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.
Definition and properties Given a random variable X ∈ R n {\displaystyle {\boldsymbol {X}}\in \mathbb {R} ^{n}} that follows a multivariate normal distribution N n ( μ , Σ ) {\displaystyle {\mathcal {N}}_{n}({\boldsymbol {\mu }},\,{\boldsymbol {\Sigma }})} , the projected normal distribution P N n ( μ , Σ ) {\displaystyle {\mathcal {PN}}_{n}({\boldsymbol {\mu }},{\boldsymbol {\Sigma }})} represents the distribution of the random variable Y = X ‖ X ‖ {\displaystyle {\boldsymbol {Y}}={\frac {\boldsymbol {X}}{\lVert {\boldsymbol {X}}\rVert }}} obtained projecting X {\displaystyle {\boldsymbol {X}}} over the unit sphere. In the general case, the projected normal distribution can be asymmetric and multimodal. In case μ {\displaystyle {\boldsymbol {\mu }}} is parallel to an eigenvector of Σ {\displaystyle {\boldsymbol {\Sigma }}} , the distribution is symmetric. The first version of such distribution was introduced in Pukkila and Rao (1988).
Support The support of this distribution is the unit (n-1)-sphere, which can be variously given in terms of a set of ( n − 1 ) {\displaystyle (n-1)} -dimensional angular spherical coordinates:
Θ = [ 0 , π ] n − 2 × [ 0 , 2 π ) ⊂ R n − 1 {\displaystyle {\boldsymbol {\Theta }}=[0,\pi ]^{n-2}\times [0,2\pi )\subset \mathbb {R} ^{n-1}}
or in terms of n {\displaystyle n} -dimensional Cartesian coordinates:
S n − 1 = { z ∈ R n : ‖ z ‖ = 1 } ⊂ R n {\displaystyle \mathbb {S} ^{n-1}=\{{\boldsymbol {z}}\in \mathbb {R} ^{n}:\lVert {\boldsymbol {z}}\rVert =1\}\subset \mathbb {R} ^{n}}
The two are linked via the embedding function, e : Θ → R n {\displaystyle e:{\boldsymbol {\Theta }}\to \mathbb {R} ^{n}} , with range e ( Θ ) = S n − 1 . {\displaystyle e({\boldsymbol {\Theta }})=\mathbb {S} ^{n-1}.} This function is defined by the formula for spherical coordinates at r = 1. {\displaystyle r=1.}
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