Introduction and Primer II

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

• Start with a unit circle U.
• Scatter that circle by applying R to U.
  
• Apply A and make an elliptical scatterplot RUA.
• RUA is a Bivariate Normal.

Notes:

• U is a n x 2 matrix containing n random draws from the uniform circle.
• When R is a n x n matrix with a random sample of size n from a $\chi$(df=2) distribution on the diagonal (and 0 off-diagonal), RU is the bivariate normal distribution G(0,I) and appears as a roughly spherical scatterplot (mvpd uses the letter G to denote Gaussian).
• RUA is the result of the matrix multiplication between n x 2 RU and 2 x 2 A, the Cholesky decomposition of a 2x2 matrix Q.
  • A makes a circle (U) into an ellipse (UA)
  • A can be thought of as a “square root” of a matrix. A’A = Q.
• RUA is the bivariate normal distribution G(0,Q) (mvpd uses the letter G to denote Gaussian).
• Changing the distribution in R changes the multivariate distribution of RUA . That’s what this package is all about!

You can change the order of application of A and R – check out the other intro.