Empirical Probability Density Function Research Articles

Visualisation and outlier detection for probability density function ensembles

AbstractExploratory data analysis (EDA) for functional data—data objects where observations are entire functions—is a difficult problem that has seen significant attention in recent literature. This surge in interest is motivated by the ubiquitous nature of functional data, which are prevalent in applications across fields such as meteorology, biology, medicine and engineering. Empirical probability density functions (PDFs) can be viewed as constrained functional data objects that must integrate to one and be nonnegative. They show up in contexts such as yearly income distributions, zooplankton size structure in oceanography and in connectivity patterns in the brain, among others. While PDF data are certainly common in modern research, little attention has been given to EDA specifically for PDFs. In this paper, we extend several methods for EDA on functional data for PDFs and compare them on simulated data that exhibit different types of variation, designed to mimic that seen in real‐world applications. We then use our new methods to perform EDA on the breakthrough curves observed in gas transport simulations for underground fracture networks.

Analysis of 5.8 GHz Network for Line of Sight (LOS) and Non-Line of Sight (NLOS) in Suburban Environment

This paper presents the findings of radio wave characterization based on the measurement data at 5.8 GHz. The measurement data were collected by a testbed channel, which links with the following scenarios: a single tree, a row of trees, a row of trees and a road, a row of trees, a road, and a building. These experiments were conducted at University Teknologi Malaysia (UTM) Skudai, Johor to represent the suburban environment. The links consist of pairs of transmitting and receiving antennas that deploy the path of a line of sight (LOS) and non-line of sight (NLOS) radio propagation wave networks. Based on the measurement data analysis, the general issue concerning the statistical probability distribution and the characteristics of LOS and NLOS are examined and discussed. Note that 5.8 GHz technology can be used in both LOS and NLOS scenarios, but its performance varies based on the presence of obstacles and signal propagation characteristics. Other prominent experimental analysis methods, such as hypothesis testing and goodness of fit tests, are implemented to consolidate the findings. The analysis found that the empirical probability density function of LOS and NLOS channels follows Gaussian, Rayleigh, and Rician distribution. Predicting specific future technological developments, such as the availability of 5.8 GHz technology, is challenging because it depends on various factors, including research and development efforts, regulatory decisions, market demand, and technological advancements.

Analysis of 5.8 GHz Network for Line of Sight (LOS) and Non-Line of Sight (NLOS) in Suburban Environment

This paper presents the findings of radio wave characterization based on the measurement data at 5.8 GHz. The measurement data were collected by a testbed channel, which links with the following scenarios: a single tree, a row of trees, a row of trees and a road, a row of trees, a road, and a building. These experiments were conducted at University Teknologi Malaysia (UTM) Skudai, Johor to represent the suburban environment. The links consist of pairs of transmitting and receiving antennas that deploy the path of a line of sight (LOS) and non-line of sight (NLOS) radio propagation wave networks. Based on the measurement data analysis, the general issue concerning the statistical probability distribution and the characteristics of LOS and NLOS are examined and discussed. Note that 5.8 GHz technology can be used in both LOS and NLOS scenarios, but its performance varies based on the presence of obstacles and signal propagation characteristics. Other prominent experimental analysis methods, such as hypothesis testing and goodness of fit tests, are implemented to consolidate the findings. The analysis found that the empirical probability density function of LOS and NLOS channels follows Gaussian, Rayleigh, and Rician distribution. Predicting specific future technological developments, such as the availability of 5.8 GHz technology, is challenging because it depends on various factors, including research and development efforts, regulatory decisions, market demand, and technological advancements.

Data-driven stochastic spectral modeling for coarsening of the two-dimensional Euler equations on the sphere

A resolution-independent data-driven subgrid-scale model in coarsened fluid descriptions is proposed. The method enables the inclusion of high-fidelity data into the coarsened flow model, thereby enabling accurate simulations also with the coarser representation. The small-scale model is introduced at the level of the Fourier coefficients of the coarsened numerical solution. It is designed to reproduce the kinetic energy spectra observed in high-fidelity data of the same system. The approach is based on a control feedback term reminiscent of continuous data assimilation implemented using nudging (Newtonian relaxation). The method relies solely on the availability of high-fidelity data from a statistically steady state. No assumptions are made regarding the adopted discretization method or the selected coarser resolution. The performance of the method is assessed for the two-dimensional Euler equations on the sphere for coarsening factors of 8 and 16 times. Applying the method at these significantly coarser resolutions yields good results for the mean and variance of the Fourier coefficients and leads to improvements in the empirical probability density functions of the attained vorticity values. Stable and accurate large-scale dynamics can be simulated over long integration times and are illustrated by capturing long-time vortex trajectories.

Stress changes during exhumation of a high-pressure nappe (Phyllite-Quartzite unit, Greece): Insights from quartz microstructures and recrystallized grain size statistics

In this study, we carried out microstructural and grain size analysis of recrystallized quartz in quartz-rich samples collected from different structural levels of a 1.5–2.0 km thick high-pressure nappe (Phyllite-Quartzite unit, Greece). Exhumation-related deformation led to recrystallization of quartz by subgrain rotation (SGR) and the formation of different generations of recrystallized quartz with different grain sizes. Applying a new, fully automated histogram-based method, we reconstructed the “hidden” empirical probability density functions included in the histograms of the initial grain size distributions. Statistical analysis showed that most grain-size histograms display bimodal distributions with mean grain sizes in cluster A ranging from 46 to 48 μm, and a cluster B with mean grain sizes between 81 and 93 μm. Few grain size histograms display trimodal distributions and show an additional cluster C with mean grain sizes ranging from 100 to 115 μm. Microstructural observations indicate that the grains of cluster C are the oldest whereas those of cluster A are the youngest. Bimodal and trimodal distributions may reflect an episodic increase in the differential stress from 17 to 19 MPa and then to 31 MPa during exhumation, accompanied by a strain rate increase of at least one order of magnitude (from 10−13 s−1 to 10−12 s−1). This stress increase likely occurred at the early stages of the exhumation and may have lasted less than 0.5 Myr. Stress increase was followed by a stress drop expressed by partial foam structures of quartz in a few samples.

Replacing Histogram with Smooth Empirical Probability Density Function Estimated by K-Moments

Whilst several methods exist to provide sample estimates of the probability distribution function at several points, for the probability density of continuous stochastic variables, only a gross representation through the histogram is typically used. It is shown that the newly introduced concept of knowable moments (K-moments) can provide smooth empirical representations of the distribution function, which in turn can yield point and interval estimates of the density function at a large number of points or even at any arbitrary point within the range of the available observations. The proposed framework is simple to apply and is illustrated with several applications for a variety of distribution functions.

Probabilistic Interval Predictor Based on Dissimilarity Functions

This work presents a new methodology to obtain probabilistic interval predictions of a dynamical system. The proposed strategy uses stored past system measurements to estimate the future evolution of the system. The method relies on the use of dissimilarity functions to estimate the conditional probability density function of the outputs. A family of empirical probability density functions, parameterized by means of two scalars, is introduced. It is shown that the proposed family encompasses the multivariable normal probability density function as a particular case. We show that the presented approach constitutes a generalization of classical estimation methods. A validation scheme is used to tune the two parameters on which the methodology relies. In order to prove the effectiveness of the presented methodology, some numerical examples and comparisons are provided.

Reducing the dimensionality of data using tempered distributions

We reformulate unsupervised dimension reduction problem (UDR) in the language of tempered distributions, i.e. as a problem of approximating an empirical probability density function by another tempered distribution, supported in a k-dimensional subspace. We show that this task is connected with another classical problem of data science — the sufficient dimension reduction problem (SDR). In fact, an algorithm for the first problem induces an algorithm for the second and vice versa.In order to reduce an optimization problem over distributions to an optimization problem over ordinary functions we introduce a nonnegative penalty function that “forces” the support of the model distribution to be k-dimensional. Then we present an algorithm for the minimization of the penalized objective, based on the infinite-dimensional low-rank optimization, which we call the alternating scheme. Also, we design an efficient approximate algorithm for a special case of the problem, where the distance between the empirical distribution and the model distribution is measured by Maximum Mean Discrepancy defined by a Mercer kernel of a certain type. We test our methods on four examples (three UDR and one SDR) using synthetic data and standard datasets.

Experimental study of molten bubble layer growth and bubble size distribution in post‐flame‐spread PMMA samples

AbstractAlthough thermoplastics possess many desirable traits as solidified materials, they are combustible and burn readily. Post burning samples of polymethylmethacrylate (PMMA) are the focus of this study. During flame spread, PMMA forms a bubble containing molten surface layer that influences the spread rate and the surface mass efflux. This study examines the formation and distribution of bubbles inside a PMMA sample that has previously been subjected to flame spread and then re‐hardened into its solid state. Experiments discussed herein were conducted in a Narrow Channel Apparatus (NCA), which reduces the influences of gravitation (buoyancy) on flame spread. Bubble sizes and counts were determined using digital image analysis (DIA). The burnt samples were analyzed by dividing sample images into eight equal area sections. Frequency distributions of bubble size (area) were compared. Distribution functions were fitted against the empirical probability density function (PDF) for the bubble size distribution. The log‐normal distribution predicts the bubble size distribution for all segments. The bubble distribution function can be used to describe physical processes inside the polymeric material as it undergoes thermal transformation by the spreading flame.

Atlas of surface currents in the Mediterranean and Canary–Iberian–Biscay waters

ABSTRACT Sea surface currents probably are the most relevant essential ocean variable affecting diverse societal challenges concerning the marine environmental (as, for example, safe and efficient navigation, marine pollution and ecological connectivity). This work introduces a climatological Atlas (monthly resolution) of currents in the Mediterranean and Canary–Iberian–Biscay basins, based on today's state of the art reanalyses of the ocean circulation. The focus is on surface and subsurface reanalyses (here understood as and m, respectively) provided by the Copernicus Marine Environment Monitoring Service (CMEMS). The climatological values are computed from the median of the empirical probability density functions and the Atlas also includes the variance matrix and a bimodality index to have quantitative information on their variability. For both domains, the subsurface climatological fields are reasonably consistent with circulation schemes proposed in the previous literature but clearly improving the time and space resolution of the emerging patterns. For the Canary–Iberian–Biscay domain, the monthly climatological surface currents capture accurately the characteristic seasonal signal and its transition between a favourable and non-favourable upwelling regime. In the Mediterranean basin, differences between the near-surface and the 15 m velocity fields suggest a non-negligible role of winds over the variability of the uppermost ocean layer, specially in the Eastern Mediterranean basin. This is, up to our knowledge, the first time that such near-surface climatological patterns are computed. It has been found that, in general, the resulting patterns agree with surface drifter trajectories. In several regions, interannual variability foster bimodal and multimodal probability distributions. The Atlas has been conceived with the purpose of providing a first quantitative assessment on the surface circulation, thus being a complementary tool of real-time ocean forecasting systems. The Atlas is distributed following the FAIR principles and is accompanied with a digital version, with enhanced visualization capabilities for both research and assessment.

Probabilistic damage localization by empirical data analysis and symmetric information measure

This article aims to address some challenges in data-based damage localization by proposing innovative methods including a hybrid algorithm for feature extraction, a symmetric information measure for feature classification, and a probabilistic approach to threshold estimation. The hybrid feature extraction combines an autoregressive (AR) model with a novel non-parametric estimator of probability density function (PDF) under empirical data analysis. The great novelty of this method is to propose a new probabilistic feature as an empirical PDF of the AR coefficients. The proposed information measure is a symmetric divergence aiming at addressing the main limitation of the classical Kullback-Leibler divergence regarding its non-symmetric characteristic. Finally, the proposed probabilistic approach exploits the concept of Markov Chain Monte Carlo to estimate a trustworthy threshold for locating damage. Vibration responses of two civil structures are used to verify the proposed methods with several comparisons. Results confirm that the methods are successful in identifying damage.

Local resolution characteristics of atmospheric turbulence.

D. L. Fried's concept of a "Lucky Image" for turbulent image streams can be seen as creating different degrees of localized resolution in images. These localized regions of resolution can be derived from Fried's equation for the probability of obtaining a Lucky Image. The existence of local resolution variations when imaging through turbulence also implies local variations in the point-spread-functions (PSFs) caused by turbulence. We characterize these local variations by using simple measures on PSFs collected in the presence of atmospheric turbulence. We also compile these variations into an empirical probability density function (PDF) that describes the different resolutions in local regions of turbulent imagery and can be used to characterize specific conditions of turbulence, e.g., the coherence diameter.

A Modified Discretized Chaotic Map and Its Generated Pseudo Binary Random Number

In this manuscript, a modified discretized chaotic map is developed which generates a pseudo-random binary number. This Modified Discretized Chaotic Map (MDCM) not only enlarges the range of system parameter of conventional chaotic map but also improves its performance in terms of key space. The numbers generated from MDCM are random in nature and their randomness is investigated with the help of empirical Probability Density Function (PDF) and Lyapunov Exponent. To investigate the randomness of the numbers, these numbers are quantized into two levels and encoded into 1-bit pseudo random binary numbers and then their randomness behavior is investigated by NIST's (National Institute of Standards and Technology) randomness testing methods. The proposed MDCM is compared with the conventional chaotic map in terms of their Lyapunov Exponent. The numbers generated from MDCM can be used not only in various applications of image processing such as image cryptography, image encryption and others, but also applicable where pseudo numbers are necessarily required.

A Laplacian-Gaussian Mixture Model for Surface EMG Signals from Upper Limbs.

Existing literature suggests that the probability density function (pdf) of surface Electromyography (sEMG) signals follows either a Gaussian or Laplacian model. In this paper, a Laplacian-Gaussian mixture model is proposed for the EMG signals extracted from the upper limbs. The model is validated using both quantitative and qualitative perspectives. Specifically, for a benchmark dataset, the Kullback-Leibler (KL) divergence is computed between the proposed model and the histogram based empirical probability density function (mpdf). For a sample signal, a goodness of fit plot with R squared value and a visual comparison between the histogram based mpdf and the estimated pdf from the proposed model are presented. Moreover, the Expectation-Maximization (EM) algorithm is derived for the estimation of the parameters of the proposed mixture model. The weight of the Laplacian component is computed for each of the signals from a benchmark dataset. It has been empirically determined that the Laplacian component has a major contribution to the mixture.

Dose Correlation of Panax ginseng and Atractylodes macrocephala Koidz. Drug Pairs in the Chinese Medicine Prescription Based on the Copula Function.

ObjectivePanax ginseng and Atractylodes macrocephala Koidz. (AMK) are widely used in treating various diseases; however, research is insufficient on measuring the relationship that exists by combining this drug pair using the copula function. Methods In this study, 279 traditional Chinese medicine prescriptions containing the Panax ginseng and AMK drug pair were extracted from the prescription database for three types of screened indications, namely, diabetes mellitus, diarrhea, and insomnia. Following the principle of dose conversion, each dynasty unit was uniformly converted into a modern unit. Then, the kernel density distribution of Panax ginseng and AMK was fitted with their empirical distribution functions. Finally, the optimal copula function was selected from the copula function family as a t-copula function. Results The empirical distribution and probability density functions of Panax ginseng and AMK were obtained. From the results, their Kendall rank correlation coefficients with indications of diabetes mellitus, insomnia, and diarrhea were 0.8689, 0.7858, and 0.7403, whereas their Spearman rank correlation coefficients were 0.9563, 0.9276, and 0.8958. Results also indicated that the use of the t-copula function can better reflect the correlation between Panax ginseng and AMK doses. Conclusion From the three indications, the dose between Panax ginseng and AMK was positively correlated. This study, therefore, confirms the medicinal principle of Chinese medicine “combining” from the perspective of mathematical statistics. Results from this study are practical to evaluate the correlation between the drug pair doses, Panax ginseng and AMK, using the copula function model, which provides a new model for the scientific explanation of compatibility for Chinese medicines. This study also provides a methodological basis for more drug measurement studies and clinical medications.

Continuous-time autoregressive models excited by semi-Lévy process for cyclostationary signal analysis

In non-stationary signal modeling, it is critical to find an appropriate theoretical model with proper properties for analysing data. Models with cyclostationary (CS) property are prominent in signal processing analysis. In this paper, we extend the stationary Lévy driven continuous-time autoregressive (CAR) model to the semi-Lévy driven CAR (SL-CAR) model, which we show that has CS property. This model provides a platform for modeling non-stationary continuous-time processes. Using sample spectral coherence test, the CS behavior of simulated data from the SL-CAR(2) model is shown. Estimation of the parameters is followed by fitting VAR-GARCH process to the discretized state process and minimizing the distance between the covariance matrices of corresponding noise vectors. Numerical simulations provide evidence that the estimators are consistent, which is followed by computing the mean square errors. By using equally spaced sampling, the modes of empirical probability density functions of the estimators converge to the true values of the parameters when the sample size increases. Finally, we apply the SL-CAR(2) for modeling electrocardiogram biomedical signals and evaluate its performance.

Advanced Data Mining for Odour Emissions Monitoring: Experimental Peak-to-mean Calculations and Spectral Analysis of Data Derived from Ioms in Two Waste Treatment Plants

A landfill for non-hazardous waste and a composting plant were equipped with an integrated odour management system made up of instrumental Odour monitoring systems and automated air samplers. Collected data by the instrumental system cover a time span of 8 months and 9 months for the two plants, respectively, and have been used for investigating spatial distribution and statistical properties of peak-to-mean ratios; moreover, a new method of data analysis, not yet employed in the field of odour dispersion modelling, has been proposed. A peak-to-mean value is defined as the ratio between an hourly mediation time and the average duration of a single human breath (few seconds). Data derived from 0.033 Hz sampling frequency by IOMS were used to calculate peak-to-mean for the monitoring period for both plants; the size of samples analysed allowed the implementation of a statistical calculation methodology for peak-to-mean estimation, based on the construction of empirical probability density functions (PDF). Moreover, data collected from instruments installed in different areas of the plants, were used to evaluate how PDF of peak-to-mean values varies with the distance from the odour emission sources. The same data were also used to develop a new method of analysis based on time series processing through the use of the Fourier Transform. Spectral analysis, sometimes used for meteorological variables, more recently has been applied in the study of air pollution data, although this type of analysis makes a decisive contribution to the understanding of cyclical behaviour in the observed data and provides information on temporal and spatial scales of such mechanisms. The aim is therefore to define a viable methodology to time series of odour concentration data collected at the fence of a plant, to be used to gather information on periodic odour variations. Interesting results, which related some periodic behaviours in odour emissions to working periods and composting phases, were observed, demonstrating the usefulness of such methodology to assess whether odour fluctuations are caused by atmospheric conditions variability or plant management operations.

Simulation and Characteristics Analysis of On-Shore OWC System Proposal as Distributed Generation Resource Considering the Irregular Wave Interaction

This article presents the simulation and characterization of an on-shore oscillating water column (OWC) system as part of a distributed generation network considering the irregular interaction of sea waves. The main issue is the adequate calculation of the power generated considering the real variations of the sea waves, employing the stochastic analysis of the wave height and period. The characterization of the wave height was carried out using the Fisher-Tippett Type 1 function, and for the wave period, an empirical probability density function to obtain the instantaneous and accumulated power in an annual period. A basic on-shore OWC system was proposed with different physical dimensions. The theoretical and numerical results present a very similar performance for both turbines (600 W and 25 kW) analyzed. Regarding the 600 W turbine, the resulting accuracy is ≈94.5%, which implies that the annual generated power is 3.13 ± 1.02 MWh/year and the overall efficiency is 23.51% ± 1.9%. However, due to the reduced power generated, the chamber dimensions were modified, achieving 160.61 ± 9.99 MWh/year with an accuracy of ≈93.2%, based on an installed power capacity proposal using a 25 kW turbine. Also, the average overall efficiency for both turbines considering the irregular wave interaction is ≈23.5% and ≈21.1% for 600 W and 25 kW turbines, respectively.

Peak value estimation for wind-induced lateral accelerations in a high-rise building

This paper discusses the uncertainty when estimating extreme values of wind-induced lateral accelerations in a high-rise building based on wind-tunnel measurements. The acceleration signals for an aeroelastic scale model under ten different velocities and three different wind angles were processed to evaluate the extreme acceleration values. The empirical cumulative distribution function (CDF) and probability density function (PDF) trends of the peaks were estimated and compared with the analytical models, which showed satisfactory fits. An effort was made for the best fit for the empirical CDF through the numerical expansion of the peak set using Polynomial Chaos Expansion (PCE). It was confirmed that in this case, the lack of a reliable fit was not due to the number of peaks. In addition, analytical models of the Gaussian and non-Gaussian processes were applied to estimate the extreme values using the entire process and the sub-processes, and this paper compares and discusses the results. Finally, the variability of the extreme acceleration values estimated using a total of ten different methods is discussed.

Modelling fish physico-thermal habitat selection using functional regression

In this paper, a new fish habitat modelling approach is introduced using the full probability density functions (PDF), rather than single measurements or central tendency metrics, to describe each predictor. To model habitat selection using PDFs, functional regression models (FRM) are used to allow for the inclusion of curves or functions (smoothed empirical PDFs) in regression models compared to scalars or vectors in classical contexts. The benefits of FRM are exemplified by comparing results with those obtained using generalized additive models (GAM), one of the most recent and performing models in the field. Abundance of juvenile Atlantic salmon sampled at 26 sites (75 m-long x river width) of the Sainte-Marguerite River (Quebec, Canada) was modelled with PDFs of four potential predictors: flow velocity, water depth, substrate size and water temperature. The latter has been less frequently used in habitat modelling and the results showed that it was the most significant predictor. Overall, FRM explained more of the variability in habitat selection than GAM (+14.9% for fry and +8.1% for 1+ parr), mainly due to their ability to use complete distributions of the habitat variables rather than aggregated values (mean). A leave-one-out cross validation showed that both GAM and FRM had similar performance to predict fish abundance. The use of FRM in fish habitat modelling is innovative and its potential should be further developed, especially in the current context where habitat variables are becoming increasingly easy to obtain due to rapid progress of remote measurement techniques.