 − 1, 2 − 2, . . . , − } ; ⬚ is k x k matrix 2 = ( − ) −1( − ), (14) (15) (16) (17) = [ ], where
= ∑ −1( − )( − ), q,r = 1,2, …, k.
It is recommended to detect and discard the outliers in an iterative way. Only one outlier should be discarded per iteration and the nonlinear regression model building should be started from the start for the new sample.
The authors have collected the sample dataset of the code metrics of 74 Data Science and
Machine
Learning
applications hosted on the
GitHub
platform
(https://github.com). The following metrics were obtained using the CK static code
analysis tool (https://github.com/mauricioaniche/ck). The sample includes lines of code
of projects (KLOC), total quantity of application classes (CLASS), total quantity of unique
method calls in a class (RFC), total quantity of class method cohesion values (LCOM), total
quantity of static methods (SMQ), total quantity of visible methods (VMQ), total quantity
of class fields (TFQ), total quantity of public class fields (PFQ), and total coupling between
objects per project (CBO). The resulting metrics are shown in Table 1. These metrics
(excepting KLOC) can be obtained at an early stage of project planning from the
conceptual data model of the application.
6. Experiment
The regression models and equations [
The result of the linear regression models [
The three-factor nonlinear regression model [
The one-factor nonlinear regression model [
The test of the model [
Thus, it is required to build the three-factor nonlinear regression model (7) to estimate the software size (in KLOC) for Data Science and Machine Learning Java applications using decimal logarithm normalizing transformation. 6.2. Building the three-factor nonlinear regression model To increase the reliability and accuracy of size estimation of Data Science and Machine Learning Java applications, the three-factor nonlinear regression model is built on the basis of decimal logarithm normalizing transformation (8) using the data sample from Table 1. The model is based on the parameters of total quantity of classes (CLASS) X1, the total quantity of visible methods (VMQ) X2, and the average quantity of fields per class (aTFQ) X3.
The model factors X1, X2 and X3 were tested for multicollinearity by the VIFs [
Let’s check the null hypothesis if the linear regression model can be applied for software size estimation in the following form
= ̂ + = ̂ 0 + ̂1 1 + ̂ 2 2 + ̂ 3 3 + . (18)
The factors of the three-factor linear regression model (18) are estimated using the least squares method: ̂ 0=-13.863468, ̂1= 0.001468, ̂ 2= 0.010861, ̂ 3= 8.149336.
The regression residuals ε are tested for normal distribution converging by Chi-Square (χ2) test with the significance level α = 0.05 and degrees of freedom v = 4. The hypothesis is rejected, because the test value χ2 = 163938.42 is greater than the quantile of ChiSquare distribution, which equals 9.4877. Thus, the building of the nonlinear regression model is justified.
To build the nonlinear regression model, the sample from Table 1 is normalized using
the decimal logarithm normalization transformation (8). Following [
Next, a linear regression model is built for the normalized data
= ̂ + = ̂ 0 + ̂1 1 + ̂ 2 2 + ̂ 3 3 + ,
The estimators for the parameters ̂ 0, ̂1, ̂ 2, ̂ 3 of the linear regression model (19) for the normalized data are calculated using the least square method.
Applying the inverse normalizing transformation to (8) the linear regression model (19), the nonlinear regression model has the form
= −1(̂ + ) = −1( ̂ 0 + ̂1 1 + ̂ 2 2 + ̂ 3 3 + ),
For the model (20), the upper and lower bounds of the prediction interval (15) are
determined for the significance level α = 0.05. The data rows 8, 44, 72 are outside of the
prediction interval. According to the technique [
The outliers are discarded iteratively with only one data row per iteration.
The estimators for the parameters of the regression model (19) at the last iteration are: ̂ 0 = −1.855315, ̂1 = 0.149619, ̂ 2 = 0.822320, ̂ 3 = 0.349549. The normal distribution was tested and confirmed for the regression residual random variable ε using Chi-Square test. The test value χ2 = 4.1411 is smaller than the value of the quantile of Chi-Square distribution 9.4877 for the significance level α = 0.05 and degrees of freedom v = 4. According to Maridia’s test, the normalized sample of 69 data rows has normal distribution since multivariate skewness (12), which is 29.39, is less than the quantile of Chi-Square distribution, which is 40.00 for 20 degrees of freedom and significance level α = 0.005. Analogically kurtosis (13), which is 22.10, is less than quantile of normal distribution, which is 28.30 for the mean of 24.
By applying the inverse transformation to decimal logarithm normalized transformation (7), the three-factor nonlinear regression model (20) has the form ̂ = 10 −1.855315 10.149619 20.822320 30.349549. (21)
The quality criteria of the nonlinear regression model (21) have the following values of indicators: R2=0.8937, MMRE=0.1867, PRED(0.25)=0.7703. This confirms the high quality of the obtained model in comparison with the existing models. The regression model (21) leads to increase the reliability and accuracy of the number of code lines estimating for
An important problem of increasing the reliability and accuracy of estimating the size of Data Science and Machine Learning Java applications is solved.
The scientific novelty of the obtained results is that the nonlinear regression model for multivariate non-Gaussian data is improved by constructing a three-factor nonlinear regression model for estimating the size of the code lines of Data Science and Machine Learning Java applications based on the decimal logarithm normalizing transformation. This model has a higher value of the multiple coefficient of determination R2, a lower value of the mean relative error MMRE, and a higher value of the percentage of prediction of the relative error level PRED(0.25) in comparison to the existing regression models and equations.
The practical significance of the obtained results is that the software that implements the built model was developed using the Kotlin programming language and the Apache Math3 mathematical package. The experimental results allow us to recommend the built model for use in practice. The model can reduce risks and cost of software development for Data Science and Machine Learning Java-applications or information systems.
Prospects for further research may include the usage of other multivariate normalizing transformations and data sets to build a nonlinear regression model.
The authors would like to express their sincere gratitude to the Doctor of Technical Sciences, Professor Sergiy B. Prikhodko, NUOS, Mykolaiv, Ukraine for his support in applying the mathematical apparatus for the research and Doctor of Technical Sciences, Professor Serhii K. Chernov, NUOS, Mykolaiv, Ukraine for the offer and motivation to participate in the conference.