Continuous Uniform Distribution Research Articles

A slow diaphragmatic breathing intervention for anxiety: How do respiration rate and inhalation/exhalation ratio influence self-reported anxiety?

The primary aim was to investigate how respiration rate and inhalation/exhalation ratio influence self-reported state anxiety during a single slow diaphragmatic breathing exercise session. Eight hundred and twenty-eight participants completed the study at two separate geographical locations (Poland and Spain). Participants performed a 10-min online guided breathing exercise. Respiration rates were sampled from a continuous uniform distribution (ranging from 6 to 12 breaths/min). Similarly, inhalation/exhalation ratios were treated as continuous variables and sampled from a uniform distribution for each participant. An application programed for this experiment displayed visual and auditory cues adjusted for each participant. Before and after the breathing exercise, each participant filled in the Current Anxiety Level Measure questionnaire. Self-trait anxiety was measured with the Clinically Useful Anxiety Outcome Scale. A linear regression model showed that respiration rate, trait anxiety, pre-test anxiety, and nationality (Polish/Spanish) were positively related to post-test anxiety levels. Adding quadratic terms of respiration rate and inhalation/exhalation ratio did not improve model fit. Polish participants exhibited higher post-test anxiety levels compared with the Spanish subsample. Age was negatively associated with post-test anxiety. No significant relationships between inhalation/exhalation ratio and post-test anxiety level were found. Slower respiration rates during a single-session breathing exercise are linearly associated with lower post-test anxiety levels in a large and varied sample. This study is the largest to date and may offer further guidance for predicting expected effect sizes for the relationships between anxiety and respiratory dynamics.

Deformation of power law in the double Pareto distribution using uniformly distributed observation time

The double Pareto distribution is a heavy-tailed distribution with a power-law tail, that is generated via geometric Brownian motion with an exponentially distributed observation time. In this study, we examine a modified model wherein the exponential distribution of the observation time is replaced with a continuous uniform distribution. The probability density, complementary cumulative distribution, and moments of this model are exactly calculated. Furthermore, the validity of the analytical calculations is discussed in comparison with numerical simulations of stochastic processes.

Big Data and Neural Networks in Smart Grid - Part 1: The Impact of Measurement Differences on the Performance of Neural Network Identification Methodologies of Overhead Low-Voltage Broadband over Power Lines Networks

Until now, the neural network identification methodology for the branch number identification (NNIM-BNI) and the neural network identification methodology for the distribution line and branch line length approximation (NNIM-LLA) have approximated the number of branches and the distribution line and branch line lengths given the theoretical channel attenuation behavior of the examined overhead low-voltage broadband over powerlines (OV LV BPL) topologies [1], [2]. The impact of measurement differences that follow continuous uniform distribution (CUDs) of different intensities on the performance of NNIM-BNI and NNIM-LLA is assessed in this paper. The countermeasure of the application of OV LV BPL topology databases of higher accuracy is here investigated in the case of NNIM-LLA. The strong inherent mitigation efficiency of NNIM-BNI and NNIM-LLA against CUD measurement differences and especially against those of low intensities is the key finding of this paper. The other two findings that are going to be discussed in this paper are: (i) The dependence of the approximation Root-Mean-Square Deviation (RMSD) stability of NNIM-BNI and NNIM-LLA on the applied default operation settings; and (ii) the proposal of more elaborate countermeasure techniques from the literature against CUD measurement differences aiming at improving NNIM-LLA approximations.

Taking Rational Numbers at Random

In this article, some prescriptions to define a distribution on the set Q0 of all rational numbers in [0,1] are outlined. We explored a few properties of these distributions and the possibility of making these rational numbers asymptotically equiprobable in a suitable sense. In particular, it will be shown that in the said limit—albeit no absolutely continuous uniform distribution can be properly defined in Q0—the probability allotted to every single q∈Q0 asymptotically vanishes, while that of the subset of Q0 falling in an interval [a,b]⊆Q0 goes to b−a. We finally present some hints to complete sequencing without repeating the numbers in Q0 as a prerequisite to laying down more distributions on it.

Statistical Study of Bisection Method for Cubic Equations with Random Coefficients

The bisection method is an iterative approach used in numerical analysis to find solutions to nonlinear equations. The main purpose of this paper is to study how the parameters of a probability distribution characterizing the coefficients of a cubic polynomial can influence the convergence of the bisection method. The study covers discrete and continuous distributions, including discrete uniform, continuous uniform, and normal distributions. It was found that for both types of uniform distribution inputs, a second-degree polynomial equation can predict the average iteration for a given parameter r, where r indicates the distribution interval [−r, r]. Interestingly, the coefficients of the second-degree polynomial are nearly identical for discrete and continuous uniform distributions. For normal distribution input, the average iteration does not depend upon the standard deviation when the mean is fixed and the standard deviation is varying. But when the standard deviation is fixed and the mean is varying, the second-degree polynomial is still the best fit. This means the average iteration depends upon the mean of the normal distribution. Overall, our paper concludes that: I. For uniform distribution input, the average iteration does not depend on whether the distribution is discrete or continuous but rather depends on the range of the distribution which is its parameter. II. For non-uniform distribution input, the average iteration depends on the mean of the distribution (location parameter) but not on the standard deviation (scale parameter). Finally, a curtain is raised in the future direction of research in which we propose to combine the bisection method with the regula falsi and Newton-Raphson methods to increase the rate of convergence.

#5770 PROPOSING A NEW RENAL OUTCOME WHEN REACHING ESTIMATED GFR OF 30—SIMULATION STUDY USING REAL-WORLD STRATIFIED DATA

Abstract Background and Aims Patients with renal diseases often have progressive decline in eGFR, with many clinical researches having outcomes of reaching renal death/renal replacement therapy. In the real life, however, patients with eGFR under 15 mL/min/1.73m2 (the same unit hereafter) lose their lives more often by cardiovascular diseases before reaching endstage renal disease. Moreover, patients with eGFR below 30 are known to have miscellaneous metabolic derangements including phosphate retention, parathyroid hormone excess, metabolic acidosis, hyperkalaemia and renal anaemia. It appears to be extremely important to recognize that, in order to avoid enfacing plethora of metabolic alterations and cardiovascular disorders that progress after renal function falls below 30, we physicians need to take a step to find ways how to keep patients' eGFR above 30 for as long time as possible. At the same time, renal clinics are fully booked already; we need to prioritize the patients who have the highest risks in reaching eGFR below 30. This simulation study was performed to draw a trajectory to clarify who are the most eligible patients for nephrologists to intervene. Method Before performing the simulations using an inverse transform sampling, a priori statistical distributions were retrieved from the real-world data, i.e., a whole-hospital-wide survey over 4 years; (a) distribution of eGFR: obtained from patients with at least 3 measurements of eGFR over 366 days or more, where a median of eGFR values was used to represent the patient's renal function, (b) distribution of eGFR slopes: similarly retrieved, which were then stratified in 6 strata with each eGFR range of [30,35)/[35,40)/[40,45)/[45,50)/[50,55)/[55,60). The simulation was performed in 4 steps. In the first step (“random number generating step”), 65,000 pairs of random numbers (U1, U2) were generated using a continuous uniform distribution, U[0,1]. In the second step (“eGFR generating step”), eGFR was generated using the clinically-retrieved distribution (a, above) from a random number U1. In the third step (“slope generating step”), a slope in eGFR decline was generated using the a priori stratified distribution (b, above) from a random number U2. In the final step (“estimation step”), renal prognosis, especially an eGFR value 10 years later, was estimated in each of the “65,000 simulated patients”, with or without an effect of nephrology consults. The primary outcome was defined as reaching the eGFR below 30 within 10 years. If this occurred, time until the eGFR 30 was also calculated. Results In the strata of simulated patients with eGFR’s [50,55) and [55,60), 35.3% and 34.9% reached their eGFR below 30, over the periods of 4.78 and 4.95 years in average. In these strata, there were marked differences in renal prognosis based on the eGFR slope, where all the patients with the fastest quartile (annual rate of decline faster than 3.09) reached the outcome, while only 4–11% of the patients with slower decline (the remaining 3/4) reached eGFR <30. The number of patients was the largest in the stratum of [55, 60). In the stratum of eGFR [45,50), 45.3% patients, including 100% of faster quartile and 24.5% of slower 3 quartiles, reached eGFR of <30. Assuming the effect of nephrology consultation as attenuating the slope by 1/3, nephrologist intervention most prominently influenced the prognosis of the patients in strata [40,45) and [45,50) in terms of proportion of patients reaching outcomes (decreases by 10.5% and 12.5%). Conclusion 1) Reaching an eGFR of <30 mL/min/1.73m2 is a newly proposed outcome in clinical research with substantial significance such as avoiding cardiovascular events and metabolic derangements that occur frequently at eGFR below 30. 2) Patients in the fastest quartile in eGFR slope, even those whose baseline eGFR 50–60, are the candidates who merit nephrology consultation.

Optimal shape for a rectangular warehouse with a lateral receive/ship dock

Purpose: This technical note provides the mathematical demonstration for obtaining the optimal aspect ratio for a rectangular storage area with a lateral receive/ship dock, representing the standard configuration of modern distribution centers and logistic warehouses. The proposed aspect ratio is the one that minimizes the travel times of operators, keeping the common assumption of a storage area having a uniform access probability.Design/methodology/approach: To obtain the optimal aspect ratio of the storage area we model the entry point of the uniformly distributed dock with a random variable with a continuous uniform distribution, and we consequently evaluate the average travel path of the operator as a function of the latter. Successively, we estimate and minimize the average roundtrip length of the operator, leading to the optimal aspect ratio of the storage area.Findings: We find that the optimal aspect ratio between the warehouse width (U) and length (V) equals 1.5. The obtained result shows that the operators’ travel times are minimized with a storage area where .Research limitations/implications: Warehouses with a dock on one side now represent modern distribution centers' standard configuration. However, no optimal aspect ratio for the storage area has been discussed. For this reason, the paper fills this lack of scientific literature in the warehouse optimization research field by providing indications on how to design this class of warehouses.Practical implications: Distribution managers may find here guidance for defining a proper design of logistics centers and evaluating the operators' actual travel times to perform a roundtrip within the storage area.Originality/value: Traditional warehouse shape optimization models assume a single input/output point to the storage area. To our knowledge, no formal demonstration has been proposed for a warehouse with a dock on one entire side.

Working principle and layout logic of closed magnetic field in sputtering

Closed magnetic field constructed by unbalanced magnetron sputtering (MS) cathodes has been a general means of developing the MS coating system. However, owing to the difficulties in characterizing the complex plasma behaviors, there are still no quantitative criteria or design bases for some critical points, such as the effective object, the working mechanism, the closure condition, the layout logic and the effectivity of the closed magnetic field. Here in this work, out of the movements of charged particles in magnetic field, the motion behaviors of electrons and ions in the vacuum chamber are studied and it is also revealed that the closed magnetic field can affect mainly the electrons and further control the distributions of ions. A Monte-Carlo collision (MCC) model of the closed magnetic field MS coating system is established by test-electron to characterize the plasma transport characteristics, and the electron constraint and coating deposition efficiency are studied by different layouts of the magnetron cathodes and the ion sources. The simulation results show that the cathode numbers and vacuum chamber size determine the constraint effect on electrons in closed magnetic field. By 8 MS cathodes and the chamber radius of 0.5 m, the proportion of the overflow electrons can decrease to 1.77%. To increase the proportion of the electrons in the coating region, four coupled magnetic fields are introduced in the center of vacuum chamber. The studies of cathode type, rotation angle and magnetic field direction reveal that the proportion of the overflow electrons is less than 3%. A local dense plasma distribution and a continuous uniform plasma distribution can be observed in the vacuum chamber, corresponding to the same and opposite layout in magnetic poles of the MS cathodes and the ion sources, and the proportion of the electrons in the coating region significantly increases to 53.41% and 42.25%, respectively.

Poisson Modeling and Predicting English Premier League Goal Scoring

The English Premier League is well-known for being not only one of the most popular professional sports leagues in the world, but also one of the toughest competitions to predict. The first purpose of this research was to verify the consistency between goal scoring in the English Premier League and the Poisson process; specifically, the relationships between the number of goals scored in a match and the Poisson distribution, the time between goals throughout the course of a season and the exponential distribution, and the time location of goals during football games and the continuous uniform distribution. We found that the Poisson process and the three probability distributions accurately describe Premier League goal scoring. In addition, Poisson regression was utilized to predict outcomes for a Premier League season, using different sets of season data and with a large number of simulations being involved. We examined and compared various soccer metrics from our simulation results, including an English club’s chances of being the champions, finishing in the top four and bottom three, and relegation points.

The Probability Distribution of Time to Extinction: A simulation study

The time to extinction (TTE) is very important topic in some fields of studies; ecology, economics, corporate competition, bacterial sciences and epidemiology. The aim of the present paper is to investigate about the empirical parametric and empirical nonparametric probability distribution of the time to extinction for two related stochastic models; Rosenzweig and Macarthur model and May model. First, we assume the amplitude of r is a random variable with a continuous uniform probability distribution on a closed interval [r 1, r 2]. Second, we assume the number of consecutive years’ during which the amplitude of r remains constant is a random variable with a discrete geometric probability distribution with parameter p which is supported on a positive integer ℤ+ and p ∈ (0,1).

The geometry of Bayesian programming

AbstractWe give two geometry of interaction models for a typed λ-calculus with recursion endowed with operators for sampling from a continuous uniform distribution and soft conditioning, namely a paradigmatic calculus for higher-order Bayesian programming. The models are based on the category of measurable spaces and partial measurable functions, and the category of measurable spaces and s-finite kernels, respectively. The former is proved adequate with respect to both a distribution-based and a sampling-based operational semantics, while the latter is proved adequate with respect to a sampling-based operational semantics.

A Local Barycentric Version of the Bak\u2013Sneppen Model

We study the behaviour of an interacting particle system, related to the Bak–Sneppen model and Jante’s law process defined in Kennerberg and Volkov (Adv Appl Probab 50:414–439, 2018). Let Nge 3 vertices be placed on a circle, such that each vertex has exactly two neighbours. To each vertex assign a real number, called fitness (we use this term, as it is quite standard for Bak–Sneppen models). Now find the vertex which fitness deviates most from the average of the fitnesses of its two immediate neighbours (in case of a tie, draw uniformly among such vertices), and replace it by a random value drawn independently according to some distribution zeta . We show that in case where zeta is a finitely supported or continuous uniform distribution, all the fitnesses except one converge to the same value.

The role of between-patch dynamics in a metapopulation: a discrete-time modelling approach

We present a single-species metapopulation model structured by population size that is discrete in time. The novel formulation of our model allows for explicit incorporation of both the local, in space, dynamics and new details of the dispersal process. To study the impact of between-patch dynamics in the model, we construct various functions to describe density-dependent dispersal, recolonization, and between-patch stochasticity. Due to the complexity of the model, numerical simulations are used to obtain the distribution of the metapopulation. Moreover, we can use our model to predict the proportion of patches occupied, expected patch density, and variance in patch density. Our results provide insight into the influence that each of these processes play in the distribution of the metapopulation. In particular, our findings emphasize the benefit of explicitly modelling these processes. We consider two density-dependent dispersal strategies based upon individuals wanting to form average-sized patches and illustrate the differences in the metapopulation distributions. For recolonization of empty suitable patches, we look at two simplistic redistribution strategies modeled as either a continuous uniform or Laplace distribution. The between-patch stochasticity is modelled using redistribution kernels where we consider both thin-tailed and fat-tailed kernels. We find that the form of density-dependent dispersal has a major influence on the distribution of the metapopulation while recolonization is a key process for metapopulation persistence. In addition, fatter tails for the between-patch stochasticity can decrease the proportion of occupied patches and increase the expected patch density and variance in patch density.

Görev Temelli Yeni Bir Stokastik Çok Kriterli Karar Verme Yaklaşımı Önerisi

In real life decision problems, the result is generally obtained by using the opinions of a limited number of experts who have sufficient knowledge and experience on the subject of the decision. In the study, a new Multi Criteria Decision Making (MCDM) approach was proposed, which enables new views to be obtained by increasing the number of experts in decision-making processes and modeling uncertainty in expert evaluations in a real-life manner. In this context, two different methods, which are based on the The Multi-Objective Optimization by Ratio Analysis (MOORA) Method, are developed using continuous uniform distribution. These approaches are MOORA Ratio and MOORA Reference Point approaches. Continuous uniform distribution was preferred because each value within a certain range had an equal chance of occurrence. This means that each expert opinion is given an equal chance. Accordingly, by using a limited number of experts' opinions, random numbers are generated with continuous uniform distribution; the results of the evaluation of criteria and alternatives can be increased. Each different assessment represents an expert. Random numbers were generated within this range, taking into account the highest and lowest assessments of current experts. The reason for this is to obtain a preliminary assessment of the decision. However, the proposed approach is a task-based approach and decision-makers make use of relevant alternatives before evaluating them. During this use, decision makers are given different practical tasks. The efficiency of performing tasks enables decision-makers to understand the needs of real users and can be evaluated from a different perspective. The task-based MCDM approach is also important in terms of universal design principles and user satisfaction criteria considered in the evaluation of shopping centers in the study. Two different MOORA approaches have been applied in order to evaluate the shopping centers considering universal design principles and user satisfaction criteria and the alternative rankings obtained have been discussed. In addition, these two different approaches were developed to calculate the importance weights of the criteria. The study involves originality in terms of the proposed new approach and the criteria considered in the evaluation of shopping centers.

Are random permutations spherically uniform?

For large $q$, does the (discrete) uniform distribution on the set of $q!$ permutations of the vector $\bar{\mathbf {x}} ^{q}=(1,2,\dots ,q)'$ closely approximate the (continuous) uniform distribution on the $(q-2)$-sphere that contains them? These permutations comprise the vertices of the regular permutohedron, a $(q-1)$-dimensional convex polyhedron. The answer is emphatically no: these permutations are confined to a negligible portion of the sphere, and the regular permutohedron occupies a negligible portion of the ball. However, $(1,2,\dots ,q)$ is not the most favorable configuration for spherical uniformity of permutations. A more favorable configuration $\hat{\mathbf {x}} ^{q}$ is found, namely that which minimizes the normalized surface area of the largest empty spherical cap among its $q!$ permutations. Unlike that for $\bar{\mathbf {x}} ^{q}$, the normalized surface area of the largest empty spherical cap among the permutations of $\hat{\mathbf {x}} ^{q}$ approaches $0$ as $q\to \infty $. Nonetheless the permutations of $\hat{\mathbf {x}} ^{q}$ do not approach spherical uniformity either. The existence of an asymptotically spherically uniform permutation sequence remains an open question.

Estimation of parameters of the continuous uniform distribution: Different classical methods

The paper is concerned with the continuous uniform distribution for the estimation of its parameters. We have defined the different methods that may be utilized for the estimation of the parameters of a continuous uniform distribution which includes least square and relative least square methods also method of ridge regression, method of moments and its modifications, method of Percentile and its modifications, method of maximum likelihood estimation. The parameters of the continuous uniform distribution are estimated using thirteen different of methods for Monte Carlo generated samples. A repeated simulation study compares these estimation methods. We use the total deviation and mean square as performance index to identify the best estimator amongst them.From the simulation results, we found that the third modified percentile estimator (using 5th and 95th percentiles) performs better for the estimation of parameters from the continuous uniform distribution.

The Uniform Distribution of Sequences Generated by Iterated Polynomials

In the paper we show that given a polynomial f over ℤ = 0, ±1, ±2, ..., deg f ⩾ 2, the sequence x, f(x), f(f(x)) = f(2)(x), ..., where x is m-adic integer, produces a uniformly distributed set of points in every real unit hypercube under a natural map of the space ℤm of m-adic integers onto unit real interval. Namely, let m, s ∈ ℕ = {1, 2, 3, ...}, m > 1, let κn have a discrete uniform distribution on the set {0, 1, ..., mn - 1. We prove that with n tending to infinity random vectors $$\left(\frac{\kappa_n}{m^n}, \frac{f(\kappa_n){\rm{mod}} m^n}{m^n}, \ldots, \frac{f^{(s-1)}(\kappa_n) {\rm{mod}} m^n}{m^n}\right)$$ weakly converge to a vector having a continuous uniform distribution in the s-dimensional unit hypercube. Analogous results were known before only for the case when s ⩽ 3 and f is a quadratic polynomial (deg f = 2).

Bounded error uniformity of the linear flow on the torus

A linear flow on the torus $$\mathbb {R}^d / \mathbb {Z}^d$$ is uniformly distributed in the Weyl sense if the direction of the flow has linearly independent coordinates over $$\mathbb {Q}$$ . In this paper we combine Fourier analysis and the subspace theorem of Schmidt to prove bounded error uniformity of linear flows with respect to certain polytopes if, in addition, the coordinates of the direction are all algebraic. In particular, we show that there is no van Aardenne–Ehrenfest type theorem for the mod 1 discrepancy of continuous curves in any dimension, demonstrating a fundamental difference between continuous and discrete uniform distribution theory.

Entropy-Based Weighting in One-Dimensional Multiple Errors Analysis of Geological Contacts to Model Geological Structure

In each step of geological modeling, errors have an impact on measurements and workflow processes and, so, have consequences that challenge accurate three-dimensional geological modeling. In the context of classical error theory, for now, only spatial positional error is considered, acknowledging that temporal, attribute, and ontological errors—and many others—are part of the complete error budget. Existing methods usually assumed that a single error distribution (Gaussian) exists across all kinds of spatial data. Yet, across, and even within, different kinds of raw data (such as borehole logs, user-defined geological sections, and geological maps), different types of positional error distributions may exist. Most statistical methods make a priori assumptions about error distributions that impact their explanatory power. Consequently, analyzing errors in multi-source and conflated data for geological modeling remains a grand challenge in geological modeling. In this study, a novel approach is presented regarding the analysis of one-dimensional multiple errors in the raw data used for model geological structures. The analysis is based on the relationship between spatial error distributions and different geological attributes. By assuming that the contact points of a geological subsurface are decided by the geological attributes related to both sides of the subsurface, this assumption means that the spatial error of geological contacts can be transferred into specific probabilities of all the related geological attributes at each three-dimensional point, which is termed the “geological attribute probability”. Both a normal distribution and a continuous uniform distribution were transferred into geological attribute probabilities, allowing different kinds of spatial error distributions to be summed directly after the transformation. On cross-points with multiple raw data with errors that follow different kinds of distributions, an entropy-based weight was given to each type of data to calculate the final probabilities. The weighting value at each point in space is decided by the related geological attribute probabilities. In a test application that accounted for the best estimates of geological contacts, the experimental results showed the following: (1) for line segments, the band shape of geological attribute probabilities matched that of existing error models; and (2) the geological attribute probabilities directly show the error distribution and are an effective way of describing multiple error distributions among the input data.

On trigonometric sums with random frequencies

We prove that if Ik are disjoint blocks of positive integers and nk are independent random variables on some probability space (Ω,F,P) such that nk is uniformly distributed on Ik, then has, with P-probability 1, a mixed Gaussian limit distribution relative to the probability space ((0, 1),B, λ), where B is the Borel σ-algebra and λ is the Lebesgue measure. We also investigate the case when nk have continuous uniform distribution on disjoint intervals Ik on the positive axis.