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Introduction and Primer II

Start with a unit circle. End with a Bivariate Normal.

Source: vignettes/articles/intro1.Rmd
intro1.Rmd

  • U is a nx2 matrix containing n random draws from the uniform circle.
  • B = sqrt(A), where A is a nx1 matrix containing n i.i.d draws from its univariate distribution.
  • BU can be thought of as the distribution of radii
  • BUR is the result of the matrix multiplication between nx2 BU and 2x2 R, the Cholesky decomposition of a 2x2 matrix Q.
    • R makes a circular (BU) into an elliptical (BUR)
    • R can be thought of as a “square root” of a matrix. R’R = Q.
  • When A ∼χ2\sim \chi^2(df=2) distribution, BUR is the bivariate normal distribution G(0,Q). Equivalently,
  • When B ∼χ\sim \chi(df=2) distribution, BUR is the bivariate normal distribution G(0,Q)