Manifestation of Energy and Entropy of Particles in a Box

The energy and entropy, expressed in free energy, determine the behavior of a system. Therefore, infinite knowledge of these two quantities leads to precise prediction of the system's trajectories. Here, we study how the energy and entropy affect the distribution of a two-component system in a box. First, using a model, we intuitively show that large particles prefer to position at contact with the wall as it accompanies an increase of the system's entropy. We intuitively show that this is a consequence of maximizing the accessible states for fluctuating degrees of freedom as a portion of excluded volumes reside outside of the box when they locate near the wall. Then we employ molecular dynamics simulations to extract the effect of entropy and energy on the binary mixture distribution and how they compete with each other to determine the system's configuration. While particle-particle and particle-wall attraction energies affect the distribution of particles, we show that the emergent entropic forces --- quasi-gravitational --- have a significant contribution to the configuration of the system. This system is realized clearly for a binary mixture of hard spheres in a box with reflective walls.

The heritability of BMI varies across the range of BMI: a heritability curve analysis in a twin cohort

The heritability of traits such as body mass index (BMI), a measure of obesity, is generally estimated using family, twin, and increasingly by molecular genetic approaches. These studies generally assume that genetic effects are uniform across all trait values, yet there is emerging evidence that this may not always be the case. This paper analyzes twin data using a recently developed measure of heritability called the heritability curve. Under the assumption that trait values in twin pairs are governed by a flexible Gaussian mixture distribution, heritability curves may vary across trait values. The data consist of repeated measures of BMI on 1506 monozygotic (MZ) and 2843 like-sexed dizygotic (DZ) adult twin pairs, gathered from multiple surveys in older Finnish Twin Cohorts. The heritability curve and BMI value-specific MZ and DZ pairwise correlations were estimated, and these varied across the range of BMI. MZ correlations were highest at BMI values from 21 to 24, with a stronger decrease for women than for men at higher values. Models with additive and dominance effects fit best at low and high BMI values, while models with additive genetic and common environmental effects fit best in the normal range of BMI. Thus, we demonstrate that twin and molecular genetic studies need to consider how genetic effects vary across trait values. Such variation may reconcile findings of traits with high heritabilities and major differences in mean values between countries or over time.

Mixture of Lindley and Lognormal Distributions: Properties, Estimation, and Application

Finite mixture models provide a flexible tool for handling heterogeneous data. This paper introduces a new mixture model which is the mixture of Lindley and lognormal distributions (MLLND). First, the model is formulated, and some of its statistical properties are studied. Next, maximum likelihood estimation of the parameters of the model is considered, and the performance of the estimators of the parameters of the proposed models is evaluated via simulation. Also, the flexibility of the proposed mixture distribution is demonstrated by showing its superiority to fit a well-known real data set of 128 bladder cancer patients compared to several mixture and nonmixture distributions. The Kolmogorov Smirnov test and some information criteria are used to compare the fitted models to the real dataset. Finally, the results are verified using several graphical methods.

Short-Term Photovoltaic Power Interval Prediction Based on the Improved Generalized Error Mixture Distribution and Wavelet Packet -LSSVM

The periodicity and non-stationary nature of photovoltaic (PV) output power make the point prediction result contain very little information, increase the difficulty of describing the prediction uncertainty, and it is difficult to ensure the most efficient operation of the power system. Effectively predicting the PV power range will greatly improve the economics and stability of the grid. Therefore, this paper proposes an improved generalized based on the combination of wavelet packet (WP) and least squares support vector machine (LSSVM) to obtain higher accuracy point prediction results. The error mixed distribution function is used to fit the probability distribution of the prediction error, and the probability prediction is performed to obtain the prediction interval. The coverage rate and average width of the prediction interval are used as indicators to evaluate the prediction results of the interval. By comparing with the results of conventional methods based on normal distribution, at 95 and 90% confidence levels, the method proposed in this paper achieves higher coverage while reducing the average bandwidth by 5.238 and 3.756%, which verifies the effectiveness of the proposed probability interval prediction method.

EM Estimation for Zero- and k-Inflated Poisson Regression Model

Count data with excessive zeros are ubiquitous in healthcare, medical, and scientific studies. There are numerous articles that show how to fit Poisson and other models which account for the excessive zeros. However, in many situations, besides zero, the frequency of another count k tends to be higher in the data. The zero- and k-inflated Poisson distribution model (ZkIP) is appropriate in such situations The ZkIP distribution essentially is a mixture distribution of Poisson and degenerate distributions at points zero and k. In this article, we study the fundamental properties of this mixture distribution. Using stochastic representation, we provide details for obtaining parameter estimates of the ZkIP regression model using the Expectation–Maximization (EM) algorithm for a given data. We derive the standard errors of the EM estimates by computing the complete, missing, and observed data information matrices. We present the analysis of two real-life data using the methods outlined in the paper.

Wind energy potential assessment based on wind speed, its direction and power data

Based on wind speed, direction and power data, an assessment method of wind energy potential using finite mixture statistical distributions is proposed. Considering the correlation existing and the effect between wind speed and direction, the angular-linear modeling approach is adopted to construct the joint probability density function of wind speed and direction. For modeling the distribution of wind power density and estimating model parameters of null or low wind speed and multimodal wind speed data, based on expectation–maximization algorithm, a two-component three-parameter Weibull mixture distribution is chosen as wind speed model, and a von Mises mixture distribution with nine components and six components are selected as the models of wind direction and the correlation circular variable between wind speed and direction, respectively. A comprehensive technique of model selection, which includes Akaike information criterion, Bayesian information criterion, the coefficient of determination R2 and root mean squared error, is used to select the optimal model in all candidate models. The proposed method is applied to averaged 10-min field monitoring wind data and compared with the other estimation methods and judged by the values of R2 and root mean squared error, histogram plot and wind rose diagram. The results show that the proposed method is effective and the area under study is not suitable for wide wind turbine applications, and the estimated wind energy potential would be inaccuracy without considering the influence of wind direction.