scholarly journals The Fréchet transform

1993 ◽  
Vol 16 (1) ◽  
pp. 155-164
Author(s):  
Piotor Mikusiński ◽  
Morgan Phillips ◽  
Howard Sherwood ◽  
Michael D. Taylor
Keyword(s):  
Distribution Function ◽  
Probability Distribution ◽  
Probability Measure ◽  
Probability Distribution Function ◽  
Random Variables ◽  
Distribution Functions ◽  
Probability Distribution Functions

LetF1,…,FNbe1-dimensional probability distribution functions andCbe anN-copula. Define anN-dimensional probability distribution functionGbyG(x1,…,xN)=C(F1(x1),…,FN(xN)). Letν, be the probability measure induced onℝNbyGandμbe the probability measure induced on[0,1]NbyC. We construct a certain transformationΦof subsets ofℝNto subsets of[0,1]Nwhich we call the Fréchet transform and prove that it is measure-preserving. It is intended that this transform be used as a tool to study the types of dependence which can exist between pairs orN-tuples of random variables, but no applications are presented in this paper.

The Method of Probabilistic Nodes Combination in Decision Making and Risk Analysis

2016 ◽  
pp. 32-60
Author(s):  
Dariusz Jacek Jakóbczak
Keyword(s):  
Decision Making ◽  
Distribution Function ◽  
Probability Distribution ◽  
Risk Analysis ◽  
Probability Distribution Function ◽  
Linear Interpolation ◽  
Distribution Functions ◽  
Probability Distribution Functions ◽  
Characteristic Points ◽  
Basic Probability

Proposed method, called Probabilistic Nodes Combination (PNC), is the method of 2D curve interpolation and extrapolation using the set of key points (knots or nodes). Nodes can be treated as characteristic points of data for modeling and analyzing. The model of data can be built by choice of probability distribution function and nodes combination. PNC modeling via nodes combination and parameter ? as probability distribution function enables value anticipation in risk analysis and decision making. Two-dimensional curve is extrapolated and interpolated via nodes combination and different functions as discrete or continuous probability distribution functions: polynomial, sine, cosine, tangent, cotangent, logarithm, exponent, arc sin, arc cos, arc tan, arc cot or power function. Novelty of the paper consists of two generalizations: generalization of previous MHR method with various nodes combinations and generalization of linear interpolation with different (no basic) probability distribution functions and nodes combinations.

scholarly journals Optimal Sampling of Parametric Families: Implications for Machine Learning

2020 ◽  
Vol 32 (1) ◽  
pp. 261-279
Author(s):  
Adrian E. G. Huber ◽  
Jithendar Anumula ◽  
Shih-Chii Liu
Keyword(s):  
Machine Learning ◽  
Distribution Function ◽  
Probability Distribution ◽  
Probability Distribution Function ◽  
Distribution Functions ◽  
Test Set ◽  
Probability Distribution Functions ◽  
Parametric Families ◽  
The Continuum ◽  
Training Sets

It is well known in machine learning that models trained on a training set generated by a probability distribution function perform far worse on test sets generated by a different probability distribution function. In the limit, it is feasible that a continuum of probability distribution functions might have generated the observed test set data; a desirable property of a learned model in that case is its ability to describe most of the probability distribution functions from the continuum equally well. This requirement naturally leads to sampling methods from the continuum of probability distribution functions that lead to the construction of optimal training sets. We study the sequential prediction of Ornstein-Uhlenbeck processes that form a parametric family. We find empirically that a simple deep network trained on optimally constructed training sets using the methods described in this letter can be robust to changes in the test set distribution.

scholarly journals Performance Probability Distribution Function for Modelling Solar Radiation in South Southern Nigeria: A Case Study of Yenagoa

2019 ◽  
pp. 1-8
Author(s):  
Chukwutem Isaac Abiodun ◽  
Obiora Emeka Anisiji
Keyword(s):  
Distribution Function ◽  
Probability Distribution ◽  
Solar Radiation ◽  
Probability Distribution Function ◽  
Distribution Functions ◽  
Logistic Distribution ◽  
Scale Parameters ◽  
Probability Distribution Functions ◽  
Statistical Distribution Function

This study has attempted to assess the performance of the most suitable statistical distribution function for modelling solar radiation over Yenagoa, Bayelsa State in Nigeria. The probability distribution functions are tested based on eleven years (2007-2017) solar radiation data obtained from National Aeronautics and Space Administration (NASA). Six probability distribution functions are tested to ascertain the most appropriate one based on four different statistical tools and fitting accuracy. The associated parameters of the most appropriate fitted probability distribution function are calculated and the trends in the characteristic of the solar radiation are deduced. The result shows that logistic distribution presents the most suitable probability distribution function for modelling solar radiation over the selected environment with RMSE of 1.500 KWh/m2/day, MAE of 1.260 KWh/m2/day, MAPE of 22.000% and R2 of 0.880.When compared with the other five distribution functions, the same trend could be seen although with different values of RMSE, MAE, MAPE and R2. The estimated distribution location and scale parameters of the model vary with month and season. The overall result will be useful for predicting future solar radiation over the studied environment. It will also be a good reference point for the design of large solar power projects in Yenagoa in particular and south southern Nigerian environments  at large.

scholarly journals Frequency analysis of low flows in intermittent and non-intermittent rivers from hydrological basins in Turkey

2018 ◽  
Vol 19 (1) ◽  
pp. 30-39 ◽  
Author(s):  
Ebru Eris ◽  
Hafzullah Aksoy ◽  
Bihrat Onoz ◽  
Mahmut Cetin ◽  
Mehmet Ishak Yuce ◽  
...  
Keyword(s):  
Distribution Function ◽  
Probability Distribution ◽  
Probability Distribution Function ◽  
Distribution Functions ◽  
Low Flow ◽  
Probability Distribution Functions ◽  
Low Flows ◽  
Intermittent Rivers ◽  
Log Normal ◽  
Best Fit

Abstract This study attempts to find out the best-fit probability distribution function to low flows using the up-to-date data of intermittent and non-intermittent rivers in four hydrological basins from different regions in Turkey. Frequency analysis of D = 1-, 7-, 14-, 30-, 90- and 273-day low flows calculated from the daily flow time series of each stream gauge was performed. Weibull (W2), Gamma (G2), Generalized Extreme Value (GEV) and Log-Normal (LN2) are selected among the 2-parameter probability distribution functions together with the Weibull (W3), Gamma (G3) and Log-Normal (LN3) from the 3-parameter probability distribution function family. Selected probability distribution functions are checked for their suitability to fit each D-day low flow sequence. LN3 mostly conforms to low flows by being the best-fit among the selected probability distribution functions in three out of four hydrological basins while W3 fits low flows in one basin. With the use of the best-fit probability distribution function, the low flow-duration-frequency curves are determined, which have the ability to provide the end-users with any D-day low flow discharge of any given return period.

scholarly journals Probabalistic modelling of the results of economic activity as a function of random values

2020 ◽  
pp. 102-112
Author(s):  
Olesya Martyniuk ◽  
Stepan Popina ◽  
Serhii Martyniuk
Keyword(s):  
Mathematical Modeling ◽  
Distribution Function ◽  
Probability Distribution ◽  
Economic Activity ◽  
Probability Distribution Function ◽  
Random Variables ◽  
Economic Structure ◽  
Random Variable ◽  
Independent Random Variables ◽  
Research Results

Introduction. Mathematical modeling of economic processes is necessary for the unambiguous formulation and solution of the problem. In the economic sphere this is the most important aspect of the activity of any enterprise, for which economic-mathematical modeling is the tool that allows to make adequate decisions. However, economic indicators that are factors of a model are usually random variables. An economic-mathematical model is proposed for calculating the probability distribution function of the result of economic activity on the basis of the known dependence of this result on factors influencing it and density of probability distribution of these factors. Methods. The formula was used to calculate the random variable probability distribution function, which is a function of other independent random variables. The method of estimation of basic numerical characteristics of the investigated functions of random variables is proposed: mathematical expectation that in the probabilistic sense is the average value of the result of functioning of the economic structure, as well as its variance. The upper bound of the variation of the effective feature is indicated. Results. The cases of linear and power functions of two independent variables are investigated. Different cases of two-dimensional domain of possible values of indicators, which are continuous random variables, are considered. The application of research results to production functions is considered. Examples of estimating the probability distribution function of a random variable are offered. Conclusions. The research results allow in the probabilistic sense to estimate the result of the economic structure activity on the basis of the probabilistic distributions of the values of the dependent variables. The prospect of further research is to apply indirect control over economic performance based on economic and mathematical modeling.

Decision Making and Data Analysis

2021 ◽  
pp. 52-81
Author(s):  
Dariusz Jacek Jakóbczak
Keyword(s):  
Distribution Function ◽  
Probability Distribution ◽  
Probability Distribution Function ◽  
Distribution Functions ◽  
Two Dimensional ◽  
Key Points ◽  
Characteristic Points ◽  
Writer Recognition ◽  
Curve Modeling ◽  
Specific Letter

The proposed method, called probabilistic nodes combination (PNC), is the method of 2D curve modeling and handwriting identification by using the set of key points. Nodes are treated as characteristic points of signature or handwriting for modeling and writer recognition. Identification of handwritten letters or symbols need modeling, and the model of each individual symbol or character is built by a choice of probability distribution function and nodes combination. PNC modeling via nodes combination and parameter γ as probability distribution function enables curve parameterization and interpolation for each specific letter or symbol. Two-dimensional curve is modeled and interpolated via nodes combination and different functions as continuous probability distribution functions: polynomial, sine, cosine, tangent, cotangent, logarithm, exponent, arc sin, arc cos, arc tan, arc cot, or power function.

scholarly journals Multiple-wavelength anomalous dispersion techniques applied to powder data: a probabilistic method for finding the substructureviajoint probability distribution functions

2008 ◽  
Vol 42 (1) ◽  
pp. 30-35 ◽  
Author(s):  
Angela Altomare ◽  
Benny Danilo Belviso ◽  
Maria Cristina Burla ◽  
Gaetano Campi ◽  
Corrado Cuocci ◽  
...  
Keyword(s):  
Probability Distribution ◽  
Probability Distribution Function ◽  
Probabilistic Method ◽  
Joint Probability ◽  
Direct Methods ◽  
Distribution Functions ◽  
Joint Probability Distribution ◽  
Probability Distribution Functions ◽  
Multiple Wavelength ◽  
Joint Probability Distribution Function

A new joint probability distribution function method is described to find the anomalous scatterer substructure from powder data. The method requires two wavelengths; the conclusive formulas provide estimates of the substructure structure factor moduli, from which the anomalous scatterer positions can be found by Patterson or direct methods. The theory has been preliminarily applied to two compounds, the first having Pt and the second having Fe as anomalous scatterer. Both substructures were correctly identified.

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