Metalog distribution
The metalog distribution is a flexible continuous probability distribution designed for ease of use in practice. Together with its transforms, the metalog family of continuous distributions is unique because it embodies all of following properties: virtually unlimited shape flexibility; a choice among unbounded, semi-bounded, and bounded distributions; ease of fitting to data with linear least squares; simple, closed-form quantile function equations that facilitate simulation; a simple, closed-form PDF; and Bayesian updating in closed form in light of new data. Moreover, like a Taylor series, metalog distributions may have any number of terms, depending on the degree of shape flexibility desired and other application needs.
Applications where metalog distributions can be useful typically involve fitting empirical data, simulated data, or expert-elicited quantiles to smooth, continuous probability distributions. Fields of application are wide-ranging, and include economics, science, engineering, and numerous other fields. The metalog distributions, also known as the Keelin distributions, were first published in 2016 by Tom Keelin.
History
The history of probability distributions can be viewed, in part, as a progression of developments towards greater flexibility in shape and bounds when fitting to data. The normal distribution was first published in 1756, and Bayes' theorem in 1763. The normal distribution laid the foundation for much of the development of classical statistics. In contrast, Bayes' theorem laid the foundation for the state-of-information, belief-based probability representations. Because belief-based probabilities can take on any shape and may have natural bounds, probability distributions flexible enough to accommodate both were needed. Moreover, many empirical and experimental data sets exhibited shapes that could not be well matched by the normal or other continuous distributions. So began the search for continuous probability distributions with flexible shapes and bounds.Early in the 20th century, the Pearson family of distributions, which includes the normal, beta, uniform, gamma, student-t, chi-square, F, and five others, emerged as a major advance in shape flexibility. These were followed by the Johnson distributions. Both families can represent the first four moments of data with smooth continuous curves. However, they have no ability to match fifth or higher-order moments. Moreover, for a given skewness and kurtosis, there is no choice of bounds. For example, matching the first four moments of a data set may yield a distribution with a negative lower bound, even though it might be known that the quantity in question cannot be negative. Finally, their equations include intractable integrals and complex statistical functions, so that fitting to data typically requires iterative methods.
Early in the 21st century, decision analysts began working to develop continuous probability distributions that would exactly fit any specified three points on the cumulative distribution function for an uncertain quantity. The Pearson and the Johnson family distributions were generally inadequate for this purpose. In addition, decision analysts also sought probability distributions that would be easy to parameterize with data. Introduced in 2011, the class of quantile-parameterized distributions accomplished both goals. While being a significant advance for this reason, the QPD originally used to illustrate this class of distributions, the Simple Q-Normal distribution, had less shape flexibility than the Pearson and Johnson families, and lacked the ability to represent semi-bounded and bounded distributions. Shortly thereafter, Keelin developed the family of metalog distributions, another instance of the QPD class, which is more shape-flexible than the Pearson and Johnson families, offers a choice of boundedness, has closed-form equations that can be fit to data with linear least squares, and has closed-form quantile functions, which facilitate Monte Carlo simulation.
Definition and quantile function
The metalog distribution is a generalization of the logistic distribution, where the term "metalog" is short for "metalogistic". Starting with the logistic quantile function,, Keelin substituted power series expansions in cumulative probability for the and the parameters, which control location and scale, respectively.Keelin's rationale for this substitution was fivefold. First, the resulting quantile function would have significant shape flexibility, governed by the coefficients. Second, it would have a simple closed form that is linear in these coefficients, implying that they could easily be determined from CDF data by linear least squares. Third, the resulting quantile function would be smooth, differentiable, and analytic, ensuring that a smooth, closed-form PDF would be available. Fourth, simulation would be facilitated by the resulting closed-form inverse CDF. Fifth, like a Taylor series, any number of terms could be used, depending on the degree of shape flexibility desired and other application needs.
Note that the subscripts of the -coefficients are such that and are in the expansion, and are in the expansion, and subscripts alternate thereafter. This ordering was chosen so that the first two terms in the resulting metalog quantile function correspond to the logistic distribution exactly; adding a third term with adjusts skewness; adding a fourth term with adjusts kurtosis primarily; and adding subsequent non-zero terms yields more nuanced shape refinements.
Rewriting the logistic quantile function to incorporate the above substitutions for and yields the metalog quantile function, for cumulative probability.
Equivalently, the metalog quantile function can be expressed in terms of basis functions:, where the metalog basis functions are and each subsequent is defined as the expression that is multiplied by in the equation for above. Note that coefficient is the median, since all other terms equal zero when. Special cases of the metalog quantile function are the logistic distribution and the uniform distribution.
Probability density function
Differentiating with respect to yields the quantile density function. The reciprocal of this quantity,, is the probability density function expressed as a p-PDF,which may be equivalently expressed in terms of basis functions as
Note that this PDF is expressed as a function of cumulative probability,, rather than variable of interest,. To plot the PDF, one can vary parametrically, and then plot on the horizontal axis and on the vertical axis.
Based on the above equations and the following transformations that enable a choice of bounds, the family of metalog distributions is composed of unbounded, semibounded, and bounded metalogs, along with their symmetric-percentile triplet special cases.
Unbounded, semi-bounded, and bounded metalog distributions
As defined above, the metalog distribution is unbounded, except in the unusual special case where for all terms that contain. However, many applications require flexible probability distributions that have a lower bound, an upper bound, or both. To meet this need, Keelin used transformations to derive semi-bounded and bounded metalog distributions. Such transformations are governed by a general property of quantile functions: for any quantile function and increasing function is also a quantile function. For example, the quantile function of the normal distribution is ; since the natural logarithm,, is an increasing function, is the quantile function of the lognormal distribution. Analogously, applying this property to the metalog quantile function using the transformations below yields the semi-bounded and bounded members of the metalog family. By considering to be metalog-distributed, all members of the metalog family meet Keelin and Powley's definition of a quantile-parameterized distribution and thus possess the properties thereof.Note that the number of shape parameters in the metalog family increases linearly with the number of terms. Therefore, any of the above metalogs may have any number of shape parameters. By contrast, the Pearson and Johnson families of distributions are limited to two shape parameters.
SPT metalog distributions
The symmetric-percentile triplet metalog distributions are a three-term special case of the unbounded, semi-bounded, and bounded metalog distributions. These are parameterized by the three points off the CDF curve, of the form,, and, where. SPT metalogs are useful when, for example, quantiles corresponding to the CDF probabilities are elicited from an expert and used to parameterize the three-term metalog distributions. As noted below, certain mathematical properties are simplified by the SPT parameterization.Properties
The metalog family of probability distributions has the following properties.Feasibility
A function of the form of or any of its above transforms is a feasible probability distribution if and only if its PDF is greater than zero for all This implies a feasibility constraint on the set of coefficients,In practical applications, feasibility must generally be checked rather than assumed. For, ensures feasibility. For , the feasibility condition is and. For, a similar closed form has been derived. For, feasibility is typically checked graphically or numerically.
The unbounded metalog and its above transforms share the same set of feasible coefficients. Therefore, for a given set of coefficients, confirming that for all is sufficient regardless of the transform in use.