Multivariate Normal Distribution - Drawing Values From The Distribution

Drawing Values From The Distribution

A widely used method for drawing a random vector x from the N-dimensional multivariate normal distribution with mean vector μ and covariance matrix Σ works as follows:

  1. Find any real matrix A such that A AT = Σ. When Σ is positive-definite, the Cholesky decomposition is typically used, and the extended form of this decomposition can be always be used (as the covariance matrix may be only positive semi-definite) in both cases a suitable matrix A is obtained. An alternative is to use the matrix A = UΛ½ obtained from a spectral decomposition Σ = UΛUT of Σ. The former approach is more computationally straightforward but the matrices A change for different orderings of the elements of the random vector, while the latter approach gives matrices that are related by simple re-orderings. In theory both approaches give equally good ways of determining a suitable matrix A, but there are differences in compuation time.
  2. Let z = (z1, …, zN)T be a vector whose components are N independent standard normal variates (which can be generated, for example, by using the Box–Muller transform).
  3. Let x be μ + Az. This has the desired distribution due to the affine transformation property.

Read more about this topic:  Multivariate Normal Distribution

Famous quotes containing the words drawing, values and/or distribution:

    “It is an inexpressible Pleasure to know a little of the World, and be of no Character or Significancy in it. To be ever unconcerned, and ever looking on new Objects with an endless Curiosity, is a Delight known only to those who are turned for Speculation: Nay, they who enjoy it, must value things only as they are the Objects of Speculation, without drawing any worldly Advantage to themselves from them, but just as they are what contribute to their Amusement, or the Improvement of the Mind.”
    —Richard Steele (1672–1729)

    “When the shingles hissed
    in the rain incendiary,
    other values were revealed to us,”
    —Hilda Doolittle (1886–1961)

    “The question for the country now is how to secure a more equal distribution of property among the people. There can be no republican institutions with vast masses of property permanently in a few hands, and large masses of voters without property.... Let no man get by inheritance, or by will, more than will produce at four per cent interest an income ... of fifteen thousand dollars] per year, or an estate of five hundred thousand dollars.”
    —Rutherford Birchard Hayes (1822–1893)