Non-central Moments Research Articles

The probability density function of the surface electromyogram and its dependence on contraction force in the vastus lateralis

IntroductionThe probability density function (PDF) of the surface electromyogram (sEMG) depends on contraction force. This dependence, however, has so far been investigated by having the subject generate force at a few fixed percentages of MVC. Here, we examined how the shape of the sEMG PDF changes with contraction force when this force was gradually increased from zero.MethodsVoluntary surface EMG signals were recorded from the vastus lateralis of healthy subjects as force was increased in a continuous manner vs. in a step-wise fashion. The sEMG filling process was examined by measuring the EMG filling factor, computed from the non-central moments of the rectified sEMG signal.Results(1) In 84% of the subjects, as contraction force increased from 0 to 10% MVC, the sEMG PDF shape oscillated back and forth between the semi-degenerate and the Gaussian distribution.(2) The PDF–force relation varied greatly among subjects for forces between 0 and ~ 10% MVC, but this variability was largely reduced for forces above 10% MVC.(3) The pooled analysis showed that, as contraction force gradually increased, the sEMG PDF evolved rapidly from the semi-degenerate towards the Laplacian distribution from 0 to 5% MVC, and then more slowly from the Laplacian towards the Gaussian distribution for higher forces.ConclusionsThe study demonstrated that the dependence of the sEMG PDF shape on contraction force can only be reliably assessed by gradually increasing force from zero, and not by performing a few constant-force contractions. The study also showed that the PDF–force relation differed greatly among individuals for contraction forces below 10% MVC, but this variability was largely reduced when force increased above 10% MVC.

The filling factor of the sEMG signal at low contraction forces in the quadriceps muscles is influenced by the thickness of the subcutaneous layer.

Introduction: It has been shown that, for male subjects, the sEMG activity at low contraction forces is normally "pulsatile", i.e., formed by a few large-amplitude MUPs, coming from the most superficial motor units. The subcutaneous layer thickness, known to be greater in females than males, influences the electrode detection volume. Here, we investigated the influence of the subcutaneous layer thickness on the type of sEMG activity (pulsatile vs. continuous) at low contraction forces. Methods: Voluntary surface EMG signals were recorded from the quadriceps muscles of healthy males and females as force was gradually increased from 0% to 40% MVC. The sEMG filling process was examined by measuring the EMG filling factor, computed from the non-central moments of the rectified sEMG signal. Results: 1) The sEMG activity at low contraction forces was "continuous" in the VL, VM and RF of females, whereas this sEMG activity was "pulsatile" in the VL and VM of males. 2) The filling factor at low contraction forces was lower in males than females for the VL (p = 0.003) and VM (p = 0.002), but not for the RF (p = 0.54). 3) The subcutaneous layer was significantly thicker in females than males for the VL (p = 0.001), VM (p = 0.001), and RF (p = 0.003). 4) A significant correlation was found in the vastus muscles between the subcutaneous layer thickness and the filling factor (p < 0.05). Discussion: The present results indicate that the sEMG activity at low contraction forces in the female quadriceps muscles is "continuous" due to the thick subcutaneous layer of these muscles, which impedes an accurate assessment of the sEMG filling process.

A novel Eulerian model based on central moments to simulate age and reactivity continua interacting with mixing processes

Abstract. In geoscientific models, simulating the properties associated with particles in a continuum can serve many scientific purposes, and this has commonly been addressed using Lagrangian models. As an alternative approach, we present an Eulerian method here: diffusion–advection–reaction type partial differential equations are derived for centralized moments, which can describe the distribution of properties associated with chemicals in reaction–transport models. When the property is age, the equations for centralized moments (unlike non-central moments) do not require terms to account for aging, making this method suitable for modeling age tracers. The properties described by the distributions may also represent kinetic variables affecting reaction rates. In practical applications, continuous distributions of ages and reactivities are resolved to simulate organic matter mineralization in surficial sediments, where macrofaunal and physical mixing processes typically dominate transport. In test simulations, mixing emerged as the predominant factor shaping reactivity and age distributions. Furthermore, the applications showcase the method's aptitude for modeling continua in mixed environments while also highlighting practical considerations and challenges.

A new univariate continuous distribution with applications in reliability

In this article, the odd Lomax Gompertz distribution has been introduced, which is derived by modifying the Gompertz distribution to serve as a baseline model in the odd generalized Lomax distribution. The newly proposed model offers enhanced flexibility and provides a promising alternative for modeling lifetime data. This study seeks to establish a solid theoretical foundation for its application through the exploration of several properties, such as non-central moments, stochastic orderings quantile function, and entropy measure, for the new model. Additionally, by conducting simulation analysis, the performance of the various estimation methods is being assessed, which enables the identification of the most reliable approach for estimating the unknown parameters of the newly developed model. The simulation analysis of the two-risk metrics, namely, value at risk and expected shortfall, revealed the ability of the distribution to capture diverse failure rate patterns, which makes it particularly relevant for assessing financial risks. Finally, the suggested model is practiced to two real-life datasets to provide the compelling evidence of superior flexibility and practical versatility compared to existing models in the literature.

Validation of the filling factor index to study the filling process of the sEMG signal in the quadriceps.

The EMG filling factor is an index to quantify the degree to which an EMG signal has been filled. Here, we tested the validity of such index to analyse the EMG filling process as contraction force was slowly increased. Surface EMG signals were recorded from the quadriceps muscles of healthy subjects as force was gradually increased from 0 to 40% MVC. The sEMG filling process was analyzed by measuring the EMG filling factor (calculated from the non-central moments of the rectified sEMG). (1) As force was gradually increased, one or two prominent abrupt jumps in sEMG amplitude appeared between 0 and 10% of MVC force in all the vastus lateralis and medialis. (2) The jumps in amplitude were originated when a few large-amplitude MUPs, clearly standing out from previous activity, appeared in the sEMG signal. (3) Every time an abrupt jump in sEMG amplitude occurred, a new stage of sEMG filling was initiated. (4) The sEMG was almost completely filled at 2-12% MVC. (5) The filling factor decreased significantly upon the occurrence of an sEMG amplitude jump, and increased as additional MUPs were added to the sEMG signal. (6) The filling factor curve was highly repeatable across repetitions. It has been validated that the filling factor is a useful, reliable tool to analyse the sEMG filling process. As force was gradually increased in the vastus muscles, the sEMG filling process occurred in one or two stages due to the presence of abrupt jumps in sEMG amplitude.

Moments of the Negative Multinomial Distribution

The negative multinomial distribution appears in many areas of applications such as polarimetric image processing and the analysis of longitudinal count data. In previous studies, general formulas for the falling factorial moments and cumulants of the negative multinomial distribution were obtained. However, despite the availability of the moment generating function, no comprehensive formulas for the moments have been calculated thus far. This paper addresses this gap by presenting general formulas for both central and non-central moments of the negative multinomial distribution. These formulas are expressed in terms of binomial coefficients and Stirling numbers of the second kind. Utilizing these formulas, we provide explicit expressions for all central moments up to the fourth order and all non-central moments up to the eighth order.

EMG probability density function: a new way to look at EMG signal filling from single motor unit potential to full interference pattern.

An analytical derivation of the EMG signal's amplitude probability density function (EMG PDF) is presented and used to study how an EMG signal builds-up, or fills, as the degree of muscle contraction increases. The EMG PDF is found to change from a semi-degenerate distribution to a Laplacian-like distribution and finally to a Gaussian-like distribution.We present a measure, the EMG filling factor, to quantify the degree to which an EMG signal has been built-up. This factor is calculated from the ratio of two non-central moments of the rectified EMG signal. The curve of the EMG filling factor as a function of the mean rectified amplitude shows a progressive and mostly linear increase during early recruitment, and saturation is observed when the EMG signal distribution becomes approximately Gaussian. Having presented the analytical tools used to derive the EMG PDF, we demonstrate the usefulness of the EMG filling factor and curve in studies with both simulated signals and real signals obtained from the tibialis anterior muscle of 10 subjects. Both simulated and real EMG filling curves start within the 0.2 to 0.35 range and rapidly rise towards 0.5 (Laplacian) before stabilizing at around 0.637 (Gaussian). Filling curves for the real signals consistently followed this pattern (100% repeatability within trials in 100% of the subjects). The theory of EMG signal filling derived in this work provides (a) an analytically consistent derivation of the EMG PDF as a function of motor unit potentials and motor unit firing patterns; (b) an explanation of the change in the EMG PDF according to degree of muscle contraction; and (c) a way (the EMG filling factor) to quantify the degree to which an EMG signal has been built-up.

Generalized Moment Generating Functions for Some Continuous Multivariate Probability Distributions

The traditional moment generating functions of random variables and their probability distributions are known to not exist for all distributions and/or at all points and, where they exist, serious difficult and tedious manipulations are needed for the evaluation of higher central and non-central moments. This paper developed the generalized multivariate moment generating function for some random vectors/matrices and their probability distribution functions with the intention to replace the traditional/conventional moment generating functions due to their simplicity and versatility. The new functions were developed for the multivariate gamma family of distributions, the multivariate normal and the dirrichlet distributions as a binomial expansion of the expected value of an exponent of a random vector/matrix about an arbitrarily chosen constant. The functions were used to generate moments of random vectors/matrices and their probability distribution functions and the results obtained were compared with those from existing traditional/conventional methods. It was observed that the functions generated same results as the traditional/conventional methods; in addition, they generated both central and non-central moments in the same simple way without requiring further tedious manipulations; they gave more information about the distributions, for instance while the traditional method gives skewness and kurtosis values of and respectively for -variate multivariate normal distribution, the new methods gives ((0))p*1 and respectively and; they could generate moments of integral and real powers of random vectors/matrices.

Astronomik bir veri kümesi uygulaması ile alfa çarpık hiperbolik sekant dağılımı

This work demonstrates the attractivity of the alpha-skew hyperbolic secant distribution, a new skewed distribution based on the alpha-skew technique and the hyperbolic secant distribution. In the first part, we determine its main features, including its cumulative distribution function, modality, non-central moments, skewness, kurtosis, moment generating function and characteristic function. The remaining part is devoted to the model applicability in a statistical context. As a first step, the parameters are estimated by maximum likelihood estimates. Then, we perform a data fitting experiment and compare the values of the Akaike and Bayesian information criteriawith those of some other similar distributions. By considering an astronomical dataset and valuable competitors also based on the alpha-skew technique, the alpha-skew hyperbolic secant distribution turns out to be the best.

Modeling the number of component failures: A Poison-geometric distribution

A two new parameter discrete distribution which generalizes the Poisson distribution is introduced. Some of the properties of the new distribution including the non-central moments, the probability generating function and the hazard rate function are derived. The unknown parameters of the new distribution are estimated using the maximum likelihood approach. Numerical example to show the applicability of the new distribution is based on a real-life data set. Comparison with Poisson distribution shows that the new distribution provided better fit on the real-life dataset. Furthermore, the likelihood ratio test indicates that the generalized parameter of the new distribution indeed, improved the potential of the new distribution in providing adequate fit for the real-life dataset.

Intervened Exponential Distribution: Properties and Applications

This manuscript aims to study the intervention-based probability model. Statistical and reliability properties such as the expressions for, cumulative density function (CDF), mean deviations about mean and median, rth order central and non-central moments, ”generation functions” for moments have been derived. Moreover, the expression for reliability function, hazard rate, reverse hazard rate, aging intensity, mean residual life function, stress-strength reliability, and entropy metrics due to R´enyi and Shannon are also derived. Monte Carlo simulation study performance of maximum likelihood estimates (MLEs) has been carried out, followed by calculations of Average Bias (ABias), and Mean Square Error (MSE). The applicability of the model in real-life situations has been discussed by analyzing the two real-life data sets.

NON-CENTRAL MOMENTS OF THE TRUNCATED NORMAL VARIABLE IN FINANCE

This paper computes the Non-central Moments of the Truncated Normal variable, i.e. a Normal constrained to assume values in the interval with bounds that may be finite or infinite. We define two recursive expressions where one can be expressed in closed form. Another closed form is defined using the Lower Incomplete Gamma Function. Moreover, an upper bound for the absolute value of the Non-central Moments is determined. The numerical results of the expressions are compared and the different behavior for high value of the order of the moments is shown. The limitations to the use of Truncated Normal distributions with a lower negative limit regarding financial products are considered. Limitations in the application of Truncated Normal distributions also arise when considering a CRRA utility function.

An extensive extension of exponentiated exponential distribution using alpha power transformation – statistical properties and applications in engineering science

Abstract In this article, an extension of exponentiated exponential distribution is familiarized by adding an extra parameter to the parent distribution using alpha power technique. The new distribution obtained is referred to as Alpha Power Exponentiated Exponential Distribution. Various statistical properties of the proposed distribution like mean, variance, central and non-central moments, reliability functions and entropies have been derived. Two real life data sets have been applied to check the flexibility of the proposed model. The new density model introduced provides the better fit when compared with other related statistical models.

On the T-X Class of Topp Leone-G Family of Distributions: Statistical Properties and Applications

This article investigates the T-X class of Topp Leone- G family of distributions. Some members of the new family are discussed. The exponential-Topp Leone-exponential distribution (ETLED) which is one of the members of the family is derived and some of its properties which include central and non-central moments, quantiles, incomplete moments, conditional moments, mean deviation, Bonferroni and Lorenz curves, survival and hazard functions, moment generating function, characteristic function and R`enyi entropy are established. The probability density function (pdf) of order statistics of the model is obtained and the parameter estimation is addressed with the maximum likelihood method (MLE). Three real data sets are used to demonstrate its application and the results are compared with two other models in the literature.

Maximum Product of Spacing Parameter Estimation of Gompertz Rayleigh Distribution and Application to Rainfall Datasets

Gompertz Rayleigh (GomR) distribution was introduced in an earlier study with few statistical properties derived and parameters estimated using only the most common traditional method, Maximum Likelihood Estimation (MLE). This paper aimed at deriving more statistical properties of the GomR distribution, estimating the three unknown parameters via a competitive method, Maximum Product of Spacing (MPS) and evaluating goodness of fit using rainfall data sets from Nigeria, Malaysia and Argentina.&#x0D; Properties of statistical distributions including distribution of smallest and largest order statistics, cumulative or integrated hazard function, odds function, rth non-central moments, moment generating function, mean, variance and entropy measures for GomR distribution were explicitly derived. The fitted data sets reveal the flexibility of GomR distribution over other distributions been compared with. Simulation study was used to evaluate the consistency, accuracy and unbiasedness of the GomR distribution parameter estimates obtained from the method of MPS. The study found that GomR distribution could not provide a better fit for Argentine rainfall data but it was the best distribution for the rainfall data sets from Nigeria and Malaysia in comparison with the distributions; Generalized Weibull Rayleigh (GWR), Exponentiated Weibull Rayleigh (EWR), Type (II) Topp Leone Generalized Inverse Rayleigh (TIITLGIR), Kumarawamy Exponential Inverse Raylrigh (KEIR), Negative Binomial Marshall-Olkin Rayleigh (NBMOR) and Exponentiated Weibull (EW). Furthermore, the estimates from MPSE were consistent as the sample size increases but not as efficient as those from MLE.

ON THE PROPERTIES AND APPLICATIONS OF A NEW EXTENSION OF EXPONENTIATED RAYLEIGH DISTRIBUTION

Statistical distributions already in existence are not the most appropriate model that adequately describes real-life data such as those obtained from experimental investigations. Therefore, there are needs to come up with their extended forms to give substitutive adaptable models. By adopting the method of Transformed-Transformer family of distributions, an extension of Exponentiated Rayleigh distribution titled Gompertz- Exponentiated Rayleigh (GOM-ER) distribution was proposed and proved to be valid. Some properties of the new distribution including random number generator, quartiles, distribution of smallest and largest order statistics, reliability function, hazard rate function, cumulative or integrated hazard function, odds function, non-central moments, moment generating function, mean, variance and entropy measures were derived. Using the methods of maximum likelihood and maximum product of spacing, the four unknown parameters were estimated. Shapes of the hazard function depicts that GOM-ER is a distribution that is strictly increasing while those of the PDF depicts that GOM-ER can be skewed or symmetrical. Two datasets were fitted to determine the flexibility of GOM-ER. Simulation study evaluates the consistency, accuracy and unbiasedness of the GOM-ER parameter estimates obtained from the two frequentist estimation methods adopted.

Reliability modeling based on Maximum Entropy and non-central moments as an alternative for RCM schemes or replaceable systems

This paper offers an alternative to determine reliability-centered maintenance (RCM) schemes for replaceable systems, when replacement times are censored and only the information that maintenance technicians, from the subjectivity of their experience, is available. Using differential entropy in information theory, and exploiting Lagrangian optimization algorithms, a Generalized Probability Density of Maximum Entropy (GPDME) is extracted. Lagrangian techniques provide a set of parameters that characterize the GPDME, the estimation of the parameters is done by first order perturbation of the integral of non-central moments, with which, the GPDME is typically built. In the emerging industry, RCM maintenance plans are not a common standard, in an attempt to put into practice, the benefits of RCM to this industrial segment, a case study, where the presented methodology was applied is provided. In the discussion and conclusions section, the areas of opportunity that are observed in the methodology presented in this work are adressed.

General Formulas for the Central and Non-Central Moments of the Multinomial Distribution

We present the first general formulas for the central and non-central moments of the multinomial distribution, using a combinatorial argument and the factorial moments previously obtained in Mosimann (1962). We use the formulas to give explicit expressions for all the non-central moments up to order 8 and all the central moments up to order 4. These results expand significantly on those in Newcomer (2008) and Newcomer et al. (2008), where the non-central moments were calculated up to order 4.

AnnuityRIR: an R-package to approximate the value of an annuity according to the non-central moments of the capitalization factor

The aim of this paper is to design the package of the R statistical software called “Annuity Random Interest Rate”, referred hereinafter as AnnuityRIR, in order to calculate the value of an n-annuity with payments of one unit each when the interest rate is random. To do this, we have employed different approaches; the two main methodologies treated in this study consider that all non-central moments of the capitalization factor are known, or contrarily some of them are unknown. Consequently, five different approaches have been developed and the practical application of the proposed methods is reflected in this paper by pricing an annuity with a random risk-free interest rate during the last ten years. The version is available from CRAN: https://cran.r-project.org/web/packages/AnnuityRIR/index.html.

The Generalized Ampadu-G Family of Distributions: Properties, Applications and Characterizations

This paper introduces a new class of distributions called the generalized Ampadu-G (GA-G for short) family of distributions, and with a certain restriction on the parameter space, the family is shown to be a life-time distribution. The shape of the density function and hazard rate function of the GA-G family is described analytically. When G follows the Weibull distribution, the generalized Ampadu-Weibull (GA-W for short) is presented along with its hazard and survival function. Several sub-models of the GA-W family are presented. The transformation technique is applied to this new family of distributions, and we obtain the quantile function of the new family. Power series representations for the cumulative distribution function (CDF) and probability density function (PDF) are also obtained. The rth non-central moments, moment generating function, and Renyi entropy associated with the new family of distributions are derived. Characterization theorems based on two truncated moments and conditional expectation are also presented. A simulation study is also conducted, and we find that using the method of maximum likelihood to estimate model parameters is adequate. The GA-W family of distributions is shown to be practically significant in modeling real life data, and is shown to be superior to some non-trivial generalizations of the Weibull distribution. A further development concludes the paper.