# Probabilistic modeling of random variables with inconsistent data

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### Resumé

The aim of the present paper was to formulate probabilistic modeling for random variables with inconsistent data to facilitate accurate reliability assessment. Traditionally, random variables have some outputs available, based on which, some distribution is identified. However, as will be illustrated, the data relevant to those extreme events might not necessarily follow the same distribution as well as the other part, but they generally have small weights in the definition of the distribution due to their small quantity. The adoption of one single probabilistic distribution to describe random variables with such inconsistent data might cause great errors in the reliability assessment, especially for extreme events. One new formulation of probabilistic modeling is proposed here for such type of random variables. The inconsistency within the data set is identified and based on how the set is divided. Each division is described by the respective distribution and finally they are unified into one framework. The relevant problems in the modeling (e.g., the identification of the boundary between the divisions, the definition of the probability distributions, and the unification of the distributions into one framework) are presented and solved. The realization of the proposed approach in the practical numerical analysis is further investigated afterwards. Finally, two examples are presented to illustrate the application from different perspectives.
Originalsprog Engelsk Applied Mathematical Modelling 73 401-411 0307-904X https://doi.org/10.1016/j.apm.2019.04.017 Udgivet - 2019

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Probabilistic Modeling
Random variables
Inconsistent
Random variable
Extreme Events
Reliability Assessment
Division
Probability distributions
Numerical analysis
Unification
Inconsistency
Numerical Analysis
Probability Distribution
Formulation
Output
Modeling

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title = "Probabilistic modeling of random variables with inconsistent data",
abstract = "The aim of the present paper was to formulate probabilistic modeling for random variables with inconsistent data to facilitate accurate reliability assessment. Traditionally, random variables have some outputs available, based on which, some distribution is identified. However, as will be illustrated, the data relevant to those extreme events might not necessarily follow the same distribution as well as the other part, but they generally have small weights in the definition of the distribution due to their small quantity. The adoption of one single probabilistic distribution to describe random variables with such inconsistent data might cause great errors in the reliability assessment, especially for extreme events. One new formulation of probabilistic modeling is proposed here for such type of random variables. The inconsistency within the data set is identified and based on how the set is divided. Each division is described by the respective distribution and finally they are unified into one framework. The relevant problems in the modeling (e.g., the identification of the boundary between the divisions, the definition of the probability distributions, and the unification of the distributions into one framework) are presented and solved. The realization of the proposed approach in the practical numerical analysis is further investigated afterwards. Finally, two examples are presented to illustrate the application from different perspectives.",
keywords = "Probability modeling, Reliability assessment, Tail behavior, Monte Carlo simulation, Data inconsistency",
author = "Jianjun Qin",
year = "2019",
doi = "10.1016/j.apm.2019.04.017",
language = "English",
volume = "73",
pages = "401--411",
journal = "Applied Mathematical Modelling",
issn = "0307-904X",
publisher = "Elsevier",

}

I: Applied Mathematical Modelling, Bind 73, 2019, s. 401-411.

Publikation: Bidrag til tidsskriftTidsskriftartikelForskningpeer review

TY - JOUR

T1 - Probabilistic modeling of random variables with inconsistent data

AU - Qin, Jianjun

PY - 2019

Y1 - 2019

N2 - The aim of the present paper was to formulate probabilistic modeling for random variables with inconsistent data to facilitate accurate reliability assessment. Traditionally, random variables have some outputs available, based on which, some distribution is identified. However, as will be illustrated, the data relevant to those extreme events might not necessarily follow the same distribution as well as the other part, but they generally have small weights in the definition of the distribution due to their small quantity. The adoption of one single probabilistic distribution to describe random variables with such inconsistent data might cause great errors in the reliability assessment, especially for extreme events. One new formulation of probabilistic modeling is proposed here for such type of random variables. The inconsistency within the data set is identified and based on how the set is divided. Each division is described by the respective distribution and finally they are unified into one framework. The relevant problems in the modeling (e.g., the identification of the boundary between the divisions, the definition of the probability distributions, and the unification of the distributions into one framework) are presented and solved. The realization of the proposed approach in the practical numerical analysis is further investigated afterwards. Finally, two examples are presented to illustrate the application from different perspectives.

AB - The aim of the present paper was to formulate probabilistic modeling for random variables with inconsistent data to facilitate accurate reliability assessment. Traditionally, random variables have some outputs available, based on which, some distribution is identified. However, as will be illustrated, the data relevant to those extreme events might not necessarily follow the same distribution as well as the other part, but they generally have small weights in the definition of the distribution due to their small quantity. The adoption of one single probabilistic distribution to describe random variables with such inconsistent data might cause great errors in the reliability assessment, especially for extreme events. One new formulation of probabilistic modeling is proposed here for such type of random variables. The inconsistency within the data set is identified and based on how the set is divided. Each division is described by the respective distribution and finally they are unified into one framework. The relevant problems in the modeling (e.g., the identification of the boundary between the divisions, the definition of the probability distributions, and the unification of the distributions into one framework) are presented and solved. The realization of the proposed approach in the practical numerical analysis is further investigated afterwards. Finally, two examples are presented to illustrate the application from different perspectives.

KW - Probability modeling

KW - Reliability assessment

KW - Tail behavior

KW - Monte Carlo simulation

KW - Data inconsistency

UR - https://authors.elsevier.com/a/1YxAp,703q0KQs

U2 - 10.1016/j.apm.2019.04.017

DO - 10.1016/j.apm.2019.04.017

M3 - Journal article

VL - 73

SP - 401

EP - 411

JO - Applied Mathematical Modelling

JF - Applied Mathematical Modelling

SN - 0307-904X

ER -