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A new probability distribution to study lifetime data in reliability is introduced in this paper. This one is a first approach to a non-homogeneous phase-type distribution. It is built by considering one cut-point in the non-negative semi-line of a phase-type distribution. The density function is defined and the main measures associated, such as th...
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... cases did not provide optimal results when a PH distribution was fitted. In particular, for the dataset related to the devices with dielectric HfAlO and subjected to a temperature of 80 • C (see Table 1 and Table 2 therein), the best PH-distribution was achieved with 128 phases. Even though the number of phases was really high, the fitting is not as satisfactory as one could expect. ...Citations
... Simultaneously, there is a shift toward more intuitive models. While the traditional analytical model, defined by its mathematical blueprint of systems and reliance on deductive processes, is esteemed for characterizing systems through structured expressions [38,39], it grapples with the challenge of authentically mirroring prevailing effects and interactions. This lacuna paves the way for functions that imbue flexibility, often eluding conventional modeling. ...
In numerous practical domains such as reliability and performance engineering, finance, healthcare, and supply chain management, a common challenge revolves around accurately modeling intricate time-based data and event duration. The inherent complexities of real-world systems often make it challenging to use conventional statistical distributions. The phase-type (PH) distributions emerge as a remarkably adaptable class of distributions suited for modeling scenarios like failure or response times. These distributions are helpful in analytical and simulation-driven system evaluation approaches and are frequently used to fit empirical datasets. This paper introduces a strategy that leverages user-friendly tools, graphical adjustment features, and integration with existing tools to streamline the process of fitting PH distributions to empirical data. The simplicity of this procedure empowers domain experts to more accurately model complex systems, resulting in enhanced decision-making, more efficient resource allocation, improved reliability assessments, and optimized system performance across an extensive spectrum of practical domains where the analysis of time-based data remains pivotal. Furthermore, this study presents a method for the automated determination of parameters within a fitted Hyper-Erlang distribution. This method utilizes the Bayesian Information Criterion (BIC) within a Bayesian optimization framework integrated into an Expectation-Maximization (EM) algorithm. Consequently, it enables deriving a given dataset’s probability density function (PDF) through a combination of Hyper-Erlang distributions. Subsequently, the PDF serves as a tool for assessing system performance.
... Here, the user can compare the graphic fit of several PHD while estimating their parameters and assess the goodness of fit. This same methodology is extended in the developed shiny app for the case of one cutpoint PHD [31]. This new class of distributions aims to reduce the number of parameters to be estimated and to improve the quality of the fit, especially in the tails of heavy distributions where the classical PHD might provoke an inaccurate fit. ...
... This class of distributions was introduced to improve the quality of the fit and to reduce the number of parameters to be estimated in comparison with the classical PHD, especially in those situations where the distribution presents heavy tails or has two modes [31]. The underlying idea is to determine a suitable cut-point that delimits properly the distribution. ...
... Finally, an interesting result associated with this new class of distributions is that the one cutpoint PHD inherits the features of classical PHD. The details about the estimation of parameters can be checked in [31]. ...
Phase-type distributions (PHDs), which are defined as the distribution of the lifetime up to the absorption in an absorbent Markov chain, are an appropriate candidate to model the lifetime of any system, since any non-negative probability distribution can be approximated by a PHD with sufficient precision. Despite PHD potential, friendly statistical programs do not have a module implemented in their interfaces to handle PHD. Thus, researchers must consider others statistical software such as R, Matlab or Python that work with the compilation of code chunks and functions. This fact might be an important handicap for those researchers who do not have sufficient knowledge in programming environments. In this paper, a new interactive web application developed with shiny is introduced in order to adjust PHD to an experimental dataset. This open access app does not require any kind of knowledge about programming or major mathematical concepts. Users can easily compare the graphic fit of several PHDs while estimating their parameters and assess the goodness of fit with just several clicks. All these functionalities are exhibited by means of a numerical simulation and modeling the time to live since the diagnostic in primary breast cancer patients.
... Simultaneously, there is a shift toward more intuitive models. While the traditional analytical model, defined by its mathematical blueprint of systems and reliance on deductive processes, is esteemed for characterizing systems through structured expressions Wang et al. [38], Acal et al. [39], it grapples with the challenge of authentically mirroring prevailing effects and interactions. This lacuna paves the way for functions that imbue flexibility, often eluding conventional modeling. ...
In numerous practical domains such as reliability and performance engineering, finance, healthcare, and supply chain management, a common and formidable challenge revolves around the accurate modeling of intricate time-based data and event durations. The inherent complexities inherent to real-world systems often render the effective application of conventional statistical distributions a formidable task. Phase-type (PH) distributions emerge as a remarkably adaptable class of distributions ideally suited for modeling scenarios like failure times or response times, thanks to their Markovian representation. These distributions find utility in both analytical and simulation-driven approaches for system evaluation , and they are frequently employed to approximate empirical datasets. This paper introduces an approach that leverages user-friendly tools, graphical adjustment features, and integration with existing tools to streamline the process of fitting PH distributions to empirical data. Simplifying this procedure empowers domain experts to more accurately model complex systems, resulting in enhanced decision-making, more efficient resource allocation, improved reliability assessments, and optimized system performance across an extensive spectrum of practical domains where the analysis of time-based data remains pivotal. Furthermore, this study presents a method for the automated determination of parameters within a fitted Hyper-Erlang distribution. This method utilizes the Bayesian Information Criterion (BIC) within a Bayesian optimization framework integrated into an Expectation-Maximization (EM) algorithm. Consequently , it enables the derivation of the probability density function (pdf) for a given dataset through a combination of Hyper-Erlang distributions. Subsequently, this pdf serves as a critical tool for the assessment of system performance.
... A third application field for this technology, connected to the natural stochasticity of these devices [1,5,[25][26][27], is associated with the implementation of security modules in hardware devoted to cryptography [27][28][29][30]. This facet of the electronic circuits is growing rapidly due to the demands of edge devices for secure data analysis and transmission on the Internet of Things. ...
... The reason behind this is that the RS physics is complex and involves different intertwined mechanisms that when put together do not follow a Weibull distribution. In this respect, Phase type distributions [26,53] achieve a much better fitting, although we do not enter into details of the statistical structure of our data since this is out of the scope of this work. See in Fig. 13 that the V reset Weibits show a more linear shape for input signals with lower ramps. ...
We present a new methodology to quantify the variability of resistive switching memories. Instead of statistically analyzing few data points extracted from current versus voltage (I-V) plots, such as switching voltages or state resistances, we take into account the whole I-V curve measured in each RS cycle. This means going from a one-dimensional data set to a two-dimensional data set, in which every point of each I-V curve measured is included in the variability calculation. We introduce a new coefficient (named two-dimensional variability coefficient, 2DVC) that reveals additional variability information to which traditional one-dimensional analytical methods (such as the coefficient of variation) are blind. This novel approach provides a holistic variability metric for a better understanding of the functioning of resistive switching memories.
A new class of distributions based on phase-type distributions is introduced in the current paper to model lifetime data in the field of reliability analysis. This one is the natural extension of the distribution proposed by Acal et al. (One cut-point phase-type distributions in reliability. An application to resistive random access memories. Mathematics 9(21):2734, 2021) for more than one cut-point. Multiple interesting measures such as density function, hazard rate or moments, among others, were worked out both for the continuous and discrete case. Besides, a new EM-algorithm is provided to estimate the parameters by maximum likelihood. The results have been implemented computationally in R and simulation studies reveal that this new distribution reduces the number of parameters to be estimated in the optimization process and, in addition, it improves the fitting accuracy in comparison with the classical phase-type distributions, especially in heavy tailed distributions. An application is presented in the context of resistive memories with a new set of electron devices for nonvolatile memory circuits. In particular, the voltage associated with the resistive switching processes that control the internal behavior of resistive memories has been modeled with this new distribution to shed light on the physical mechanisms behind the operation of these memories.
