scholarly journals Flexible Power-Normal Models with Applications

Mathematics ◽  
2021 ◽  
Vol 9 (24) ◽  
pp. 3183
Author(s):  
Guillermo Martínez-Flórez ◽  
Diego I. Gallardo ◽  
Osvaldo Venegas ◽  
Heleno Bolfarine ◽  
Héctor W. Gómez
Keyword(s):  
Probability Density Function ◽  
Probability Density ◽  
Density Function ◽  
Gaussian Model ◽  
Real Data ◽  
Simulation Studies ◽  
New Model ◽  
Asymmetric Model

The main object of this paper is to propose a new asymmetric model more flexible than the generalized Gaussian model. The probability density function of the new model can assume bimodal or unimodal shapes, and one of the parameters controls the skewness of the model. Three simulation studies are reported and two real data applications illustrate the flexibility of the model compared with traditional proposals in the literature.

scholarly journals On a new distribution based on the arccosine function

2021 ◽  
Author(s):  
Christophe Chesneau ◽  
Lishamol Tomy ◽  
Jiju Gillariose
Keyword(s):  
Probability Density Function ◽  
Probability Density ◽  
Density Function ◽  
Hazard Rate ◽  
Survival Function ◽  
Real Data ◽  
Cumulative Distribution ◽  
Data Sets ◽  
Power Functions ◽  
New Distribution

AbstractThis note focuses on a new one-parameter unit probability distribution centered around the inverse cosine and power functions. A special case of this distribution has the exact inverse cosine function as a probability density function. To our knowledge, despite obvious mathematical interest, such a probability density function has never been considered in Probability and Statistics. Here, we fill this gap by pointing out the main properties of the proposed distribution, from both the theoretical and practical aspects. Specifically, we provide the analytical form expressions for its cumulative distribution function, survival function, hazard rate function, raw moments and incomplete moments. The asymptotes and shape properties of the probability density and hazard rate functions are described, as well as the skewness and kurtosis properties, revealing the flexible nature of the new distribution. In particular, it appears to be “round mesokurtic” and “left skewed”. With these features in mind, special attention is given to find empirical applications of the new distribution to real data sets. Accordingly, the proposed distribution is compared with the well-known power distribution by means of two real data sets.

scholarly journals A Probability Density Function Generator Based on Deep Learning

2018 ◽  
Author(s):  
Chi-Hua Chen ◽  
Fangying Song ◽  
Feng-Jang Hwang ◽  
Ling Wu
Keyword(s):  
Deep Learning ◽  
Probability Density Function ◽  
Probability Density ◽  
Density Function ◽  
Probability Distributions ◽  
Real Data ◽  
Learning Model ◽  
Cumulative Distribution ◽  
Function Generator ◽  
Deep Learning Model

To generate a probability density function (PDF) for fitting probability distributions of real data, this study proposes a deep learning method which consists of two stages: (1) a training stage for estimating the cumulative distribution function (CDF) and (2) a performing stage for predicting the corresponding PDF. The CDFs of common probability distributions can be adopted as activation functions in the hidden layers of the proposed deep learning model for learning actual cumulative probabilities, and the differential equation of trained deep learning model can be used to estimate the PDF. To evaluate the proposed method, numerical experiments with single and mixed distributions are performed. The experimental results show that the values of both CDF and PDF can be precisely estimated by the proposed method.

Event Location with Sparse Data: When Probabilistic Global Search is Important

2020 ◽  
Author(s):  
Stephen Arrowsmith ◽  
Junghyun Park ◽  
Il-Young Che ◽  
Brian Stump ◽  
Gil Averbuch
Keyword(s):  
Probability Density Function ◽  
Probability Density ◽  
Density Function ◽  
Arrival Time ◽  
Real Data ◽  
Parameter Estimates ◽  
Model Parameter ◽  
Event Location ◽  
Seismic Location ◽  
Probability Density Function Pdf

Abstract Locating events with sparse observations is a challenge for which conventional seismic location techniques are not well suited. In particular, Geiger’s method and its variants do not properly capture the full uncertainty in model parameter estimates, which is characterized by the probability density function (PDF). For sparse observations, we show that this PDF can deviate significantly from the ellipsoidal form assumed in conventional methods. Furthermore, we show how combining arrival time and direction-of-arrival constraints—as can be measured by three-component polarization or array methods—can significantly improve the precision, and in some cases reduce bias, in location solutions. This article explores these issues using various types of synthetic and real data (including single-component seismic, three-component seismic, and infrasound).

scholarly journals Qualitative Properties of Randomized Maximum Entropy Estimates of Probability Density Functions

Mathematics ◽  
2021 ◽  
Vol 9 (5) ◽  
pp. 548
Author(s):  
Yuri S. Popkov
Keyword(s):  
Probability Density Function ◽  
Probability Density ◽  
Maximum Entropy ◽  
Density Function ◽  
Lagrange Multipliers ◽  
Real Data ◽  
Model Parameters ◽  
Lake Area ◽  
Data Set ◽  
Maximum Entropy Estimation

The problem of randomized maximum entropy estimation for the probability density function of random model parameters with real data and measurement noises was formulated. This estimation procedure maximizes an information entropy functional on a set of integral equalities depending on the real data set. The technique of the Gâteaux derivatives is developed to solve this problem in analytical form. The probability density function estimates depend on Lagrange multipliers, which are obtained by balancing the model’s output with real data. A global theorem for the implicit dependence of these Lagrange multipliers on the data sample’s length is established using the rotation of homotopic vector fields. A theorem for the asymptotic efficiency of randomized maximum entropy estimate in terms of stationary Lagrange multipliers is formulated and proved. The proposed method is illustrated on the problem of forecasting of the evolution of the thermokarst lake area in Western Siberia.

scholarly journals Partial Histogram Bayes Learning Algorithm for Classification Applications

2018 ◽  
Vol 7 (4.11) ◽  
pp. 126
Author(s):  
Haider O. Lawend ◽  
Anuar M. Muad ◽  
Aini Hussain
Keyword(s):  
Probability Density Function ◽  
Probability Density ◽  
Density Function ◽  
Learning Algorithm ◽  
Gaussian Function ◽  
Real Data ◽  
Distribution Functions ◽  
Factors Affecting ◽  
Probability Distribution Functions ◽  
Large Memory

This paper presents a proposed supervised classification technique namely partial histogram Bayes (PHBayes) learning algorithm. Conventional classifier based on Gaussian function has limitation when dealing with different probability distribution functions and requires large memory for large number of instance. Alternatively, histogram based classifiers are flexible for different probability density function. The aims of PHBayes are to handle large number of instances in datasets with lesser memory requirement, and fast in training and testing phases. The PHBayes depends on portion of the observed histogram that is similar to the probability density function. PHBayes was analyzed using synthetic and real data. Several factors affecting classification accuracy were considered. The PHBayes was compared with other established classifiers and demonstrated higher accurate classification, lesser memory even when dealing with large number of instance, and faster in training and testing phases.  

scholarly journals Optimization of the Photon Path Length Probability Density Function-Simultaneous (PPDF-S) Method and Evaluation of CO2 Retrieval Performance Under Dense Aerosol Conditions

Sensors ◽  
2019 ◽  
Vol 19 (5) ◽  
pp. 1262 ◽  
Author(s):  
Chisa Iwasaki ◽  
Ryoichi Imasu ◽  
Andrey Bril ◽  
Sergey Oshchepkov ◽  
Yukio Yoshida ◽  
...  
Keyword(s):  
Probability Density Function ◽  
Probability Density ◽  
Density Function ◽  
Atmospheric Aerosol ◽  
Path Length ◽  
Stable Solution ◽  
Simulation Studies ◽  
Path Modification ◽  
Aerosol Types ◽  
Photon Path

The photon path length probability density function-simultaneous (PPDF-S) algorithm is effective for retrieving column-averaged concentrations of carbon dioxide (XCO2) and methane (XCH4) from Greenhouse gases Observing Satellite (GOSAT) spectra in Short Wavelength InfraRed (SWIR). Using this method, light-path modification attributable to light reflection/scattering by atmospheric clouds/aerosols is represented by the modification of atmospheric transmittance according to PPDF parameters. We optimized PPDF parameters for a more accurate XCO2 retrieval under aerosol dense conditions based on simulation studies for various aerosol types and surface albedos. We found a more appropriate value of PPDF parameters referring to the vertical profile of CO2 concentration as a measure of a stable solution. The results show that the constraint condition of a PPDF parameter that represents the light reflectance effect by aerosols is sufficiently weak to affect XCO2 adversely. By optimizing the constraint, it was possible to obtain a stable solution of XCO2. The new optimization was applied to retrieval analysis of the GOSAT data measured in Western Siberia. First, we assumed clear sky conditions and retrieved XCO2 from GOSAT data obtained near Yekaterinburg in the target area. The retrieved XCO2 was validated through a comparison with ground-based Fourier Transform Spectrometer (FTS) measurements made at the Yekaterinburg observation site. The validation results showed that the retrieval accuracy was reasonable. Next, we applied the optimized method to dense aerosol conditions when biomass burning was active. The results demonstrated that optimization enabled retrieval, even under smoky conditions, and that the total number of retrieved data increased by about 70%. Furthermore, the results of the simulation studies and the GOSAT data analysis suggest that atmospheric aerosol types that affected CO2 analysis are identifiable by the PPDF parameter value. We expect that we will be able to suggest a further improved algorithm after the atmospheric aerosol types are identified.

On Estimation of a Functional of a Probability Density Function

1993 ◽  
Vol 43 (1-2) ◽  
pp. 13-24
Author(s):  
L. O. Odongo ◽  
M. Samanta
Keyword(s):  
Probability Density Function ◽  
Probability Density ◽  
Density Function ◽  
Mean Square Error ◽  
Regularity Conditions ◽  
Mean Square ◽  
Simulation Studies ◽  
Kernel Estimate ◽  
Primary 62G07 ◽  
The Mean

The problem of estimating the integral of the square of a probability density function is considered, It is shown that under some regularity conditions the kernel estimate of this functional is asymptotically normally distributed. An expression for the smoothing parameter that minimizes the mean square error of the estimate is derived. Results of simulation studies are included. AMS (1980) Subject Classification: Primary 62G07 Secondary 60FOS.

scholarly journals PROBABILITY DENSITY FUNCTION OF WAVE HEIGHTS OFF THE WESTERN COAST OF TAIWAN

1982 ◽  
Vol 1 (18) ◽  
pp. 23
Author(s):  
Frederick L.W. Tang ◽  
Jea-Tzyy Juang
Keyword(s):  
Probability Density Function ◽  
Analytical Solution ◽  
Probability Density ◽  
Density Function ◽  
Western Coast ◽  
Convolution Integral ◽  
Local Wind ◽  
New Model ◽  
Wave Heights ◽  
Better Than

A new probability density function of wave heights off the western coast of Taiwan is submitted in this paper. According to the bathymetry of this area, waves from the central part of Taiwan Strait refract to the point of measurement and minor waves generated by local wind add the energy on the major ones; So an analytical solution is to be worked out by assuming that the wave energies are the linear sum of these two sources and convolution integral is adopted. The new model approaches reality better than Ray'leigh's.

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