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.

  • 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.

  • Glossary

    • Parallel box plots are very useful for comparing distributions.
    • A linear transformation is any transformation of a variable that can be achieved by multiplying it by a constant, and then adding a second constant.
    • If Y is the transformed value of X, then $Y=aX+b$.
    • Two or more box plots drawn on the same Y-axis.
    • Two or more box plots drawn on the same Y-axis.

  • Exploratory Data Analysis (EDA)

    • EDA is different from initial data analysis (IDA), which focuses more narrowly on checking assumptions required for model fitting and hypothesis testing, handling missing values, and making transformations of variables as needed.

  • Introduction to data solutions

    • 1.39 The histogram shows that the distribution is bimodal, which is not apparent in the box plot.
    • The box plot makes it easy to identify more precise values of observations outside of the whiskers.
    • Since the distribution is already unimodal and symmetric, a log transformation is not necessary.