scholarly journals Determination of partition surface of grained material by means of non-classical approximation methods of distributions functions of particle size and density

2016 ◽  
Vol 32 (1) ◽  
pp. 137-154
Author(s):  
Tomasz Niedoba
Keyword(s):  
Particle Size ◽  
Distribution Function ◽  
Statistical Methods ◽  
Particle Density ◽  
Distribution Functions ◽  
Alternative Methods ◽  
Hard Coal ◽  
Classical Distribution ◽  
Partition Surface ◽  
Non Parametric

Abstract In this paper, the grained material analyzed was hard coal collected from one of the mines located in Upper Silesia. Material was collected from a dust jig where it was separated in industrial conditions by concentrate and waste. It was then screened in sieves and it was separated in dense media into density fractions. Both particle size distribution and particle density distribution for feed and concentrate were approximated by several classical distribution functions. The best results were obtained by means of the Weibull (RRB) distribution function. However, because of the unsatisfying quality of approximations it was decided to apply non-parametric statistical methods, which became more and more popular alternative methods in conducting statistical investigations. In the paper, the kernel methods were applied to this purpose and the Gauss kernel was accepted as the kernel function. Kernel method, which is relatively new, gave much better results than classical distribution functions by means of the least squared method. Both classical and non-parametric obtained distribution functions were evaluated by means of mean standard error, the values of which proved that they sufficiently well approximate the empirical data. Such function forms were then applied to determine the theoretical distribution function for vector (D, P), where D is the random variable describing particle size and P – its density. This approximation was sufficiently acceptable. That is why it served to determine the equation of partition surface dependent on particle size and particle density describing researched material. The obtained surface proves that it is possible to evaluate material separation which occurs during mineral processing operations, such as jigging, by means of more than one feature of researched material. Furthermore, its quality confirms that it is justified to apply non-parametric statistical methods instead of commonly used classical ones.

Convergence rate of perturbed empirical distribution functions

1979 ◽  
Vol 16 (01) ◽  
pp. 163-173 ◽  
Author(s):  
B. B. Winter
Keyword(s):  
Distribution Function ◽  
Convergence Rate ◽  
Uniform Distribution ◽  
Empirical Distribution ◽  
Distribution Functions ◽  
Unit Mass ◽  
Common Distribution ◽  
Common Distribution Function ◽  
Empirical Distribution Functions ◽  
Non Parametric

Given an i.i.d. sequenceX1,X2, … with common distribution function (d.f.) F, the usual non-parametric estimator ofFis the e.d.f.Fn;whereUois the d.f. of the unit mass at zero. Anadmissible perturbation of the e.d.f., say, is obtained ifUois replaced by a d.f., whereis a sequence of d.f.'s converging weakly toUo.Suchperturbed e.d.f.′s arise quite naturally as integrals of non-parametric density estimators, e.g. as. It is shown that if F satisfies some smoothness conditions and the perturbation is not too drastic then‘has the Chung–Smirnov property'; i.e., with probability one,1. But if the perturbation is too vigorous then this property is lost: e.g., if F is the uniform distribution andHnis the d.f. of the unit mass atn–αthen the above lim sup is ≦ 1 or = ∞, depending on whetheror

scholarly journals Factor Analysis and Mathematical Modeling in Determining the Quality of Coal

2020 ◽  
Vol 1 (1) ◽  
Author(s):  
Tomasz NIEDOBA ◽  
Agnieszka SUROWIAK ◽  
Paulina PIĘTA
Keyword(s):  
Particle Size ◽  
Factor Analysis ◽  
Mean Squared Error ◽  
Particle Density ◽  
Hard Coal ◽  
Combustion Heat ◽  
Coking Coal ◽  
Particle Size Fractions ◽  
Density Fractions ◽  
Sphericity Test

The separation of coal material of three types of coals originating from three various Polish hard coal mines (types 31, 34.2 and 35, according to Polish nomenclature, which were steam coal, semi-coking coal and coking coal) into particle size fractions and then into particle density fractions was done and then the following parameters were measured for each particle size-density fraction: combustion heat, ash contents, sulfur contents, volatile parts contents, analytic moisture. In this way a 7-dimensional vector of data was created. Using methods of factor analysis the important features of coal were selected, which decide about their membership to individual types. To evaluate the appropriateness of the applied method the Bartlett’s sphericity test as well coefficient of Kaiser-Mayer-Olkin (KMO) were used. To select important factors the Kaiser criterion and Cattell’s scree test were used. The obtained results were compared with the results obtained in previous works by means of observation tunnels method. The results showed which particular features are crucial to define the type of coal what is also important to select appropriate method of its enrichment. Furthermore, the construction of a mathematical model presenting the relations between these properties and particle size and density is presented. Because of the fact that particles of certain size or density may occur in neighboring fractions three sorts of relations were examined basing on regression analysis.The analysis was conducted for all three coal types. Because of the fact that the models contain various amounts of independent variables R2 coefficient, mean squared error (MSE) and Mallow’s statistics Cp were applied to evaluate and compare obtained results.

Convergence rate of perturbed empirical distribution functions

1979 ◽  
Vol 16 (1) ◽  
pp. 163-173 ◽  
Author(s):  
B. B. Winter
Keyword(s):  
Distribution Function ◽  
Convergence Rate ◽  
Uniform Distribution ◽  
Empirical Distribution ◽  
Distribution Functions ◽  
Unit Mass ◽  
Common Distribution ◽  
Common Distribution Function ◽  
Empirical Distribution Functions ◽  
Non Parametric

Given an i.i.d. sequence X1,X2, … with common distribution function (d.f.) F, the usual non-parametric estimator of F is the e.d.f. Fn; where Uo is the d.f. of the unit mass at zero. An admissible perturbation of the e.d.f., say , is obtained if Uo is replaced by a d.f. , where is a sequence of d.f.'s converging weakly to Uo. Such perturbed e.d.f.′s arise quite naturally as integrals of non-parametric density estimators, e.g. as . It is shown that if F satisfies some smoothness conditions and the perturbation is not too drastic then ‘has the Chung–Smirnov property'; i.e., with probability one, 1. But if the perturbation is too vigorous then this property is lost: e.g., if F is the uniform distribution and Hn is the d.f. of the unit mass at n–α then the above lim sup is ≦ 1 or = ∞, depending on whether or

Scaling

2018 ◽  
Author(s):  
Stefan Thurner ◽  
Rudolf Hanel ◽  
Peter Klimekl
Keyword(s):  
Distribution Function ◽  
Dynamical Systems ◽  
Complex Systems ◽  
Power Law ◽  
Scaling Laws ◽  
Power Laws ◽  
Distribution Functions ◽  
Temporal Process ◽  
Approximate Power

Scaling appears practically everywhere in science; it basically quantifies how the properties or shapes of an object change with the scale of the object. Scaling laws are always associated with power laws. The scaling object can be a function, a structure, a physical law, or a distribution function that describes the statistics of a system or a temporal process. We focus on scaling laws that appear in the statistical description of stochastic complex systems, where scaling appears in the distribution functions of observable quantities of dynamical systems or processes. The distribution functions exhibit power laws, approximate power laws, or fat-tailed distributions. Understanding their origin and how power law exponents can be related to the particular nature of a system, is one of the aims of the book.We comment on fitting power laws.

Probability and Random Processes

2018 ◽  
Author(s):  
Stefan Thurner ◽  
Rudolf Hanel ◽  
Peter Klimekl
Keyword(s):  
Random Processes ◽  
Distribution Functions ◽  
Basic Logic ◽  
Heavy Tailed Distributions ◽  
Random Components ◽  
Classical Distribution ◽  
Heavy Tailed ◽  
Basic Facts ◽  
Stochastic Events ◽  
Stochastic Event

Phenomena, systems, and processes are rarely purely deterministic, but contain stochastic,probabilistic, or random components. For that reason, a probabilistic descriptionof most phenomena is necessary. Probability theory provides us with the tools for thistask. Here, we provide a crash course on the most important notions of probabilityand random processes, such as odds, probability, expectation, variance, and so on. Wedescribe the most elementary stochastic event—the trial—and develop the notion of urnmodels. We discuss basic facts about random variables and the elementary operationsthat can be performed on them. We learn how to compose simple stochastic processesfrom elementary stochastic events, and discuss random processes as temporal sequencesof trials, such as Bernoulli and Markov processes. We touch upon the basic logic ofBayesian reasoning. We discuss a number of classical distribution functions, includingpower laws and other fat- or heavy-tailed distributions.

scholarly journals CHARACTERISTICS AND ADSORPTION PERFORMANCE OF FORMULATED TRIKOTAC FILTER AIDS

2019 ◽  
Vol 81 (3) ◽  
Author(s):  
N. Masdiana ◽  
M. Rashid ◽  
S. Hajar ◽  
M. R. Ammar
Keyword(s):  
Particle Size ◽  
Activated Carbon ◽  
Particle Density ◽  
Adsorption Properties ◽  
The Other ◽  
Bet Surface Area ◽  
Adsorption Performance ◽  
Filtration System ◽  
Filter Aids ◽  
The Relationship

TrikotAC filter aids is a combination of a pre-coating material PreKot™ with two adsorbents; activated carbon and lime and their characteristics were investigated in this study. TrikotAC was formulated into three different weight ratios of 5:1:94, 10:1:89 and 10:5:85, respectively. The relationship between adsorption properties and characteristics of the formulated materials particle size distribution, particle density, bulk density, and BET surface area were investigated. The results showed that the adsorption capacity for TrikotAC 10:5:85 (11.88 mg/g) was higher than for the other formulated filter aids samples, and the formulated filter aids material TrikotAC showed promising characteristic as a filter aids and adsorbent for organic compound in fabric filtration system.

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