Fitted Probability Density Function Research Articles

On Sum and Geometric Sum of independent New Quasi Lindley Random Variables and its Applications

The Laplace transformation method is used to drive the distribution of the sum Sn of n-fixed random variables, which has a new quasi Lindley distribution with two parameters θ and α, NQLD (θ,α). The sum of NQLD (SUNQLD) distribution is obtained in pdf and cdf formats. It is discussed how to calculate the random sum SN of a random number of NQLD random variables. The random sum of the NQLD distribution's pdf and cdf are calculated. When N has a geometric distribution, the geometric sum of NQLD distribution (GSN QLD) was introduced as an example of a random number of NQLD random variables. For all cases, some statistical measures are determined. The distribution's parameters are estimated using the maximum likelihood method. To test the viability and efficiency of the proposed distributions SNQLD and GSNQLD, lifetime count data sets from acute myeloid leukaemia are fitted. The results should become accepted knowledge in the fields of probability theory and its allied sciences. In addition, the histogram, fitted probability density function (pdf), and P-P plots for the analyzed real data set are presented.

Improved Gaussian mixture model to map the flooded crops of VV and VH polarization data

Accurate and timely monitoring of flooded crop areas is crucial for disaster rescue and loss assessment. However, most flooded crop monitoring methods based on synthetic aperture radar (SAR) imagery were developed for rice, which is probably inappropriate for crops with complex canopy structures that strongly attenuate SAR signals. Additionally, these methods often rely on empirical thresholds and region-specific reference samples, limiting their reliability and applicability on a larger spatial scale. To address these issues, we developed a novel flooded crop mapping approach at a regional scale using Sentinel-1 time-series data and an unsupervised Gaussian Mixture Model (GMM). Our approach leverages a Flood Separability Index (FSI) derived from the fitted probability density function of flooded and non-flooded crop areas in a GMM. This allows us to overcome the limitations of manual input selection in previous studies. The multi-temporal GMM was constructed using the time-series images with optimal polarization to estimate the flooded crop extents on a regional scale. We also investigated the scattering mechanisms of three typical crop disaster structures within an agricultural landscape area. Our results indicate that the proposed multi-temporal GMM is robust in crop planting areas with complex canopy structures. The performance of both single-temporal and multi-temporal GMMs surpasses that of baseline methods such as Otsu and K-means. Compared with VV polarization, VH polarization exhibits greater potential for accurately mapping flooded crops in complex agricultural regions. Our approach does not require labeled samples or many predefined parameters, making it fast and feasible for mapping flooded crops with complex canopy structures in large spatial areas.

Characterizing the Anchoring Effects of Official Forecasts on Private Expectations

The paper proposes a method for simultaneously estimating the treatment effects of a change in a policy variable on a numerable set of interrelated outcome variables (different moments from the same probability density function). Firstly, it defines a non-Gaussian probability density function as the outcome variable. Secondly, it uses a functional regression to explain the density in terms of a set of scalar variables. From both the observed and the fitted probability density functions, two sets of interrelated moments are then obtained by simulation. Finally, a set of difference-in-difference estimators can be defined from the available pairs of moments in the sample. A stylized application provides a 29-moment characterization of the direct treatment effects of the Peruvian Central Bank’s forecasts on two sequences of Peruvian firms’ probability densities of expectations (for inflation -π- and real growth -g-) during 2004-2015.

Comparing Weibull distribution method and Gram–Charlier series method within the context of estimating low-occurrence strong wind speed of idealized building cases

Understanding low-occurrence strong wind speed (LOSWS) distributions at the pedestrian level is important. However, the robustness of the statistical methods for estimating LOSWSs under different conditions remains uncertain. In this study, the performances of the Weibull distribution method and Gram–Charlier series (GCS) method were compared. Their accuracies for isolated building and building array cases were analyzed. It was found that the constant peak factor (PF) of the Gaussian distribution showed acceptable accuracy only when the exceedance probability q = 10%. For q = 0.1%, the PF of the three-parameter Weibull distribution (3W) was more accurate than that of GCS-3rd. For fitting probability density functions, GCS-6th exhibited better flexibility than the two-parameter Weibull distribution (2W) and 3W. However, large leading terms leaded to oscillations at several points in GCS-6th. Regarding the estimation accuracy of LOSWSs, the 2W and 3W methods are superior to the GCS methods when the available orders of statistics are equal. Although the GCS methods showed higher accuracy than the 3W method in some regions, the oscillations at specific points in the GCS methods may lead to lower accuracy on average. The present findings can serve as an illuminating reference for further applications of these statistical methods.

Deep Learning-Based Channel Modeling for Free Space Optical Communications

Instead of conventional knowledge-driven modeling, this study introduced data-driven channel modeling methods based on deep learning (DL) in a free-space optical (FSO) communication system to achieve low complexity, less modeling difficulty, high fidelity, and differentiability. The FSO channel was modeled via three DL algorithms: generative adversarial network (GAN), bidirectional long short-term memory (BiLSTM), and Bayesian neural network (BNN). Consequently, their performance was comprehensively compared in terms of optical amplitude waveforms, Kullback-Leibler (KL) divergence, 95% confidence interval, data distribution, and fitted probability density functions. The results indicated that the GAN-based channel exhibited optimal performance, implying its suitability for learning the stochastic distribution of the FSO channel. In addition, the performance of GAN-based channels was examined under different conditions: turbulence intensities (weak, moderate, and strong), transmission distances (3, 4, and 5 km), launch powers (0, 10, and 20 dBm), modulation formats (on-off keying, pulse position modulation, and pulse amplitude modulation 4), and the adaptability to channel variation. The test results were satisfactory and confirmed the ability of the GAN to learn the approximate distribution function of the FSO channel. Thus, the primary aim was to prove that for FSO channels with statistical distribution characteristics, a data-driven DL model can not only serve as an effective surrogate model for specific scenarios but also provide a potential tool for future modeling methods.

Long-Distance Pollen Dispersal in Urban Green Roof and Ground-Level Habitats

Long-distance pollen dispersal is critical for gene flow in plant populations, yet pollen dispersal patterns in urban habitats such as green roofs have not been extensively studied. Pollen dispersal patterns typically are assessed either by fitting non-linear models to the relationship between the degree of pollen dispersal and distance to the pollen source (i.e., curve fitting), or by fitting probability density functions (PDFs) to pollen dispersal probability histograms (i.e., PDF fitting). Studies using curve fitting typically report exponential decay patterns in pollen dispersal. However, PDF fitting typically produces more fat-tailed distributions, suggesting the exponential decay may not be the best fitting model. Because the two approaches may yield conflicting results, we used both approaches to examine pollen dispersal patterns in the wind-pollinated Amaranthus tuberculatus and the insect-pollinated Solanum lycopersicum at two green roof and two ground-level sites in the New York (NY, United States) metropolitan area. For the curve fitting analyses, the exponential decay and inverse power curves provided good fits to pollen dispersal patterns across both green roof and ground-level sites for both species. Similar patterns were observed with the PDF fitting analyses, where the exponential or inverse Gaussian were the top PDF at most sites for both species. While the curve fitting results are consistent with other studies, the results differ from most studies using PDF fitting, where long-distance pollen dispersal is more common than we observed. These results highlight the need for further research to compare curve and PDF fitting for predicting pollen dispersal patterns. And, critically, while long-distance pollen dispersal may be an important component of overall pollen dispersal for A. tuberculatus and S. lycopersicum in both urban green roof and ground-level sites, our results suggest it potentially may occur to a lesser extent compared with plants in less-urban areas.

Space Target Anomaly Detection Based on Gaussian Mixture Model and Micro-Doppler Features

With the dramatic increase in human space activities, anomaly detection becomes an important issue in passive space target surveillance. In this article, an anomaly detection algorithm based on Gaussian mixture model and radar micro-Doppler features is proposed to detect the abnormal motion status of the space target. By coherent sampling and time-frequency analysis on the radar echo with additive white Gaussian noise corresponding to the normal motion statuses of the target, four micro-Doppler features are extracted and tested for normal distribution. Furthermore, the distribution of the multi-dimensional features and the corresponding parameters are fitted and estimated by Gaussian mixture model and expectation-maximization algorithm. Then, an anomaly detector is derived by solving for decision region using the fitted probability density function and a preset confidence level. Experimental results show that the average anomaly detection rate of the proposed method is 16.7%, 19.1%, and 34.0% higher than the one-class support vector machine, the convex hull, and the convolutional autoencoder-based methods, respectively.

Interhemispheric variability of the electron density and derived parameters by the Swarm satellites during different solar activity

With the data from the Swarm satellite mission, we study the variability of ionospheric plasma parameters for low and high solar activities. We focus on the electron density measured by Swarm and derived parameters and analyze the variability of these parameters in the contexts of the Northern and Southern hemispheres, specific latitudinal regions, and the solar activity level. We consider two time intervals: with high solar activity (HSA) from August 2014 to July 2015 and with low solar activity (LSA) from January to December 2018. We show that the electron density is described mainly by three probability density functions (PDF): exponentiated Weibull, lognormal, and chi distributions. These results with PDFs can be applied to the modeling or prediction of ionospheric parameters in different regions. The best fit of PDFs was obtained for low and mid-latitudes, while at high latitudes and in the polar caps, the double-peaked features of the distribution require the fit of multiple PDFs. The electron density (Ne) distribution at low latitudes follows more a lognormal distribution, while in the high latitude region, the chi distribution prevails. Different results were obtained for the rate of change of density index (RODI) with two fitting PDFs: lognormal or exponentiated Weibull, where the best fits are for high latitudes and polar caps. We demonstrate high variability in the electron density and derived parameters at low latitudes and in the polar caps. Comparing both hemispheres, we obtained higher values of these parameters during the solar minimum in the Southern hemisphere at high latitudes and polar caps, while for the Northern hemisphere, higher values were obtained at low latitudes. The dependence on the satellite’s altitude was also considered. The main patterns in the diurnal variation of parameters in different regions do not depend on the level of solar activity (which affects only the maximum values). The largest asymmetry between both hemispheres inNediurnal distribution was obtained for the polar cap regions. Here a 50% decrease inNewas observed in the Northern hemisphere during HSA in the early morning sector (04–07 Magnetic Local Time), which has not yet been observed in the Southern hemisphere. For the first time, such a global statistical characterization of the ionospheric plasma density based on the in situ data is presented.

A statistical homogenization approach for incorporating fiber aspect ratio distribution in large area polymer composite deposition additive manufacturing property predictions

Assessing the material stiffness of fiber reinforced polymer composites deposited in Large Area Additive Manufacturing (LAAM) is needed to define the process-structure-property mapping for the LAAM technology. While the screw-extrusion-based LAAM systems yield a distribution of fiber aspect ratio (i.e., length to diameter ratio for cylindrical inclusions) within deposited beads, most composite micromechanical models ignore the fiber geometry variation and instead assume a single value of fiber aspect ratio. This paper presents a statistics-based homogenization approach for including the fiber aspect ratio distribution in the prediction of the elastic properties of an extruded polymer composite bead. The fiber length distribution of a 13 wt% Carbon Fiber reinforced Acrylonitrile Butadiene Styrene (CF-ABS) processed through a LAAM deposition system is measured using high resolution optical microscopy. The Weibull probability distribution function is employed to statistically describe the measured values. The fitted probability density function is then incorporated into a fiber orientation homogenization approach to compute the variability of the elastic properties of the extruded composite. Elastic properties predicted by our proposed method are shown to differ from those presented in prior studies that ignored fiber length variability. The fiber aspect ratio reduction from fiber length attrition during the LAAM single screw-extrusion is shown to decrease the predicted flow-direction effective elastic modulus by 7%. Elastic moduli computed using our measured fiber aspect ratio distribution and proposed homogenization approach compare well to reported data from previously associated experimental studies.

Dynamics of initial spacing on the diameter of hybrid Eucalyptus grandis x Eucalyptus urophylla in a systematic design

We evaluated the effect of initial spacing on the distribution of Eucalyptus grandis x Eucalyptus urophylla tree diameters in a Nelder wheel design. The study area was in west-central Brazil. A Nelder wheel design with three repetitions was used. This design provides 432 planting spots per plot/repetition. Planting density varied from 0.50 m² to 41.25 m². The diameter at breast height was measured for all plants every four months until 36 months of age, totaling eight measurements. The description of the diameter distribution was developed by fitting probability density functions for each spacing and age. The quality of fit was evaluated using the root mean square error percentage, Pearson’s correlation coefficient, and the Kolmogorov-Smirnov test (α < 0.05). The spacings up to 1.17 m² must be harvested in less than two years, since it is noted that in the third year the increment in diameter is static and the mortality rate increases. The spacing highly affects the diameter structure of the population, impelling the forester to consider different uses for the wood. For wood requiring larger diameters, spacing above 8.69 m² per tree is recommended

Streamflow regionalization for the Mortes River Basin upstream from the Funil Hydropower Plant, MG

Maximum and minimum streamflow are fundamental for water resource management, especially for water rights. However, lack of monitoring and scarce streamflow data limit such studies. Streamflow regionalization is a useful tool to overcome these limitations. The study developed models for regionalization of the maximum and minimum reference streamflows for the Mortes River Basin (MRB) (Water Resources Planning and Management Unit - GD2), Southern Minas Gerais State. The study used long-term streamflow historical series provided by the Brazilian National Water Agency (ANA). Previous exploratory analysis was performed, and it was observed that the streamflow series are stationary according to the Mann-Kendall test. The estimation of the streamflow for different return periods (RP) was performed by fitting Probability Density Functions (PDFs) that were tested by the Anderson-Darling (AD) test. The Generalized Extreme Values (GEV) and Wakeby were the most appropriate PDFs for maximum and minimum streamflows, respectively. The streamflow models were fitted using a power regression procedure, considering the drainage area of the watersheds as inputs. The fittings reached the coefficient of determination (R2) greater than 0.90. Thus, the streamflow regionalization models demonstrated good performance and are a potential tool to be used for water resource management in the studied basin.

Modeling the seasonality of extreme coastal water levels with mixtures of circular probability density functions

Understanding when floods occur is fundamental to reducing flood risk, yet depictions of flood seasonality rarely address two key issues: multiple seasons and the periodicity of seasonal data. We analyze the timing of extreme water levels at 47 presently operational tide gages located along the US coastlines with periods of record ranging from 29 to 118 years. Because the dates of extreme water levels are naturally multimodal periodic data, we model them with weighted mixtures of circular von Mises distributions. Fitting probability density functions on the circumference of a circle allow periodicity to be handled implicitly, while mixture models provide insight into the timing and importance of the various modes in the overall distribution. To our knowledge, this is the first application of circular statistical modeling to coastal water levels. We find clear spatial patterns in extreme water level seasonality along the US coasts. Gages located on the West Coast experience their main flood season in the winter and a secondary season in the summer, while those on the East and Gulf coasts have more uniform distributions, fewer seasons, and maximum flood likelihood in the late summer. Analysis of water level after removing tides and sea level change reduces the importance of secondary and higher seasons and yields greater uniformity compared with raw water levels. Finally, peaks-over-threshold datasets reveal secondary flood seasons that are obscured in annual maxima series, which may become increasingly impactful as sea levels rise. These findings are pertinent to coastal infrastructure design, risk sharing via financial instruments, and emergency preparedness and response.

Estimating the Outage Due to Polarization Dependent Loss Based on the Bit-Wise Achievable Information Rate for Probabilistically Shaped 64-QAM

The implication of per-span polarization dependent loss (PDL) on the statistical behaviour of the bit-wise achievable information rate (BW-AIR) is investigated for probabilistically shaped and uniform 32 Gbaud, dual-polarization 64-ary quadrature amplitude modulation. For constellation entropies ranging from 5.6 to 6 bits/symbol, and the number of spans ranging from 4 to 24, a simulation model that accurately captures the effect of PDL on both the signal and amplified spontaneous emission noise is used to characterize the statistical properties of the BW-AIR. Fitted probability density functions are employed to evaluate an outage probability based on a specified value of the normalized, generalized mutual information. A sensitivity metric is used to quantify a trade-off between the BW-AIR and resilience to PDL for cases of practical interest.

Characterizing the statistical distribution for transmission coefficient of turbulent optical orbital-angular-momentum channels.

The instantaneous transmission coefficient, i.e., instantaneous transmittance, of a turbulent optical orbital-angular-momentum (OAM) channel is mathematically formulated as a weighted integration and is found to range between 0 and 1. Common probability distribution models for optical irradiance fluctuations with a support from 0 to ∞ are not strictly proper for statistical description of the fluctuating transmission coefficient. The novel dual Johnson S B distribution is proposed to model the statistical behavior of the fluctuating transmission coefficient. Its applicability is verified by making comparisons between the histograms of transmission-coefficient samples generated by Monte Carlo simulations and the corresponding fitted probability density functions; the values for its four independent control parameters under different conditions are obtained by the fit of the dual Johnson S B distribution to relevant simulated transmission-coefficient samples. It is found that each of the four independent control parameters of the dual Johnson S B distribution can be considered as a function of three quantities, viz., the OAM index, the Fried's atmospheric coherence width, and the ratio of the root-mean-square (RMS) OAM-beam radius to the Fried's atmospheric coherence width. The results demonstrate that the statistical distribution of the fluctuating transmission coefficient depends less on the first two quantities than on the last one. Finding a model for direct mapping from these three quantities to the four control parameters of the dual Johnson S B distribution deserves future study.

Convex Combination of SISO Equalization and Blind Source Separation for MIMO Blind Equalization

For Multiple-Input Multiple-Output (MIMO) system the received signal will suffer not only inter-symbol interference but also inter-antenna interference under frequency selective fading channel. This paper proposes a MIMO blind equalization algorithm consisting of the convex combination of Single-Input Single-Output (SISO) blind equalization algorithm and blind source separation (BSS). The main purpose of the SISO equalization algorithm is to convert the convolution channel into multiplicative channel, and the BSS algorithm is mainly used to separate the different sources. The SISO equalization algorithm used in the paper is the modified constant modulus algorithm (MCMA) because of its simplicity and effectiveness. The BSS algorithm is the constraint fitting probability density function algorithm (CFPA). The MIMO blind equalization algorithm is named as MCMA–CFPA. Moreover, a low complexity MCMA–CFPA and a dual mode algorithm based on soft switching are presented. Simulation results show that the proposed algorithms can simultaneously equalize and separate all transmission signals.

Auto-extraction of stratified interface in the underground space based on Bayesian detection algorithm with statistical fitting of probability density by actual data

There are many man-made or natural complex layered structures in underground space. The corresponding dielectric constants of different layers are different. The phase and amplitude changes of electromagnetic wave appear on the interface when these electromagnetic waves propagate in these layered media. Efficient extraction of these layered interfaces is one of the most important applications for non-destructive detection. It is very complicated to calculate phase and amplitude changes corresponding to the electromagnetic wave propagation near the interface in the medium using electromagnetic wave propagation and scattering theory. Based on a large number of measured data, we statistically analyze the amplitude and phase characteristics of the echo in different media, and give the fitted probability density function. The probability density function is used to calculate the likelihood ratio, so that the Bayesian algorithm is utilized to extract the interfaces of different media automatically. Taking the automatic extraction of the interface between the ice and the underground space as an example to verify the algorithm, we first perform the decorrelation filtering, and then the fitting of probability density function with lots of measured data, followed by Bayesian detection. And finally the layered media interface is extracted while suppressing the related noises.

Statistical approach for the optimum selection of photovoltaic modules on a specific site

This paper introduces a probabilistic approach (PA) for the optimum selection of various photovoltaic (PV) modules manufactured by different suppliers to select the most suited module at a specific site location. This approach is based on fitting the probability distributions of the irradiance data measured at a specific hour of a typical day over a long term period (7 years). The goodness of fit test is employed to determine the best fitted probability density function which is then used in the calculation of the average output power and capacity factor (CF) of each module. Thus, the module with the highest average CF over the year is therefore considered as the best suited module for the given site location. The source files of solar irradiance database are approved by the Egyptian Meteorological Authority. Detailed modeling of the PV system characteristics using MATLAB/Simulink has been introduced, which shows complete stepping procedures and respective results. The combination of the detailed PV modeling along with the PA facilitates precise prediction of the best module in terms of output power under varying operating conditions and partial shading for the specified site.

Statistical modelling of CG interdistance across multiple organisms

BackgroundStatistical approaches to genetic sequences have revealed helpful to gain deeper insight into biological and structural functionalities, using ideas coming from information theory and stochastic modelling of symbolic sequences. In particular, previous analyses on CG dinucleotide position along the genome allowed to highlight its epigenetic role in DNA methylation, showing a different distribution tail as compared to other dinucleotides.In this paper we extend the analysis to the whole CG distance distribution over a selected set of higher-order organisms. Then we apply the best fitting probability density function to a large range of organisms (>4400) of different complexity (from bacteria to mammals) and we characterize some emerging global features.ResultsWe find that the Gamma distribution is optimal for the selected subset as compared to a group of several distributions, chosen for their physical meaning or because recently used in literature for similar studies. The parameters of this distribution, when applied to our larger set of organisms, allows to highlight some biologically relavant features for the considered organism classes, that can be useful also for classification purposes.ConclusionsThe quantification of statistical properties of CG dinucleotide positioning along the genome is confirmed as a useful tool to characterize broad classes of organisms, spanning the whole range of biological complexity.

Object-to-group probabilistic distance measure for uncertain data classification

Uncertain objects, where each feature is represented by multiple observations or a given or fitted probability density function, arise in applications such as sensor networks, moving object databases and medical and biological databases. We propose a methodology to classify uncertain objects based on a new probabilistic distance measure between an uncertain object and a group of uncertain objects. This object-to-group probabilistic distance measure is unique in that it accounts separately for the correlations among the features within each class and within each object. We compare the proposed object-to-group classifier to two existing classifiers, namely, the K-Nearest Neighbor classifier on object means (certain-KNN) and the uncertain-naïve Bayes classifier. In addition, we compare the object-to-group classifier to an uncertain K-Nearest Neighbor classifier (uncertain-KNN), also proposed here, that uses existing probabilistic distance measures for object-to-object distances. We illustrate the advantages of the proposed classifiers with both simulated and real data.

A Framework to Analyze the Stochastic Harmonics and Resonance of Wind Energy Grid Interconnection

This paper addresses a modeling and analysis methodology for investigating the stochastic harmonics and resonance concerns of wind power plants (WPPs). Wideband harmonics from modern wind turbines (WTs) are observed to be stochastic, associated with real power production, and they may adversely interact with the grid impedance and cause unexpected harmonic resonance, if not comprehensively addressed in the planning and commissioning of the WPPs. These issues should become more critical as wind penetration levels increase. We thus propose a planning study framework comprising the following functional steps: First, the best fitted probability density functions (PDFs) of the harmonic components of interest in the frequency domain are determined. In operations planning, maximum likelihood estimations (MLEs) followed by a chi-square test are used once field measurements or manufacturers’ data are available. Second, harmonic currents from the WPP are represented by randomly-generating harmonic components based on their PDFs (frequency spectrum) and then synthesized for time domain simulations via inverse Fourier transform. Finally, we conduct a comprehensive assessment by including the impacts of feeder configurations, harmonic filters and the variability of parameters. We demonstrate the efficacy of the proposed study approach for a 100-MW offshore WPP consisting of 20 units of 5-MW full converter turbines, a realistic benchmark system adapted from a WPP under development in Korea and discuss lessons learned through this research.