Most Probable Value Research Articles

Bayesian Calibration of Hysteretic Parameters with Consideration of the Model Discrepancy for Use in Seismic Structural Health Monitoring

Bayesian model calibration techniques are commonly employed in the characterization of nonlinear dynamic systems, as they provide a conceptual and effective framework to deal with model uncertainties, experimental errors and procedure assumptions. This understanding has resulted in the need to introduce a model discrepancy term to account for the differences between model-based predictions and real observations. Indeed, the goal of this work is to investigate model-driven seismic structural health monitoring procedures based on a Bayesian uncertainty quantification framework, and thus make relevant considerations for its use in the seismic structural health monitoring, focusing on masonry structures. Specifically, the Bayesian inference has been applied to the calibration of nonlinear hysteretic systems to both provide: (i) most probable values (MPV) of the parameters following the calibration; and (ii) estimates of the model discrepancy posterior distribution. The effect of the model discrepancy in the calibration is first illustrated recurring to a single degree of freedom using a Bouc–Wen type oscillator as a numerical benchmark. The model discrepancy is then introduced for calibrating a reference nonlinear Bouc–Wen model derived from real data acquired on a monitored masonry building. The main novelty of this study is the application of the framework of uncertainty quantification on models representing data measured directly on masonry structures during seismic events.

A new probabilistic frequency-domain approach for influence line extraction from static transmissibility measurements under unknown moving loads

On the basis of the amazing theoretical finding that there is equivalence between the static transmissibility subjected to moving loads and the ratio of two influence lines in the frequency domain, a new approach is proposed to extract the influence lines for a beam-like structure under moving loads. To accommodate the uncertainties involved in the measurements as well as modelling error, the relationship between response measurements and the Fourier transform of influence lines is embedded in the framework of Bayesian inference with the aid of complex-valued probabilistic model of prediction error. The formulas are presented for closed-form transformation between the solutions of FFT coefficients and those of inverse FFT coefficients. Analytical solutions of the Most Probable Values (MPVs) as well as posterior uncertainties of the influence lines in both frequency domain and spatial domain are derived. Two applications are conducted to verify the efficiency and accuracy of the fast Bayesian scheme. It is shown that the new approach can be realized by avoiding the ill-poseness nature of inverse problem. Due to the introduction of the concept of static transmissibility, given that the reference influence line is known in advance, this method possesses an obvious advantage in avoiding using the knowledge of the moving loads. As a frequency-domain approach, it can reduce the computational complexity of influence line extraction by avoiding complicated matrix manipulation.

Comparison of total nitrogen data from direct and Kjeldahl‐based approaches in integrated data sets

AbstractThere are multiple protocols for determining total nitrogen (TN) in water, but most can be grouped into direct approaches (TN‐d) that convert N forms to nitrogen‐oxides (NOx) and combined approaches (TN‐c) that combine Kjeldahl N (organic N +NH3) and nitrite+nitrate (NO2+NO3‐N). TN concentrations from these two approaches are routinely treated as equal in studies that use data derived from multiple sources (i.e., integrated data sets) despite the distinct chemistries of the two methods. We used two integrated data sets to determine if TN‐c and TN‐d results were interchangeable. Accuracy, determined as the difference between reported concentrations and the most probable value (MPV) of reference samples, was high and similar in magnitude (within 3.5–4.5% of the MPV) for both methods, although the bias was significantly smaller at low concentrations for TN‐d. Detection limits and data flagged as below detection suggested greater sensitivity for TN‐d for one data set, while patterns from the other data set were ambiguous. TN‐c results were more variable (less precise) by many measures, although TN‐d data included a small fraction of notably inaccurate results. Precision of TN‐c was further compromised by propagated error, which may not be acknowledged or detectable in integrated data sets unless complete metadata are available and inspected. Finally, concurrent measures of TN‐c and TN‐d in lake samples were extremely similar. Overall, TN‐d tended to be slightly more accurate and precise, but similarities in accuracy and the near 1 : 1 relationship for concurrent TN‐d and TN‐c measurements support careful use of data interchangeably in analyses of heterogeneous, integrated data sets.

Study on the detection efficiency and uniformity of a Water Cherenkov Detector based on Geant4

The Water Cherenkov Detector (WCD) is an important and widely used detector in cosmic ray physics, mainly used to measure the electromagnetic component of extensive air showers. Generally, a full detector simulation based on Geant4 for large arrays would require too much computing time and memory, because it would be a very CPU-intensive job to track all the Extensive Air Shower particles (mainly electrons and photons) hitting a very large area WCD detector. Therefore we developed a code simulating the behavior of muons, gamma-rays and electrons crossing a cell of a full WCD array. The results of this simulation can be collected in look-up table that can be used by fast simulation codes for large arrays. In this note we describe the results obtained simulating a single cell of cubic geometry (5×5×5 m3). The code we have developed for this particular geometry can easily be adapted to other detector designs. Firstly, the parameters in simulation are determined analysing the properties of the WCDs. Then, we calculate the most probable value (MPV) and sigma of the Landau function describing the distribution of the photoelectron numbers detected by the WCD photomultipliers. The results of the simulation are tested by a comparison with experimental results available in the literature: the relative error between their MPVs is only 1.37% for vertical muon- signal. We show that photoelectron distributions do not only depend on particle types and energies, but also on the incidence zenith angles and on the distance between the particle impact position and the center of the cell (i.e. the photomultiplier).

An innovative Bayesian system identification method using autoregressive model

Abstract This paper proposes an innovative Bayesian method for system identification based on autoregressive (AR) model. The dynamics of a structure is first modeled by an AR model. Due to measurement noise and modeling errors in practical problems, it is important to quantify uncertainties of the model. The posterior PDF of the parameters of the AR model is then formulated following Bayes’ theorem. New formulations of the most probable values (MPVs) and the posterior uncertainties of the AR model parameters are derived in closed form. It is shown that the model of a vibrating structure can be transformed to an AR model, so the modal parameters of the structure can be extracted from the parameter matrices of the AR model. For assessing the posterior uncertainties of the modal parameters, original analytical formulations are derived to propagate the uncertainties of AR model parameters to the modal parameters. The proposed method is verified by measured ambient vibration data of a 20-story building. Working directly on the measured accelerations, the proposed method can make use of the original information in the data to identify all modal parameters of interest together with corresponding uncertainties in a few minutes. The contribution of this paper is that the algebraically involved derivation is resolved to develop new formulations for the MPVs and associated uncertainties, reveal the complicated relationship between the uncertainties of modal parameters and those of AR model parameters, and provide a mathematically manageable algorithm for efficient practical applications.

An expectation-maximization algorithm for Bayesian operational modal analysis with multiple (possibly close) modes

The Bayesian FFT method has gained attention in operational modal analysis of civil engineering structures. Not only the most probable value (MPV) of modal parameters can be computed efficiently, but also the identification uncertainty can be rigorously quantified in terms of posterior covariance matrix. A recently developed fast algorithm for general multiple (possibly close) modes was found to work well in most cases, but convergence could be slow or even fail in challenging situations. The algorithm is also tedious to computer-code. Aiming at resolving these issues, an expectation-maximization (EM) algorithm is developed by viewing the modal response as a latent variable. The parameter-expansion EM and the parabolic-extrapolation EM are further adopted, allowing mode shape norm constraints to be incorporated and accelerating convergence, respectively. A robust implementation is provided based on the QR and Cholesky decompositions, so that the computation can be done efficiently and reliably. Empirical studies verify the performance of the proposed EM algorithm. It offers a more efficient and robust (in terms of convergence) alternative that can be especially useful when the existing algorithm has difficulty to converge. In addition, it opens a way to compute the MPV in the Bayesian FFT method for other unexplored cases, e.g., multi-mode multi-setup problem.

Many-body localization in the presence of long-range interactions and long-range hopping

We study many-body localization (MBL) in a one-dimensional system of spinless fermions with a deterministic aperiodic potential in the presence of long-range interactions or long-range hopping. Based on perturbative arguments there is a common belief that MBL can exist only in systems with short-range interactions and short-range hopping. We analyze effects of power-law interactions and power-law hopping, separately, on a system which has all the single particle states localized in the absence of interactions. Since delocalization is driven by proliferation of resonances in the Fock space, we mapped this model to an effective Anderson model on a complex graph in the Fock space, and calculated the probability distribution of the number of resonances up to third order. Though the most-probable value of the number of resonances diverge for the system with long-range hopping ($t(r) \sim t_0/r^\alpha$ with $\alpha < 2$), there is no enhancement of the number of resonances as the range of power-law interactions increases. This indicates that the long-range hopping delocalizes the many-body localized system but in contrast to this, there is no signature of delocalization in the presence of long-range interactions. We further provide support in favor of this analysis based on dynamics of the system after a quench starting from a charge density wave ordered state, level spacing statistics, return probability, participation ratio and Shannon entropy in the Fock space. We demonstrate that MBL persists in the presence of long-range interactions though long-range hopping with $1<\alpha <2$ delocalizes the system partially, with all the states extended for $\alpha <1$. Even in a system which has single-particle mobility edges in the non-interacting limit, turning on long-range interactions does not cause delocalization.

Bayesian operational modal analysis and assessment of a full-scale coupled structural system using the Bayes-Mode-ID method

This paper presents a structural assessment project for a construction training center in Hong Kong. The training center consists of coupled main and complementary buildings. Due to the vigorous activities in the training center, the objectives of the project include conducting operational modal analysis (OMA) of the buildings to assess their structural performance and the coupling of vibrations between the two buildings. OMA is conducted using Bayes-Mode-ID recently developed by the authors, which is an efficient Bayesian modal-component-sampling system identification method for field testing of civil engineering structures under ambient vibrations. Implementation issues for Bayes-Mode-ID are discussed in detail for the full-scale coupled structural system. Due to the large number of measurement points (high resolution mode shapes are desired for understanding the coupling behavior of the system) and limited number of sensors, the measurements were divided into 21 setups in order to properly characterize the dynamics of the building. Each setup covered one portion of the training center and the partial mode shapes from different setups were assembled to provide the global mode shapes. By following a Bayesian approach, not only the most probable values (MPVs) of the modal parameters (modal frequencies, modal damping ratios and mode shapes) but also their associated uncertainties can be obtained. The identified modal parameters reveal interesting dynamic behaviors of the coupled-building and they will be helpful for structural assessment and structural health monitoring (SHM) of the training center in the future.

Calculation of Hessian under constraints with applications to Bayesian system identification

In Bayesian system identification with globally identifiable models, the posterior (i.e., given data) probability density function (PDF) of model parameters can be approximated by a Gaussian PDF. The most probable value (MPV) of the parameters is equal to the mean of the Gaussian PDF. It maximises the posterior PDF, or equivalently, minimises the negative of logarithm (NL) of the posterior PDF. The covariance matrix of the Gaussian PDF is equal to the Hessian of the NL at the MPV. Model parameters can be subjected to constraints, which must be accounted for in the calculation of the posterior covariance matrix. In applications such as modal identification, existing strategies define a set of free parameters and map them to the model parameters so that the constraints are always satisfied. The Hessian of the NL with respect to the free parameters is obtained and then transformed to give the posterior covariance matrix of the model parameters where constraints are accounted for. Analytical expressions for this Hessian are complicated because of the composite actions of the NL and the mapping; and this creates significant burden in computer coding. In this work, a theoretical framework is developed for evaluating the Hessian of a function under constraints in a systematic manner. It is applied to obtain new analytical expressions for evaluating the posterior covariance matrix in Bayesian operational modal analysis. The resulting expressions are simpler than existing ones based on direct differentiation. They allow problems with similar mathematical structures to be computer-coded in a coherent manner. Numerical examples are presented to illustrate consistency and computational aspects.

Operational modal identification of a boat-shaped building by a Bayesian approach

In model updating, damage detection and structural health monitoring (SHM), modal identification is usually the first step to calculate the modal properties, such as natural frequencies, damping ratios and mode shapes, of the targeted structure utilizing the measured time-domain responses. Ambient vibration test provides an efficient and economical mean to collect structural vibration responses for the purpose of modal identification since no artificial excitation is required. This paper presents the work on the ambient vibration test and modal identification of a recently constructed boat-shaped building situated in Hong Kong. The building is not an isolated structure, and it is connected with a secondary structure in its fourth and fifth floors. To identify detail mode shapes, sixty-two measurement points were desired. They were located in the three staircases (to minimize the effect on the normal operation of the building) and on a typical floor of the building. Due to the large number of measurement points coving almost all floors of the building, multiple setups with moving references approach were designed. Some common problems encountered in the field test were discussed in this paper. A recently developed fast Bayesian FFT method was employed to perform the modal identification. This method could identify the most probable value (MPV) of modal parameters efficiently and also calculate the associated posterior uncertainty analytically, providing an important tool to evaluate the accuracy of modal parameters. The first nine modes were identified including five translational modes, two torsional modes, one combined mode and one vertical mode. Due to the connections with other building, torsional behavior was observed in some translations modes. Based on the MPV and the associated uncertainty, the variability of modal parameters across different setups is also addressed. Some interesting feathers of this structure were studied and investigated, providing the baseline for the future SHM and condition assessment of this building.

Stochastic sensitivity analysis of large-eddy simulation predictions of the flow around a 5:1 rectangular cylinder

A stochastic analysis of the sensitivity to grid resolution and modeling of large-eddy simulations (LES) results is carried out for the flow around a 5:1 rectangular cylinder, which is the object of an international benchmark (BARC) collecting experimental and numerical flow realizations. Significant dispersion of the BARC predictions was observed for some quantities, also in LES, and deterministic sensitivity analyses were not conclusive. LES are carried out here by using a spectral-element numerical method. An explicit quadratic low-pass filter in the modal space is used, characterized by a cut-off value and by a weight function, which provides dissipation of the modes higher than the cut off and acts as a SGS dissipation. The uncertain parameters are the size of the spectral elements in the spanwise direction and the weight of the explicit filter. The impact of the uncertainty in these parameters is evaluated through generalized polynomial chaos. The analysis is repeated for two different grid resolutions in the streamwise and lateral directions. The most-probable values and the stochastic variance of the results are compared with the ensemble average and with the overall dispersion of the BARC predictions respectively.

Bayesian Operational Modal Analysis of a Pedestrian Bridge Using a Field Test with Multiple Setups

An ambient vibration test was conducted for the pedestrian bridge that links the City University (CityU) of Hong Kong and the adjacent subway station. In this test, 100 locations forming a net of measurement were selected. Due to the limited number of sensors available, 10 setups were designed to cover all the degrees of freedom (dofs) of interest. A recently developed Bayesian method incorporating multiple setups was used to analyze the collected data. Besides the most probable values (MPVs) of modal parameters, the associated posterior uncertainty was also calculated analytically without resorting to the finite difference method. Six modes were identified, including some local modes of specified spans and global modes, and compared with those identified in a previous study. The difficulty encountered in the field test is discussed, along with some interesting features. Practical interpretation of the statistics of modal parameters calculated from frequentist and Bayesian context is also discussed. After the field test, a lift was installed between the ground and the bridge deck, which changed the modal properties of the original structure. This study provides a baseline of the dynamic characteristics of the original structure for future safety assessment.

Mode identifiability of a cable-stayed bridge based on a Bayesian method

Modal identification based on ambient vibration data has attracted extensive attention in the past few decades. Since the excitation for ambient vibration tests is mainly from the environmental effects such as wind and traffic loading and no artificial excitation is applied, the signal to noise (s/n) ratio of the data acquired plays an important role in mode identifiability. Under ambient vibration conditions, certain modes may not be identifiable due to a low s/n ratio. This paper presents a study on the mode identifiability of an instrumented cable-stayed bridge with the use of acceleration response data measured by a long-term structural health monitoring system. A recently developed fast Bayesian FFT method is utilized to perform output-only modal identification. In addition to identifying the most probable values (MPVs) of modal parameters, the associated posterior uncertainties can be obtained by this method. Likewise, the power spectral density of modal force can be identified, and thus it is possible to obtain the modal s/n ratio. This provides an efficient way to investigate the mode identifiability. Three groups of data are utilized in this study: the first one is 10 data sets including six collected under normal wind conditions and four collected during typhoons; the second one is three data sets with wind speeds of about 7.5 m/s; and the third one is some blind data. The first two groups of data are used to perform ambient modal identification and help to estimate a critical value of the s/n ratio above which the deficient mode is identifiable, while the third group of data is used to perform verification. A couple of fundamental modes are identified, including the ones in the vertical and transverse directions respectively and coupled in both directions. The uncertainty and s/n ratio of the deficient mode are investigated and discussed. A critical value of the modal s/n ratio is suggested to evaluate the mode identifiability of the deficient mode. The work presented in this paper could provide a base for the vibration-based condition assessment in future.

Series of Full-Scale Field Vibration Tests and Bayesian Modal Identification of a Pedestrian Bridge

Many spectacular pedestrian bridges were designed and constructed recently. Owing to their special shapes, it is expected that various types and a wide range of vibration frequency components will be induced by pedestrians. To avoid accidents and reduce risk, the vibration characteristics of pedestrian bridges during their service life must be carefully assessed. The most direct and reliable way to study the vibration characteristics of a structural system is through field vibration tests. In this paper, a series of full-scale field vibration tests (including ambient, forced, and free vibration tests) were carried out on a pedestrian bridge at City University of Hong Kong (CityU). The difficulties encountered in the field tests are reported. The recently developed Bayesian methods were utilized to determine the modal parameters of the bridge based on measurements from all three kinds of tests. In addition to the most probable values (MPVs) of modal parameters, the associated posterior uncertainties were also analytically calculated. Four modes were identified, including three vertical bending modes and one torsional mode. The accuracy of the identified modal parameters was assessed through the posterior uncertainty. Because the amplitudes of the vibration in the three kinds of tests were different, the modal parameters determined from these kinds of tests were compared and discussed. Suggestions related to the proper use and potential vibration problems during the lifecycle of pedestrian bridges were provided based on the analysis results.

Fast Bayesian approach for modal identification using free vibration data, Part II—Posterior uncertainty and application

A Bayesian statistical framework has been developed for modal identification using free vibration data in the companion paper (Zhang et al., Mech. Syst. Sig. Process. (2015)). Efficient strategies have been developed for evaluating the most probable value (MPV) of the modal parameters in both well-separated mode and general multiple mode cases. This paper investigates the posterior uncertainty of the modal parameters in terms of their posterior covariance matrix, which is mathematically equal to the inverse of the Hessian of the negative log-likelihood function (NLLF) evaluated at the MPVs. Computational issues associated with the determination of the posterior covariance matrix are discussed. Analytical expressions are derived for the Hessian so that it can be evaluated accurately and efficiently without resorting to finite difference method. The proposed methods are verified with synthetic data and then applied to field vibration test data.

Fast Bayesian approach for modal identification using free vibration data, Part I – Most probable value

The identification of modal properties from field testing of civil engineering structures is becoming economically viable, thanks to the advent of modern sensor and data acquisition technology. Its demand is driven by innovative structural designs and increased performance requirements of dynamic-prone structures that call for a close cross-checking or monitoring of their dynamic properties and responses. Existing instrumentation capabilities and modal identification techniques allow structures to be tested under free vibration, forced vibration (known input) or ambient vibration (unknown broadband loading). These tests can be considered complementary rather than competing as they are based on different modeling assumptions in the identification model and have different implications on costs and benefits. Uncertainty arises naturally in the dynamic testing of structures due to measurement noise, sensor alignment error, modeling error, etc. This is especially relevant in field vibration tests because the test condition in the field environment can hardly be controlled. In this work, a Bayesian statistical approach is developed for modal identification using the free vibration response of structures. A frequency domain formulation is proposed that makes statistical inference based on the Fast Fourier Transform (FFT) of the data in a selected frequency band. This significantly simplifies the identification model because only the modes dominating the frequency band need to be included. It also legitimately ignores the information in the excluded frequency bands that are either irrelevant or difficult to model, thereby significantly reducing modeling error risk. The posterior probability density function (PDF) of the modal parameters is derived rigorously from modeling assumptions and Bayesian probability logic. Computational difficulties associated with calculating the posterior statistics, including the most probable value (MPV) and the posterior covariance matrix, are addressed. Fast computational algorithms for determining the MPV are proposed so that the method can be practically implemented. In the companion paper (Part II), analytical formulae are derived for the posterior covariance matrix so that it can be evaluated without resorting to finite difference method. The proposed method is verified using synthetic data. It is also applied to modal identification of full-scale field structures.

Ionization efficiency study for low energy nuclear recoils in germanium

We used the internal conversion (E0 transition) of germanium-72 to indirectly measure the low energy nuclear recoils of germanium. Together with a reliable Monte Carlo package, in which we implement the internal conversion process, the data was compared to the Lindhard (k=0.159) and Barker–Mei models. A shape analysis indicates that both models agree well with data in the region of interest within 4%. The most probable value (MPV) of the nuclear recoils obtained from the shape analysis is 17.5±0.12 (sys) ±0.035 (stat) keV with an average path-length of 0.014μm.

Influencing factors of geometrical structure of surface shrinkage cracks in clayey soils

In order to investigate the effects of temperature, thickness of soil layer, wetting and drying cycles and soil types on geometrical structure of surface shrinkage cracks in clayey soils, special software Crack Image Analysis System (CIAS) for analyzing shrinkage crack patterns was developed. Eight groups of soil samples were prepared and subjected to drying to crack in laboratory. The number of crack segments and intersections, average crack length, width and aggregate area, crack intensity factor (CIF), and the corresponding probability density functions (PDF) of these parameters were determined by analyzing several crack patterns derived from different experimental conditions. The results show that the soil cracking behavior and the geometrical structure of crack patterns are significantly influenced by these considered factors. There is a tendency of crack length, width, aggregate area and their most probable value (MPV) related to the PDF increases with temperature increase. With thicker soil layers, the average crack length, width, aggregate area and CIF are increased, and the main distribution ranges of crack length, width and aggregate area are increased also. When the soil is subjected to multiple wetting–drying cycles, the soil surface generates more irregular and coarse cracks. The number of short and narrow crack segments increases significantly, and the CIF decreases with an increase in wetting–drying cycles. It is also observed that the extent of cracking is directly related to the soil fines fraction and its plasticity index ( I P). The greatest CIF and crack width are observed in the soils with the largest fines fraction and highest I P. In addition, the ratio of numbers of crack segments to intersections ranges from 1.5 to 2, and cracking mainly takes place in three stages: main-cracks initiation stage; sub-cracks initiation stage; terminal stable stage.

Money in Gas-Like Markets: Gibbs and Pareto Laws

We consider the ideal-gas models of trading markets, where each agent is identified with a gas molecule and each trading as an elastic or money-conserving (two-body) collision. Unlike in the ideal gas, we introduce a saving propensity λ of agents, such that each agent saves a fraction λ of its money and trades with the rest. We show that the steady-state money or wealth distribution in a market is Gibbs-like for λ = 0, has got a non-vanishing most-probable value for λ ≠ 0 and Pareto-like when λ is widely distributed among the agents. We compare these results with observations on wealth distributions of various countries.

Vertical Metallicity Gradient of the Galaxy Based on UBV Starcount Data

The spatial distribution of metal abundance in the Galaxy has been analyzed using the UBV starcount data recently obtained in two high-latitude regions of the North Galactic Pole (NGP) and Selected Area 54 (SA54). A least-squares analysis was performed to determine the vertical metallicity gradient for each of the thin and thick disk components that gives a reasonable fit to the observed U-B and B-V colour distributions to V = 18 mag. The most probable value of the vertical gradient is obtained as d[Fe/H]/dz = −0.5 kpc−1 for the thin disk, and −0.1 kpc−1 for the thick disk.