Symmetric Probability Density Function Research Articles

Inverse power XLindley distribution with statistical inference and applications to engineering data

In this article, a new two-parameter distribution is constructed using the inverse transformation technique on the power XLindley distribution. It is called the inverse power XLindley distribution. As an attractive property, it can generate symmetric and asymmetric probability density functions, ideal for modeling lifetime phenomena. In addition, it is suitable for various real data since the corresponding hazard rate function has an increasing, decreasing, reverse J-shape or J-shape. Essential characteristics and features of our study include quantiles, moments, inverse moments, probability-weighted moments, incomplete moments, and inequality measures. The inferences from the distribution are explored. In particular, the parameters are determined using twelve efficient estimation methods. These methods are maximum likelihood, Anderson-Darling, right-tailed Anderson-Darling, left-tailed Anderson-Darling, Cramér-von Mises, least squares, weighted least squares, maximum product of spacing, minimum spacing absolute distance, minimum spacing absolute-log distance, percentiles, and Kolmogorov. The performance of the resulting estimates is analyzed using Monte Carlo simulation. The numerical results and graphical presentation indicate that the maximum product of spacing estimation approach has the highest accuracy and precision. Using three real data sets and comparisons with other distributions, the effectiveness of the proposed distribution is demonstrated and visually presented.

In Situ Measurement of Track Shape in Cold Spray Deposits

Cold spray is a material deposition technology with a high deposition rate and attractive material properties that has great interest for additive manufacturing (AM). Successfully cold spraying free-form parts that are close to their intended shape, however, requires knowing the fundamental shape of the sprayed track, so that a spray path can be planned that builds up a part from a progressively overlaid sequence of tracks. Several studies have measured track shape using ex situ or quasi-in situ approaches, but an in situ measurement approach has, to the authors’ knowledge, not yet been reported. Furthermore, most studies characterize the track cross section as a symmetric Gaussian probability density function (PDF) with fixed shape parameters. The present study implements a novel in situ track shape measurement technique using a custom-built nozzle-tracking laser profilometry system. The shape of the track is recorded throughout the duration of a spray, allowing a comprehensive investigation of how the track shape evolves as the deposit is built up. A skewed track shape is observed—likely due to the side-injection design of the applicator used—and a skewed Gaussian PDF—a more generalized version of the standard Gaussian PDF—is fit to the track profile. The skewed Gaussian fit parameters are studied across two principal nozzle path parameters: nozzle traverse speed and step size. Empirical relationships between the fit parameters and the nozzle path parameters are derived, and a physics-based inverse relationship between nozzle speed and powder mass deposition rate is obtained. One of the fit parameters is shown to be an effective means of monitoring deposition efficiency during spraying. Overall, the approach presents a promising means of measuring track shape, in situ, as well as modeling it using a more general shape function.

Statistical analysis of the inverse power Zeghdoudi model: Estimation, simulation and modeling to engineering and environmental data

Abstract In this research, we investigate a brand-new two-parameter distribution as a modification of the power Zeghdoudi distribution (PZD). Using the inverse transformation technique on the PZD, the produced distribution is called the inverted PZD (IPZD). Its usefulness in producing symmetric and asymmetric probability density functions makes it the perfect choice for lifetime phenomenon modeling. It is also appropriate for a range of real data since the relevant hazard rate function has one of the following shapes: increasing, decreasing, reverse j-shape or upside-down shape. Mode, quantiles, moments, geometric mean, inverse moments, incomplete moments, distribution of order statistics, Lorenz, Bonferroni, and Zenga curves are a few of the significant characteristics and aspects explored in our study along with some graphical representations. Twelve effective estimating techniques are used to determine the distribution parameters of the IPZD. These include the Kolmogorov, least squares (LS), a maximum product of spacing, Anderson-Darling (AD), maximum likelihood, minimum absolute spacing distance, right-tail AD, minimum absolute spacing-log distance, weighted LS, left-tailed AD, Cramér-von Mises, AD left-tail second-order. A Monte Carlo simulation is used to examine the effectiveness of the obtained estimates. The visual representation and numerical results show that the maximum likelihood estimation strategy regularly beats the other methods in terms of accuracy when estimating the relevant parameters. The usefulness of the recommended distribution for modelling data is illustrated and displayed visually using two real data sets through comparisons with other distributions.

A New Comprehensive Cloud Macrophysics Scheme With a Prognostic Dual‐Triangular PDF

AbstractTo improve cloud simulation in a general circulation model, we develop a new cloud macrophysics scheme that treats detrained cumulus generated by convective detrainment process separately from pure stratus; prognoses two symmetric triangular probability density functions of total specific humidity for each detrained cumulus and pure stratus; and diagnoses both cloud fraction and cloud condensate in all liquid, ice, and mixed‐phases in a consistent and unified way without any adjustment to remove empty or very dense cloud. Supersaturation is allowed within ice cloud. The new scheme (“NEW”) is compared with the previous model (“OLD”) using single‐column simulations for subtropical marine stratocumulus (DYCOMS2) and continental deep convection (ARM97) cases. In DYCOMS2, both NEW and OLD produce vertical profiles of grid‐mean cloud condensate similar to large‐eddy simulation. In ARM97, compared with OLD, NEW simulates less sporadic vertical profiles of in‐cloud condensates, due to consistent diagnosis of cloud fraction and cloud condensate; more continuously‐varying detrained cumulus with time, due to prognostic treatment of detrained cumulus; and ice cloud fraction and ice condensate similar to those of OLD, in spite of completely different treatment of ice cloud processes. The global performance of NEW is similar to OLD with improved relative humidity. Compared to OLD, NEW simulates more and improved cloud condensate, but less and degraded cloud fraction, particularly, in the lower troposphere. Detrained cumulus is moister and colder and has a larger moisture variance than pure stratus. Overall, NEW simulates stronger condensation‐deposition rates than OLD, due in part to the separate treatment of detrained cumulus and pure stratus.

Spontaneous and explicit symmetry breaking of thermoacoustic eigenmodes in imperfect annular geometries

This article deals with the symmetry breaking of azimuthal thermoacoustic modes in annular combustors. Using a nominally symmetric annular combustor, we present experimental evidence of a predicted spontaneous reflectional symmetry breaking, and also an unexpected explicit rotational symmetry breaking in the neighbourhood of the Hopf bifurcation which separates linearly stable azimuthal thermoacoustic modes from self-oscillating modes. We derive and solve a multidimensional Fokker–Planck equation to unravel a unified picture of the phase space topology. We demonstrate that symmetric probability density functions of the thermoacoustic state vector are elusive, because the effect of asymmetries, even imperceptible ones, is magnified close to the bifurcation. This conclusion implies that the thermoacoustic oscillations of azimuthal modes in real combustors will systematically exhibit a statistically dominant orientation of the mode in the vicinity of the Hopf bifurcation.

Distributed Remote Estimation Over the Collision Channel With and Without Local Communication

Internet of Things networks are the large-scale distributed systems consisting of a massive number of simple devices communicating, typically, over a shared wireless medium. This new paradigm requires novel ways of coordinating access to limited communication resources without introducing unreasonable delays. Herein, the optimal design of a remote estimation system with <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$n$</tex-math></inline-formula> sensors communicating with a fusion center via a collision channel of limited capacity <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$k\leq n$</tex-math></inline-formula> is considered. In particular, for independent and identically distributed observations with a symmetric probability density function, we show that the problem of minimizing the mean-squared error with respect to a threshold strategy is quasi-convex. When coordination among sensors via a local communication network is available, the online learning of possibly unknown parameters of the probabilistic model is possible, enabling each sensor to optimize its own threshold autonomously. We propose two strategies for remote estimation with local communication: 1) one strategy swiftly reaches the performance of the optimal decentralized threshold policy and 2) the second strategy approaches the performance of the optimal centralized scheme with a slower convergence rate. A hybrid scheme that combines the best of both approaches is proposed, offering fast convergence and excellent performance.

Scaling exponent analysis and fidelity of the tunable discrete quantum walk in the noisy channel

We studied the behavior of tunable one-dimensional discrete-time quantum walk (DTQW) in the presence of decoherence, modeled as a noisy quantum channel with bit-flip, phase-flip, and general amplitude damping effects. The fidelity of the walker based on the angle parametrized coin operator is examined for different tunable conditions. DTQW with bit-flip and phase-flip or dephasing gave rise to the symmetrical probability density function profile while showing different convergence characteristics on their variance behavior. Meanwhile under decoherence with the general amplitude channel, the walker exhibits the non-symmetric profile of probability density function while retaining faster spread than a normal walk under extreme decoherence which is captured by their scaling exponent. These indicate the existence of the broad class of transport behavior of the walker. We showed that the fidelity of the walker can be optimized by adjusting the parameter of the coin rotation angle θ. This work may be useful for the optimization or better control of the quantum walk under noisy channels.

Impact of Gaussian uncertainty assumptions on probabilistic optimization in particle therapy

Range and setup uncertainties in charged particle therapy may induce a discrepancy between the planned and the delivered dose. Countermeasures based on probabilistic (stochastic) optimization usually assume a Gaussian probability density to model the underlying range and setup error. While this standard assumption is generally taken for granted, this study explicitly investigates the dosimetric consequences if the actual range and setup errors obey a different probability density function (PDF) over the course of treatment to the one used during the probabilistic treatment plan optimization. Discrete random sampling was performed for conventionally and probabilistically optimized proton and carbon ion treatment plans utilizing various PDFs that modeled the setup and range error. This method allowed us to assess the treatment plan robustness against different PDFs of conventional and probabilistic plans, which both explicitly assume Gaussian uncertainties. The induced uncertainty in dose was quantified by estimating the expectation value and standard deviation of the RBE-weighted dose for each PDF on the basis of 2500-5000 random dose samples. Probabilistic dose metrics and standard deviation volume histograms were computed to quantify treatment plan robustness of both optimization approaches. It was shown that the classical planning target volume margin extension concept did not compensate the influence of range and setup errors and consequently resulted in a non-negligible average standard deviation in dose of 7.3% throughout the clinical target volume (CTV). In contrast, probabilistic optimization on normally distributed errors yielded treatment plans that not only entailed a lower standard deviation against normally distributed errors accounted for during optimization, but also lower standard deviations for other symmetric PDFs. It was shown that the impact of an incorrect probability distribution assumption is of lower importance after probabilistic optimization as the average uncertainty in the CTV drops to 3.9%. Probabilistic optimization is an effective tool to create robust particle treatment plans. Normally distributed range and setup error assumptions for probabilistic optimization are a reasonable first approximation and yield treatment plans that are also robust against other PDFs.

An extension of Levin\u2013Ste\u010dkin\u2019s theorem to uniformly convex and superquadratic functions

In this paper, some integral inequalities for uniformly convex functions are studied by using unordered submajorization for cumulative functions. Strongly convex functions and superquadratic functions are considered, too. A Levin–Stečkin like theorem is obtained for such functions. As applications, some bounds for the Fejér functional are derived. A result on the Schur-convexity of averages of convex functions is extended to uniformly convex functions. Some specifications for symmetric functions are also given. A corollary for symmetric probability density functions is established. A Levin–Stečkin type inequality for generalized psi -uniformly convex functions is provided. Some interpretations for Simpson distributions are presented.

Quantization Bias for Digital Correlators

In radio interferometry, the quantization process introduces a bias in the magnitude and phase of the measured correlations which translates into errors in the measurement of source brightness and position in the sky, affecting both the system calibration and image reconstruction. In this paper, we investigate the biasing effect of quantization in the measured correlation between complex-valued inputs with a circularly symmetric Gaussian probability density function (PDF), which is the typical case for radio astronomy applications. We start by calculating the correlation between the input and quantization error and its effect on the quantized variance, first in the case of a real-valued quantizer with a zero mean Gaussian input and then in the case of a complex-valued quantizer with a circularly symmetric Gaussian input. We demonstrate that this input-error correlation is always negative for a quantizer with an odd number of levels, while for an even number of levels, this correlation is positive in the low signal level regime. In both cases, there is an optimal interval for the input signal level for which this input-error correlation is very weak and the model of additive uncorrelated quantization noise provides a very accurate approximation. We determine the conditions under which the magnitude and phase of the measured correlation have negligible bias with respect to the unquantized values: we demonstrate that the magnitude bias is negligible only if both unquantized inputs are optimally quantized (i.e. when the uncorrelated quantization error model is valid), while the phase bias is negligible when (1) at least one of the inputs is optimally quantized, or when (2) the correlation coefficient between the unquantized inputs is small. Finally, we determine the implications of these results for radio interferometry.

ON EXTROPY PROPERTIES OF MIXED SYSTEMS

An expression of the extropy of a mixed system's lifetime was given firstly. Based on this expression, two mixed systems with same signature but with different components were compared. It was shown that the extropy of lifetime of a mixed system equals to that of its dual system if the lifetimes of the components have symmetric probability density function. Moreover, some bounds of the extropy of lifetimes of mixed systems were obtained and the concept of Jensen–extropy (JE) divergence of mixed systems was proposed. The JE divergence is non-negative and it can be used as an alternative information criteria for comparing mixed systems with homogeneous components. To illustrate the applications of JE divergence, some examples are addressed at the end of this paper.

Performance analysis of diffusion least mean fourth algorithm over network

The diffusion least mean fourth (DLMF) algorithm was initially presented for distributed estimate over network with the sub-Gaussian node noise, which can obtain better performance than diffusion least mean square (DLMS) algorithm. Unfortunately, due to the higher order moment of the weight error in the DLMF algorithm and complexity of the network, it is difficult to analyze the mean square performance of the DLMF algorithm. In this paper, the stochastic behaviors including mean behavior and mean square behavior are investigated when the input signals of all nodes are Gaussian processes and the background noises have symmetric probability density functions. The analytical models of the mean weight error and the mean square deviation (MSD) are derived by using a novel manner, which can accurately predict the algorithm behavior. The simulations are carried out to verify the validity of the analysis and compare the performance of DLMF algorithm with that of the DLMS algorithm under different background noise distributions.

Concentration function for the skew-normal and skew-$t$ distributions, with application in robust Bayesian analysis

Data from many applied fields exhibit both heavy tail and skewness behavior. For this reason, in the last few decades, there has been a growing interest in exploring parametric classes of skew-symmetrical distributions. A popular approach to model departure from normality consists of modifying a symmetric probability density function in a multiplicative fashion, introducing skewness. An important issue, addressed in this paper, is the introduction of some measures of distance between skewed versions of probability densities and their symmetric baseline. Different measures provide different insights on the departure from symmetric density functions: we analyze and discuss $L_{1}$ distance, $J$-divergence and the concentration function in the normal and Student-$t$ cases. Multiplicative contaminations of distributions can be also considered in a Bayesian framework as a class of priors and the notion of distance is here strongly connected with Bayesian robustness analysis: we use the concentration function to analyze departure from a symmetric baseline prior through multiplicative contamination prior distributions for the location parameter in a Gaussian model.

Optimal search strategy for a three-dimensional randomly located target

In this paper, we formulate a new search model that finds a three-dimensional randomly located target in a known zone by a single searcher. The searcher wishes to find the target's position, that is given by the value of the three independent trivariate random variables (X, Y, Z) and they have joint symmetric probability density function f(x, y, z). The purpose is to obtain an optimal search strategy that minimises the expected value of the time required to detect the target, assuming trivariate standard normal distributed estimates of its position. An illustrative example is given to demonstrate the applicability of this technique.

An optimal two-stages search plan for a random walk target motion in the plane

This paper addresses the problem of searching for a moving target in the plane by a single autonomous sensor platform unmanned air vehicle (UAV). This sensor consists of a search team from two-searchers. The target moves in the plane with autocollimator linear motion (one-dimensional random walk) either for x-axis or y-axis. The search consists of two stages, the broad search and investigating search. In a broad search the sensor wishes to find the target's initial position, which is given by the value of the two independent random variables (X, Y ) and they have joint symmetric probability density function f(x, y). In an investigation search stage, the search team will be unmanned to detect the target on one of two real lines intersected at the target's initial position. It is desired to search in an optimal manner in each stage to minimise the expected value of the first meeting time between one of the searchers and the target, assuming circular normal distributed estimates of its initial position.

Optimal search strategy for a three-dimensional randomly located target

In this paper, we formulate a new search model that finds a three-dimensional randomly located target in a known zone by a single searcher. The searcher wishes to find the target's position, that is given by the value of the three independent trivariate random variables (X, Y, Z) and they have joint symmetric probability density function f(x, y, z). The purpose is to obtain an optimal search strategy that minimises the expected value of the time required to detect the target, assuming trivariate standard normal distributed estimates of its position. An illustrative example is given to demonstrate the applicability of this technique.

Исследование точности аппроксимации симметричных плотностей вероятности ортогональными представлениями по полиномам Эрмита

Исследование точности аппроксимации симметричных плотностей вероятности ортогональными представлениями по полиномам Эрмита

Modeling of Correlated Two-Dimensional Non-Gaussian Noises

The article describes and analyzes mathematical models of multiplicative and additive non-Gaussian noises affecting the useful signals. For synthesis and analysis, and, hence, the effective design of information systems and radio devices operating in conditions of intense perturbations it is necessary to choose not only the adequate mathematical model of the useful signals and information processes, but also the corresponding models of random effects, possessing in general non-Gaussian multiplicative and additive character. To describe the arbitrary non-Gaussian noises, which are quasiharmonic processes whose spectrum is close (or narrowband) to the band of the desired signal, the authors used a two-dimensional elliptic symmetric probability density function, including two extreme cases: Gaussian processes and a sinusoidal signal with random initial phase distributed uniformly in the interval [0, 2 π]. Model of correlated non-Gaussian narrowband noises of elliptically symmetric two-dimensional probability density function allows you to make a synthesis of information systems and devices based only on a priori knowledge of one-dimensional probability density function and the correlation function. Because knowing one-dimensional probability density function of the instantaneous values, we can determine the probability density function of the envelope; this makes it possible to use the elliptically symmetric probability density function to describe not only additive, but multiplicative (baseband) noises. To describe the real density of probability density function of the non-Gaussian process, the authors propose to approximate its a priori known one-dimensional probability density function and a specially designed transitional probability density function, and show the adequacy of this approximation of the real two-dimensional probability density function of correlated noises.

Statistical regularities of Carbon emission trading market: Evidence from European Union allowances

As an emerging financial market, the trading value of carbon emission trading market has definitely increased. In recent years, the carbon emission allowances have already become a way of investment. They are bought and sold not only by carbon emitters but also by investors. In this paper, we analyzed the price fluctuations of the European Union allowances (EUA) futures in European Climate Exchange (ECX) market from 2007 to 2011. The symmetric and power-law probability density function of return time series was displayed. We found that there are only short-range correlations in price changes (return), while long-range correlations in the absolute of price changes (volatility). Further, detrended fluctuation analysis (DFA) approach was applied with focus on long-range autocorrelations and Hurst exponent. We observed long-range power-law autocorrelations in the volatility that quantify risk, and found that they decay much more slowly than the autocorrelation of return time series. Our analysis also showed that the significant cross correlations exist between return time series of EUA and many other returns. These cross correlations exist in a wide range of fields, including stock markets, energy concerned commodities futures, and financial futures. The significant cross-correlations between energy concerned futures and EUA indicate the physical relationship between carbon emission and energy production process. Additionally, the cross-correlations between financial futures and EUA indicate that the speculation behavior may become an important factor that can affect the price of EUA. Finally we modeled the long-range volatility time series of EUA with a particular version of the GARCH process, and the result also suggests long-range volatility autocorrelations.

A Modified Computational Scheme for the Stochastic Perturbation Finite Element Method

A MODIFIED COMPUTATIONAL SCHEME OF THE STOCHASTIC PERTURBATION FINITE ELEMENT METHOD (SPFEM) IS DEVELOPED FOR STRUCTURES WITH LOW-LEVEL UNCERTAINTIES. THE PROPOSED SCHEME CAN PROVIDE SECOND-ORDER ESTIMATES OF THE MEAN AND VARIANCE WITHOUT DIFFERENTIATING THE SYSTEM MATRICES WITH RESPECT TO THE RANDOM VARIABLES. WHEN THE PROPOSED SCHEME IS USED, IT INVOLVES LIMITED ANALYSES OF DETERMINISTIC SYSTEMS. IN THE CASE OF ONE RANDOM VARIABLE WITH A SYMMETRIC PROBABILITY DENSITY FUNCTION, THE PROPOSED COMPUTATIONAL SCHEME CAN EVEN PROVIDE A RESULT WITH FIFTH-ORDER ACCURACY. COMPARED WITH THE TRADITIONAL COMPUTATIONAL SCHEME OF SPFEM, THE PROPOSED SCHEME IS MORE CONVENIENT FOR IMPLEMENTATION. THREE NUMERICAL EXAMPLES DEMONSTRATE THAT THE PROPOSED SCHEME CAN BE USED IN LINEAR OR NONLINEAR STRUCTURES WITH CORRELATED OR UNCORRELATED RANDOM VARIABLES