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Johnson System of Distributions Johnson Curves

Johnson (1949) developed a flexible system of distributions, based on three families of transformations, that translate an observed, non-normal variate to one conforming to the standard normal distribution. The exponential, logistic, and hyperbolic sine transformations are used to generate log-normal (SL), unbounded (SU), and bounded (SB) distributions, respectively. The coefficients defining a Johnson distribution consist of two shape (γ, ?), a location (ξ), and a scale (λ) parameter. This allows a unique distribution to be derived for whatever combination of mean, standard deviation, skewness, and kurtosis occurs for a given set of observed data. Once a variate is appropriately transformed, probability densities and percentage points may be derived based on the standard normal curve.

Johnson’s (1949) original procedure for determining the transformation coefficients was based on moments derived from the observed data and he used a graphical calculator (i.e., an abaque) to perform his calculations. Draper (1952) suggested algebraic formulae to replace the abaque for increased accuracy. Hill et al. (1976) provided a FORTRAN algorithm to fit Johnson curves based on moments and Hill (1976) published a companion program for transforming observed (Johnson) variates to their standard normal counterparts, and vice versa. Wheeler (1980) derived an alternative method of fitting Johnson distributions to data based on quantiles instead of moments.

The flexibility inherent in the Johnson system of distributions offers a compelling alternative to the conventional distributions routinely employed in the analysis of real-world data sets. It has potential for widespread use in a variety of disciplines, including aerospace engineering (Tielrooij et al. 2015), atmospheric chemistry (Mage, 1980), bioinformatics (George & Ramachandran, 2008; Marko & Weil, 2012), biomechanics (Stanfield et al., 1996), biomedical engineering (Breton & Kovatchev, 2008), climate modeling (Liu, 2012), econometrics (Lu, et al., 2008; Simonato, 2011), engineering (Farnum, 1996), forest science (Hafley & Schreuder, 1977), management science (Alexopoulos et al., 2008), materials science (Matthews et al., 2006), occupational hygiene (Flynn, 2007), psychometrics (den Oord, 2005), and remote sensing (Ben-David & Davidson, 2012).

The Johnson Curve Toolbox for Matlab is a set of Matlab functions for working with the Johnson family of distributions to analyze non-normal, univariate data sets. Portions of it are based on my port of the AS 99 (Hill et al., 1976) and AS 100 (Hill, 1976) FORTRAN-66 code. The Toolbox provides support for fitting Johnson curves to data based on moments or quantiles; using Johnson transformations to convert Johnson variates to normal variates (and vice versa); generating random numbers from Johnson distributions; calculating probability densities (PDF), cumulative probability densities (CDF), and inverse CDF’s; and calculating likelihoods and goodness-of-fit measures. Examples of fitting Johnson curves to biological, environmental, demographic, and financial data are also provided.

Citation

Jones, D. L. 2014. Johnson Curve Toolbox for Matlab: analysis of non-normal data using the Johnson family of distributions. College of Marine Science, University of South Florida, St. Petersburg, Florida, USA.



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