Probability Density Function Of Parameters Research Articles

Probabilistic Identification of Multi-DOF Structures Subjected to Ground Motion Using Manifold-Constrained Gaussian Processes

Bayesian uncertainty quantification has a pivotal role in structural identification, yet the posterior distribution estimation of unknown parameters and system responses is still a challenging task. This study explores a novel method, named manifold-constrained Gaussian processes (GPs), for the probabilistic identification of multi-DOF structural dynamical systems, taking shear-type frames subjected to ground motion as a demonstrative paradigm. The key idea of the method is to restrict the GPs (priorly defined over system responses) on a manifold that satisfies the equation of motion of the structural system. In contrast to widely used Bayesian probabilistic model updating methods, the manifold-constrained GPs avoid the numerical integration when formulating the joint probability density function of unknown parameters and system responses, hence achieving an accurate and computationally efficient inference for the posterior distributions. An eight-storey shear-type frame is analyzed as a case study to demonstrate the effectiveness of the manifold-constrained GPs. The results indicate the posterior distributions of system responses, and unknown parameters can be successfully identified, and reliable probabilistic model updating can be achieved.

Voltage Restoration After Unforeseen Disturbances in Weakly Observable Distribution Systems

Following an unforeseen disturbance in a power system, the system state should be known to extract the emergency voltage restoration strategy. Such a condition does not usually hold in distribution systems due to the lack of enough measurements. Here, the joint probability density function ( <sc xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">pdf</small> ) of uncertain parameters is extracted from historical data. This <sc xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">pdf</small> is updated to comply with the measurements. A set of samples is then generated that best models the updated <sc xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">pdf</small> . These data reconciliation and sampling techniques enable the functioning of the emergency voltage control ( <sc xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">evc</small> ) with measurement scarcity. They are designed to be fast enough to meet the quasi-real-time requirements of the intended application. The <sc xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">evc</small> problem is cast as a stochastic programming problem. The resultant problem is a mixed-integer non-convex (and hence, NP-hard) optimization problem. It is solved using five distinct <sc xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">sm</small> s based on relaxation/approximation of the non-convex constraints. The results show that a novel combination of relaxation and approximation aimed at respectively mitigating the emergency under- and over-voltages outperforms the others. The propounded stochastic <sc xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">evc</small> is validated through numerical studies.

Probability density function of whole-body vibration in passenger car

This study identified the probability density function (PDF) model of the driver/passenger whole body vibration (WBV) in the passenger car based on wide in-situ measurements. The WBV was evaluated at the seat surface and feet of twenty-eight test vehicles travelled on test sections of different quality of total travel distance over 2,600 km. The WBV was quantified by the frequency-weighted acceleration at the driver’s/passenger’s seat surface and feet according to the ISO 2631-1 in three perpendicular axes. The lognormal PDF was identified as the most suitable model to fit empirical WBV data. Vibration data were divided in the groups according to the road roughness, vehicle speed, road and vehicle category. The results showed the marked influence of the road roughness and the lower influence of the vehicle speed, road and vehicle category on the PDF parameters. The linear relation between scale and shape parameters of PDF and road roughness level was estimated. The identified parameters of lognormal PDF can be useful for the basic statistics estimation of the WBV in passenger car and estimation of human ride comfort and health.

A Monte-Carlo based 3-D ballistics model for guiding bat carcass surveys using environmental and turbine operational data

Wind turbines cause direct mortality of bats through collision with turbine blades. However, accurate bat fatality estimates are unavailable for many facilities and regions due to lack of physically consistent guidance for surveying bat carcasses. We develop a Monte-Carlo based 3-D ballistics model to address this important gap to guide carcass surveys, help improve fatality estimates by combining it with statistical models and offer post-construction impact assessment guidelines. Monte-Carlo simulations account for variability of environmentally relevant parameters and subsequently the variability of the generated carcass fall zone distributions. The input parameters are bat characteristics, including mass, size, and drag coefficient, wind speed, turbine RPM, yaw, bat flight speed, bat strike angle, and bat strike locations on the rotor. Using turbine SCADA (Supervisory Control and Data Acquisition), bat biometrics data, and carcass drag coefficient range, probability density functions of the input parameters were computed. With the help of carcass survey and SCADA details, bat fatalities were selected, and Monte-Carlo based ballistics model simulations were performed using corresponding values of the input parameters of the ballistics model. The surveyed carcass location for a bat found on the same day was in good agreement with the modelled fall zone distribution, providing a measure of model validation. Next, simulations were conducted for a bat fatality for which the carcass was found four days after collision. It was found that the observed location was within the modeled fall zone distribution showing the model capacity for planning surveys. Further, Monte-Carlo based 3-D ballistics model simulations were performed for the full migration season and the resulting 1-D (radial) and 2-D (surface) histograms were compared with the survey 1-D and 2-D histograms, respectively. The modelled 2-D fall zone distribution hot spot overlaps with surveyed carcass positions indicating the model robustness over a wide range of meteorological conditions and time scales. However, by comparison, the modeled 1-D fall zone distribution was found to be different from the surveyed 1-D fall zone distribution. We hypothesize that uncertainty in bat radial strike location (distribution of bat collisions on turbine blades in radial direction) distribution is primary source of difference between the surveyed and modelled 1-D fall zone distribution. There are no measurements available for this key input parameter. As a next step, the sensitivity of carcass fall zone distributions with respect to bat radial strike location distributions were tested. It was found that the probability density function for bat radial strike location affects the modelled fall zone distributions significantly. There is a need to measure bat radial and angular strike locations which can be used as input in the ballistics modeling framework to improve prediction of carcass fall zone distributions. The proposed model is useful for carcass survey planning, improving survey estimates, designing turbine curtailment studies, and collision risk modeling.

Prognosis of growth and production of Eucalyptus urograndis planted at different spacings

A topic of great interest in forestry research is recognizing the behavior of stands planted at different spacings to optimize the potential of producing multiple products. This study aimed to assess the future production of multiple products from Eucalyptus sp. planted at 12 different spacings. The study area was in central Brazil. A Nelder systematic design with three replications was used. This design yielded 432 planting spots per plot per replicate. Diameter at breast height (DBH) and total height were measured for all trees from 4 to 36 months of age. In addition, the volume of five trees for the spacings of 16, 24, and 36 months of age was measured. Subsequently, an equation system was fitted for each area to project the Weibull 3P probability density function parameters: minimum diameter, maximum diameter, and the number of trees. The current and mean annual increments were assessed based on the total volume. Trees of 36 and 48 months of age were compared between the population designed for multiple products (Scenario I) and the population designed for only one product (Scenario II). The results showed that spacing affects the technical harvest age and that 0.77 m² and 1.76 m² are optimal for a short-term regime

On the Combination of Random Matrix Theory With Measurements on a Single Structure

Abstract An approach is proposed for the evaluation of the probability density functions (PDFs) of the modal parameters for an ensemble of nominally identical structures when there is only access to a single structure and the dispersion parameter is known. The approach combines the Eigensystem realization algorithm on sets of dynamic data, with an explicit nonparametric probabilistic method. A single structure, either a mathematical model or a prototype, is used to obtain simulated data or measurements that are employed to build a discrete time state-space model description. The dispersion parameter is used to describe the uncertainty due to different sources such as the variability found in the population and the identification errors found in the noisy measurements from the experiments. With this approach, instead of propagating the uncertainties through the governing equations of the system, the distribution of the modal parameters of the whole ensemble is obtained by randomizing the matrices in the state-space model with an efficient procedure. The applicability of the approach is shown through the analysis of a two degrees-of-freedom mass-spring-damper system and a cantilever system. The results show that if the source of uncertainty is unknown and it is possible to specify an overall level of uncertainty, by having access to a single system's measurements, it is possible to evaluate the resulting PDFs on the modal parameters. It was also found that high values of the dispersion parameter may lead to nonphysical results such as negative damping ratios values.

Few observation binary orbit solver (fobos) from two (or more) astrometric observations

ABSTRACT We have developed a new, fast method of estimating the orbital properties of a binary or triple system using as few as two epochs of astrometric data. fobos (Few Observation Binary Orbit Solver) uses a flat prior brute force Monte Carlo method to produce probability density functions of the likely orbital parameters. We test the code on fake observations and show that it can (fairly often) constrain the semi-major axis to within a factor of 2–3, and the inclination to within ∼20° from only two astrometric observations. We also show that the 68 and 95 per cent confidence intervals are statistically reliable. Applying this method to triple systems allows the relative inclination of the secondary and tertiary star orbits to be constrained. fobos can usually find a statistically significant number of possible matches in CPU minutes for binary systems, and CPU hours for triple systems.

General strategies for modeling joint probability density function of wind speed, wind direction and wind attack angle

The joint probability density function (JPDF) of the wind field parameters is essential for wind load analysis or energy assessment, but one key parameter of the wind attack angle is often excluded in current studies. Three different methodologies are proposed, aiming to estimate the JPDF of wind speed, wind direction and wind attack angle. Strategy I consists of the vine approach with the modified binary Bernstein copula adopted for nonparametric estimation. The modified ternary Bernstein copula is recommended to estimate the ternary copula directly in Strategy II. In strategy III, the Offset Elliptical Normal mixture model is extended to evaluate the three-dimensional case, where the wind vector is resolved into three orthogonal components. A typical mountainous bridge site with strongly inhomogeneous wind characteristics is selected to inspect the performance of these strategies. Satisfactory results are given by all these strategies in which the strategy II has better goodness-of-fit. Strategy I is slightly less reliable in estimating the dependencies of higher tree variables while avoids the time-consuming nonparametric estimation of three-dimensional copula. The strategy III is based on the physical model and performs better in the conditional probability estimation when the number of samples is small.

Determination of Relationships Between Stand Variables and Parameters of Weibull Function for Fagus orientalis Libsky Stands

Aim of study: Information about the diameter distribution of a stand is a key resource to determine planning strategies, silvicultural treatment options and product variety. In this study, the diameter distributions of Fagus orientalis Libsky stands located in Karabük region were researched, and relationship between parameters of Weibull function and stand variables was investigated. Area of study: Data used in this study were obtained from pure Oriental beech (Fagus orientalis Libsky) stands located in the Karabük region, north-central Turkey. Material and methods: For this study, sixty-two sample plots from pure Fagus orientalis Lipsky stands located in Karabük region were taken. Maximum likelihood method was used to estimate parameters of the two-parameter Weibull probability density function. The parameters estimated were then expressed as linear functions of stand variables such as mean diameter, basal area, minimum and maximum diameters etc. Main results: The regression model using arithmetic mean diameter as an independent variable and the model using maximum diameter of the stand as an independent variable were found superior for estimation of scale and shape parameters, respectively. Highlights: While the Weibull distributions determined by both methods give close results, the method of determining the distribution parameters with the developed regression models seems to be superior in terms of examining the diameter distribution changes according to different stand structure simulations.

A review and assessment of importance sampling methods for reliability analysis

This paper reviews the mathematical foundation of the importance sampling technique and discusses two general classes of methods to construct the importance sampling density (or probability measure) for reliability analysis. The paper first explains the failure probability estimator of the importance sampling technique, its statistical properties, and computational complexity. The optimal but not implementable importance sampling density, derived from the variational calculus, is the starting point of the two general classes of importance sampling methods. For time-variant reliability analysis, the optimal but not implementable stochastic control is derived that induces the corresponding optimal importance sampling probability measure. In the first class, the optimal importance sampling density is directly approximated by a member of a family of parametric or nonparametric probability density functions. This approximation requires defining the family of approximating probability densities, a measure of distance between two probability densities, and an optimization algorithm. In the second class, the approximating importance sampling density has the general functional form of the optimal solution. The approximation amounts to replacing the limit-state function with a computationally convenient surrogate. The paper then explores the performances of the two classes of importance sampling methods through several benchmark numerical examples. The challenges and future directions of the importance sampling technique are also discussed.

Evaluation of Eleven Numerical Methods for Determining Weibull Parameters for Wind Energy Generation in the Caribbean Region of Colombia

The two-parameter Weibull probability density function (PDF) is widely utilized by different researchers and engineers to fit wind speed data for statistical analysis and modeling. The characterization of wind resources in the frequency and probability domain is necessary to estimate the power output potential of new wind energy projects. Considering that exist a variety of Weibull equations evidenced in the literature review, this article evaluates 11 different methods to calculate the shape and scale parameters of the Weibull PDF. In this sense, it was written an algorithm within a Matlab function that solves the 11 methods for calculating the Weibull PDF parameters. Wind speed data extracted from the ERA5 database was used as input data for applying the proposed algorithm, and statistical parameters such as the Root Mean Square Error (RMSE), the Relative Root Mean Square Error (RRMSE), and chi-square test (X2) we utilized for assessing the performance of each one of the 11 methods for modeling the wind distribution. The statistical results pointed that the numerical iteration methods (e.g. maximum likelihood method) showed better results than parameterized equations such as the Graphical Method, hence, this research recommends the implicit methods for determining Weibull PDF parameters of wind speed data.

Estimation of Hydropower Potential Using Bayesian and Stochastic Approaches for Streamflow Simulation and Accounting for the Intermediate Storage Retention

Hydropower is the most widely used renewable power source worldwide. The current work presents a methodological tool to determine the hydropower potential of a reservoir based on available hydrological information. A Bayesian analysis of the river flow process and of the reservoir water volume is applied, and the estimated probability density function parameters are integrated for a stochastic analysis and long-term simulation of the river flow process, which is then used as input for the water balance in the reservoir, and thus, for the estimation of the hydropower energy potential. The stochastic approach is employed in terms of the Monte Carlo ensemble technique in order to additionally account for the effect of the intermediate storage retention due to the thresholds of the reservoir. A synthetic river flow timeseries is simulated by preserving the marginal probability distribution function properties of the observed timeseries and also by explicitly preserving the second-order dependence structure of the river flow in the scale domain. The synthetic ensemble is used for the simulation of the reservoir water balance, and the estimation of the hydropower potential is used for covering residential energy needs. For the second-order dependence structure of the river flow, the climacogram metric is used. The proposed methodology has been implemented to assess different reservoir volume scenarios offering the associated hydropower potential for a case study at the island of Crete in Greece. The tool also provides information on the probability of occurrence of the specific volumes based on available hydrological data. Therefore, it constitutes a useful and integrated framework for evaluating the hydropower potential of any given reservoir. The effects of the intermediate storage retention of the reservoir, the marginal and dependence structures of the parent distribution of inflow and the final energy output are also discussed.

Data-driven approach for post-earthquake condition and reliability assessment with approximate Bayesian computation

Estimating the reliability of civil structures after earthquake events is a challenging problem because the uncertainties of random variables are usually unknown. This paper presents a data-driven approach for post-earthquake reliability assessments of civil structures. The measured vibration data are used to update the probability density functions of random variables. Two approximate Bayesian computation techniques are introduced to generate the posterior probability density functions of structural parameters. One is the likelihood-free Bayesian inference method, in which the weighted sum of squared errors term in the likelihood function is approximated by a Gaussian process model and the posterior probability density functions is generated using the Metropolis–Hastings algorithm. The second method is the variational inference approach, in which the posterior probability density function is approximated by a Gaussian mixture model. Both the proposed techniques can provide an approximation of the posterior probability density functions of the unknown parameters. Furthermore, the updated probability density functions are applied for reliability assessments. Numerical studies on a steel frame structure and a reinforced concrete structure are conducted to verify the accuracy and efficiency of the proposed techniques. Results show that the posterior probability density functions of unknown structural parameters can be updated and the reliability of the structure and components can be estimated.

A probabilistic approach for determining Monte Carlo beam source parameters: II. Impact of beam modeling uncertainties on dosimetric functions and treatment plans

Objective. The Monte Carlo method is recognized as a valid approach for the evaluation of dosimetric functions for clinical use. This procedure requires the accurate modeling of the considered linear accelerator. In Part I, we propose a new method to extract the probability density function of the beam model physical parameters. The aim of this work is to evaluate the impact of beam modeling uncertainties on Monte Carlo evaluated dosimetric functions and treatment plans in the context of small fields. Approach. Simulations of output factors, output correction factors, dose profiles, percent-depth doses and treatment plans are performed using the CyberKnife M6 model developed in Part I. The optimized pair of electron beam energy and spot size, and eight additional pairs of beam parameters representing a 95% confidence region are used to propagate the uncertainties associated to the source parameters to the dosimetric functions. Main results. For output factors, the impact of beam modeling uncertainties increases with the reduction of the field size and confidence interval half widths reach 1.8% for the 5 mm collimator. The impact on output correction factors cancels in part, leading to a maximum confidence interval half width of 0.44%. The impact is less significant for percent-depth doses in comparison to dose profiles. For these types of measurement, in absolute terms and in comparison to the reference dose, confidence interval half widths less than or equal to 1.4% are observed. For simulated treatment plans, the impact is more significant for the treatment delivered with a smaller field size with confidence interval half widths reaching 2.5% and 1.4% for the 5 and 20 mm collimators, respectively. Significance. Results confirm that AAPM TG-157's tolerances cannot apply to the field sizes studied. This study provides an insight on the reachable dose calculation accuracy in a clinical setup.

Risk assessment of sectional water quality based on deterioration rate of water quality indicators: A case study of the main canal of the Middle Route of South-to-North Water Diversion Project

The Middle Route of South-to-North Water Diversion Project of China (MRSNWDPC) is the most important inter-basin water diversion project. This study screened out 8 water quality indicators, including pH, ammonia nitrogen (NH3-N), fecal coliform (F. coli), dissolved oxygen (DO), permanganate index (PI), five-day biochemical oxygen demand (BOD5), fluoride (F-), sulfate (SO42-), which were from monthly monitoring data of 29 sections between January 2016 and December 2018, and analyzed the probability distribution characteristics of water quality deterioration rate (WQDR) samples, compared the advantages of three kinds of probability density function parameter optimization method, then the probability density model of each water quality indicator is constructed. By defining the section water quality risk and setting the threshold, the risk level of harmful water quality events (“ degradation ”or“ standard exceedance ”) in the next one month was calculated based on the real-time monitoring data. The main conclusions include: (1) The probability density of WQDR samples in the MRSNWDPC basically corresponded to the p-III distribution, and the optimum line method can be used as the parameter optimization method of the distribution function. (2) The water quality risk assessment method proposed in this paper could calculate sectional risk based on real-time monitoring data of water quality and play an early warning role in the occurrence of harmful water quality events in the next one month. (3) From 2016 to 2018, water quality deterioration existed in the MRSNWDPC, and the probability of pH, DO and BOD5 with higher risk level was higher than other indicators. However, further research is needed to confirm whether a stable trend of water quality deterioration exists. This study redefines the concept of risk of sectional water quality from the perspective of the probability of occurrence of harmful events and sets the threshold value of risk level in combination with the water body function, so as to predict the future water quality risk level in the next month, which plays a good warning role in the water quality management of the MRSNWDPC and can help relevant management departments to make response plans in advance. It is worth mentioning that the method proposed in this paper has been applied to the water quality monitoring and early warning integrated management platform of the main canal of the MRSNWDPC.

Bayesian Operational Modal Analysis with Genetic Optimization for Structural Health Monitoring of the Long-Span Bridge

Modal parameters, including natural frequencies, damping ratios, and mode shapes, are vital for evaluating the in-service condition of long-span bridges subjected to performance degradation or damages. Operational modal identification techniques requiring only ambient vibration responses are usually utilized to identify the modal parameters of in-service bridges. Due to the measurement incompleteness and modeling errors, uncertainties would inevitably be involved in the identification. The changing operational loads of long-span bridges, e.g. vehicles and winds, could also increase uncertainties. These uncertainty factors could decrease the precision of the operational modal identification and undermine the reliability of structural health monitoring (SHM) for long-span bridges. This study introduces an improved Bayesian modal identification approach using the scaled Fast Fourier Transform data for uncertainty quantification. The proposed method integrates the genetic optimization with the posterior probability density function of modal parameters to ensure the accuracy and robustness of the iteration process. Additionally, an asymptotic estimate interval with the assumption of a high signal-to-noise ratio is defined to improve the computational efficiency. The acceleration responses from a numerical example and a full-scale long-span bridge are adopted to validate the performance of the proposed method. Results show that the improved Bayesian approach can accurately identify the modal parameters and efficiently quantify the uncertainties. The method enhances the reliability of modal tracking for SHM of operational long-span bridges.

Electromagnetic Multi–Gaussian Speckle

Generalizing our prior work on scalar multi-Gaussian (MG) distributed optical fields, we introduce the two-dimensional instantaneous electric-field vector whose components are jointly MG distributed. We then derive the single-point Stokes parameter probability density functions (PDFs) of MG-distributed light having an arbitrary degree and state of polarization. We show, in particular, that the intensity contrast of such a field can be tuned to values smaller or larger than unity. We validate our analysis by generating an example partially polarized MG field with a specified single-point polarization matrix using two different Monte Carlo simulation methods. We then compute the joint PDFs of the instantaneous field components and the Stokes parameter PDFs from the simulated MG fields, while comparing the results of both Monte Carlo methods to the corresponding theory. Lastly, we discuss the strengths, weaknesses, and applicability of both simulation methods in generating MG fields.

Normal Inverse Gaussian Features for EEG-Based Automatic Emotion Recognition

Electroencephalography (EEG)-based emotion recognition is crucial in the domain of Human-Computer Interaction (HCI), which gained significant attention in recent years. However, the non-stationarity and chaotic nature of the EEG signals pose challenges and restrict the state-of-the-art techniques from precisely identifying distinct emotional states from the EEG data and hence offer limited emotion recognition performance. To capture the underlying non-linear characteristics of EEG, this study employed a novel local mean decomposition (LMD) algorithm which decomposes EEG signals into product functions (PFs). Further PFs are modeled by Normal Inverse Gaussian (NIG) probability density function (PDF) parameters. Thus these PDF features are fed to an optimized Adaboost classifier developed with the help of a cross-validation approach. The novelty of the work lies in the NIG modeling of LMD domain PFs to identify specific emotions from the EEG signals. The significance of the NIG parameters is illustrated by qualitative, pictorial, and statistical analyses. To assess the efficiency of the proposed approach, intensive experiments are conducted on open-source datasets, SJTU Emotion EEG Dataset (SEED), SEED-IV, and Database for Emotion Analysis of Physiological Signals (DEAP). The emotion recognition performance is evaluated in terms of heat maps, receiver operating characteristics (ROC), and accuracy. The proposed emotion recognition system outperformed the state-of-art methods and achieved a maximum accuracy of 97.3%, 98%, and 98.6% with the cross-validation approach and 93.23%, 94.87%, and 95.58% for the cross-subject validation approach using DEAP and SEED, SEED-IV datasets, respectively.

Robust Sliding Window CFAR Detection Based on Quantile Truncated Statistics

In this paper, the concept of quantile is introduced and elaborately related to the truncation depth, based on which quantile truncated statistics (QTS) is put forward. The QTS gives a reasonable explanation of truncation depth and makes the selection of truncation depth well-founded and controllable. In addition, maximum likelihood estimation based on QTS (QTS-MLE) for the probability density function (PDF) parameters is derived. We start the analysis from Weibull background assuming that the shape parameter is known, and then extend it to the case where the shape parameter is unknown. By analyzing the variance and mean square error (MSE) of the estimated parameters in Weibull background, it is found that QTS-MLE has better estimation performance than the MLE based on truncated statistics (TS-MLE). On this basis, the constant false alarm rate (CFAR) detector based on QTS-MLE, i.e. QTS-CFAR, is proposed. The analytic expressions of the false alarm rate and detection probability of QTS-CFAR are derived under the Weibull background with known shape parameter. The full CFAR characteristics of TS- and QTS-CFAR detectors in Weibull background with unknown shape parameter are proved by invariant theory. Monte Carlo simulations show that QTS-CFAR detector has better anti-interference performance and false alarm control ability in multiple-target environment. Furthermore, the superiority of QTS-CFAR detector is verified by the real data collected by skywave over-the-horizon radar. Finally, we present the expression of QTS-MLE for the scale parameter in the Gamma background.

State of health estimation of lithium-ion batteries based on the regional frequency

The state of health (SOH) estimation of lithium-ion batteries at low data sampling frequencies has important practical significance in engineering application. This paper proposes a new SOH evaluation method for lithium-ion batteries under the framework of probability density function (PDF). The concepts of regional voltage and regional frequency are introduced, and the SOH models are established based on the experimental cycle data of the nickel-cobalt-aluminium (NCA) battery cell and lithium iron phosphate (LFP) battery module. The results show that the SOH is a simple linear function of the regional frequency in the obtained model, and the fitting R -squared ( R 2 ) of two battery SOH models can reach more than 0.99 with appropriate regional voltage. This method does not directly use the characteristic parameters of PDF and is insensitive to the sampling frequency as low as 1/60 Hz. It is effective for accurate online SOH evaluation in occasions where the sampling frequency cannot be too high, such as large energy storage power station. Moreover, this method is more suitable for module SOH evaluation, which can overcome the poor SOH evaluation caused by the inconsistency of cells in a module. • An improved PDF method for SOH evaluation is proposed based on regional frequency. • The regional frequency method can be used at the sampling frequency as low as 1/60 Hz. • The regional frequency method is suitable for SOH modeling of cells and modules. • A good linear positive correlation between battery SOH and regional frequency.