Comparison of selected mathematical functions for the analysis of growth behavior of items and physical interpretation of AvramiWeibull function

Empirical data of sigmoidal-shaped y(t) growth behavior of different types of items, such as papers and citations earned by individual and all successively published papers of selected top-cited authors, germination of tomato seeds and three different bacteria, are analyzed and compared by Avrami-Weibull, Verhulst (logistic) and Gompertz functions. It was found that: (1) Avrami-Weibull function describes different types of the data better than Gompertz and Verhulst funtions, and (2), in comparison with Verhulst and Gompertz functions, Avrami-Weibull function, expressed in the form: y(t)/ymax = 1-exp[(t/Q)q] (where ymax is the maximum value of y(t) when t ® ¥, and Q and q are constants), is equally very versatile in explaining the generation rate dy(t)/dt of items in terms of its parameters Q and q. Using the basic concepts involved in the derivation of Avrami-Weibull function for overall crystallization from melt and supersaturated solution, the growth behavior of cumulative number y(t) of items produced at time t by individual (simple) sources and collectives or groups of simple sources (i.e. complex or composite sources) is presented. Comparison of the process of receiving of citations by papers with the processes of occurrence of chemical reactions and crystallization of solid phases from melts and supersaturated solutions shows that this process is similar to that of overall crystallization of solid phases from melts and solutions. Analysis of growth of citations using Avrami-Weibull function to individual papers published by different authors shows that 1 < q < 4 for most cases. This suggests that the process of citations to individual articles is mainly determined by progressive nucleation mode involving both diffusion and integration of published knowledge.

Strong temporal variation in treefall and branchfall rates in a tropical forest is related to extreme rainfall: results from 5 years of monthly drone data for a 50 ha plot

A mechanistic understanding of how tropical-tree mortality responds to climate variation is urgently needed to predict how tropical-forest carbon pools will respond to anthropogenic global change, which is altering the frequency and intensity of storms, droughts, and other climate extremes in tropical forests. We used 5 years of approximately monthly drone-acquired RGB (red–green–blue) imagery for 50 ha of mature tropical forest on Barro Colorado Island, Panama, to quantify spatial structure; temporal variation; and climate correlates of canopy disturbances, i.e., sudden and major drops in canopy height due to treefalls, branchfalls, or the collapse of standing dead trees. Canopy disturbance rates varied strongly over time and were higher in the wet season, even though wind speeds were lower in the wet season. The strongest correlate of monthly variation in canopy disturbance rates was the frequency of extreme rainfall events. The size distribution of canopy disturbances was best fit by a Weibull function and was close to a power function for sizes above 25 m2. Treefalls accounted for 74 % of the total area and 52 % of the total number of canopy disturbances in treefalls and branchfalls combined. We hypothesize that extremely high rainfall is a good predictor because it is an indicator of storms having high wind speeds, as well as saturated soils that increase uprooting risk. These results demonstrate the utility of repeat drone-acquired data for quantifying forest canopy disturbance rates at fine temporal and spatial resolutions over large areas, thereby enabling robust tests of how temporal variation in disturbance relates to climate drivers. Further insights could be gained by integrating these canopy observations with high-frequency measurements of wind speed and soil moisture in mechanistic models to better evaluate proximate drivers and with focal tree observations to quantify the links to tree mortality and woody turnover.

Characterizing Stand and Biomass Tables from Diameter Distribution Models: A Case Study for Mixed-Hardwood Forests in Eastern Tennessee, USA

Accurately assessing forest structure and productivity is critical to making timely management decisions and monitoring plant communities. This study aims to evaluate the prediction accuracy of site-level stand and biomass tables from the diameter distribution models. The efficacy of the single Weibull function and two finite mixture models was compared for six species groups on three mixed-hardwood sites in eastern Tennessee, USA. To evaluate model performance, two types of stand/biomass tables were generated. The first type was constructed from all species on a given site (site-specific), whereas the second type was built for a single species from all sites (species-specific). Results indicate that both types of stand and biomass tables were consistently well quantified by the two-component mixture model in terms of goodness of fit, parsimony and robustness. The two-component mixture model better characterized the complex, multimodal diameter distributions than the single Weibull model, which underpredicted the upper portion of the distributions. The three-component model tends to overfit the data, which results in lower prediction accuracy. Among the three models examined, the two-Weibull mixture model is suggested to construct site-level stand/biomass tables, which provides more reliable and accurate predictions to assess forest structure and product class. Study Implications Compared to pine monocultures, diameter distribution models for upland mixed-hardwood forests in the Southeastern United States have not been widely explored. Mixed-hardwood forests not only supply high-quality timber for domestic and international uses, but also provide various ecosystem services and essential habitats for wildlife. The finite mixture model has been proposed for characterizing the irregular forms of diameter distribution curves, but the reliability of this method has not been explicitly examined for a wide variety of species. This study provided insights for natural resources managers to select appropriate models when modeling stand and biomass tables for mixed-hardwood forests.

APPLICATION OF STATISTICAL TECHNIQUES IN THE PRACTICE OF BUILDING A DEFILIENCE SCALE

The main substantial features of the PD curve (default probability) formed in practical modeling are substantiated in the articles. It is proved that the main characteristics of the PD curve are that it is based on data on the actually restored default rate in each of the risk classes over a period of time and has a shape that approximate for coincides with the exposure function. It is shown that the best aspect that affects the calibration is the number of rating classes and ways to build them. It is determined that the slope of the curve demonstrates the classification model of efficiency. It is determined that the slope of the curve demonstrates the classification efficiency of the model. Models with high discriminant properties are characterized by a curve shape that has a slow increase in the rating classes of the upper part of the scale and a significant acceleration of growth in the last risk classes. Two main approaches to determining the number of risk classes are analyzed: the percentile-based approach and the equal score range approach. It is shown that when forming classes, it is necessary to take into account the total amount of sample observations, the proportion of «good» and «bad», and choose the number of classes so that it is not too large and not too small. Calibration practice shave been shown to be influenced by data, purpose, and study limitations. The application of the least squares method and the extrapolation method is considered on practical examples. The least squares method and in particular the derived extrapolation method allow to build a calibration curve on the basis of data on the relative frequency of defaults. It is determined that the mathematical apparatus of the family of nonlinear curves allows to model the process of exponential growth with different levels of intensity. The exponential curve and related functions may be useful in modeling more conservative PD estimates or for models with highly discriminatory properties, while the Weibull function, S-curve, and power function may be better adapted to moderate growth processes. The application of practical methods of constructing the PD scale is important for many domestic banking professionals who deal with internal models of credit risk. Keywords: Calibration, Default, Probability, Curves, Probability of default curve calibration, Least squares method, Extrapolation method. JEL Classіfіcatіon С44 Formulas: 21; fig.: 1; tabl.: 7; bibl.: 10.

Viscosity Approximation of PDMS Using Weibull Function

The viscosity of a fluid is one of its basic physico-chemical properties. The modelling of this property as a function of temperature has been the subject of intensive studies. The knowledge of how viscosity and temperature variation are related is particularly important for applications that use the intrinsic friction of fluids to dissipate energy, for example viscous torsional vibration dampers using high viscosity poly(dimethylsiloxane) as a damping factor. This article presents a new method for approximating the dynamic viscosity of poly(dimethylsiloxane). It is based on the three-parameter Weibull function that far better reflects the relationship between viscosity and temperature compared with the models used so far. Accurate mapping of dynamic viscosity is vitally important from the point of view of the construction of viscous dampers, as it allows for accurate estimation of their efficiency in the energy dissipation process.

Exploring Wind Energy Potential as a Driver of Sustainable Development in the Southern Coasts of Iran: The Importance of Wind Speed Statistical Distribution Model

Wind energy as a clean and inexhaustible source of renewable energy can be a key element of sustainable development that decreases dependence of countries on fossil fuels. Therefore, implementing accurate and comprehensive feasibility studies in countries with a high level of consumption of traditional energy resources is vital; an approach encouraged and supported by green funds and climate change action. It is also crucial to helping spur economic and sustainable growth of these countries. In this regard, this study aims at accurate evaluation of onshore wind energy potential in seven coastal cities in the south of Iran. Six Probability Distribution Functions (PDFs) were examined over representative stations. It was deduced that the Weibull function, which is the most used PDF in similar studies, was only applicable to one station. Here, Gamma distribution offered the best fit for three stations and for the other ones, Generalized Extreme Value (GEV) performed better. Considering the ranking of six examined PDFs and the simplicity of Gamma, it was identified as the effective function in the southern coasts of Iran bearing in mind the geographic distribution of stations. Moreover, six wind energy converter power curve functions contributed to investigating the capacity factor. It is found that, using only one function could cause under- or over-estimation. Then, stations were classified based on the National Renewable Energy Laboratory system. Last but not least, examining a range of wind energy converters enabled scholars to extend this study into practice and prioritize the development of stations considering budget limits.

An Adapted Weibull Function for Agricultural Applications

The Weibull function is applied extensively in the life sciences and engineering, but under-used in agriculture. The function was consequently adapted to include parameters and metrics that increase its utility for characterizing agricultural processes. The parameters included initial and final dependent variables (Y0 and YF, respectively), initial independent variable (x0), a scale constant (k), and a shape constant (c). The primary metrics included mode, integral average, domain, skewness and kurtosis. Nested within the Weibull function are the Mitscherlich and Rayleigh functions where c is fixed at 1 and 2, respectively. At least one of the three models provided an excellent fit to six example agricultural datasets, as evidenced by large adjusted coefficient of determination (RA2 ≥ 0.9266), small normalized mean bias error (MBEN ≤ 1.49 %), and small normalized standard error of regression (SERN ≤ 8.08 %). The Mitscherlich function provided the most probable (PX) representation of corn (Zea Mays L.) yield (PM = 87.2 %), Rayleigh was most probable for soil organic carbon depth profile (PR = 96.4 %), and Weibull was most probable for corn seedling emergence (PW = 100 %), nitrous oxide emissions (PW = 100 %), nitrogen mineralization (PW = 58.4 %), and soil water desorption (PW = 100 %). The Weibull fit to the desorption data was also equivalent to those of the well-established van Genuchten and Groenevelt-Grant desorption models. It was concluded that the adapted Weibull function has good potential for widespread and informative application to agricultural data and processes.

A python program to model and analyze wind speed data

A python program has been developed to analyze wind distributions using the Weibull density function. A two-parameter Weibull function is frequently used to model and assess wind potential and wind distribution. This python program finds first Weibull parameters from the recorded wind data by five different methods, namely, Empirical Method(EPM), Method of Moment (MoM), Energy Pattern Factor Method (EPFM), Maximum Likelihood Method (MLM), Modified Maximum Likelihood Method (MMLM), the parameters are then used to find theoretically fitted pdfs. The program is implemented on wind distribution of two cities of Pakistan (Chakri and Sadiq Abad). The program-generated pdfs were plotted with the histogram of recorded data, the fitting was excellent. To check the validity of the fitted pdfs, statistical errors Root Mean Square (RMSE), MeanAbsolute Percent Error (MAPE), Mean Absolute Error (MABE), and Chi-square statistic are calculated. In all cases,these statistical errors are well below the acceptance range. Both pictorial results and numerical values of statistical errors indicate the performance of the python program to analyze wind speed data