Statistical Distribution Research Articles

Minimum Night Flow Estimation in District Metered Areas

The residential minimum water demand characterisation is of fundamental importance for water distribution system management. During the minimum consumption, indeed, maximum pressures are on network pipes, and at the same time, tank levels rise. The water consumption analysis during the period of low demand and high pressure is thus of great interest for leakage estimation due to the increase in water loss with pressure. In order to contribute to the study of and forecast the daily minimum residential water demand, some probability distributions were tested by means of statistical inferences on a data set collected from different District Metering Areas (DMAs), showing that the stochastic minimum flow demand is defined by the Log-Normal (LN), Gumbel (Gu) and Log-Logistic (LL) distributions, as an extreme minimum value. With reference to the analysed DMAs, the parameters of such statistical distributions were estimated and the relationships are provided as a function only of the supplied users for different DMAs. The data were analysed with 1 h intervals of discretisation, with the aim of providing a useful guide to water utilities, which usually manage water distribution system data with such a resolution time. Indeed, once the minimum residential flow consumption at a 1 h interval was estimated as a function of the user number, by subtracting it to the inflow measured, it is possible to estimate the leakages rate at the DMA.

NUMERICAL-ANALYTICAL METHOD OF DECOMPOSITION-AUTOCOMPENSATION FOR SOLVING THE SIGNAL RECOGNITION PROBLEM BASED ON INACCURATE OBSERVATIONS

A numerical-analytical method is developed to solve the problem of optimal recognition of a set of possible signals observed as an additive mixture containing not only a fluctuation observation error (with an unknown statistical distribution) but also a singular interference (with parametric uncertainty). The method allows for both the detection of signals present in the mixture and the estimation of their parameters within a specified quality criterion and associated constraints. The proposed method, based on the idea of generalized invariant-unbiased estimation of linear functional values, enables decomposition of the computational procedure and autocompensation of singular interference without resorting to the traditional expansion of the state space. Linear spectral decompositions in specified functional bases are used for the parametric finite-dimensional representation of signals and interference, while knowledge of the correlation matrix of the observation error is sufficient for error description. Random and systematic errors are analyzed, and an illustrative example is provided.

Human visual clustering of point arrays.

Although the importance of unsupervised learning has been recognized since William James's "blooming, buzzing confusion," it has received less attention in the literature than supervised learning. An important form of unsupervised learning is clustering, which involves determining the groups of distinct objects that belong together. Visual clustering is foundational for ensemble perception, numerosity judgments, spatial problem-solving, understanding information visualizations, and other forms of visual cognition, and yet surprisingly few researchers have directly investigated this human ability. In this study, participants freely clustered arrays that varied in the number of points (10-40) and cluster structure of the stimuli, which was defined based on the statistical distribution of points. We found that clustering is a reliable ability: Participants' clusterings of the same stimulus on two occasions were highly similar. With respect to the objective properties of the clusterings that people produce, points of individual clusters tend to follow a Gaussian distribution. With respect to processing, we identified five visual attributes that characterize the clusters that participants draw-cluster numerosity, area, density, and linearity and also percentage of points on the convex hull. We also discovered evidence for sequential strategies, with some attributes dominating when drawing the initial clusters of a stimulus and others guiding the final clusters. Collectively, these findings offer a comprehensive picture of human visual clustering and serve as a foundation for the development of new models of this important ability. (PsycInfo Database Record (c) 2024 APA, all rights reserved).

Mueller matrix polarimetry for quantitative evaluation of the Achilles tendon injury recovery

Achilles tendon injuries, as a widely existing disease, have attracted a lot of research interest. Mueller matrix polarimetry, as a novel label-free quantitative imaging method, has been widely used in various applications of lesion identification and pathological diagnosis. However, focusing on the recovery process of Achilles tendon injuries, current optical imaging methods have not yet achieved the label-free precise identification and quantitative evaluation. In this study, using Mueller matrix polarimetry, various Achilles tendon injury samples were characterized specifically, and the efficacy of different recovery schemes was evaluated accordingly. Experiments indicate that injured Achilles tendons show less phase retardance, larger diattenuation, and relatively disordered orientation. The combination of experiments with Monte Carlo simulation results illustrate the microscopic mechanism of the Achilles tendon recovery process from three aspects, that is, the increased fiber diameter, a more consistent fiber orientation, and greater birefringence induced by more collagen protein. Finally, based on the statistical distribution of polarization measurements, a polarization specific characterization parameter was extracted to construct a label-free image, which cannot only intuitively show the injury and recovery of Achilles tendon samples, but also give a quantitative evaluation of the treatment.Graphical

Aspects of Modeling Coal Enrichment Processes by Gravity Methods

This study examines the challenges associated with processing hard coal, with a specific focus on gravitational enrichment methods and the utilization of jigs for coal separation. The research involves the simulation and modeling of physical property distributions and the analysis of both the feed density distribution and the characteristics of the enrichment products. Findings indicate that the resultant density distributions are influenced not only by the gravitational enrichment process but also by the preceding procedures and the inherent properties of the coal, such as particle size, sulfur content, and ash content, all of which significantly affect the quality of the outcomes. In modeling and optimization efforts, the study emphasizes approximating grain density using selected statistical distributions—specifically, the Weibull, logistic, and Gaudin–Schuhmann–Andreyev (GSA) distributions—before and after the enrichment process. Statistical analyses demonstrate that the GSA distribution most accurately fits the grain density distribution in the feed, while the Weibull distribution provides the best approximation for the separation products. The quality of these approximations was assessed using the coefficient of determination (R2) and the Mean Squared Error (MSE). The best quality of approximation for feed was obtained by means of the GSA distribution function, and the MSE was approximately 3.1 for two analyzed values of feed flow rates. In the case of concentrates and tailings, the results are not unequivocal.

A Fish-Counting Method Using Fusion of Spatial Sensing and Temporal Information

In modern aquaculture, accurate and efficient fish counting is crucial for the optimization of resource management and the enhancement of production profitability. Acoustic methods, known for their low energy consumption and extensive detection range, are widely utilized for underwater fish counting. However, traditional acoustic echo methods heavily rely on prior knowledge of fish schools and specific distribution models, leading to complexity and limited adaptability in practical applications. This paper introduces a fish-counting approach that integrates spatial sensing with temporal information. Initially, a spatial sensing matrix is constructed using ultrasonic Frequency-Modulated Continuous Wave (FMCW) technology, which facilitates the extraction of multidimensional features from fish echoes and reduces reliance on prior knowledge of fish schools. Subsequently, temporal information is extracted from echo signals using a Long Short-Term Memory (LSTM) network model, preventing missed detections caused by obstructions in single fish echoes during echo sessions. Finally, by fusing spatial and temporal feature information and employing a data-driven approach, we achieve fish counting while avoiding potential issues arising from improper selection of statistical distribution models. Tests on real fish datasets show that our proposed method consistently outperforms conventional statistical echo methods across all metrics, demonstrating its effectiveness in accurate fish counting.

An evaluation of two-way coupled Euler–Lagrange methodology through direct comparison with particle-resolved simulations

The two-way coupled Euler–Lagrange (EL) methodology is an efficient computational tool for investigating multiphase flows, enabling simulations with tens of millions of particles without Reynolds number limitations. This method resolves the fluid motion on scales larger than a filter length scale, which typically exceeds the particle size and the inter-particle spacing. However, EL simulations require closure models to account for unresolved scales. This work compares particle-resolved (PR) and EL simulations to assess the accuracy of EL solutions. We examine how well EL simulations capture the statistical distribution of particle and fluid quantities in multiphase flows by comparing them with PR solutions. The focus is on modeling the force on particles and understanding the influence of the filter scale on EL simulation accuracy. The results show that, due to variations in feedback forces at individual particle locations, the forces computed in the EL method do not consistently correlate with those obtained from the PR simulations.

Characterization and application of the uncertainties in fracture toughness KIc based on the Weibull statistical distribution and master curve method

Fracture toughness at different temperatures is important in engineering for the integrity assessment of reactor pressure vessels. However, the large dependence on temperature and scatter in the ductile-to-brittle transition temperature region, along with the application of different indexing parameters, results in mass uncertainties in fracture toughness. In this paper, these uncertainties were quantified and characterized based on the Oak Ridge National Laboratory KIc database. The reference nil-ductility transition temperature RTNDT and reference temperature T0 were used as indices for the epistemic uncertainties. The quantity RTNDT – T0 was characterized by a three-parameter Weibull distribution using the commonly used parameter estimation methods. Combining the Master Curve method, a Weibull statistical model of the plain strain fracture toughness KIc was established to account for the aleatory uncertainties. A procedure was proposed for treating the epistemic and aleatory uncertainties for probabilistic fracture mechanics analysis. The effects of fracture toughness uncertainties on the crack initiation probability were evaluated for a cracked pressure vessel. The results show that considering both epistemic and aleatory uncertainties can reduce over-conservatism, and epistemic uncertainties strongly affect the crack initiation probability.

Statistical modeling of soft magnetic composites’ permeability

Soft magnetic composites (SMC) also known as powder cores are widely used in many applications because of their linearity, controllable permeability and isotropy. Because of the distributed air gap, the permeability is governed by the inner demagnetizing fields. To model the permeability dependence on the filling fraction, essentially two approaches are used both based on spatial periodicity hypothesis: the non magnetic grain boundary model and the effective medium theory. Actually, the first one works only for dense materials and the second works at low concentration. Both need fitting of two parameters, the inner demagnetizing factor and the particle susceptibility, just because the periodicity hypothesis is never verified for magnetic filling factors larger than 20%. A breakaway model is proposed based on the computation of mathematical esperance of the statistical distribution of magnetic chains and the subsequent determination of demagnetizing coefficient of an ellipsoid of equivalent aspect ratio. The model is collated with permeability data from the literature for spherical or non-spherical particles based SMCs and shows a excellent agreement with only one or even without fitting parameter in the whole concentration range.

Successful sensitization of 2.5-year-olds to other-race faces through bimodal training

The present study investigated the potential for sensitizing 2.5-year-old Caucasian infants to other-race faces (Asian faces). In the domain of face perception, infants become less sensitive to facial distinctions of other-race faces through perceptual narrowing at the end of the first year of life. Nevertheless, infants around 12 months can regain their sensitivity to other-race faces. For instance, exposing them to a specific statistical distribution and employing the mechanisms of statistical learning is one way to enhance their discriminatory abilities towards other-race faces. Following this idea, we investigated if even older infants around 2.5 years can be sensitized to other-race faces. We trained the infants with a bimodal distribution of a morphed continuum of Asian female faces with faces closer to the endpoints presented most frequently. We assessed infants’ discrimination of Asian faces by measuring their looking times after the training phase. The 2.5-year-olds showed a difference in looking times after the training, indicating that the exposure to a bimodal frequency distribution led to a successful discrimination between Asian faces. These findings demonstrate that 2.5-year-olds can be sensitized to other-race faces by exposing them to a bimodal distribution of such faces, underlining the plasticity of face perception in childhood.

Investigating Hailstorm Updrafts and Nowcasting Hail Size Using a Novel Radar-Based Updraft Detection

Abstract Nowcasting hail size poses a major challenge in operational practice due to physical limitations of weather radar technology once hailstones are sufficiently large to enter the resonance scattering regime. Numerous radar-based hail size proxies have been derived in recent decades, but their performance is generally poor in identifying giant hail (≥10 cm). Using a novel thunderstorm updraft detection method, we examine the updraft characteristics of hailstorms in the U.S. Great Plains based on a NEXRAD dataset of 114 hail events between 2013 and 2023. We find that some radar-derived variables within the detected updraft are well suited for discriminating between small hail (1.0–3.0 cm) and severe hail (≥3.5 cm), e.g., minimum copolar cross-correlation coefficient in the midlevel updraft, whereas other radar metrics such as the area of reflectivity > 50 dBZ in the upper portion of the updraft suggest the presence of giant hail. However, the statistical distributions of each variable overlap for different hail sizes, and there is no single metric which performs well across the entire hail size spectrum. Therefore, we trained a random forest model to nowcast hail size categories using a multitude of these radar metrics. The model shows promising performance for discriminating hail sizes > 5 cm but requires further refinement for smaller hail. We showcase the model’s capabilities for a set of hailstorms in the Great Plains. Significance Statement Large hail, which may reach sizes up to 20 cm in extreme cases, is a severe threat in thunderstorms and responsible for significant damage in thunderstorm-prone areas around the world. Weather radars are generally used for issuing warnings for the threat of large hail, but it is surprisingly difficult to apply their data to distinguish between hail sizes. We utilize a new approach that detects the thunderstorm updraft region, which is where hailstones grow, and analyze the radar data therein. Based on these data, we then develop a model for assessing hail sizes that could be used for warnings and illustrate the model results for six hailstorms.

The Logistic Distribution as a Limit Law for Random Sums and Statistics Constructed from Samples with Random Sizes

In the present paper, based on the representation of the logistic distribution as a normal scale mixture obtained by L. Stefanski in 1990, it is demonstrated that the logistic distribution can be limiting for sums of a random number of random variables and other statistics that admit (at least asymptotically) additive representation and are constructed from samples with random sizes. These results complement a theorem proved by B. V. and D. B. Gnedenko in 1982 that established convergence of the distributions of extreme order statistics in samples with geometrically distributed random sizes to the logistic distribution. Hence, along with the normal law, this distribution can be used as an asymptotic approximation of the distributions of observations that can be assumed to have an additive structure, for example, random-walk-type time series. An approach is presented for the definition of the new asymmetric generalization of the logistic distribution as a special normal variance–mean mixture.

Single-Scene SAR Image Data Augmentation Based on SBR and GAN for Target Recognition

High-performance neural networks for synthetic aperture radar (SAR) automatic target recognition (ATR) often encounter the challenge of data scarcity. The lack of sufficient labeled SAR image datasets leads to the consideration of using simulated data to supplement the dataset. On the one hand, electromagnetic computation simulations provide high amplitude accuracy but are inefficient for large-scale datasets due to their complex computations and physical models. On the other hand, ray tracing simulations offer high geometric accuracy and computational efficiency but struggle with low amplitude correctness, hindering accurate numerical feature extraction. Furthermore, the emergence of generative adversarial networks (GANs) provides a way to generate simulated datasets, trying to balance computational efficiency with image quality. Nevertheless, the simulated SAR images generated based on random noise lack constraints, and it is also difficult to generate images that exceed the parameter conditions of the real image’s training set. Hence, it is essential to integrate physics-based simulation techniques into GANs to enhance the generalization ability of the imaging parameters. In this paper, we present the SingleScene-SAR Simulator, an efficient framework for SAR image simulation that operates under limited real SAR data. This simulator integrates rasterized shooting and bouncing rays (SBR) with cycle GAN, effectively achieving both amplitude correctness and geometric accuracy. The simulated images are appropriate for augmenting datasets in target recognition networks. Firstly, the SingleScene-SAR Simulator employs a rasterized SBR algorithm to generate radar cross section (RCS) images of target models. Secondly, a specific training pattern for cycle GAN is established to translate noisy RCS images into simulated SAR images that closely resemble real ones. Finally, these simulated images are utilized for data augmentation. Experimental results based on the constructed dataset show that with only one scene SAR image containing 30 target chips, the SingleScene-SAR Simulator can efficiently produce simulated SAR images that exhibit high similarity in both spatial and statistical distributions compared with real images. By employing simulated SAR images for data augmentation, the accuracy of target recognition networks can be consistently and significantly enhanced.

The Conditional Bias of Extreme Precipitation in Multi‐Source Merged Data Sets

AbstractMulti‐source data merging via weighted average (WA) is widely applied to enhance large‐scale precipitation estimates. However, these data sets usually contain substantial conditional biases with respect to extreme precipitation (EP) events—undermining their utility for extreme event analysis. Nevertheless, the main source of such EP biases remains unknown. Here, we demonstrate that WA algorithms are responsible for less than 1% of total EP biases. Instead, EP biases originate from the multi‐source precipitation inputs, which are not adequately adjusted prior to WA. Specifically, current data‐merging frameworks only correct the monthly means or statistical distributions of the remote sensing/reanalysis precipitation inputs prior to WA. Such procedures are insufficient for adjusting EP timing uncertainties, which eventually propagate into the WA‐based merged data set as an EP bias. Therefore, developing algorithms that iteratively adjust EP timing and intensity errors should be prioritized in future precipitation merging frameworks.

Characterization of the Blue Straggler Star Populations, through Statistical, Photometric, and Spectral Energy Distribution Analysis, in the Old Open Cluster: NGC 2243

We present a statistical, photometric, and spectral energy distribution (SED) analysis of the poorly studied old open cluster: NGC 2243, to characterize its blue straggler star (BSS) population. We applied ensemble-based unsupervised machine learning methods to estimate the membership probabilities using Gaia Data Release 3 (DR3) astrometric data. NGC 2243 is an open cluster that is 3.67 Gyr old with a metallicity of −0.375 dex, situated at a distance of 3.65 kpc. By analyzing the position of cluster members on the color–magnitude diagram using MIST isochrones, we have identified 12 potential BSSs in NGC 2243. We fitted the radial surface density profile and investigated the dynamical state and mass segregation effect of the cluster. It is found that the BSSs are significantly concentrated within the central region. We used data from Swift/UVOT, Gaia DR3, Pan-STARRS1 DR2, 2MASS, and WISE to fit the SEDs of the 12 identified BSSs using VOSA. We estimated the masses of the BSSs from the Hertzsprung–Russell diagram and found that they ranged from 1.25 to 2.22 M ☉. Consequently, we concluded that the BSSs likely gained 0.11–1.08 M ☉ through the mass transfer or merger channels. We discovered a hot companion associated with one BSS candidate, which has a temperature of 19,000 K, a luminosity of 0.55 L ☉, and a radius of 0.065 R ☉. The hot companion is probably a white dwarf, with its mass estimated to be approximately 0.18–0.20 M ☉ and an age of 186 Myr, suggesting it is a post-mass-transfer (Case A or Case B) system.

The European Fault-Source Model 2020 (EFSM20): geologic input data for the European Seismic Hazard Model 2020

Abstract. Earthquake hazard analyses rely on seismogenic source models. These are designed in various fashions, such as point sources or area sources, but the most effective is the three-dimensional representation of geological faults. We here refer to such models as fault sources. This study presents the European Fault-Source Model 2020 (EFSM20), which was one of the primary input datasets of the recently released European Seismic Hazard Model 2020. The EFSM20 compilation was entirely based on reusable data from existing active fault regional compilations that were first blended and harmonized and then augmented by a set of derived parameters. These additional parameters were devised to enable users to formulate earthquake rate forecasts based on a seismic-moment balancing approach. EFSM20 considers two main categories of seismogenic faults: crustal faults and subduction systems, which include the subduction interface and intraslab faults. The compiled dataset covers an area from the Mid-Atlantic Ridge to the Caucasus and from northern Africa to Iceland. It includes 1248 crustal faults spanning a total length of ∼95 100 km and four subduction systems, namely the Gibraltar, Calabrian, Hellenic, and Cyprus arcs, for a total length of ∼2120 km. The model focuses on an area encompassing a buffer of 300 km around all European countries (except for Overseas Countries and Territories) and a maximum of 300 km depth for the subducting slabs. All the parameters required to develop a seismic source model for earthquake hazard analysis were determined for crustal faults and subduction systems. A statistical distribution of relevant seismotectonic parameters, such as faulting mechanisms, slip rates, moment rates, and prospective maximum magnitudes, is presented and discussed to address unsettled points in view of future updates and improvements. The dataset, identified by the DOI https://doi.org/10.13127/efsm20 (Basili et al., 2022), is distributed as machine-readable files using open standards (Open Geospatial Consortium).

Survey of Whistler‐Mode Wave Amplitudes and Frequency Spectra in Jupiter's Magnetosphere

AbstractWe present statistical distributions of whistler‐mode chorus and hiss waves at frequencies ranging from the local proton gyrofrequency to the equatorial electron gyrofrequency (fce,eq) in Jupiter's magnetosphere based on Juno measurements. The chorus wave power spectral densities usually follow the fce,eq variation with major wave power concentrated in the 0.05fce,eq–fce,eq frequency range. The hiss wave frequencies are less dependent on fce,eq variation than chorus with major power concentrated below 0.05fce,eq, showing a separation from chorus at M < 10. Our survey indicates that chorus waves are mainly observed at 5.5 < M < 13 from the magnetic equator to 20° latitude, consistent with local wave generation near the equator and damping effects. The hiss wave powers extend to 50° latitude, suggesting longer wave propagation paths without attenuation. Our survey also includes the whistler‐mode waves at high latitudes which may originate from the Io footprint, auroral hiss, or propagating hiss waves reflected to high M shells.

Effect of filter type on the topology of the eigenframe of the subfilter stress tensor and interscale energy transfer

Effects of different filter kernels, namely, spectral cutoff ( $\mathcal {S}$ -filter) and Gaussian ( $\mathcal {G}$ -filter), on the geometrical properties of the subfilter stress (SFS) tensor and the filtered strain-rate (FSR) tensor are analysed in a forced homogeneous isotropic turbulence. Utilizing the Euler angle–axis methodology, it is observed that despite similar mean behaviour, the eigenframe alignment between SFS and FSR exhibits a non-trivially different statistical distribution for two different filters. Besides the eigenframe alignment, the eigenstructure of these tensors is also investigated. It is found that in contrast to the eigenstructure of the FSR which does not show sensitive dependence on the filter kernel type, the eigenstructure of the SFS tensor is significantly influenced by the filter type. Subsequently, the impact of different filter kernels on the subfilter energy flux (SFEF) is investigated. It is observed that energy transfer in $\mathcal {G}$ -filtering is preferably distributed over the forward region, whereas for the $\mathcal {S}$ -filter, the SFEF is more evenly distributed over both forward–backward regions, leading to a heavy energy transfer cancellation. Additionally, by decomposing the SFEF into different partial energy fluxes, it is found that the impact of the $\mathcal {S}$ -filtering on the eigenstructure of the SFS leads to the amplification of the backward energy transfer. Conversely, the $\mathcal {G}$ -filtering amplifies the forward energy transfer by producing a more pronounced alignment between the contractive–extensive eigenvectors.

Augmented machine learning for sewage quality assessment with limited data

Physical, chemical, and biological processes within sewers significantly alter sewage composition during conveyance. This leads to the formation of sulfide and methane—compounds that contribute to sewer corrosion and greenhouse gas emissions. Reliable modeling of these compounds is essential for effective sewer management, but the development of machine learning (ML) models is hindered by differences in data accessibility and sampling frequencies of water quality variables. Here we present a mechanistically enhanced hybrid (ME-Hybrid) model that combines mechanistic modeling with data-driven approaches. This model harmonizes datasets with varying sampling frequencies and generates synthetic samples for ML training, thereby enhancing the monitoring of methane and sulfide in sewers. The optimal ME-Hybrid model integrates the backpropagation neural network with mechanistic frequency harmonization. We demonstrate that the ME-Hybrid model outperforms pure ML and linear interpolation in capturing fluctuating trends and extremes of sulfide concentrations, achieving a coefficient of determination (R2) of 0.94. Synthetic samples generated through mechanistic augmentation closely approximate real samples in modeling performance, statistical distribution, and data structure. This enables the model to maintain high predictive accuracy (R2 > 0.76) for sulfide even when trained on only 50 % of the dataset. Additionally, the ME-Hybrid model successfully assesses sewer methane concentrations with an R2 of 0.94, validating its applicability and generalization ability. Our results provide a reliable methodological framework for modeling and prediction under data scarcity. By facilitating better monitoring and management of sewer systems, the ME-Hybrid model aids in the development of strategies that minimize environmental impacts, enhance urban resilience, and ultimately lead to sustainable urban water systems.

A Comprehensive Microstructure-Aware Electromigration Modeling Framework; Investigation of the Impact of Trench Dimensions in Damascene Copper Interconnects.

As electronic devices continue to shrink in size and increase in complexity, the current densities in interconnects drastically increase, intensifying the effects of electromigration (EM). This renders the understanding of EM crucial, due to its significant implications for device reliability and longevity. This paper presents a comprehensive simulation framework for the investigation of EM in nano-interconnects, with a primary focus on unravelling the influential role of microstructure, by considering the impact of diffusion heterogeneity through the metal texture and interfaces. As such, the resulting atomic flux and stress distribution within nano-interconnects could be investigated. To this end, a novel approach to generate microstructures of the conductor metal is presented, whereby a predefined statistical distribution of grain sizes obtained from experimental texture analyses can be incorporated into the presented model, making the model predictive under various scales and working conditions with no need for continuous calibration. Additionally, the study advances beyond the state-of-the-art by comprehensively simulating all stages of electromigration including stress evolution, void nucleation, and void dynamics. The model was employed to study the impact of trench dimensions on the dual damascene copper texture and its impact on electromigration aging, where the model findings were corroborated by comparing them to the available experimental findings. A nearly linear increase in normalized time to nucleation was detected as the interconnect became wider with a fixed height for aspect ratios beyond 1. However, a saturation was detected with a further increase in width for lines of aspect ratios below 1, with no effective enhancement in time to nucleation. An aspect ratio of 1 seems to maximize the EM lifetime for a fixed cross-sectional area by fostering a bamboo-like structure, where about a 2-fold of increase was estimated when going from aspect ratio 2 to 1.