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Software reliability prediction plays a very important role in the analysis of software quality and balance of software cost. The data during software lifecycle is used to analyze and predict software reliability. However, predicting the variability of software reliability with time is very difficult. Recently, support vector regression (SVR) has been widely applied to solve nonlinear predicting problems in many fields and has obtained good performance in many situations; however it is still difficult to optimize SVR's parameters. Previously, some optimization algorithms have been used to find better parameters of SVR, but these existing algorithms usually are not fully satisfactory. In this paper, we first improve estimation of distribution algorithms (EDA) in order to maintain the diversity of the population, and then a hybrid improved estimation of distribution algorithms (IEDA) and SVR model, called IEDA-SVR model, is proposed. IEDA is used to optimize parameters of SVR, and IEDA-SVR model is used to predict software reliability. We compare IEDA-SVR model with other software reliability models using real software failure datasets. The experimental results show that the IEDA-SVR model has better prediction performance than the other models.

Reliability is the ability of software system to perform its required functions under stated conditions for a specified period of time, and it is an important characteristic inherent in the concept of software quality. It is intimately connected with defects and faults. As more and more faults are encountered, the software reliability will decrease. Software reliability generally changes with time, and these changes can be treated as a time series process. Artificial neural networks (ANN) have general nonlinear mapping capabilities, and have increasingly attracted attention in the field of time series predicting [1], [2] and [3]. In [4], the reliability of the systems can be predicted by feed-forward multi-layer ANN and radial basis function ANN respectively. The ANN technology has better prediction performance than the autoregressive integrated moving average (ARIMA) approach. In [5], ANN has contributed significantly to software reliability prediction, and which achieved better prediction performance than traditional statistical models. In [6], the counter-propagation and back-propagation ANN models were used to estimate parameters of a reliability distribution with only a small dataset. The experimental results show that the proposed approach improves the accuracy of reliability predicting. In [7], the system reliability may be predicted by a hybrid learning neural fuzzy system. Numerical results demonstrate that the proposed model achieved more accurate predicting results than ARIMA and generalized regression ANN model (GRNN). However, the ANN suffers from a number of weaknesses, e.g., it is based on gradient descent, and it is easy to local minima. Recently, support vector machines (SVMs) [8], [9], [10] and [11] have been widely applied to solve nonlinear predicting problems in many fields. With the introduction of ɛ-insensitive loss function, it has been also extended to solve nonlinear regression estimation problems, such as new techniques known as support vector regression (SVR) [12]. In [13], the SVM was used to solve financial time series problems. The experimental results demonstrate that SVM forecasts better than back propagation (BP) algorithm. In [14], a two-step kernel learning method based on SVR was proposed for predicting financial time series. The results confirm the advantage of SVR. However, although SVR has very good learning performance and generalization ability, there is no structured way to determine the parameters of SVR. Estimation of distribution algorithms (EDA) [15], sometimes called probabilistic model-building genetic algorithm (GA) [16], have emerged as a generalization of GA, for overcoming the two main problems: poor performance in certain deceptive problems and the difficulty of mathematically modeling a huge number of algorithm variants [17]. In GA, a population of candidate solutions to a problem is maintained as part of the search for an optimum solution. This population is typically represented explicitly as an array of objects. Depending on the specifics of the GA, the objects might be bit strings, vectors of real numbers or some custom representation. In EDA, this explicit representation of the population is replaced with a probability distribution over the choices available at each position in the vector that represents a population member. Moreover, in GA, new candidate solutions are often generated by combining and modifying existing solutions in a stochastic way. The underlying probability distribution of new solutions over the space of possible solutions is usually not explicitly specified. In EDA, a population may be approximated with a probability distribution and new candidate solutions can be obtained by sampling this distribution. Compared with traditional GA, EDA can solve nonlinear variable coupling problems for complex optimization. Software reliability predictions are used for various purposes, such as software planning, reliability assessment, detecting faults in manufacturing processes, and evaluating risks. As reliability prediction plays an increasingly important role in assessing the performance of software systems, intensive studies have been carried out to ensure software reliability. The rest of this paper is organized as follows. Section 2 describes SVR model, expressing it as a combinatorial optimization problem with constraints. Section 3 explains the improved EDA (IEDA) and gives a model of optimizing SVR parameters based on IEDA. In Section 4, we give some assessing methods of the software reliability. Section 5 describes the numerical experiments and the results. Finally, Section 6 shows the conclusions from the experiment results.

In this paper, we proposed a hybrid IEDA-SVR. In order to maintain the population diversity, the chaos mutation was introduced into traditional EDA. IEDA is used to optimize parameters of SVR, and IEDA-SVR is used to predict software reliability. We used two real software failure datasets in the experiments. We compare prediction performance of proposed model with other various models. The experimental results demonstrate that it is very effective of utilizing IEDA-SVR to predict software reliability, and IEDA-SVR has better prediction performance than the other compared models and a fairly accurate prediction capability. The experimental results also show that to maintain the population diversity can improve the performance of the prediction model. In addition, IEDA-SVR does not require priori knowledge of parameters and as such can give useful results. The contribution of this paper is as follows: (1) To maintain the population diversity can improve the performance of EAD, which confirms the software reliability prediction model based on IEAD is effective. (2) In this paper, a method to optimize the SVR parameters is proposed. The proposed method can avoid the blindness of the SVR parameter selection and provide a technical means for the better use of SVR. (3) Software reliability prediction is very important for software engineering. In this paper, an intelligent approach is adopted in order to achieve software reliability prediction and obtain good performance. We note that the sensitivity of predefined parameters is an important issue. Our future work is to analyze the impact of these parameters for performance of software reliability predict model.