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To handle large variation data, an interval piecewise regression method with automatic change-point detection by quadratic programming is proposed as an alternative to Tanaka and Lee's method. Their unified quadratic programming approach can alleviate the phenomenon where some coefficients tend to become crisp in possibilistic regression by linear programming and also obtain the possibility and necessity models at one time. However, that method can not guarantee the existence of a necessity model if a proper regression model is not assumed especially with large variations in data. Using automatic change-point detection, the proposed method guarantees obtaining the necessity model with better measure of fitness by considering variability in data. Without piecewise terms in estimated model, the proposed method is the same as Tanaka and Lee's model. Therefore, the proposed method is an alternative method to handle data with the large variations, which not only reduces the number of crisp coefficients of the possibility model in linear programming, but also simultaneously obtains the fuzzy regression models, including possibility and necessity models with better fitness. Two examples are presented to demonstrate the proposed method.

This study proposes fuzzy multiple objective programming to determine the measure of fitness and the number of change-points in an interval piecewise regression model. To increase the measure of fitness, Tanaka and Lee proposed a conceptual procedure, which is a heuristic approach and becomes complicated for determining the proper polynomial. Therefore, a multiple objective approach is adopted to obtain a compromise solution among three objectives — maximizing the measure of fitness, minimizing the number of change-points and minimizing the width to obtain the interval regression models. By using the proposed method, a better measure of fitness can be obtained. Two numerical examples are used as demonstrations to illustrate our approach in more detail.

This article presents a robust fuzzy procedure for estimating change-point regression models. We propose incorporating the fuzzy change-point algorithm with the M-estimation technique for robust estimations. The fuzzy c partitions concept is embedded into the change-point regression model so the fuzzy c-regressions and fuzzy c-means clustering can be employed to obtain the estimates of change-points and regression parameters. The M estimation with a robust criterion is used to make the estimators robust to the presence of outliers and heavy-tailed distributions. We create two robust algorithms named FCH and FCT by using Huber’s and Tukey’s functions as the robust criterion respectively. Extensive experiments with numerical and real examples are provided for demonstrating the effectiveness and the superiority of the proposed algorithms. The experimental results show the proposed algorithms are resistant to atypical observations and outperform the existing methods. The proposed FCH and FCT are generally comparable but FCT performs better in the presence of extremely high leverage outliers and heavy-tailed distributions. Real data applications show the practical usefulness of the proposed method.

The reliability of a repairable system that is either improving or deteriorating depends on the system's chronological age. If such a system undergoes "minimal repair" at the occurrence of each failure so that the rate of system failures is not disturbed by the repair, then a nonhomogeneous Poisson process (NHPP) may be used to model the "age-dependent" reliability of the system. The power law process (PLP) is a model within the class of NHPP models and is commonly used as a model for describing the failure times of a repairable system. We introduce a new model that is an extension of the PLP model: the power law process change-point model. This model is capable of describing the failure times of particular types of repairable systems that experience a single change in their rates of occurrence of failures. Bayesian inference procedures for this model are developed.

Most of software reliability growth models (SRGMs) describe a software reliability growth process depending on only testing-time. However, it is said that a software reliability growth process in an actual testing-phase of a software development process depends on not only testing-time but also testing-effort factors. And we often observe a phenomenon that stochastic characteristics of the software failure-occurrence time or the software failure-occurrence time-interval changes notably in an actual testing-phase. The testing-time when such phenomenon is observed is called change-point. It is said that the effect of change-point on the software reliability growth process influences accuracy for software reliability assessment based on conventional SRGMs. This paper discusses a two-dimensional software reliability growth modeling with change-point for describing an actual phenomenon being related to the software reliability growth process. Further, we show examples of the applications of software reliability assessment based on our two-dimensional SRGM by using actual data.

We discuss software reliability measurement with change of testing-environment by developing software reliability growth models. It is known that such change influences the accuracy for the software reliability assessment based on a software reliability growth model. This paper additionally shows numerical illustrations for software reliability measurement based on our software reliability growth models by using actual data.

A software hazard rate model is known as one of the important and useful mathematical models for describing the software failure occurrence phenomenon observed in a testing phase. It is difficult to say that the testing environment always constant during a testing phase due to changing the specification and fault target and so forth. Therefore, taking into consideration of the effect of the change in software reliability growth modeling is expected to conduct more accurate software reliability assessment. In this paper, we develop extended software hazard rate models based on well-known Jelinski–Moranda and Moranda models, by considering with a change of testing environment. Especially in this paper, we incorporate the uncertainty of the effect of the change on the software reliability growth process into the software hazard rate modeling. Finally, we show numerical examples for our models and results of model comparisons by using actual data.

Software failure-occurrence or the fault-detection phenomenon is changed notably in an actual testing phase or an operational phase due to the changes of factors influencing the software reliability growth process. Taking into consideration of the effect of the change in software reliability growth modeling is expected to conduct more accurate software reliability assessment because it is said that such approach enables us to conduct more plausible software reliability assessment reflecting an actual testing-environment. We develop a framework for developing software hazard rate models with effect of change-point, and discuss change-point detection methods for applying our model to quantitative software reliability assessment. Additionally, we discuss an optimal software release problem for estimating optimal shipping time with the effect of the change as one of the application problems of our model in software project management. Finally, we show numerical examples for software reliability assessment and our optimal software release policy based on our change-point model by using actual data.

We discuss software hazard rate modeling with a change of testing-environment and a software reliability assessment method based on the proposed software hazard rate models. A software hazard rate model is known as one of the important and useful mathematical models for describing the software failure-occurrence phenomenon and conducting quantitative software reliability assessment. Taking into consideration of the effect of the change in software reliability growth modeling is expected to conduct more accurate software reliability assessment because it is said that such approach enables us to conduct more plausible software reliability assessment reflecting the actual testing-environment. Especially in this paper, we develop exponential-type software hazard rate models with effect of change-point and a software reliability assessment method based on our models. Finally, we show numerical examples for our models and results of model comparisons with existing software hazard rate models by using actual data.

Existing optimal software release problems have been discussed by using an evaluation criterion such as cost, reliability, delivery. When we use the methods by those evaluation criteria, the optimal release time is determined by an evaluation criterion. However, it is more realistic that we determine the optimal release time with multiple attributes. Therefore, in this study, we estimate the optimal release time by using multi-attribute utility theory (MAUT). Since MAUT is one of utility theories, we can estimate an optimal release time and change-point from the perspective of utility by maximizing the multi-attribute utility function. Especially, we consider the both of two attributes: cost and reliability. Then, we apply a software reliability growth model (SRGM) with change-point to represent the cost and reliability attributes. Concretely, we use an exponential SRGM with change-point. That is, we can estimate not only the optimal release time but also change-point. Finally, we show numerical examples by using actual data sets. Especially, we check the behavior of the optimal release time, change-point, total software cost and utility.

We have no doubt that the software reliability growth process in the testing phase depends on the test environment factors, such as the attained testing coverage, the number of test-runs and the debugging skills, which affect the software failure occurrence or fault detection phenomenon in the testing phase. In this paper, we propose software reliability models that consider the effects of the testing environment factors. Our models are developed by a program size-dependent discrete binomial-type software reliability modeling approach. This modeling approach is also consistent with software reliability data collection and enables us to consider the effect of the program size. Finally, we compare the accuracy of our models in terms of mean square errors (MSE) and Akaike’s information criterion (AIC) with the existing corresponding model by using actual data.

We discuss a Markovian modeling approach for software reliability assessment with the effects of change-point and imperfect debugging environment. Testing-time when the characteristic of the software failure-occurrence or fault-detection phenomenon changes notably is called change-point. Taking into account the effect at change-point in software reliability growth modeling is important to improve the accuracy of software reliability assessment. Our modeling approach describes a software reliability growth process with not only the effect of change-point but also the imperfect debugging activities based on a semi-Markov process for reflecting actual situation of debugging activities. Finally, we show numerical examples of our model for software reliability analysis and check the performance of our model with an existing Markovian software reliability growth model by using actual data.

This study provides an analytical model to predict the fixing pattern of issues in the open-source software (OSS) packages to assist developers in software development and maintenance. Moreover, the continuous evolution of software due to bugs removal, new features addition or existing features modification results in the source code complexity. The proposed model quantifies the complexity in the source code using the Shannon entropy measure. In addition, the issues fixing growth behavior is viewed as a function of continuation time of the software in the field environment and amount of uncertainty or complexity present in the source code. Therefore, a two-dimensional function called Cobb–Douglas production function is applied to model the intensity function of the issues fixing rate. Furthermore, the rate of fixing the different issue types is considered variable that may alter after certain time points. Thus, this study incorporates the concept of multiple change-points to predict and assess the fixing behavior of issues in the software system. The performance of the proposed model is validated by fitting the proposed model to the actual issues data of three open-source projects. Findings of the data analysis exhibit excellent prediction and estimation capability of the model.

A Markovian software reliability model is known as one of the useful models which enable us to describe the details on the dynamic behavior of software reliability growth process with imperfect debugging activities. As the application of this model for software development management, we discuss software shipping time estimation based on the extended Markovian software reliability model considering with the effect of change-point, which is the testing-time when the characteristic of the software failure-occurrence or fault-detection phenomenon changes notably. Especially in our discussion on the software shipping time estimation, we derive an analytical software shipping policy under certain software reliability objective and show numerical examples of the application of the software shipping policy by using actual software reliability data. Finally, we give some useful consideration on the relationship between the software shipping time and the software reliability growth process for conducing quality-oriented software development management.

The class of locally stationary wavelet processes is a wavelet-based model for covariance nonstationary zero mean time series. This paper presents an algorithm for the pointwise adaptive estimation of their time-varying spectral density. The performance of the procedure is evaluated on simulated and real time series. Two applications of the procedure are also presented and evaluated on real data. The first is a test of local existence for the coefficients of the so-called wavelet periodogram. The second is a new test of covariance stationarity.

Modeling diffusion dynamics of multi-generation innovation requires a critical examination of external factors that may affect its diffusion process. It has been observed that due to companies continuously varying marketing strategies, the adoption rate of an innovation alters with time. However, there are other factors such as the launch of a new competitive product or improved product generation, which may affect the growth of an innovation. The time-instance at which these changes are observed is called change-point. Motivated by this phenomenon, the present research identifies the launch of a new generation as a change-point where adoption function of the previous generation experiences a structural change. The objective of the current research is to improve the forecasting accuracy of a diffusion model for technological innovations by integrating essential factors that affect the diffusion process. From the findings of empirical analysis, it can be inferred that the proposed two-generational diffusion model illustrates the diffusion pattern of Dynamic Random Access Memory (DRAM) semiconductors remarkably well. In fact, the computed results show that the suggested model has better forecasting ability than previously established multi-generation models.

The analysis of the consequences of land use (in particular forest use) may be considered as a partial step towards an integrated modeling of a land system. In the paper a forest territory is considered, where a gap-cut is made, and after a given time period the eventual change in the spatial distribution of undergrowth plants and tree seedlings is to be detected. Floristic data are collected along a line transect. A method for the detection of the change in the plant distributions along the transect is proposed to see whether this occurs at the geometric frontier of the human intervention.

Since in the considered case the distribution of the change-point estimate is not known, as a substitute of its confidence interval, the so-called change-interval is calculated, using an adaptation of the bootstrap method. As an illustration, for a concrete plant species, the maximum likelihood estimation of the change-point and the calculation of the above mentioned change-interval is presented. Finally, the validation of the proposed method against some typical ecological situations is also presented, which provides a justification of the used algorithms.

We consider a density estimation problem with a change-point. The contribution of the paper is theoretical: we develop an adaptive estimator based on wavelet block thresholding and we evaluate these performances via the minimax approach under the 𝕃^p risk with p ≥ 1 over a wide range of function classes: the Besov classes, (with no particular restriction on the parameters π and r). Under this general framework, we prove that it attains near optimal rates of convergence.

Inference is considered about the point in a sequence of random variables at which the probability distribution changes. In particular, we consider the case a Poisson distribution with changing in parameter.

An application of the maximization technique in the synthesis of polynomial adaptive algorithms for retrospective (a posteriori) estimation of the change-point of the mean value and standard deviation (uncertainty) of the non-Gaussian random sequences is presented. Statistical simulation shows a significant increase in the accuracy of polynomial estimates, which is achieved by taking into account the higher-order statistics (cumulant coefficients) of handled statistical data.