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Projected normal distribution

In directional statistics, the projected normal distribution (also known as offset normal distribution, angular normal distribution or angular Gaussian distribution) is a probability distribution over directions that describes the radial projection of a random variable with n-variate normal distribution over the unit (n-1)-sphere.

Measurement, Density function & Angular Central Gaussian Distribution

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Notation
P N n ( μ , Σ ) {\displaystyle {\mathcal {PN}}_{n}({\boldsymbol {\mu }},{\boldsymbol {\Sigma }})}
Parameters
μ ∈ R n {\displaystyle {\boldsymbol {\mu }}\in \mathbb {R} ^{n}} (location) Σ ∈ R n × n {\displaystyle {\boldsymbol {\Sigma }}\in \mathbb {R} ^{n\times n}} (scale)
PDF
complicated, see text
Support
Unit n-sphere, with angular or Cartesian coordinates: Θ = [ 0 , π ] n − 2 × [ 0 , 2 π ) {\displaystyle {\boldsymbol {\Theta }}=[0,\pi ]^{n-2}\times [0,2\pi )} S n − 1 = { z ∈ R…

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Overview

Definition and properties

Density function

Angular Central Gaussian Distribution

Sources

  • Doi Doi (identifier)
  • PMC PMC (identifier)
  • PMID PMID (identifier)
  • JSTOR JSTOR (identifier)
  • ArXiv ArXiv (identifier)

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Projected normal distribution

Nodes59
Edges58
Triples77
Avg. degree1.97
Density0.033898
Components1

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Projected normal distribution

Top relations

related to Sources · 52
Projected normal distribution → Alan, Arbitrary Dimension, Armand, BA989, Bayesian Analysis, Bayesian Inference, Biometrika, Breidt, Daniel, David, Directional, Distribution, Draxler, Elsevier, Fangpo, Felix, Free-Form Flows, Gaussian, Gelfand, Hernandez-Stumpfhauser
related to Angular Central Gaussian Distribution · 8
Projected normal distribution → ACG, Cartesian, Gaussian, In, Let, Notice, Sigma, We
related to Definition and properties · 7
Projected normal distribution → Given, In, PN, Pukkila, Rao, Sigma, The
related to Density function · 6
Projected normal distribution → In, PN, Sigma, The, Theta, To
Notation · 1
Projected normal distribution → P N n ( μ , Σ ) {\displaystyle {\mathcal {PN}}_{n}({\boldsymbol {\mu }},{\boldsymbol {\Sigma }})}
Parameters · 1
Projected normal distribution → μ ∈ R n {\displaystyle {\boldsymbol {\mu }}\in \mathbb {R} ^{n}} (location) Σ ∈ R n × n {\displaystyle {\boldsymbol {\Sigma }}\in \mathbb {R} ^{n\times n}} (scale)
PDF · 1
Projected normal distribution → complicated, see text
Support · 1
Projected normal distribution → Unit n-sphere, with angular or Cartesian coordinates: Θ = [ 0 , π ] n − 2 × [ 0 , 2 π ) {\displaystyle {\boldsymbol {\Theta }}=[0,\pi ]^{n-2}\times [0,2\pi )} S n − 1 = { z ∈ R…

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Important terminology

displaystyle boldsymbol density mathbf distribution n-1 theta sigma measure function normal mathbb acg mu angular lebesgue space operatorname mathcal det

Entity relationships Subject–Predicate–Object triples

SubjectPredicateObjectConfidenceSrc
Projected normal distributionNotationP N n ( μ , Σ ) {\displaystyle {\mathcal {PN}}_{n}({\boldsymbol {\mu }},{\boldsymbol {\Sigma }})}1.00infobox
Projected normal distributionParametersμ ∈ R n {\displaystyle {\boldsymbol {\mu }}\in \mathbb {R} ^{n}} (location) Σ ∈ R n × n {\displaystyle {\boldsymbol {\Sigma }}\in \mathbb {R} ^{n\times n}} (scale)1.00infobox
Projected normal distributionPDFcomplicated, see text1.00infobox
Projected normal distributionSupportUnit n-sphere, with angular or Cartesian coordinates: Θ = [ 0 , π ] n − 2 × [ 0 , 2 π ) {\displaystyle {\boldsymbol {\Theta }}=[0,\pi ]^{n-2}\times [0,2\pi )} S n − 1 = { z ∈ R…1.00infobox
Projected normal distributionrelated to Angular Central Gaussian DistributionIn0.60section
Projected normal distributionrelated to Angular Central Gaussian DistributionGaussian0.60section
Projected normal distributionrelated to Angular Central Gaussian DistributionACG0.60section
Projected normal distributionrelated to Angular Central Gaussian DistributionCartesian0.60section
Projected normal distributionrelated to Angular Central Gaussian DistributionLet0.60section
Projected normal distributionrelated to Angular Central Gaussian DistributionSigma0.60section
Projected normal distributionrelated to Angular Central Gaussian DistributionWe0.60section
Projected normal distributionrelated to Angular Central Gaussian DistributionNotice0.60section

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