Probabilistic Results Research Articles

A performance-based approach to quantify atmospheric river flood risk

Abstract. Atmospheric rivers (ARs) are a class of meteorologic phenomena that cause significant precipitation and flooding on the US West Coast. This work presents a new Performance-based Atmospheric River Risk Analysis (PARRA) framework that adapts existing concepts from probabilistic risk analysis and performance-based engineering for application in the context of AR-driven fluvial flooding. The PARRA framework is a chain of physically based models that link the atmospheric forcings, hydrologic impacts, and economic consequences of AR-driven fluvial flood risk together at consistent “pinch points”. Organizing around these pinch points makes the framework modular, meaning that models between pinch points can be updated without affecting the rest of the model chain, and it produces a probabilistic result that quantifies the uncertainty in the underlying system states. The PARRA framework can produce results beyond analyses of individual scenario events and can look toward prospective assessment of events or system changes that have not been seen in the historic record. The utility of the PARRA framework is demonstrated through a series of analyses in Sonoma County, CA, USA. Individual component models are fitted and validated against a historic catalog of AR events occurring from 1987 to 2019. Comparing simulated results from these component model implementations against observed historic ARs highlights what we can learn about the drivers of extremeness in different flood events by taking a probabilistic perspective. The component models are then run in sequence to generate a first-of-its-kind AR flood loss exceedance curve for Sonoma County. The prospective capabilities of the PARRA framework are presented through the evaluation of a hypothetical mitigation action. Elevating 200 homes, selected based on their proximity to the Russian River, was sufficient to reduce the average annual loss by half. Although expected benefits were minimal for the smallest events in the stochastic record, the larger, more damaging ARs were expected to see loss reductions of approximately USD 50–75 million per event. These results indicate the potential of the PARRA framework to examine other changes to flood hazard, exposure, and vulnerability at the community level.

Interrogating Subsurface Structures Using Probabilistic Tomography: An Example Assessing the Volume of Irish Sea Basins

AbstractThe ultimate goal of a scientific investigation is usually to find answers to specific, often low‐dimensional questions: what is the size of a subsurface body? Does a hypothesized subsurface feature exist? Existing information is reviewed, an experiment is designed and performed to acquire new data, and the most likely answer is estimated. Typically the answer is interpreted from geological and geophysical data or models, but is biased because only one particular forward function is considered, one inversion method is applied, and because human interpretation is a biased process. Interrogation theory provides a systematic way to answer specific questions by combining forward, design, inverse, and decision theories. The optimal answer is made more robust since it balances multiple possible forward models, inverse algorithms and model parametrizations, probabilistically. In a synthetic test, we evaluate the area of a low‐velocity anomaly by interrogating Bayesian tomographic results. By combining the effect of four inversion algorithms, the optimal answer is very close to the true answer, even on a coarsely gridded parametrization. In a field data test, we evaluate the volume of the East Irish Sea basins using probabilistic 3D shear wave speed depth inversion results. This example shows that interrogation theory provides a useful way to answer realistic questions about the Earth. A key revelation is that while the majority of computation may be spent solving inverse problems, much of the skill and effort involved in answering questions may be spent defining and calculating target function values in a clear and unbiased manner.

A Gaussian Process State Space Model Fusion Physical Model and Residual Analysis for Fatigue Evaluation.

Residual stress is closely related to the evolution process of the component fatigue state, but it can be affected by various sources. Conventional fatigue evaluation either focuses on the physical process, which is limited by the complexity of the physical process and the environment, or on monitored data to form a data-driven model, which lacks a relation to the degenerate process and is more sensitive to the quality of the data. This paper proposes a fusion-driven fatigue evaluation model based on the Gaussian process state–space model, which considers the importance of physical processes and the residuals. Through state–space theory, the probabilistic space evaluation results of the Gaussian process and linear physical model are used as the hidden state evaluation results and hidden state change observation function, respectively, to construct a complete Gaussian process state–space framework. Then, through the solution of a particle filter, the importance of the residual is inferred and the fatigue evaluation model is established. Fatigue tests on titanium alloy components were conducted to verify the effectiveness of the fatigue evaluation model. The results indicated that the proposed models could correct evaluation results that were far away from the input data and improve the stability of the prediction.

A probabilistic Q-system using the Markov chain to predict rock mass quality in tunneling

Uncertainty in rock mass conditions is mainly caused by the inherently inhomogeneous nature of rock masses. Assessment of rock mass quality without accounting for inherent uncertainty often leads to unnecessary conservatism in design and construction, resulting in excessive cost and schedule overrun. In this study, to advance rock mass quality assessment, a rock mass classification Q-based prediction model to assess probabilistic rock mass quality has been proposed using the Markov chain framework with Monte Carlo simulation. In addition, an analytical approximation approach has also been developed to derive the statistics (mean, standard deviation, and coefficient of variation) of the Q value given statistics of Q parameters in the Markov chain model. The proposed prediction model and analytical calculation approach were applied to a water tunnel. The probabilistic prediction results have been validated by the field recorded Q data during tunnel construction in a probabilistic framework with the use of the accuracy plot where uncertainties are explicitly quantified in the predicted probabilistic model. The proposed Q-based prediction model can assess the rock mass quality in unexcavated tunnel sections using a probabilistic approach to serve as a preliminary site condition indicator.

Reliability analysis of an earth dam in operating conditions using direct coupling

In this study, we present a numerical investigation of the direct coupling between the deterministic GeoStudio package (Seepage/W and Slope/W software) and the StRAnD reliability package for normal operating conditions of an existing old earth dam over time. Direct coupling avoids errors associated to point estimate methods and to using response surface surrogates. One relevant feature of the study is the use of realistic pore water pressures for each equilibrium analysis, accomplished by considering a long-term steady-state analysis as an initial condition, followed by four years of equilibrium analysis, before each transient seepage analysis. All reliability analysis were performed using the first-order reliability method. The most important random parameters found in sensitivity analyses are four dam body parameters (saturated hydraulic conductivity, ks; specific weight, γ; effective cohesion, c′; and friction angle, ϕ′) and two filter parameters (ks and ϕ′). A range of values of the relationship between the reliability index (β) and the factor of safety (FS) was found for all probabilistic and deterministic results. Finally, a large difference in terms of critical deterministic and probabilistic slip surfaces is identified for the same time of analysis.Article highlightsRealistic pore water pressures used in dam equilibrium analysis, by considering random seepage analysis in previous 5 to 30 days.Direct coupling of deterministic and reliability softwares avoids errors associated to point estimate and response surface surrogate methods.Existing old earth dam studied in Normal Operating Condition.For same analysis time, large differences are observed between critical deterministic and probabilistic slip surfaces.

Geometric and probabilistic results for the observability of the wave equation

Given any measurable subset $\omega$ of a closed Riemannian manifold $(M,g)$ and given any $T>0$, we define $\ell^T(\omega)\in[0,1]$ as the smallest average time over $[0,T]$ spent by all geodesic rays in $\omega$. This quantity appears naturally when studying observability properties for the wave equation on $M$, with $\omega$ as an observation subset: the condition $\ell^T(\omega)>0$ is the well known \emph{Geometric Control Condition}. In this article we establish two properties of the functional $\ell^T$, one is geometric and the other is probabilistic. The first geometric property is on the maximal discrepancy of $\ell^T$ when taking the closure. We may have $\ell^T(\mathring{\omega})<\ell^T(\overline\omega)$ whenever there exist rays grazing $\omega$ and the discrepancy between both quantities may be equal to $1$ for some subsets $\omega$. We prove that, if the metric $g$ is $C^2$ and if $\omega$ satisfies a slight regularity assumption, then $\ell^T(\overline\omega) \leq \frac{1}{2} \left( \ell^T(\mathring{\omega}) + 1 \right)$. We also show that our assumptions are essentially sharp; in particular, surprisingly the result is wrong if the metric $g$ is not $C^2$. As a consequence, if $\omega$ is regular enough and if $\ell^T(\overline\omega)>1/2$ then the Geometric Control Condition is satisfied and thus the wave equation is observable on $\omega$ in time $T$. The second property is of probabilistic nature. We take $M=\mathbb{T}^2$, the flat two-dimensional torus, and we consider a regular grid on it, a regular checkerboard made of $n^2$ square white cells. We construct random subsets $\omega_\varepsilon^n$ by darkening each cell in this grid with a probability $\varepsilon$. We prove that the random law $\ell^T(\omega_\varepsilon^n)$ converges in probability to $\varepsilon$ as $n\rightarrow+\infty$. As a consequence, if $n$ is large enough then the Geometric Control Condition is satisfied almost surely and thus the wave equation is observable on $\omega_\varepsilon^n$ in time $T$.

Intrinsic Decomposition Method Combining Deep Convolutional Neural Network and Probability Graph Model.

With the rapid development of computer vision and artificial intelligence, people are increasingly demanding image decomposition. Many of the current methods do not decompose images well. In order to find the decomposition method with high accuracy and accurate recognition rate, this study combines convolutional neural network and probability map model, and proposes a single-image intrinsic image decomposition method that is on both standard dataset images and natural images. Compared with the existing single-image automatic decomposition algorithm, the visual effect comparable to the user interaction decomposition algorithm is obtained, and the method of this study also obtains the lowest error rate in the quantitative comparison on the standard dataset image. The multi-image collaborative intrinsic image decomposition method proposed in this study obtains the decomposition result of consistent foreground reflectivity on multiple sets of image pairs. In this study, the eigenimage decomposition is applied to the illumination uniformity in the small change detection, and the promising reflectivity layer image obtained by the decomposition helps to improve the accuracy of the cooperative saliency detection. This study proposes an algorithm for the cooperation between CNN and probability graph model, and introduces how to combine the probability graph model with the traditional CNN to accomplish the pixel-level eigendecomposition task. This study also designs a single-image and multi-image intrinsic image decomposition results analysis experiments, then analyzes the probabilistic graphical model coordination intrinsic image decomposition results, and finally analyzes the convolutional neural network coordination intrinsic decomposition performance to draw the conclusion of this study. The effect on the Msrc-v2 dataset was increased by 0.8% over the probability plot model.

A k‐nearest neighbor‐based averaging model for probabilistic PV generation forecasting

AbstractProbabilistic forecasting of PV generation is crucial in uncertainty management to reinforce PV‐integrated power systems for long‐term planning. In this context, developing a reliable probabilistic forecast model is challenging due to weather conditions' stochastic nature and varying daily PV production patterns at multiple time instants. Due to varying probability distribution patterns, a nonparametric approach, such as quantile regression, is challenging to approximate the forecast error distribution. A forecast combination concept that yields impressive probabilistic forecasting results has rarely been utilized for PV generation forecasting. The use of complementary and sensible point forecasters characterizing different aspects of intricate PV production patterns in a k‐nearest neighbor quantile regression averaging framework is the proposed probabilistic forecasting model. The proposed forecasting model's accuracy is assessed with real‐world PV generation data from the USA for multi‐time instants. The probabilistic forecast performance of the proposed quantile k‐nearest neighbors regression averaging model is evaluated against the traditional quantile regression averaging, quantile regression forests, quantile k‐nearest neighbors, basic quantile regression, and regression bootstrapping in terms of the forecast (quantiles and prediction intervals) accuracy, using relevant error measures and scores. The proposed model has superior performance compared to the other models, as indicated by the lowest forecast errors and scores.

A Wind Speed Correction Method Based on Modified Hidden Markov Model for Enhancing Wind Power Forecast

Short-term wind power forecast (WPF) depends highly on the wind speed forecast (WSF), which is the prime contributor to the forecasting error. To achieve more accurate WPF results, this article proposes a wind speed correction method to improve the WSF result obtained by using the weather research and forecasting (WRF) model. First, the WRF model is constructed to forecast the wind speed, and its performance is analyzed. Second, a novel hidden Markov model (HMM) is developed to explore both the temporal autocorrelation of WSF error and the nonlinear correlation between the WSF result and the error. In the model, the fuzzy C-means cluster is introduced to properly divide the hidden state space of HMM and the emission probability of HMM is improved as continuous by the kernel density estimation (KDE) to make full use of the observation information. The proposed HMM model is better at wind speed correction through modification. Third, the HMM is solved by the Viterbi algorithm and the minimum mean-square error regulation to correct the predicted wind speed. Finally, the deterministic and probabilistic WPF results are obtained by using another KDE model, the proposed method is demonstrated to be superior to the benchmarks in case studies.

Multiobjective Reliability‐Based Design Optimization of Recycled Aggregates Used in Corrosive Environment Based on Response Surface Modelling

In this study, durability properties of concrete samples including rapid chloride migration and electrical resistance as corrosion evaluation indicators were tested. The effects of some environmental parameters including humidity percentage, temperature, and chloride concentration were also examined on concrete electrical resistance. The prediction models for mechanical and durability parameters of concrete were obtained using the response surface method. These models were then evaluated using a metaheuristic optimization algorithm coupled with a simple and fast usability new method of reliability evaluation. Eventually, the probabilistic optimal values of using recycled aggregates were calculated for achieving environmentally friendly concrete. Probabilistic multiobjective optimization results revealed that, in an environment with humidity of 70%, temperature of 23°C, and chloride ion concentrations of 3%, 5%, and 8%, use of recycled aggregates for above different chloride ion concentrations was limited to (33.56%, 13.23%), (32.14%, 5.56%), and (13.23%, 2.86%), recycled coarse and fine aggregates, respectively. Furthermore, optimization procedures were performed for the environment or precontaminant recycled aggregate with the chloride concentration of 10%, but the analysis procedure did not converge to an optimum design point.

Assessing uncertainty propagation in hybrid models for daily streamflow simulation based on arbitrary polynomial chaos expansion

Accurate streamflow prediction is of significant importance in watershed management and has been attracting a lot of research interest. Hybrid hydrological models that employ advantages of physically-based models (PBMs) and data-driven models (DDMs) emerge recently as reliable tools for analyzing hydrological processes. Both PBMs and DDMs have uncertain parameters that could affect the hybrid modeling results. Conventional computational algorithms that rely on repeated sampling, such as Monte Carlo Simulation (MCS), have been recognized as robust tools for assessing the propagation of parameter uncertainties. However, the efficiency of these computational algorithms can decrease exponentially as the number of parameters increases. Meanwhile, although parameter uncertainty of PBMs has been widely studied, that of DDMs in a hybrid modeling framework has not been well investigated. In this study, a hybrid model framework is proposed for streamflow prediction, and an effective arbitrary polynomial chaos expansion (aPCE) method is implemented to assess the propagation of DDM parameter uncertainty within the hybrid modeling framework. The hybrid hydrological model integrating fully-distributed MIKE SHE and four machine-learning techniques increases the coefficient of determination (R2) from 0.70 to 0.82 for two validation periods. The aPCE method generates highly similar probabilistic results as MCS and achieves a 90% efficiency enhancement in uncertainty assessment. The results prove that aPCE is an efficient alternative to MCS for assessing uncertainties of hyperparameters in hybrid models. This study helps improve the performance of the streamflow prediction and also provide an insight into the uncertainty propagation involved in the hybrid models.

A quantitative reliability metric for querying large database

A redesigned quantitative reliability metric based on the F-distribution (QRMf) is reported for evaluating the reliability of library search. The QRMf provides orthogonal information to the comparison metric (e.g., dot product) and yields a probabilistic result. An intralibrary search can be considered as an idealized search because the top hit, i.e., the closest matching object, will match perfectly. If the search of an unknown object yields the same hit list as the intralibrary search, it would indicate good reliability. For each object in the hit list, a QRMf compares the order of an intralibrary and interlibrary search results and calculates a variance of interlibrary similarity metrics between the records of the intralibrary search and records in the corresponding positions of the interlibrary search. This variance that measures the discordance of the intra and interlibrary search can simply be compared to the variance of the similarity metrics within the interlibrary search results. The ratio of these variances follows an F-distribution that can be used to determine if the discordance is statistically significant and generates the probability based on the cumulative distribution function. The QRMf works for both similarity and dissimilarity and can be used for any queried object and comparison metric that is searched against a database. In this work, the QRMf was used along with the dot product similarity to query the mass spectra of novel synthetic opioids measured by gas chromatography-mass spectrometry (GC/MS). An automated pipeline was devised that used a basis set correction to assist peak detection. The basis was constructed by mass spectra obtained from the blank measurement preceding the analytical run to remove interferences from column bleed and septum degradation. After peak detection, the pipeline applied multivariate curve resolution to the chromatographic peak window to remove background components from the mass spectra. The corrected mass spectra were searched against a customized library for identification. The QRMf can be used along with the similarity metric to detect misidentifications and assist in finding the correct identification when it is not the closest match.

An efficient method for probabilistic finite element analysis with application to contact assessment of CANDU fuel channels

In CANDU reactors, irradiation induced creep and growth of the fuel channel (FC) assembly leads to Pressure Tube (PT) sag and to a reduction in the gap between the PT and the Calandria Tube (CT) (PT–CT gap). Excessive PT sag can lead to contact between the PT and the CT. Uncertainties in the parameters that influence PT–CT contact require probabilistic assessment methods to assess risk of PT-CT contact and demonstrate adherence to CSA Standard N285.8 provisions. Improved models in the prediction of PT–CT contact require computationally expensive 3D finite element analysis (3D FEA), a tool that is prohibitively difficult to use in probabilistic assessments. Current probabilistic analyses rely on 1D FEA models for the prediction of PT-CT contact estimates that are generally less accurate. Therefore, an improved and computationally more efficient approach for conducting probabilistic finite element analyses (PFEA) of CANDU FCs is highly desirable. In this paper, such a PFEA approach is proposed by coupling the multiplicative dimensional reduction method (M-DRM) and the polynomial chaos expansion (PCE) method. The analysis that follows studies the PT–CT gap predictions using deterministic 3D FEA by considering two different PT configurations present within the population of the FCs in a CANDU reactor core. The PFEA is then carried out using 1D and 3D FEA by employing the ABAQUS software suite. The accuracy of the probabilistic analysis results using the proposed approach is initially checked against Monte Carlo Simulation (MCS) results using 1D FEA. Following a good agreement, the proposed approach was applied for PFEA based on 3D FEA. The results indicate that the two different PT orientations significantly influence the probabilistic contact results. They also show that the probabilistic results based on 1D FEA, which are currently used by the nuclear industry, are not reliable. Moreover, it is also shown that a sensitivity analysis can be performed without any additional effort. The low computational cost, predictive capability and the generality of the proposed method are important attributes for adopting it in the future in probabilistic assessments of an entire reactor core.

Probabilistic Analysis of Tunnel Roof Deflection under Sequential Excavation Using ANN-Based Monte Carlo Simulation and Simplified Reliability Approach

Tunnel roof deflection is an important measure to control the safety of excavation activity, especially for large-section tunnels using the sequential construction method. A three-dimensional (3D) model of a sequential excavation project in the Lijiaping metro tunnel, Chongqing, China, was developed and the maximum tunnel roof deflection was calculated deterministically at the end of each excavation stage. By defining the characteristics of the soft rock layers surrounding the tunnel section as the random variables, a probabilistic analysis was conducted and the problem was formulated as a reliability model. Two different approaches were used to solve the established reliability problem. One entails an artificial neural network (ANN) trained by a large data set obtained from numerical simulations and then accompanied by the Monte Carlo (MC) sampling method to calculate the probability. The other is a simplified reliability approach using a small data set to approximate the exceedance probability via a regression-based algorithm. The proposed ANN-based metamodel used in the first method could accurately predict tunnel roof deflection and replace the software simulator. Subsequently, the probabilistic results obtained from this method following MC sampling could well converge and provide the probability with enough accuracy. Interestingly, the simplified approach with more than 40 random samples can also provide acceptable results, which provides an economical approach to estimate the exceedance probability.

Genetic Counseling and Testing for Pulmonary Arterial Hypertension in the United States

Genetic Counseling and Testing for Pulmonary Arterial Hypertension in the United States

Liver-Based Probabilistic Risk Assessment of Exposure to Organophosphate Esters via Dust Ingestion Using a Physiologically Based Toxicokinetic (PBTK) Model.

Organophosphate esters (OPEs) are widely used and harmful to organisms and human health. Dust ingestion is an important exposure route for OPEs to humans. In this study, by integrating ToxCast high-throughput in vitro assays with in vitro to in vivo extrapolation (IVIVE) via physiologically based Toxicokinetic (PBTK) modeling, we assessed the hepatocyte-based health risk for humans around the world due to exposure to two typical OPEs (TPHP and TDCPP) through the dust ingestion exposure route. Results showed that the health guidance value of TPHP and TCDPP obtained in this study was lower than the value obtained through animal experiments. In addition, probabilistic risk assessment results indicate that populations worldwide are at low risk of exposure to TPHP and TDCPP through dust ingestion due to low estimated daily intakes (EDIs) which are much lower than the reference dose (RfDs) published by the US EPA, except in some regional cases. Most margin of exposure (MOE) ranges of TDCPP for children are less than 100, which indicates a moderately high risk. Researchers should be concerned about exposure to TDCPP in this area. The method proposed in this study is expected to be applied to the health risk assessment of other chemicals.

Spatial variability characteristics of the effective friction angle of Crag deposits and its effects on slope stability

This study investigated the spatial variability characteristics of the effective friction angle of Crag deposits which are granular soils occur in the east of England. Cone Penetration Test data were obtained at 26 locations and interpreted statistically. The distribution characteristic of the effective friction angle of Crag deposits was derived with the mean value, the standard deviation and the correlation length calibrated. Illustrations were also shown on how factors such as ground water pressures and the existence of soft/organic soil zones affect the measurement of the autocovariance function and thus the correlation length. Bayesian inference technique was adopted alongside the method of moments to determine the correlation length. Based on the obtained statistical parameters, both semi-deterministic (based on standard geotechnical design codes) and probabilistic finite element limit analyses were carried out to investigate the stability of slopes in Crag deposits. Slopes of various inclined angles were considered and comparisons between the semi-deterministic and probabilistic results were conducted to improve the understanding of the stability of Crag slopes and to provide insight into the slope stability code used in practice.

Metamodel-Based Slope Reliability Analysis—Case of Spatially Variable Soils Considering a Rotated Anisotropy

A rotation of the anisotropic soil fabric pattern is commonly observed in natural slopes with a tilted stratification. This study investigates the rotated anisotropy effects on slope reliability considering spatially varied soils. Karhunen–Loève expansion is used to generate the random fields of the soil shear strength properties (i.e., cohesion and friction angle). The presented probabilistic analyses are based on a meta-model combining Sparse Polynomial Chaos Expansion (SPCE) and Global Sensitivity Analysis (GSA). This method allows the number of involved random variables to be reduced and then the computational efficiency to be improved. Two kinds of deterministic models, namely a discretization kinematic approach and a finite element limit analysis, are considered. A variety of valuable results (i.e., failure probability, probability density function, statistical moments of model response, and sensitivity indices of input variables) can be effectively provided. Moreover, the influences of the rotated anisotropy, autocorrelation length, coefficient of variation and cross-correlation between the cohesion and friction angle on the probabilistic analysis results are discussed. The rotation of the anisotropic soil stratification has a significant effect on the slope stability, particularly for the cases with large values of autocorrelation length, coefficient of variation, and cross-correlation coefficient.

Uncertainty Evaluation Algorithm in Power System Dynamic Analysis With Correlated Renewable Energy Sources

The variation and uncertainty of renewable energy potentially impact power system stability. This article proposes an efficient uncertainty evaluation algorithm to evaluate the influence of renewable energy uncertainty on power system dynamic performance. This algorithm applies probabilistic collocation method (PCM), which greatly reduces the simulation burden without compromising the result accuracy, to approximate the dynamic simulation results. The proposed algorithm also considers the correlation among adjacent renewable generators by using the Copula function, which is suitable to model nonlinear correlations. Additionally, this article extends the utilization of PCM. The actual historical data of renewable generation productions can be utilized directly, and kernel density estimation (KDE) is used to capture the nonparametric distribution of renewable generation production. To improve the accuracy of the approximation results, the proposed method adopts K-means clustering technique to select the approximation samples of input variables. The proposed probabilistic dynamic simulation algorithm is compared with Monte Carlo simulations (MCS) with the probabilistic results on the IEEE 39-bus system with multiple renewable generators, and the accuracy and efficiency of the proposed algorithm are validated.

Randomness of Möbius coefficients and Brownian motion: growth of the Mertens function and the Riemann hypothesis

The validity of the Riemann hypothesis (RH) on the location of the non-trivial zeros of the Riemann ζ-function is directly related to the growth of the Mertens function , where μ(k) is the Möbius coefficient of the integer k; the RH is indeed true if the Mertens function goes asymptotically as M(x) ∼ x 1/2+ϵ , where ϵ is an arbitrary strictly positive quantity. We argue that this behavior can be established on the basis of a new probabilistic approach based on the global properties of the Mertens function, namely, based on reorganizing globally in distinct blocks the terms of its series. With this aim, we focus attention on the square-free numbers and we derive a series of probabilistic results concerning the prime number distribution along the series of square-free numbers, the average number of prime divisors, the Erdős–Kac theorem for square-free numbers, etc. These results point to the conclusion that the Mertens function is subject to a normal distribution as much as any other random walk. We also present an argument in favor of the thesis that the validity of the RH also implies the validity of the generalized RH for the Dirichlet L-functions. Next we study the local properties of the Mertens function, i.e. its variation induced by each Möbius coefficient restricted to the square-free numbers. Motivated by the natural curiosity to see how closely to a purely random walk any sub-sequence is extracted by the sequence of the Möbius coefficients for the square-free numbers, we perform a massive statistical analysis on these coefficients, applying to them a series of randomness tests of increasing precision and complexity; together with several frequency tests within a block, the list of our tests includes those for the longest run of ones in a block, the binary matrix rank test, the discrete Fourier transform test, the non-overlapping template matching test, the entropy test, the cumulative sum test, the random excursion tests, etc, for a total of 18 different tests. The successful outputs of all these tests (each of them with a level of confidence of 99% that all the sub-sequences analyzed are indeed random) can be seen as impressive ‘experimental’ confirmations of the Brownian nature of the restricted Möbius coefficients and the probabilistic normal law distribution of the Mertens function analytically established earlier. In view of the theoretical probabilistic argument and the large battery of statistical tests, we can conclude that while a violation of the RH is strictly speaking not impossible, it is however extremely improbable.