Box–Muller transformation

(noun)

A pseudo-random number sampling method for generating pairs of independent, standard, normally distributed (zero expectation, unit variance) random numbers, given a source of uniformly distributed random numbers.

Related Terms

Examples of Box–Muller transformation in the following topics:

  • The Uniform Distribution

    • A general method is the inverse transform sampling method, which uses the cumulative distribution function (CDF) of the target random variable.
    • The normal distribution is an important example where the inverse transform method is not efficient.
    • However, there is an exact method, the Box–Muller transformation, which uses the inverse transform to convert two independent uniform random variables into two independent normally distributed random variables.

  • Box-Cox Transformations

    • George Box and Sir David Cox collaborated on one paper (Box, 1964).
    • The Box-Cox transformation of the variable x is also indexed by λ, and is defined as
    • Rewriting the Box-Cox formula as
    • Examples of the Box-Cox transformation xʹ versus x for λ = −1, 0, 1.
    • Examples of the Box-Cox transformation versus log(x) for −2 < λ< 3.

  • References

    • Box, G.
    • An analysis of transformations, Journal of the Royal Statistical Society, Series B, 26, 211-252.

  • The Payne Rearrangement

    • The first diagram below illustrates three such reactions, and a general mechanism is written in the gray-shaded box.
    • As expected for an SN2, process, these transformations are stereospecific.
    • The course of reaction in the absence of the Payne rearrangement is displayed in the gray-shaded box.

  • Domesticity and "Domestics"

    • Part of the separate spheres ideology, the cult of domesticity identified the home as women's "proper sphere. " Prescriptive literature advised women on how to transform their homes into domestic sanctuaries for their husbands and children.
    • These laws, as well as subsequent Supreme Court rulings such as Muller v.

  • Prostaglandins

    • A rough outline of some of the transformations that take place is provided below.
    • It is helpful to view arachadonic acid in the coiled conformation shown in the shaded box.

  • Class Cubozoa and Class Hydrozoa

    • Cubozoans live as box-shaped medusae while Hydrozoans are true polymorphs and can be found as colonial or solitary organisms.
    • Class Cubozoa includes jellies that have a box-shaped medusa: a bell that is square in cross-section; hence, they are colloquially known as "box jellyfish."
    • Polyp forms then transform into the medusoid forms.

  • Examples

    • Suppose we want the compute the Fourier transform of a box-shaped function.
    • Here is a result which is a special case of a more general theorem telling us how the Fourier transform scales.
    • Now compute the Fourier transform of $f$ :
    • Thus, making the function more peaked in the space/time domain makes the Fourier transform more broad; while making the function more broad in the space/time domain, makes it more peaked in the Fourier domain.

  • Transforming data values

    • Transform>Recode is a very flexible and general purpose tool for recoding values in any UCINET data structure. its dialog box has two tabs: "Files" and "Recode. "
    • Transform>Reverse recodes the values of selected rows, columns, and matrices so that the highest value is now the lowest, the lowest is now the highest, and all other values are linearly transformed.
    • The choices that are available in the Transform>Symmetrize tool are:
    • This transformation, though it may seem odd at first, is quite helpful.
    • >Upper > Lower or >Upper (and similar functions available in the dialog box) compare the values in cell AB and BA, and return one or the other based on the test function.

  • Examining numerical data exercises

    • Create a box plot for the data given in Exercise 1.30.
    • What features are apparent in the box plot but not in the histogram?
    • Describe the distribution and comment on whether or not a log transformation may be advisable for these data.
    • Describe the distribution and comment on why we might want to use log-transformed values in analyzing or modeling these data.
    • Since the dis- tribution is already unimodal and symmetric, a log transformation is not necessary. 1.45 Answers will vary.