Probability Function Research Articles

The Aemulus Project. VI. Emulation of Beyond-standard Galaxy Clustering Statistics to Improve Cosmological Constraints

There is untapped cosmological information in galaxy redshift surveys in the nonlinear regime. In this work, we use the Aemulus suite of cosmological N-body simulations to construct Gaussian process emulators of galaxy clustering statistics at small scales (0.1–50 h −1 Mpc) in order to constrain cosmological and galaxy bias parameters. In addition to standard statistics—the projected correlation function w p(r p), the redshift-space monopole of the correlation function ξ 0(s), and the quadrupole ξ 2(s)—we emulate statistics that include information about the local environment, namely the underdensity probability function P U(s) and the density-marked correlation function M(s). This extends the model of Aemulus III for redshift-space distortions by including new statistics sensitive to galaxy assembly bias. In recovery tests, we find that the beyond-standard statistics significantly increase the constraining power on cosmological parameters of interest: including P U(s) and M(s) improves the precision of our constraints on Ωm by 27%, σ 8 by 19%, and the growth of structure parameter, f σ 8, by 12% compared to standard statistics. We additionally find that scales below ∼6 h −1 Mpc contain as much information as larger scales. The density-sensitive statistics also contribute to constraining halo occupation distribution parameters and a flexible environment-dependent assembly bias model, which is important for extracting the small-scale cosmological information as well as understanding the galaxy–halo connection. This analysis demonstrates the potential of emulating beyond-standard clustering statistics at small scales to constrain the growth of structure as a test of cosmic acceleration.

Study on earthquake and tsunami hazard: evaluating probabilistic seismic hazard function (PSHF) and potential tsunami height simulation in the coastal cities of Sumatra Island

This study uses integrated geological, geodesy, and seismology data to assess the potential tsunami and Probabilistic Seismic Hazard Function (PSHF) near Sumatra’s coastal cities. It focuses on estimating the possible level of ground shaking due to the seismic activity within the Sumatran Fault Zone (SFZ) and subduction zone. It uses the Peak Ground Acceleration (PGA) as a measure. An amplification factor that is based on the previous study is used. It is calculated through the Horizontal-Vertical Spectral Ratio (HVSR), which measures possible surface ground shaking. The Seismic Hazard Function (SHF) is calculated considering magnitudes 6.5 to 9.0 for subduction sources and 6.5 to 7.8 for SFZ sources. Also, the PGA based on the Maximum Possible Earthquake (MPE) magnitude is estimated, and tsunami heights are simulated to assess the possible hazard risk. The tsunami source model in this study is characterized by considering the possibility of the long-term perspectives on giant earthquakes and tsunamis that might occur in subduction zones around the off-coast of southern Sumatra Island. The potentiality source zone is characterized based on the utilization of the cross-correlation of correlation dimension (DC) based on the shallow earthquake catalog of 2010 to 2022 and the SHmax-rate of surface strain rate. Based on the MPE, the relatively high estimated PGA at the base rock was found around Mentawai and Pagai Utara islands at about 0.224 g and 0.328 g, with the largest estimated PGA based on the MPE at the surface with values of about 0.5 g and 0.6 g. The possible maximum tsunami height (Hmax) estimated based on source scenarios position around the west coast of Sumatera Island, such as for Kota Padang and Kota Bungus, reaches up to 12.0 m and 22.0 m, respectively. The findings provide valuable insight into seismic and tsunami hazards, benefiting future mitigation strategies.

A Framework for Valuation and Portfolio Optimization of Venture Capital Deals with Contractual Terms

Venture capitalists invest not only in the business aspect of a deal but also in its contractual terms. Therefore, the selection of deals and the combination of contractual terms pose challenging decisions for them. This paper consists of two main sections. The first section introduces a novel framework for the valuation of venture capital (VC) deals, including startups and their contractual terms. By taking into account risk situations, this section presents the valuation of combined contractual terms, including call options, liquidity preference, and participant rights. In the second section, a new multiobjective mathematical model for VC deals and contractual terms portfolio selection is developed using right-tail probability, strategy alignment, and a utility function. To solve the proposed model, three metaheuristic algorithms—Non-Dominated Sorting Genetic Algorithm (NSGA-II), Multi-Objective Binary Harmony Search Algorithm, and Dynamic Tuning Parameter Binary Harmony Search Algorithm (DTPBHS)—are applied. Based on numerical examples, DTPBHS outperforms other algorithms in the “Mean Ideal Distance” index, but NSGA-II demonstrates the best performance in the “Rate of Achievement of two objectives simultaneously” index. Furthermore, we demonstrate that the proposed utility function is more robust than the right-tail probability function under default deals conditions.

Optimized Sigmoid Functions for Speech Presence Probability and Gain Function in Speech Enhancement

Speech presence probability (SPP) and gain functions such as Wiener filter or MMSE estimators require an estimate of the a-priori signal-to-noise ratio (SNR). However, the estimation of the a-priori SNR is computationally involved and sensitive to noise variations. This paper proposes to approximate the SPP and the overall gain function of a speech enhancement system by using sigmoid functions to reduce the need of estimating the a-prior SNR. By applying an approximation via the sigmoid functions it is shown that only the a-posteriori estimate of SNR is needed, resulting in a low complexity system. The sigmoid function is designed with an optimization algorithm to optimize its parameters with respect to speech quality measures. The optimization algorithm is based on the idea that the solution obtained for a given problem should move towards the best solution and avoid the worst solution. The proposed algorithm requires minimal control parameters and does not require any algorithm specific parameters. Simulation results show that the proposed sigmoid functions achieve good results in terms of speech quality measures when compared with existing methods while providing significantly lower complexity for implementation.

Peale's dolphins (Lagenorhynchus australis) are acoustic mergers between dolphins and porpoises

Most dolphin species produce broadband clicks for echolocation (i.e., biosonar pulses), and whistles and burst-pulsed calls for communication purposes. A few dolphin species in the southern hemisphere are reported to only produce clicks of a more narrowband high-frequency (NBHF) type, very similar to the clicks produced by porpoises. Here, we use underwater acoustic recordings of Peale's dolphins (Lagenorhynchus australis) in the Falkland Islands to show that they also can produce broadband clicks and tonal whistle sounds similar to other whistling odontocete species in addition to NBHF clicks. Whistles and broadband clicks co-occurred in seven out of twelve acoustic encounters with Peale's dolphins. The co-occurrence of whistles and broadband signals, which were predominately burst-pulses, produced by Peale's dolphins in this study points to a probable function as communication signals. Furthermore, the measured frequency and time parameters of Peale's dolphin whistles were comparable with whistles recorded from four additional species in the Lagenorhynchus genus in various parts of their ranges. This is the only species of Lagenorhynchus recorded to date that can relax acoustic crypsis in a similar manner as Heaviside's (Cephalorhynchus heavisidii), Commerson's (C. commersonii), and Hector's (C. hectori) dolphins. Our findings make it likely that additional NBHF species have the ability to generate both pulsed signal types and whistles, and further challenges the acoustic classification of NBHF species.

Attributes Relevance in Content-Based Music Recommendation System

The possibility of recommendations of musical songs is becoming increasingly required because of the millions of users and songs included in online databases. Therefore, effective methods that automatically solve this issue need to be created. In this paper, the mentioned task is solved using three basic factors based on genre classification made by neural network, Mel-frequency cepstral coefficients (MFCCs), and the tempo of the song. The recommendation system is built using a probability function based on these three factors. The authors’ contribution to the development of an automatic content-based recommendation system are methods built with the use of the mentioned three factors. Using different combinations of them, four strategies were created. All four strategies were evaluated based on the feedback score of 37 users, who created a total of 300 surveys. The proposed recommendation methods show a definite improvement in comparison with a random method. The obtained results indicate that the MFCC parameters have the greatest impact on the quality of recommendations.

Entropy Estimators for Markovian Sequences: A Comparative Analysis.

Entropy estimation is a fundamental problem in information theory that has applications in various fields, including physics, biology, and computer science. Estimating the entropy of discrete sequences can be challenging due to limited data and the lack of unbiased estimators. Most existing entropy estimators are designed for sequences of independent events and their performances vary depending on the system being studied and the available data size. In this work, we compare different entropy estimators and their performance when applied to Markovian sequences. Specifically, we analyze both binary Markovian sequences and Markovian systems in the undersampled regime. We calculate the bias, standard deviation, and mean squared error for some of the most widely employed estimators. We discuss the limitations of entropy estimation as a function of the transition probabilities of the Markov processes and the sample size. Overall, this paper provides a comprehensive comparison of entropy estimators and their performance in estimating entropy for systems with memory, which can be useful for researchers and practitioners in various fields.

Numerical characterization of dual-frequency capacitively coupled plasmas modulated by electron beam injection

The modulated approach of electron beam (EB) injection can achieve favorable parameters for capacitive coupled plasmas (CCP). In this work, a one-dimensional particle-in-cell/Monte Carlo collision (PIC/MCC) model is used to simulate the stable dual-frequency CCP with EB injection. First, when the parameters of EB are kept constant at 0.01 A and 30 eV, the results demonstrate significant enhancements in electron density, self-bias voltage, and ion flux. Furthermore, the electron energy probability function (EEPF) appears to have a transition from a typical bi-Maxwellian distribution to a Maxwellian distribution, and the dominant heating mode shifts from the α-mode to the α-γ-mode. Secondly, when the EB current and energy are all changed, the basic parameters of DF-CCP can be achieved by different modulations. Furthermore, we also discuss the transition of the electron heating mode as the current increases from 0.001 A to 1 A and the energy increases from 10 eV to 490 eV. In particular, we conduct a comparative study among different cases of EB injection. According to these results, the modulation capability of EB injection in DF-CCP is thoroughly investigated, which can greatly benefit atom-scale etching in practical applications.

Chemical characteristics, morphology and source apportionment of PM10 over National Capital Region (NCR) of India.

The present study frames the physico-chemical characteristics and the source apportionment of PM10 over National Capital Region(NCR) of India using the receptor model's Positive Matrix Factorization (PMF) and Principal Momponent Mnalysis/Absolute Principal Component Score-Multilinear Regression (PCA/APCS-MLR). The annual average mass concentration of PM10 over the urban siteof Faridabad, IGDTUW-Delhi and CSIR-NPL of NCR-Delhi were observed to be 195 ± 121, 275 ± 141 and 209 ± 81µgm-3, respectively. Carbonaceous species (organic carbon (OC), elemental carbon (EC) and water-soluble organic carbon (WSOC)), elemental constituents (Al, Ti, Na, Mg, Cr, Mn, Fe, Cu, Zn, Br, Ba, Mo Pb) and water-soluble ionic components (F-, Cl-, SO42-, NO3-, NH4+, Na+, K+, Mg2+, Ca2+) of PM10were entrenched to the receptor models to comprehend the possible sourcesof PM10. The PMF assorted sources over Faridabad were soil dust (SD 15%), industrial emission (IE 14%), vehicular emission (VE 19%), secondary aerosol (SA 23%) and sodium magnesium salt (SMS 17%). For IGDTUW-Delhi, the sources were SD (16%), VE (19%), SMS (18%), IE (11%), SA (27%) and VE + IE (9%). Emission sources like SD (24%), IE (8%), SMS (20%), VE + IE (12%), VE (15%) and SA + BB (21%) were extracted over CSIR-NPL, New Delhi, which are quite obvious towards the sites. PCA/APCS-MLR quantified the similar sources with varied percentage contribution. Additionally, catalogue the Conditional Bivariate Probability Function (CBPF) for directionality of the local source regions and morphology as spherical, flocculent and irregular were imaged using a Field Emission-Scanning Electron Microscope (FE-SEM).

The infimum values of two probability functions for the Gamma distribution

Let α, β be positive real numbers and let Xα,β\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$X_{\\alpha ,\\beta}$\\end{document} be a Gamma random variable with shape parameter α and scale parameter β. We study infimum values of the function (α,β)↦P{Xα,β≤κE[Xα,β]}\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$(\\alpha ,\\beta )\\mapsto P\\{X_{\\alpha ,\\beta}\\le \\kappa E[X_{\\alpha ,\\beta}] \\}$\\end{document} for any fixed κ>0\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$\\kappa >0$\\end{document} and the function (α,β)↦P{|Xα,β−E[Xα,β]|≤Var(Xα,β)}\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$(\\alpha ,\\beta )\\mapsto P\\{|X_{\\alpha ,\\beta}-E[X_{\\alpha ,\\beta}]| \\le \\sqrt{\\operatorname{Var}(X_{\\alpha ,\\beta})}\\}$\\end{document}. We show that infα,βP{Xα,β≤E[Xα,β]}=12\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$\\inf_{\\alpha ,\\beta}P\\{X_{\\alpha ,\\beta}\\le E[X_{\\alpha ,\\beta}]\\}= \\frac{1}{2}$\\end{document} and infα,βP{|Xα,β−E[Xα,β]|≤Var(Xα,β)}=P{|Z|≤1}≈0.6827\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$\\inf_{\\alpha ,\\beta}P\\{|X_{\\alpha ,\\beta}-E[X_{\\alpha ,\\beta}]|\\le \\sqrt{\\operatorname{Var}(X_{\\alpha ,\\beta})}\\}=P\\{|Z|\\le 1\\}\\approx 0.6827$\\end{document}, where Z is a standard normal random variable.

A Streptomyces species from the ginseng rhizosphere exhibits biocontrol potential.

Plants and their associated microbes live in complicated, changeable, and unpredictable environments. They usually interact with each other in many ways through multidimensional, multiscale, and multilevel coupling manners, leading to challenges in the coexistence of randomness and determinism or continuity and discreteness. Gaining a deeper understanding of these diverse interaction mechanisms can facilitate the development of data-mining theories and methods for complex systems, coupled modeling for systems with different spatiotemporal scales and functional properties, or even a universal theory of information and information interactions. In this study, we use a "closed-loop" model to present a plant-microbe interaction system and describe the probable functions of microbial natural products. Specifically, we report a rhizosphere species, Streptomyces ginsengnesis G7, which produces polyketide lydicamycins and other active metabolites. Interestingly, these distinct molecules have the potential to function both as antibiotics and as herbicides for crop protection. Detailed laboratory experiments conducted in Arabidopsis (Arabidopsis thaliana), combined with a comprehensive bioinformatics analysis, allow us to rationalize a model for this specific plant-microbe interaction process. Our work reveals the benefits of exploring otherwise neglected resources for the identification of potential functional molecules and provides a reference to better understand the system biology of complex ecosystems.

Parameter estimation in single-phase transformers via the generalized normal distribution optimizer while considering voltage and current measurements

This research addresses, from a perspective of metaheuristic optimization, the problem regarding parametric estimation in single-phase transformers while considering voltage and current measures at the terminals of the transformer and weighing linear loads. Transformer parametric estimation is modeled as a nonlinear problem in order to minimize the mean square error between the calculated voltage and current variables and the measurements taken. The nonlinearities are associated with Kirchhoff's first and second laws applied to the equivalent electrical circuit of the single-phase transformer. The nonlinear optimization problem is solved by applying a metaheuristic optimization algorithm known as the generalized normal distribution optimizer (GNDO), which uses evolution rules that allow exploring and exploiting the solution space via the classical probability function based on normal distributions. Numerical results in three test transformers of 20, 45, and 112.5 kVA demonstrate the effectiveness and robustness of the proposed GNDO approach when compared to other optimizers reported in the literature, such as the crow search algorithm, the coyote optimization algorithm, and the exact solution of the nonlinear optimization model using the fmincon solver of the MATLAB software. All numerical simulations confirm the potential of the GNDO approach to deal with complex optimization problems in engineering and science with promising results and low computational effort.

Switchable tetraplex elements in the heterogeneous nuclear ribonucleoprotein K promoter: micro-environment dictated structural transitions of G/C rich elements

We have elucidated the hnRNP K promoter as a hotspot for tetraplex-based molecular switches receptive to micro-environmental stimuli. We have characterised the structural features of four tetraplex-forming loci and identified them as binding sites of transcription factors. These segments form either G-quadruplex or i-motif structures, the structural dynamicity of which has been studied in depth via several biophysical techniques. The tetraplexes display high dynamicity and are influenced by both pH and KCl concentrations in vitro. The loci complementary to these sequences form additional non-canonical secondary structures. In the cellular context, the most eminent observation of this study is the binding of hnRNP K to the i-motif forming sequences in its own promoter. We are the first to report a probable transcriptional autoregulatory function of hnRNP K in coordination with higher-order DNA structures. Herein, we also report the positive interaction of the endogenous tetraplexes with Sp1, a well-known transcriptional regulator. Treatment with tetraplex-specific small molecule ligands further uncovered G-quadruplexes’ functioning as repressors and i-motifs as activators in this context. Together, our findings strongly indicate the critical regulatory role of the identified tetraplex elements in the hnRNP K promoter. Communicated by Ramaswamy H. Sarma

Causal Artificial Intelligence Models of Food Quality Data.

Causal Artificial Intelligence Models of Food Quality Data.

Data-driven nonlinear parametric model order reduction framework using deep hierarchical variational autoencoder

A data-driven parametric model order reduction (MOR) method using a deep artificial neural network is proposed. The present network, a least-squares hierarchical variational autoencoder (LSH-VAE), is capable of performing nonlinear MOR for the parametric interpolation of a nonlinear dynamic system with a significant number of degrees of freedom. LSH-VAE differs from existing networks in two major respects: a deep hierarchical structure and a hybrid weighted, probabilistic loss function. The enhancements result in significantly improved accuracy and stability compared with conventional nonlinear MOR methods, autoencoders, and variational autoencoders. In LSH-VAE, the parametric MOR framework is based on the spherically linear interpolation of the latent manifold. The present framework is validated and evaluated on three nonlinear and multiphysics dynamic systems. First, the present framework is evaluated on the fluid–structure interaction benchmark problem to assess its efficiency and accuracy. Then, a highly nonlinear aeroelastic phenomenon, limit cycle oscillation, is analyzed. Finally, the framework is applied to three-dimensional fluid flow to demonstrate its capability to efficiently analyze a significantly large number of degrees of freedom. The superior performance of LSH-VAE is emphasized by comparing its results against those of widely used nonlinear MOR methods, a convolutional autoencoder, and β\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\beta $$\\end{document}-VAE. The proposed framework exhibits significantly enhanced accuracy compared with that of conventional methods, while the computational efficiency continues to remain significantly higher.

Geometry-based cross-linking algorithm for modeling resorcinol resins through united-atom molecular dynamics simulations

The addition-condensation polymerization (ACP) of resorcinol with formaldehyde and the electrochemical polymerization (ECP) of resorcinol were desirable to improve the performances of resins. To achieve computational design, a new geometry-based cross-linking modeling algorithm for three-dimensional (3D) cross-linked network structures of resorcinol resins was developed for molecular dynamics (MD) simulations using a united-atom model. The ACP and ECP of resorcinol were successfully modeled using one algorithm by changing the parameters of the reaction rate constants of methylene, ether, and biphenyl bond formations. Furthermore, reaction judgment was performed under a reaction probability function comprising the Gaussian functions based on the interatomic distance of the reaction candidate atom pairs and interior angles regarding the spatial arrangement of two resorcinol rings. This geometry-based cross-linking algorithm successfully generated strainless 3D cross-linked network structures of the ACP and ECP resorcinol resins without the formation of highly bent unstable structures, such as U-shaped biphenyls. The MD simulation results indicated that the cross-linking of the ACP resin comprising methylene linkages proceeded with a network structure formation mechanism similar to that of phenolic resins, where the cross-linked network structure was determined at the gel point and the reactions after the gel point were to fix and freeze the network structure. For the ECP resins comprising ether and biphenyl linkages, the molar ratio of ether to biphenyl bonds affected the elastic property of the resins, and the incorporation of rigid biphenyl structures into ether-linkage networks increased the elastic modulus. Moreover, an increase in rigidity was observed as a decrease in the Poisson's ratio.

A Conflict Evidence Fusion Method Based on Bray–Curtis Dissimilarity and the Belief Entropy

The D-S evidence theory is extensively applied to manage uncertain information. However, the theory encounters challenges related to conflicts during the fusion process, impeding the precise identification of multi-subset focal elements. This paper introduces a novel method for conflicting evidence fusion that incorporates the Bray–Curtis dissimilarity, cosine distance of the included angle, and belief entropy. The method comprehensively evaluates three aspects—evidence similarity, evidence distance, and the amount of information—while considering factors like the credibility and uncertainty of evidence. Initially, the evidence undergoes conversion into single-subset focal element evidence through the improved Pignistic probability function. Subsequently, the credibility between pieces of evidence is established using the Bray–Curtis dissimilarity and angle cosine distance, while the uncertainty of the evidence is computed using belief entropy. The weighted correction coefficient of the evidence is determined by integrating the credibility and uncertainty of the evidence. Subsequently, the corrected evidence is fused using the D-S evidence theory to derive the final judgment. An analysis of two sets of arithmetic examples, considering both single-subset and multi-subset focal elements, demonstrates the faster convergence and enhanced accuracy and reliability of the proposed method in comparison to existing approaches.

Grey comprehensive evaluation of development performance of provinces in China based on spatiotemporal probability function and variable weight strategy

In the new stage of promoting high-quality economic development, the effect of the transformation of development momentum and the ability of sustainable development has become the key factors for the competitiveness of provinces in China. Especially in the context of the impact of COVID-19 and the obstacles of world trade protectionism, the sustainability of development performance is increasingly important. In the past, when evaluating the development performance of various provinces in China, a single index weight was usually used. In view of evaluation criteria, the lack of consideration of regional differentiation factors would result in the evaluation results deviating from reality. This paper introduces the entropy weight method to determine the weight of regional indicators of differentiated development. Based on the space-time probability function, a grey clustering evaluation model of regional development performance is constructed to conduct a comprehensive grey evaluation of the development performance of various provinces in China from 2009 to 2019. It is found that the new evaluation model can correct the deficiencies of similar probability functions and single index weight and obtain more accurate evaluation results. It’s found that the development performance evaluation results of each province are always in a dynamic adjustment process, which needs to be verified with the help of subsequent expansion analysis.

On Neutrosophic Truncation

Neutrosophic have found their place in neutrosophic studies due to the prevalence of indeterminacy in the world. We present the novel notion of neutrosophic truncated distribution, which is highly significant in analyzing events that involve the exclusion of certain data from the original dataset, particularly where there is a presence of indeterminacy in data. Unsure or ambiguous information, which is disregarded in classical logic, is incorporated within neutrosophic logic due to its focus on both certain and uncertain data. In this paper, the approach of neutrosophic truncation, and truncated distribution of neutrosophic random variable have been introduced, in addition to deriving some of its properties. And other cases discussed neutrosophic truncation depends on the neutrosophic probability function, a classical probability function, and studies neutrosophic probability and neutrosophic interval together. It studies the neutrosophic left truncated and neutrosophic right truncated. Some illustrative examples and statistical properties such as the cumulative function, the moment generating function, the order statistic, and the rth moment are presented.

Comparing maximum likelihood and maximum pseudolikelihood estimators for the Ising model

The Ising model is one of the most popular models in network psychometrics. However, statistical analysis of the Ising model is difficult due to the presence of its intractable normalizing constant in the probability function. As a result, maximum likelihood estimation using the exact likelihood is only possible for small graphs, and approximation methods are needed for larger graphs. Two popular approximations of the exact likelihood are the joint pseudolikelihood (JPL) and the disjoint pseudolikelihood (DPL). These approximations yield consistent estimators, but we do not know how well they perform for finite data. In this paper, we investigate the relative performance of parameter estimation methods based on the two approximations and compare them to maximum likelihood estimation using the exact likelihood. We perform an extensive simulation study comparing the estimators in terms of bias and variance. We show that maximum pseudolikelihood estimation based on the JPL is a stable estimation method that is able to accurately approximate the maximum likelihood estimates, but that maximum pseudolikelihood estimation based on the DPL only works well for large sample sizes.