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1
100%
EN
The article deals with the kernel estimation of the probability density function. The main subject of the research is the optimization of parameters of the estimator. In the particular case, the research focused on the estimation of the univariate, unimodal data representative for the normal distribution.
2
A Novel Multimodal Probability Model for Cluster Analysis
100%
EN
Cluster analysis is a tool for data analysis. It is a method for finding clusters of a data set with most similarity in the same group and most dissimilarity between different groups. In general, there are two ways, mixture distributions and classification maximum likelihood method, to use probabilitymodels for cluster analysis. However, the corresponding probability distributions to most clustering algorithms such as fuzzy c-means, possibilistic c-means, mode-seeking methods, etc., have not yet been found. In this paper, we construct a multimodal probability distribution model and then present the relationships between many clustering algorithms and the proposed model via the maximum likelihood estimation. Moreover, we also give the theoretical properties of the proposed multimodal probability distribution.
3
On the instantaneous frequency of Gaussian stochastic processes
80%
EN
We study the instantaneous frequency (IF) of continuoustime, complex-valued, zero-mean, proper, mean-square differentiable, nonstationary Gaussian stochastic processes. We compute the probability density function for the IF for fixed time, which generalizes a result known for wide-sense stationary processes to nonstationary processes. For a fixed point in time, the IF has either zero or infinite variance. For harmonizable processes, we obtain as a consequence the result that the mean of the IF, for fixed time, is the normalized first-order frequency moment of the Wigner spectrum.
EN
The paper presents results of experimental investigation probability density function as tool in identify pattern in upward two-phase flow in vertical minichannels.
5
Hysteresis Modeling Using a Preisach Operator
80%
EN
The aim of this paper is to present the analysis and modeling of the hysteresis phenomenon using a Preisach operator. The fundamentals of parameterized hysteresis modeling are introduced, by utilizing three probability density functions. Then, the Preisach operator and its characteristics are defined. Subsequently, results of simulations obtained by means of the aforementioned functions are presented and compared to the ones obtained by other authors.
EN
In this paper the problem of non-parametric estimation of the probability density function for hydrological data is considered. For a given random sample X1, X2, ..., Xn we define an estimator fˆ n of the density function ƒ based on a function K of a real variable – the so-called kernel of a distribution – and a properly chosen number sequence {hn} from the interval (0, ∞). This estimator of density function of a random variable X under more general assumptions is known in the statistical literature as the Parzen-Rosenblatt estimator or the kernel estimator. The method of kernel estimation presented in the paper has been applied to determine the probability distribution of the groundwater level based on long-term measurements made in the melioration research carried out at the foothill object Długopole.
PL
W pracy rozważono statystyczny opis przepływów turbulentnych przy wykorzystaniu metody funkcji gęstości rozkładu prawdopodobieństwa charakterystyk przepływu. W szczególności otrzymano równanie ewolucyjne dla jednopunktowej funkcji gęstości prawdopodobieństwa pulsacji koncentracji dyfundującej substancji w zadanym polu prędkości cieczy. Rozważono bliżej przypadek statystycznie stacjonarnego i horyzontalnie jednorodnego przepływu turbulentnego, dla którego znaleziono ścisłe, zależne od dwóch parametrów, rozwiązanie zagadnienia dyfuzji, wyrażone przez 3-konfluentną funkcję Heuna.
EN
In the paper the use of probabiiity density functions metod to the statistical description of turbulent flows is considered. This formulation is less generał than the characteristic functionals approach, but primary in the relation to the Friedmann-Keller hierarchy equations. An evolution equation for single-point probability density function of pulsation of concentration of the diffusing substance in prescribed fluid velocity field, is derived. The case of statistical stationary and horizontal homogeneous turbulent flow is considered, for which an exact, new, two-parameter solution of diffusion problem in terms of Triconfluent Heun function, is obtained.
EN
The article presents usage of microseismic monitoring for location of microseismic events in polish geological conditions. For location of one synthetic microseismic event, two methods of acquisition were applied: surface and downhole monitoring array. Downhole microseismic monitoring is a technique of recording induced seismicity using receivers placed in the monitoring well near to the treatment well. In case of surface monitoring receivers are placed at the surface. For determination of hypocenter location probability density function was used. Based on provided analysis it is concluded that for polish conditions it is better to use downhole microseismic monitoring. Event located with usage of this technique was located correctly and uncertainty of this location was lower.
EN
The Probability Density Function (PDF) is a key concept in statistics. Constructing the most adequate PDF from the observed data is still an important and interesting scientific problem, especially for large datasets. PDFs are often estimated using nonparametric data-driven methods. One of the most popular nonparametric method is the Kernel Density Estimator (KDE). However, a very serious drawback of using KDEs is the large number of calculations required to compute them, especially to find the optimal bandwidth parameter. In this paper we investigate the possibility of utilizing Graphics Processing Units (GPUs) to accelerate the finding of the bandwidth. The contribution of this paper is threefold: (a) we propose algorithmic optimization to one of bandwidth finding algorithms, (b) we propose efficient GPU versions of three bandwidth finding algorithms and (c) we experimentally compare three of our GPU implementations with the ones which utilize only CPUs. Our experiments show orders of magnitude improvements over CPU implementations of classical algorithms.
10
Exact response probability density functions of some uncertain structural systems
70%
EN
This paper has the goal of defining a class of uncertain structural systems for which it is possible to consider an approach able to give the exact response in terms of the probability density function (PDF). The uncertain structures have been identified in the discretized statically determined ones and the approach has been identified in the coupling of the approximated principal deformation modes method (APDM) and of the probability transformation method (PTM). The first one gives the explicit relationships between the response variables and the uncertainty ones, that are exact when the structures are statically determined. The second method allows to determine the explicit relationship between the PDFs of the response and of the uncertainty variables. The results of some applications have confirmed the goodness of these choices and that the proposed approach gives always exact results for both correlated and uncorrelated uncertainty random variables.
EN
This paper presents statistical analysis of RSSI readouts recorded in indoor environment. Many papers concerning indoor location, based on RSSI measurement, assume its normal probability density function (PDF). This is partially excused by relation to PDF of radio-receiver's noise and/or together with influence of AWGN (average white Gaussian noise) radio-channel – generally modelled by normal PDF. Unfortunately, commercial (usually unknown) methods of RSSI calculations, typically as "side-effect" function of receiver's AGC (automatic gain control), results in PDF being far different from Gaussian PDF. This paper presents results of RSSI measurements in selected ISM bands: 433/868 MHz and 2.4/5 GHz. The measurements have been recorded using low-cost integrated RF modules (at 433/868 MHz and 2.4 GHz) and 802.11 WLAN access points (at 2.4/5 GHz). Then estimated PDF of collected data is shown and compared to normal (Gaussian) PDF.
EN
The aim of this study was to recognize the possibility of downscaling probability density function (PDF) of daily precipitation by means of canonical correlation analysis (CCA). Sea level pressure (SLP) over Europe and the North Atlantic was used as a predictor. A skilful statistical model could be used to generate projections of future changes of precipitation PDF driven by GCM (General Circulation Model) simulations. Daily precipitation totals from 8 stations located on the Polish coast of the Baltic Sea covering the period 1961-2010 were used to estimate the gamma distribution parameters, and only wet days (i.e. ≥0.1mm) were taken in the analysis. The results of the Kolmogorov-Smirnov test and comparison of empirical and theoretical (gamma-distributed) quantiles proved that gamma distribution gives a reliable description of daily precipitation totals. The validation of CCA models applied to gamma parameters revealed that the reliable reconstruction of precipitation PDF is possible only for average long-term conditions. In the case of individual months/seasons the agreement between empirical and reconstructed quantiles is poor. This study shows the potential of modelling of precipitation PDF, however efforts should be made to improve model performance by establishing more reliable links between regional forcing and the variability of the gamma parameters.
EN
The article presents the probability density functions as well as characteristic functions of selected periodic and random signals. On their basis, original models of the bias of the mean square value digital estimator have been designed. These models were employed in investigating estimation errors caused by analog to digital conversion and analog to digital conversion with a dither signal. Selected graphical model representations and their analyses are demonstrated. It has been shown that for a triangular probability density function random signal with the amplitude At = kq, k∈N \ {0}, mean square value reconstruction occurs on the basis of a signal quantized with the accuracy of Sheppard's correction. Whereas for periodic signals as well as for the sum of periodic and random signals, the δ component of bias due to the nonsatisfaction of the reconstruction condition is a suppressed oscillating function of the quotient of the amplitude A and the quantization step size q. It has been proved that by adding, prior to quantization, a triangular distribution random signal with zero mean and the amplitude At = kq (k = 1, 2, ...) in the mean square value measurement of any periodic signal, this bias component can be brought to zero.
EN
In this paper, analytical expressions for the distribution of the envelope and phase of linearly modulated signals such as BPSK, M-PSK, and M-QAM in AWGN are presented. We perform numerical simulations for different orders of signal constellations. The results show that the proposed theoretical models are in excellent agreement with the estimated distributions from various numerical experiments.
EN
The problem of query selectivity estimation for database queries is critical for efficient query execution by database management systems. A query execution method strongly depends on early estimated size of a query result. This estimation determines a data access method used later during the query execution. The selectivity parameter is a fraction of table rows that satisfy a single-table query condition. For a selection condition of a range query where an attribute has a continuous domain, the selectivity is equivalent to a definite integral form probability density function (PDF) of attribute values distribution. For a compound selection condition based on many attributes we need a multidimensional space-efficient non-parametric estimator of multivariate PDF of attribute values distribution. A known approach based on Discrete Cosine Transform (DCT) spectrum as an representation of multidimensional PDF is considered. The energy compaction property of DCT lets omit a region of spectrum coefficients with small absolute values without significant losing an accuracy of selectivity estimation. An area of relevant spectrum coefficients is called a sampling zone. Results of experiments from previous works shows that applying the reciprocal shape of the sampling zone gives the least selectivity estimation error subject to a predetermined size of the zone. The main result of this work is a theoretical confirmation of only experimental results from previous works. The paper presents the proof of the theorem that the reciprocal shape of the sampling zone is asymptotically error-optimal. The proof is based on calculus of variations and the isoperimetric problem.
PL
Szacowanie selektywności zapytań jest krytyczne dla efektywnej realizacji zapytań w systemach zarządzania bazami danych. Sposób realizacji zapytania zależy od wstępnego oszacowania rozmiaru danych spełniających kryteria zapytania. Takie oszacowanie pozwala wybrać metodę dostępu do danych użytą później podczas realizacji zapytania. Selektywność dla zapytań jednotablicowych to stosunek liczby wierszy spełniających kryteria zapytania do liczby wszystkich wierszy tablicy. Dla zakresowych warunków zapytania, określonych na atrybutach z ciągła dziedziną, selektywność jest całką oznaczoną z funkcji gęstości prawdopodobieństwa (PDF), określającej rozkład wartości tego atrybutu. Dla złożonych warunków zapytania, opartych na kilku atrybutach, istnieje potrzeba użycia nieparametrycznego estymatora wielowymiarowej PDF, którego reprezentacja powinna być oszczędna pod względem zajętości pamięci. Jedno ze znanych podejść do konstrukcji takiego estymatora oparte jest na dyskretnej transformacie kosinusowej (DCT) - tzn. widmie z histogramu wielowymiarowego. Własność kompakcji energii pozwala na pominięcie nieznaczących współczynników widma DCT bez istotnej utraty oszacowania selektywności. Obszar znaczących współczynników widma nazywany jest strefą próbkowania. Wyniki prac eksperymentalnych innych autorów wskazują, że dla zadanego rozmiaru reprezentacji widma, optymalną strefą próbkowania (kształtem strefy o najmniejszym błędzie oszacowania selektywności) jest tzw. strefa odwrotnie proporcjonalna. Głównym wynikiem tego opracowania jest teoretyczne potwierdzenie tych eksperymentów. Artykuł przedstawia dowód twierdzenia o asymptotycznej optymalności strefy odwrotnie proporcjonalnej dla przypadku dwuwymiarowego. Dowód opiera się na elementach rachunku wariacyjnego i zagadnieniu izoperymetrycznym.
EN
The article presents selected issues in the field of stochastic simulation of production processes. Attention was drawn to the possibility of including, in this type of models, the risk accompanying the implementation of processes. Probability density functions that can be used to characterize random variables present in the model are presented. The possibility of making mistakes while creating this type of models was pointed out. Two selected examples of the use of stochastic simulation in the analysis of production processes on the example of the mining process are presented.
EN
The aim of this study was to recognize the possibility of downscaling probability density function (PDF) of daily precipitation by means of canonical correlation analysis (CCA). Sea level pressure (SLP) over Europe and the North Atlantic was used as a predictor. A skilful statistical model could be used to generate projections of future changes of precipitation PDF driven by GCM (General Circulation Model) simulations. Daily precipitation totals from 8 stations located on the Polish coast of the Baltic Sea covering the period 1961-2010 were used to estimate the gamma distribution parameters, and only wet days (i.e. ≥0.1mm) were taken in the analysis. The results of the Kolmogorov-Smirnov test and comparison of empirical and theoretical (gamma-distributed) quantiles proved that gamma distribution gives a reliable description of daily precipitation totals. The validation of CCA models applied to gamma parameters revealed that the reliable reconstruction of precipitation PDF is possible only for average long-term conditions. In the case of individual months/seasons the agreement between empirical and reconstructed quantiles is poor. This study shows the potential of modelling of precipitation PDF, however efforts should be made to improve model performance by establishing more reliable links between regional forcing and the variability of the gamma parameters.
18
Modeling of polymer/clay nanocomposites by an iterative micromechanical approach
70%
EN
An iterative micromechanical method is presented in order to predict the elastic constants of composites and nanocomposites including arbitrarily oriented reinforcement particles. The proposed method is capable of introducing into the matrix any kind of heterogeneity based on its dimension, orientation, mechanical properties and volume fraction. The efficiency and convergence of solution method is studied by computing the elasticity tensor of a unidirectional particulate composite. It is then applied to model the elastic behavior of nylon-6/clay nanocomposite with taking into consideration the probability distribution of aspect ratio and orientation of effective particles. The results are validated by comparison with available experimental data.
XX
The aim of the research was to identify the potential for the use of probability density functions (PDF) in modeling of near-surface wind speed. The approaches of Empirical Orthogonal Functions (EOF) and Canonical Correlation Analysis (CCA) are used in combination with 2-parametric Weibull distribution. The downscaling model was built using a diagnosed relationship between sea level pressure (SLP) patterns over Europe and the Northern Atlantic and estimated monthly values of Weibull parameters at 9 stations along the Polish Baltic Coast. The obtained scale (A) and shape (k) parameters make it possible to describe temporal variations of wind fields and their theoretical probability values. This may have further application in the modeling of extreme wind speeds for seasonal forecasting, climate prediction or in historical reconstructions. The model evaluation was done separately for the calibration (1971-2000) and validation periods (2001-2010). The scale parameter was reconstructed reasonably, while there were some problematic issues with the shape parameter, especially in the validation period. The quality of the developed models is generally higher for the winter season, due to larger SLP gradients, whereas the results for the spring and summer seasons were less satisfactory. Despite this, the 99th percentile of theoretical wind speeds are in most cases satisfactory, due to the lesser importance of the shape parameter for typical distributions in the analyzed region.
20
Pseudo-Gaussian Density Functions for Gap Height Between Two Surfaces
60%
EN
The presented paper formulates the sequence of pseudo-Gaussian distributions where are presented the mathematical results of standard deviation and probability density function derivations for bio-and micro-bearing gap height on the basis of Mow, Dowson, Cwanek, Wierzcholski experimental results. The mathematical derivation of the optimum probability density functions and least standard deviation are illustrated in two cases. The first case relates to the height of the gap if vibration and unsteady load causes the random changes. The second case relates to the height of the gap if the asperities of the cartilage surface roughness cause the random changes. Author formulates one Theorem and some Corollaries where the sequence of pseudo-Gaussian-distributions are derived. The presented proof shows, that such sequence is convergent to the Gaussian distribution in infinity.
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