Power Law Process Research Articles

Memory-type time-between-events charts using nonhomogeneous Poisson process

The traditional time-between-events (TBE) control charts are developed in non-adaptive fashion assuming the Poisson process, where the TBE follows the exponential distribution. However, in many situations we need an adaptive strategy, especially for healthcare or system monitoring. The focus of this study is to introduce new exponentially weighted moving average and cumulative sum adaptive TBE control charts for high-quality processes based on nonhomogeneous Poisson process by assuming the power law intensity. Charts assuming the nonhomogeneous Poisson process are dynamic and can be used where the underlying failure rate may be changing over time. The performance of the proposed control charts is evaluated using the average run length and coefficient of variation of the run length distribution. Three real data examples which correspond to different shape parameters of the power law process are also discussed in the article.

Using the Non-Homogeneous Poisson Process (Duane's Model) to Analyze the Number of Failures in Industrial Equipment

Objective: The aim of this paper is to show the application of Duane's Chart to analyze the time between failures in a company in the South of Rio de Janeiro. Theoretical framework: A Duane chart is a scatter plot of the cumulative failure rate versus time. Use a Duane chart to do the following: •Assess whether the data follows a power-law process or a homogeneous Poisson process. •Determine whether a repairable system is improving, getting worse or remaining stable. The fitted line on the Duane chart plots the best fitted line when the power-law process assumption is valid and the shape and scale are estimated using the least squares method. The Duane plot should be approximately linear if the power-law process or homogeneous Poisson process is appropriate. A negative slope shows an improvement in reliability, a positive slope shows a deterioration in reliability and no slope (a horizontal line) shows a stable system (Zanin et al., 2010). Method: Data was taken from a company in the South of Rio de Janeiro and a case study was carried out using Duane's Chart. Final Considerations: This result indicates that Duane's Model is a very viable option when the aim is to monitor the failure rate as a function of time to check whether a system is improving or not. In this specific Case Study, it is clear that there is a significant improvement in the system over time and that the work carried out in this company is being very effective in reducing and even eliminating failures in the long term. Implications of the research: The use cases of Duane's Chart are multiplying in scientific literature and are proving to be highly effective for interpreting data in the area of reliability. Originality/value: Despite being a well-known Statistical Tool, Duane's Chart has a specific application in monitoring the number of failures over time in a system.

Accelerated failure time frailty model for modeling multiple systems subject to minimal repair

AbstractThis article presents accelerated failure time models with and without frailty for modeling multiple systems subject to minimal repair. The study considers the conventional accelerated failure time model, the accelerated failure time model with Gamma frailty, and proposes the accelerated failure time model with weighted Lindley frailty, which has attractive properties such as a closed‐form Laplace transform. The proposed model extends the accelerated failure time model with the intensity function of a power law process. It retains the direct physical interpretation of the original accelerated failure time model, in which the role of covariates is to accelerate or decelerate the time to each repair. This framework includes parametric approaches to model fitting, which we consider for estimating the vector of regression parameters under this model and the parameter in the baseline intensity functions. The methodology is illustrated with a simulation study and a toy example to demonstrate the applicability of these models in the industrial context.

Statistical Inference for Generalized Power-Law Process in repairable systems

Repairable systems are often used to model the reliability of restored components after a failure is observed. Among various reliability growth models, the power law process (PLP) or Weibull process has been widely used in industrial problems and applications. In this article, we propose a new class of model called the generalized PLP (GPLP), based on change points. These can be treated as known or unknown parameters, or interpreted as failure times. Herein, we consider the impact of all or some fixes on the failure intensity function. In this context, unlike the usual PLP, the GPLP is not restricted to the assumption of minimal repair (MR). Other situations, such as perfect, efficient, and harmful repair, can be considered. We present some special cases of the GPLP, such as the main models used to analyze repairable systems under the assumption of imperfect repair. The estimators of the proposed model parameters were obtained using the maximum likelihood method. We evaluated the performance of the parameter estimators through Monte Carlo (MC) simulations. The proposed approach is fully illustrated using two real failure time datasets.

Analysis of meteorological drought periods based on the Standardized Precipitation Evapotranspiration Index (SPEI) using the Power Law Process approach

Analysis of meteorological drought periods based on the Standardized Precipitation Evapotranspiration Index (SPEI) using the Power Law Process approach

Modified Imperfect Repair Model (ARAM1) and new PLP parameterization

The class of models known as Arithmetic Reduction of Age (ARA) has found widespread use in modeling equipment maintenance data, assuming that the system continuously degrades between failures, as indicated by β > 1 in the Power Law Process (PLP), commonly employed for fitting such data. However, there are situations where system improvement between failures may occur, as signaled by β < 1 in the PLP. Systems or equipment that enhance over time to a certain degree are less common, as degradation often proves to be a more typical feature. Nevertheless, some systems may exhibit this initial improvement due to various factors such as adaptation, maturation, or updates. In this context, the study introduces the modified ARA1 model (ARAM1), enabling the modeling of systems in a state of reduction or degradation, thereby overcoming this limitation. Additionally, it proposes a new reparameterization of the PLP as a time truncation while preserving the original interpretation of the PLP parameters. An example with a real-world dataset illustrates the potential applications of the proposed model. From the results, it is evident that the ARAM1 model serves as a valuable tool for enhancing the understanding of the reliability of systems subject to imperfect repairs.

Repairable System Analysis Using the Discrete Weibull Distribution

In many practical circumstances, a repair can be performed when a system fails to restore to a condition before a failure. This type of repair is known as minimal repair and one of the most used models is the power law process (PLP). It is common to consider equipment failure time as a continuous variable in this model, assuming a high degree of accuracy in the measurement tool. However, in practical situations, failures are usually observed and recorded as integer numbers of time units, such as the number of days and hours, indicating a discrete process. In this study, we used a discrete Weibull distribution instead of a continuous Weibull distribution. Since both Weibull models have similar complexities, some benefits are observed in the usage of the discrete model, such as a lower standard deviation of the parameter related to the system deterioration and also a lower Akaike's information criterion. For illustrative purposes, the use of a discrete Weibull distribution was applied to a dataset related to the failures of concrete mixer trucks. Moreover, an approach using a Markov chain with discrete states to obtain the average number of failures and the respective optimum maintenance policy was included. Additionally, six cases (found in the literature) fitted by the PLP model were reanalyzed and compared under the discrete model perspective.

Annual Mean Temperature and Rain Precipitation in North America Using NHPP to Detect Climate Changes

In this study, non-homogeneous Poisson processes (NHPP) are assumed to analyze annual average temperatures and rain precipitations, considering climate data for some regions of North America reported for a long period. A power law process (PLP) is assumed for the intensity function (derivative of the mean value function) or rate \(\lambda\) (t), t \(\ge\) 0 of the NHPP which the Poisson events occur considering data (accumulated number of years in a given time interval [0,t) where the climate measure is above a threshould given by the overal average in the assumed period) in presence or not of a change-point. The parameters of the assumed model are estimated under a Bayesian approach and using MCMC (Markov Chain Monte Carlo) methods. Alternatively to the use of a PLP process, we also assume a polynomial parametrical form for the mean value function of the NHPP process where a simple Bayesian inference approach is proposed to get better fit for the intensity and mean value functions of the NHPP process. From the fitted models it was possible to to detect the years where climate changes occurred.

Optimal block replacement based on expert judgement method

Applying the theory of uncertainty is a good approach to study the reliability of a system when there is a little frequency of suitable data. Based on this theory, the policy of block replacement with minimal repair is considered assuming the system lifetime follows a power law process. The unknown parameters are estimated according to evidence theory that affects the epistemic uncertainty. The Dempster-Shafer as well as Yager rules are applied to aggregate the judgements and mental estimates of two or more experts. After determining the unknown parameters, a judgement interval is derived for the optimal replacement time using the long-run cost criterion.

Mathematical model for conversion of groundwater flow from confined to unconfined aquifers with power law processes

Abstract In this work, we propose a mathematical model to depict the conversion of groundwater flow from confined to unconfined aquifers. The conversion problem occurs due to the heavy pumping of confined aquifers over time, which later leads to the depletion of an aquifer system. The phenomenon is an interesting one, hence several models have been developed and used to capture the process. However, one can point out that the model has limitations of its own, as it cannot capture the effect of fractures that exist in the aquitard. Therefore, we suggest a mathematical model where the classical differential operator that is based on the rate of change is substituted by a non-conventional one including the differential operator that can represent processes following the power law to capture the memory effect. Moreover, we revise the properties of the aquitard to evaluate and capture the behaviors of flow during the process in a different aquitard setting. Numerical analysis was performed on the new mathematical models and numerical solutions were obtained, as well as simulations for various fractional order values.

The ENBIS‐21 quality and reliability engineering international special issue

The ENBIS-21 Quality and Reliability Engineering International Special Issue is related to the contributions to the 21st Annual Conference of the European Network for Business and Industrial Statistics (ENBIS) held online from 13 to 15 September, 2021. The topics covered by the 12 papers included in this special issue fall within the scope of ENBIS, whose ultimate goal is to bridge the gap between methodology and application in the field of business and industrial statistics and foster networking between statisticians in academic and commercial settings. It becomes evident from the papers on this issue that classical methods of industrial statistics are still having a wide range of applications and expansions towards complex and big data structures as well as they are increasingly cross-fertilized by artificial intelligence and machine learning perspectives and approaches. The issue is opened by the contribution of Yang et al.1 in the Statistical Process Monitoring (SPM) of data in the form of tensors. The authors develop a hierarchical in-situ process monitoring approach for anomaly detection, which is applied to thermal video streams acquired during a metal additive manufacturing process for highly complex parts. A more traditional SPM approach based on generalized likelihood ratio (GLR) control charts is elaborated by Rizzo and Di Bucchianico2 to set up tailor-made monitoring for high-purity (high-quality) processes and deals with issues arising in GLR control charts for discrete processes. Along the line of developing statistical methods for complex data structures, Gril et al.3 set up a tensor-on-tensor regression model able to dynamically predict repetitive human motions helpful in avoiding unwanted collisions at the interface with robots in industrial assembly tasks. Chuquin et al.4 present a straightforward extension of the K-means clustering to spatially correlated functional data, with an application in the analysis of the normalized difference vegetation index in a large region of Ecuador. From a more general perspective, Vanacore et al.5 address the problem of assessing prediction performance in the multi-class classification of imbalanced data sets. In the optimal Design of Experiments (DoE) Pesce et al.6 offer new results that are oriented to bias reduction in large datasets and protection against confounders. Whereas, Kirchhoff et al.7 use DoE as the initial solution for the optimization of shift planning in high-bay warehouse operations based on Gaussian process models with low-rank correlation kernels. Risk assessment and management is the topic of the following three papers. Testik and Unlu8 compare traditional and fuzzy failure modes and effect analysis to rank risks in different areas of testing and calibration laboratories. Meynaoui et al.9 deal with an accidental scenario in a sodium-cooled fast nuclear reactor by means of a global sensitivity analysis of the input parameter distribution used in numerical simulator outputs to model physical phenomena. Lai and Zwetsloot10 offer a data-driven ensemble ranking system of product quality, which is validated in the identification of high-risk manufacturers in a solar industry case study. The final two papers are devoted to the Reliability and Maintenance of repairable systems. El-Aroui and Gaudoin11 propose a general goodness-of-fit test for imperfect repair models that has the favorable property of being asymptotically distribution-free for renewal and power law processes. Fouladirad et al.12 develop a new bounded transformed gamma process model to describe and predict degradation phenomena by relaxing the typical assumption that the degradation level can grow indefinitely, which is often unrealistic for technological units. The proposed model is also applied to a set of wear measures of cylinder liners that equip a diesel engine for marine propulsion. We are deeply grateful to the journal staff for their professional assistance in the editorial process, as well as to the anonymous referees for the time and effort spent to provide careful and constructive comments, and to all the authors, who made revisions on a very tight timescale.

Correlation versus co-fractality: Evidence from foreign-exchange-rate variances

The concept of correlation appears to be the cornerstone of modern finance as it is applied in almost all finance-related research studies. However, Fama (1963) argued that “if the [population] variance is infinite, other statistical tools (e.g., least-squares regression) which are based on the assumption of finite variance will, at best, be considerably weakened and may in fact give very misleading answers” (p. 421). This study shows variances of foreign exchange rates to be governed by power laws with a tail exponent of α < 3, suggesting infinite second moments. We derive a new concept to measure dependencies between power-law processes with this tail exponent, which we term co-fractality. We show that risk diversification based on the concept of correlation indeed gives misleading results. Notably, foreign-exchange-rate variances lacking co-fractality in our earlier subsample do not show evidence for co-fractality in our later subsample. We argue that co-fractality, as opposed to correlation, should be used to measure the dependency between processes governed by power laws.

Piecewise differential equations: theory, methods and applications

&lt;abstract&gt;&lt;p&gt;Across many real-world problems, crossover tendencies are seen. Piecewise differential operators are constructed by using different kernels that exhibit behaviors arising in several real-world problems; thus, crossover behaviors could be well modeled using these differential and integral operators. Power-law processes, fading memory processes and processes that mimic the generalized Mittag-Leffler function are a few examples. However, the use of piecewise differential and integral operators cannot be applied to all processes involving crossovers. For instance, a considerable alteration eventually manifests when groundwater over-abstraction causes it to flow from confined to unconfined aquifers. The idea of piecewise differential equations, which can be thought of as an extension of piecewise functions to the framework of differential equations, is introduced in this work. While we concentrate on ordinary differential equations, it is important to note that partial differential equations can also be constructed with the same technique. For both integer and non-integer instances, piecewise differential equations have been introduced. We have explained the usage of the Laplace transform for the linear case and demonstrated how a new class of Bode diagrams could be produced. We have provided some examples of numerical solutions as well as conditions for the existence and uniqueness of their solutions. We discussed a few scenarios in which we used chaos and non-linear ordinary differential equations to produce novel varieties of chaos. We believe that this idea could lead to some significant conclusions in the future.&lt;/p&gt;&lt;/abstract&gt;

수리 가능한 체계에 대한 Upside-Down 욕조 곡선 형태의 삼분할된 고장 강도 모형

Purpose: This study proposes an upside-down bathtub-shaped three-segmented failure intensity model for repairable systems. The procedure for estimating the parameters for the proposed model is presented under certain assumptions regarding the change points of the model.BRMethods: The proposed model comprises three failure intensity functions - one for each of the three model segments. The model is therefore a P-H-P model, with the power law process model, the homogeneous Poisson process model, and the power law process model that follows the non-homogeneous Poisson process as the failure intensity functions. The change points and the parameters of each failure intensity function were determined using maximum likelihood estimation and least squares estimation.BRResults: The case study shows that compared with other extant models, the proposed P-H-P model is more suitable in terms of log-likelihood function value, Akaike information criterion corrected, and mean squared error.BRConclusion: The proposed P-H-P model is expected to better explain the failure intensities of similar systems and to allow for more accurate reliability analysis.

Emergency shutdowns of propylene production plants: Root cause analysis and availability modeling

Unplanned shutdown events in chemical process plants cause revenue losses due to production curtailments, repair costs, and harmful environmental and safety effects on plant personnel and the surrounding community. Several recent studies have used news articles and databases to perform statistical analyses and modeling of the causes and consequences of chemical process accidents, but these studies do not analyze the operating deviations that led to shutdowns. This work structures narratives about unplanned shutdown events into data, analyzes the causes of those events, and uses the collected data to model the failure probability of process plants. A case study comparing two propylene production technologies, steam cracking and propane dehydrogenation is developed. We use a set of 118 shutdown events reported in four sources across a five-year span for 7 propylene production plants. Using this data, the three leading causes for unplanned shutdowns in both propylene production technologies were determined to be: equipment trips/shutdowns, utility supply interruptions, and weather-related events. A power law process (PLP) for data from multiple repairable systems is used to model the failure behavior for both technologies. For the data collected, this procedure indicated an increasing failure intensity function for both technologies, with propane dehydrogenation increasing more significantly. In addition, both maximum likelihood estimates (MLEs) and confidence intervals for the PLP parameters are presented. Since these are repairable systems, an estimated inherent availability is determined for both technologies. Finally, recommendations that address how to reduce the occurrence of these events and mitigate their consequences are discussed.

Imperfect repair models: Sequential goodness‐of‐fit testing based on predictive performances

AbstractImperfect repair (IR) modelling has attracted a lot of attention during the last decades. To assess and predict repairable systems' reliability, analysts have to choose among a plethora of models: renewal processes, non‐homogeneous Poisson processes (NHPPs), Brown–Proschan, Kijima, ARA, ARI, Quasi‐Renewal (QR), Trend‐Renewal and so forth. Choosing an appropriate model for a given failures, dataset is an important practical issue. The fit of an IR model can be assessed using goodness‐of‐fit (GoF) tests but very few have been proposed and a good fit to past data does not guarantee good reliability predictions. This work proposes general GoF tests for IR models based on sequential (or on‐line) assessment of times‐to‐failures forecasts. The suggested predictive‐sequential (or prequential) GoF tests have a low computational complexity and a common test statistic for several IR models. These tests have been proved to be asymptotically distribution‐free (ADF) for renewal and power‐law processes (PLPs). Our numerical simulations and preliminary theoretical results highly suggest that this ADF property still holds for several other IR models. The prequential tests are much easier to use than the already known bootstrap tests, and a simulation study shows that they are also slightly more powerful. The simulations also show that the prequential tests are powerful to identify classes of similar appropriate models but are much less powerful to distinguish models belonging to the same class. A comparison of MTTF estimates show that models from the same class give close reliability predictions and that the tests are able to reject models which would yield to dramatic errors in reliability predictions.

Non-homogeneous Poisson and linear regression models as approaches to study time series with change-points

In this study we consider some stochastic models and a Bayesian approach to analyze count time series data in the presence of one or more change-points. When the observed count data are all different of zero, it is possible to perform an analysis using linear regression models with normal distribution errors for the log-transformed data. When we have the presence of zero counts at different times, the statistical model based on the log-transformation is not suitable. Hence, in this case, it is possible to consider non-homogeneous Poisson processes (NHPP) models with a suitable rate function. In the present work we consider both models (linear regression and Poisson) to analyze three data sets. These sets are data recording the monthly visitors to New Zealand with the purpose of education, yearly tuberculosis incidence data from New York City, and monthly measles incidence data from Brazil. When the NHPP model is used a PLP (power law process) model is assumed for the intensity function. Additionally, in both models, the presence of change-points will be allowed.

A new approach considering temporal correlations for GPS campaign time series

GPS campaign-mode surveys are periodically collected measurements and their time series has a considerable percentage of data gaps unlike continuous time series. Studying error characteristics of a time series is imperative to compute reliable parameter uncertainties. The power-law process can best describe the background noise for a GPS continuous time series and may be introduced into the analysis through well-known methods. However, the power-law process cannot be applied successfully over GPS campaign time series due to the constraints mentioned above. Here we demonstrate a new approach enabling one to project the stochastic properties of campaign time series into the stochastic domain of the permanent stations of International GNSS Service (IGS) network. The stochastic domain, the combination of white noise (WN) and flicker noise (FN), was obtained from the power spectrum of a continuous time series consisting of truncated daily RINEX observations. We implemented it in the Python3 environment as an open-source package called GPS/GNSS campaign time series analysis (GCTS v1.0). The analysis showed that the velocity uncertainties may be, on average, underestimated by a factor of 2 when assuming the WN-only noise model using campaign data from the UNAVCO data archive collected in San Bernardino and its vicinity. Moreover, to reveal the progress of the new approach, we modelled buried screw dislocations of the San Jacinto, the San Andreas, and parallel faults in their vicinity in elastic half-space. When using the new approach with the combination of WN and FN instead of the WN-only model, we achieved a significant improvement of 15% on average in the chi-square value. Our outcomes showed that the noise process should be considered even for campaign time series. The proposed approach facilitates the analysis of GPS campaign time series even if they include considerable data gaps.

Statistical modeling and reliability analysis of multiple repairable systems with dependent failure times under perfect repair

In repairable systems, a fundamental aspect to be considered is to predict the reliability of the systems under study. Establishing reliability models for multiple repairable systems, however, is still a challenging problem when considering the dependency or unobserved heterogeneity of component failures. This paper focuses on a special repair assumption, called perfect repair, for repairable systems with dependent failures, where only the failed component is restored to as good as new condition. A frailty model is proposed to capture the statistical dependency and unobserved heterogeneity. The classical inferential method for estimating parameters and the reliability predictors will be shown for models in repairable systems under the assumption of perfect repair. An extensive simulation study is conducted under different scenarios to verify the suitability of the model and the asymptotic consistency and efficiency properties of the maximum likelihood estimators. The proposed methodology can be especially useful for industries that operate with repairable systems subjected to the replacement of parts after failures and to non-quantifiable factors that can interfere with the failure times of these parts. We illustrate the practical relevance of the proposed model on two real data sets. The first deals with a set of 9 sugarcane harvesters, observed during a fixed period of time, whose cutting blades failed several times in this interval. The other deals with a set of 5 dump trucks whose diesel engines also had recurring failures during the observation period. Parametric inference is carried out under the power-law process model that has found significant attention in industrial applications.

A Swift study of long-term changes in the X-ray flaring properties of Sagittarius A

ABSTRACT The radiative counterpart of the supermassive black hole at the Galactic Centre, Sagittarius A*, displays flaring emission in the X-ray band atop a steady, quiescent level. Flares are also observed in the near-infrared band. The physical process producing the flares is not fully understood and it is unclear if the flaring rate varies, although some recent works suggest it has reached unprecedented variability in recent years. Using over a decade of regular X-ray monitoring of Neil Gehrels Swift Observatory, we studied the variations in count rate of Sgr A* on time-scales of years. We decomposed the X-ray emission into quiescent and flaring emission, modelled as a constant and power-law process, respectively. We found that the complete, multiyear data set cannot be described by a stationary distribution of flare fluxes, while individual years follow this model better. In three of the ten studied years, the data is consistent with a purely Poissonian quiescent distribution, while for 5 yr, only an upper limit of the flare flux distribution parameter could be determined. We find that these possible changes cannot be explained fully by the different number of observations per year. Combined, these results are instead consistent with a changing flaring rate of Sgr A*, appearing more active between 2006–2007 and 2017–2019, than between 2008–2012. Finally, we discuss this result in the context of flare models and the passing of gaseous objects, and discuss the extra statistical steps taken, for instance, to deal with the background in the Swift observations.