Statistical Distributions

Forbes, Catherine; Evans, Merran; Hastings, Nicholas Anthony John; Peacock, Brian

doi:10.1002/9780470627242

Cited by 576 publications

(614 citation statements)

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“…The lognormal distribution (Forbes et al, 2011) is applicable to random variables that are constrained by zero but have a few very large values. The resulting distribution is asymmetrical and positively skewed.…”

Section: Lognormal Distributionmentioning

confidence: 99%

“…x ,    , we obtain a uniform distribution (Forbes et al, 2011). In probability theory the uniform distribution is a probability distribution continues which is uniform on a set, or that attaches the same probability to all the points belonging to a given interval [a, b] The main statistic quantities of the Cauchy distribution are reported in Table 17.…”

Section: Uniform Distributionmentioning

confidence: 99%

“…The Cauchy distribution (Forbes et al, 2011) is of mathematical interest due to the absence of defined moments. Its probability density function takes the following form: …”

Section: Cauchy Distributionmentioning

confidence: 99%

“…In the theory of probability the triangular is a probability distribution continues whose probability density function describes a triangle (Forbes et al, 2011), or that it is nothing on the two extreme values and is linear between these and an intermediate value (the mode). In statistics is used as a model when the sample available is very limited.…”

Section: Triangular Distributionmentioning

confidence: 99%

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Non Gaussian multi-parameter Bayesian estimation through the mechanical equivalent of logical inference

Bolognani¹,

Verzobio²,

Cappello³

et al. 2017

Proceedings of the Joint COST TU1402 - COST TU1406 - IABSE WC1 Workshop: The Value of Structural Health Monitoring for the Reli

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Abstract. Structural Health Monitoring requires engineers to understand the state of a structure from its observed response. When this information is uncertain, Bayesian probability theory provides a consistent framework for making inferences. However, structural engineers are often unenthusiastic about Bayesian logic, finding its application complicated and onerous, and prefer to make inference using heuristics. Here, we propose a quantitative method for logical inference based on a formal analogy between linear elastic mechanics and Bayesian inference with linear Gaussian variables. To start, we investigate the case of single parameter estimation, where the analogy is stated as follows: the value of the parameter is represented by the position of a cursor bar with one degree of freedom; uncertain pieces of information on the parameter are modelled as linear elastic springs in series or parallel, connected to the bar and each with stiffness equal to its accuracy; the posterior mean value and the accuracy of the parameter correspond respectively to the position of the bar in equilibrium and to the resulting stiffness of the mechanical system composed of the bar and the set of springs. Similarly, a multi-parameter estimation problem is reproduced by a mechanical system with as many degrees of freedom as the number of unknown parameters. In this case, the inverse covariance matrix of the parameters corresponds to the Hessian of the potential energy, while the posterior mean values of the parameters coincide with the equilibrium -or minimum potential energy -position of the mechanical system. We use the mechanical analogy to estimate, in the Bayesian sense, the drift of elongation of a bridge cable-stay undergoing continuous monitoring. We demonstrate how we can solve this in the same way as any other linear Bayesian inference problem, by simply expressing the potential energy of the equivalent mechanical system, with a few trivial algebraic steps and with the same methods of structural mechanics. We finally discuss the extension of the method to non-Gaussian estimation problems.

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Section: Lognormal Distributionmentioning

confidence: 99%

Section: Uniform Distributionmentioning

confidence: 99%

“…The Cauchy distribution (Forbes et al, 2011) is of mathematical interest due to the absence of defined moments. Its probability density function takes the following form: …”

Section: Cauchy Distributionmentioning

confidence: 99%

Section: Triangular Distributionmentioning

confidence: 99%

See 2 more Smart Citations

Non Gaussian multi-parameter Bayesian estimation through the mechanical equivalent of logical inference

Bolognani¹,

Verzobio²,

Cappello³

et al. 2017

Proceedings of the Joint COST TU1402 - COST TU1406 - IABSE WC1 Workshop: The Value of Structural Health Monitoring for the Reli

View full text Add to dashboard Cite

show abstract

“…Based on Hertzian indentation tests it has been suggested that flaw size in glass can be closely fitted by a Pareto distribution (Poloniecki and Wilshaw 1971;Poloniecki 1974;Tandon et al 2013). The Pareto distribution has the scale and shape parameters a 0 > 0 and c > 0 and the distribution function is (Forbes et al 2010)…”

Section: Introductionmentioning

confidence: 99%

A numerical method for analysis of fracture statistics of glass and simulations of a double ring bending test

Kinsella

Persson

2018

Glass Struct Eng

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The results from a new numerical method for simulating the strength and fracture locations of small glass specimens subjected to double ring bending are compared with experimental data. The method implements the weakest-link principle while assuming the existence of Griffith flaws. A Weibull distribution for the strength is simulated based on a single population of Pareto distributed crack sizes. The effect of using different fracture criteria is investigated. An alternative distribution is simulated based on two populations of flaws. This distribution models the apparent bimodality in the empirical data set. The numerical method is dependent on a representation of the surface flaws condition in glass. As new techniques become available for examining the surface characteristics, this numerical method is promising as a means for modelling the strength better than current methods do.

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Physician Characteristics Associated With Ordering 4 Low-Value Screening Tests in Primary Care

et al. 2018

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Key Points Question Do physicians who order a high frequency of 1 low-value screening test also order a high frequency of other low-value screening tests? Findings In this cohort study of 2394 primary care physicians, 18.4% of the physicians were in the top ordering quintile of at least 2 of 4 low-value screening tests. These physicians ordered 39.2% of all low-value screening tests. Meaning The study findings suggest that efforts to reduce low-value care should consider strategies that focus on physicians who order a high frequency of low-value care.

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Statistical Distributions

Cited by 576 publications

References 0 publications

Non Gaussian multi-parameter Bayesian estimation through the mechanical equivalent of logical inference

Non Gaussian multi-parameter Bayesian estimation through the mechanical equivalent of logical inference

A numerical method for analysis of fracture statistics of glass and simulations of a double ring bending test

Physician Characteristics Associated With Ordering 4 Low-Value Screening Tests in Primary Care

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