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:

    “What is the good of drawing conclusions from experience? I don’t deny we sometimes draw the right conclusions, but don’t we just as often draw the wrong ones?”
    —G.C. (Georg Christoph)

    “We must be physicists in order ... to be creative since so far codes of values and ideals have been constructed in ignorance of physics or even in contradiction to physics.”
    —Friedrich Nietzsche (1844–1900)

    “The man who pretends that the distribution of income in this country reflects the distribution of ability or character is an ignoramus. The man who says that it could by any possible political device be made to do so is an unpractical visionary. But the man who says that it ought to do so is something worse than an ignoramous and more disastrous than a visionary: he is, in the profoundest Scriptural sense of the word, a fool.”
    —George Bernard Shaw (1856–1950)