quantile-parameterized distribution

The development of quantile-parameterized distributions was inspired by the practical need for flexible continuous probability distributions that are easy to fit to data. Historically, the PearsonJohnson NL, Kotz S, Balakrishnan N. Continuous univariate distributions, Vol 1, Second Edition, John Wiley & Sons, Ltd, 1994, pp. 15–25. and Johnson{{cite journal | url=https://www.jstor.org/stable/2332539 | jstor=2332539 | title=Systems of Frequency Curves Generated by Methods of Translation | last1=Johnson | first1=N. L. | journal=Biometrika | year=1949 | volume=36 | issue=1/2 | pages=149–176 | doi=10.2307/2332539 | pmid=18132090 | url-access=subscription }}{{cite journal | url=https://www.jstor.org/stable/2335422 | jstor=2335422 | title=Systems of Frequency Curves Generated by Transformations of Logistic Variables | last1=Tadikamalla | first1=Pandu R. | last2=Johnson | first2=Norman L. | journal=Biometrika | year=1982 | volume=69 | issue=2 | pages=461–465 | doi=10.1093/biomet/69.2.461 | url-access=subscription }} families of distributions have been used when shape flexibility is needed. That is because both families can match the first four moments (mean, variance, skewness, and kurtosis) of any data set. In many cases, however, these distributions are either difficult to fit to data or not flexible enough to fit the data appropriately.

For example, the beta distribution is a flexible Pearson distribution that is frequently used to model percentages of a population. However, if the characteristics of this population are such that the desired cumulative distribution function (CDF) should run through certain specific CDF points, there may be no beta distribution that meets this need. Because the beta distribution has only two shape parameters, it cannot, in general, match even three specified CDF points. Moreover, the beta parameters that best fit such data can be found only by nonlinear iterative methods.

Practitioners of decision analysis, needing distributions easily parameterized by three or more CDF points (e.g., because such points were specified as the result of an expert-elicitation process), originally invented quantile-parameterized distributions for this purpose. Keelin and Powley (2011){{Cite journal |doi=10.1287/deca.1110.0213 |title=Quantile-Parameterized Distributions |year=2011 |last1=Keelin |first1=Thomas W. |last2=Powley |first2=Bradford W. |journal=Decision Analysis |volume=8 |issue=3 |pages=206–219 }} provided the original definition. Subsequently, Keelin (2016){{Cite journal |doi=10.1287/deca.2016.0338 |title=The Metalog Distributions |year=2016 |last1=Keelin |first1=Thomas W. |journal=Decision Analysis |volume=13 |issue=4 |pages=243–277 }} developed the metalog distributions, a family of quantile-parameterized distributions that has virtually unlimited shape flexibility, simple equations, and closed-form moments.

Keelin and Powley define a quantile-parameterized distribution as one whose quantile function (inverse CDF) can be written in the form

: $$

F^{-1} (y)= \left\{

\begin{array}{cl}

L_0 & \text{for } y=0\\

\sum_{i=1}^n a_i g_i(y) & \text{for } 0

L_1 & \mbox{for } y=1

\end{array}\right.

where

: $$

\begin{array}{rcl}

L_0 &=& \lim_{y\rarr 0^+} F^{-1}(y) \\

L_1 &=& \lim_{y\rarr 1^-} F^{-1}(y)

\end{array}

and the functions $g\_i(y)$ are continuously differentiable and linearly independent basis functions. Here, essentially, $L\_0$ and $L\_1$ are the lower and upper bounds (if they exist) of a random variable with quantile function $F^\{-1\}(y)$. These distributions are called quantile-parameterized because for a given set of quantile pairs $\backslash \{(x\_i,\; y\_i)\; \backslash mid\; i=1,\backslash ldots,n\backslash \}$, where $x\_i=F^\{-1\}(y\_i)$, and a set of $n$ basis functions $g\_i(y)$, the coefficients $a\_i$ can be determined by solving a set of linear equations. If one desires to use more quantile pairs than basis functions, then the coefficients $a\_i$ can be chosen to minimize the sum of squared errors between the stated quantiles $x\_i$ and $F^\{-1\}(y\_i)$. Keelin and Powley illustrate this concept for a specific choice of basis functions that is a generalization of quantile function of the normal distribution, $x=\backslash mu+\backslash sigma\; \backslash Phi^\{-1\}\; (y)$, for which the mean $\backslash mu$ and standard deviation $\backslash sigma$ are linear functions of cumulative probability $y$:

: $\backslash mu(y)=a\_1+a\_4\; y$

: $\backslash sigma(y)=a\_2+a\_3\; y$

The result is a four-parameter distribution that can be fit to a set of four quantile/probability pairs exactly, or to any number of such pairs by linear least squares. Keelin and Powley call this the Simple Q-Normal distribution. Some skewed and symmetric Simple Q-Normal PDFs are shown in the figures below.

File:SymmetricSimpleQNormalFree.png

File:Skewed_Simple_Q_Normal_Distributions.png

QPD’s that meet Keelin and Powley’s definition have the following properties.

Differentiating $x=F^\{-1\}\; (y)=\backslash sum\_\{i=1\}^n\; a\_i\; g\_i\; (y)$ with respect to $y$ yields $dx/dy$. The reciprocal of this quantity, $dy/dx$, is the probability density function (PDF)

: $f(y)\; =\; \backslash left(\; \backslash sum\_\{i=1\}^n$

a_i {{d g_i(y)}\over{dy}} \right)^{-1}

where

A function of the form of $F^\{-1\}\; (y)$ is a feasible probability distribution if and only if $f(y)>0$ for all $y\; \backslash in\; (0,1)$. This implies a feasibility constraint on the set of coefficients $\backslash boldsymbol\; a=(a\_1,\backslash ldots,a\_n)\; \backslash in\; \backslash R^n$:

: $\backslash sum\_\{i=1\}^n\; a\_i\; \{\{d\; g\_i(y)\}\backslash over\{dy\}\}\; >0$ for all $y\; \backslash in\; (0,1)$

In practical applications, feasibility must generally be checked rather than assumed.

A QPD’s set of feasible coefficients $S\_\backslash boldsymbol\; a=\backslash \{\backslash boldsymbol\; a\backslash in\backslash R^n\; \backslash mid\; \backslash sum\_\{i=1\}^n\; a\_i\; d\; g\_i\; (y)/dy\; >\; 0$ for all $y\backslash in\; (0,1)\backslash \}$ is convex. Because convex optimization requires convex feasible sets, this property simplifies optimization applications involving QPDs.

The coefficients $\backslash boldsymbol\; a$ can be determined from data by linear least squares. Given $m$ data points $(x\_i,y\_i)$ that are intended to characterize the CDF of a QPD, and $m\; \backslash times\; n$ matrix $\backslash boldsymbol\; Y$ whose elements consist of $g\_j\; (y\_i)$, then, so long as $\backslash boldsymbol\; Y^T\; \backslash boldsymbol\; Y$ is invertible, coefficients' column vector $\backslash boldsymbol\; a$ can be determined as $\backslash boldsymbol\; a=(\backslash boldsymbol\; Y^T\; \backslash boldsymbol\; Y)^\{-1\}\; \backslash boldsymbol\; Y^T\; \backslash boldsymbol\; x$, where $m\backslash geq\; n$ and column vector $\backslash boldsymbol\; x=(x\_1,\backslash ldots,x\_m)$. If $m=n$, this equation reduces to $\backslash boldsymbol\; a=\backslash boldsymbol\; Y^\{-1\}\; \backslash boldsymbol\; x$, where the resulting CDF runs through all data points exactly. An alternate method, implemented as a linear program, determines the coefficients by minimizing the sum of absolute distances between the CDF and the data subject to feasibility constraints.{{Cite thesis |url=https://searchworks.stanford.edu/view/13257318 |title=Cyber risk management :AI-generated warnings of threats |year=2019 |publisher=Stanford University |last1=Faber |first1=Isaac Justin |last2=Paté-Cornell |first2=M. Elisabeth |last3=Lin |first3=Herbert |last4=Shachter |first4=Ross D. }}

A QPD with $n$ terms, where $n\backslash ge\; 2$, has $n-2$ shape parameters. Thus, QPDs can be far more flexible than the Pearson distributions, which have at most two shape parameters. For example, ten-term metalog distributions parameterized by 105 CDF points from 30 traditional source distributions (including normal, student-t, lognormal, gamma, beta, and extreme value) have been shown to approximate each such source distribution within a K–S distance of 0.001 or less.{{Cite journal |doi=10.1287/deca.2016.0338|at=Table 8 |title=The Metalog Distributions |year=2016 |last1=Keelin |first1=Thomas W. |journal=Decision Analysis |volume=13 |issue=4 }}

QPD transformations are governed by a general property of quantile functions: for any quantile function $x=Q(y)$ and increasing function $t(x),\; x=t^\{-1\}\; (Q(y))$ is a quantile function.Gilchrist, W., 2000. Statistical modelling with quantile functions. CRC Press. For example, the quantile function of the normal distribution, $x=\backslash mu+\backslash sigma\; \backslash Phi^\{-1\}\; (y)$, is a QPD by the Keelin and Powley definition. The natural logarithm, $t(x)=\backslash ln(x-b\_l)$, is an increasing function, so $x=b\_l+e^\{\backslash mu+\backslash sigma\; \backslash Phi^\{-1\}\; (y)\}$ is the quantile function of the lognormal distribution with lower bound $b\_l$. Importantly, this transformation converts an unbounded QPD into a semi-bounded QPD. Similarly, applying this log transformation to the unbounded metalog distribution{{Cite journal |doi=10.1287/deca.2016.0338|at=Section 3, pp. 249–257 |title=The Metalog Distributions |year=2016 |last1=Keelin |first1=Thomas W. |journal=Decision Analysis |volume=13 |issue=4 }} yields the semi-bounded (log) metalog distribution;{{Cite journal |doi=10.1287/deca.2016.0338|at=Section 4 |title=The Metalog Distributions |year=2016 |last1=Keelin |first1=Thomas W. |journal=Decision Analysis |volume=13 |issue=4 }} likewise, applying the logit transformation, $t(x)=\backslash ln((x-b\_l)/(b\_u-x))$, yields the bounded (logit) metalog distribution with lower and upper bounds $b\_l$ and $b\_u$, respectively. Moreover, by considering $t(x)$ to be $F^\{-1\}\; (y)$ distributed, where $F^\{-1\}\; (y)$ is any QPD that meets Keelin and Powley’s definition, the transformed variable maintains the above properties of feasibility, convexity, and fitting to data. Such transformed QPDs have greater shape flexibility than the underlying $F^\{-1\}\; (y)$, which has $n-2$ shape parameters; the log transformation has $n-1$ shape parameters, and the logit transformation has $n$ shape parameters. Moreover, such transformed QPDs share the same set of feasible coefficients as the underlying untransformed QPD.[http://metalogdistributions.com/images/Powley_Dissertation_2013-augmented.pdf Powley, B.W. (2013). “Quantile Function Methods For Decision Analysis”. Corollary 12, p 30. PhD Dissertation, Stanford University]

The $k^\{th\}$ moment of a QPD is:

: $E[x^k]\; =\; \backslash int\_0^1\; \backslash left(\; \backslash sum\_\{i=1\}^n\; a\_i\; g\_i(y)\; \backslash right)^k\; dy$

Whether such moments exist in closed form depends on the choice of QPD basis functions $g\_i\; (y)$. The unbounded metalog distribution and polynomial QPDs are examples of QPDs for which moments exist in closed form as functions of the coefficients $a\_i$.

Since the quantile function $x=F^\{-1\}(y)$ is expressed in closed form, Keelin and Powley QPDs facilitate Monte Carlo simulation. Substituting in uniformly distributed random samples of $y$ produces random samples of $x$ in closed form, thereby eliminating the need to invert a CDF expressed as $y=F(x)$.

The following probability distributions are QPDs according to Keelin and Powley’s definition:

• The quantile function of the normal distribution, $x=\backslash mu+\backslash sigma\; \backslash Phi^\{-1\}\; (y)$.
• The quantile function of the Gumbel distribution, $x=\backslash mu\; -\; \backslash beta\; \backslash ln(-\backslash ln(y))$.
• The quantile function of the Cauchy distribution, $x=x\_0+\backslash gamma\; \backslash tan[\backslash pi(y-0.5)]$.
• The quantile function of the logistic distribution, $x=\backslash mu+s\; \backslash ln(y/(1-y)\; )$.
• The unbounded metalog distribution, which is a power series expansion of the $\backslash mu$ and $s$ parameters of the logistic quantile function.
• The semi-bounded and bounded metalog distributions, which are the log and logit transforms, respectively, of the unbounded metalog distribution.
• The SPT (symmetric-percentile triplet) unbounded, semi-bounded, and bounded metalog distributions, which are parameterized by three CDF points and optional upper and lower bounds.
• The Simple Q-Normal distribution{{Cite journal |doi=10.1287/deca.1110.0213|at=pp. 208–210 |title=Quantile-Parameterized Distributions |year=2011 |last1=Keelin |first1=Thomas W. |last2=Powley |first2=Bradford W. |journal=Decision Analysis |volume=8 |issue=3 }}
• The metadistributions, including the meta-normal{{Cite journal |page=253 |doi=10.1287/deca.2016.0338 |title=The Metalog Distributions |year=2016 |last1=Keelin |first1=Thomas W. |journal=Decision Analysis |volume=13 |issue=4 }}
• Quantile functions expressed as polynomial functions of cumulative probability $y$, including Chebyshev polynomial functions.

Like the SPT metalog distributions, the Johnson Quantile-Parameterized Distributions{{cite journal | url=https://pubsonline.informs.org/doi/abs/10.1287/deca.2016.0343 | doi=10.1287/deca.2016.0343 | title=Johnson Quantile-Parameterized Distributions | year=2017 | last1=Hadlock | first1=Christopher C. | last2=Bickel | first2=J. Eric | journal=Decision Analysis | volume=14 | pages=35–64 | url-access=subscription }}{{cite journal | url=https://pubsonline.informs.org/doi/abs/10.1287/deca.2018.0376 | doi=10.1287/deca.2018.0376 | title=The Generalized Johnson Quantile-Parameterized Distribution System | year=2019 | last1=Hadlock | first1=Christopher C. | last2=Bickel | first2=J. Eric | journal=Decision Analysis | volume=16 | pages=67–85 | s2cid=159339224 | url-access=subscription }} (JQPDs) are parameterized by three quantiles. JQPDs do not meet Keelin and Powley’s QPD definition, but rather have their own properties. JQPDs are feasible for all SPT parameter sets that are consistent with the rules of probability.

The original applications of QPDs were by decision analysts wishing to conveniently convert expert-assessed quantiles (e.g., 10th, 50th, and 90th quantiles) into smooth continuous probability distributions. QPDs have also been used to fit output data from simulations in order to represent those outputs (both CDFs and PDFs) as closed-form continuous distributions.Keelin, T.W. (2016), Section 6.2.2, pp. 271–274. Used in this way, they are typically more stable and smoother than histograms. Similarly, since QPDs can impose fewer shape constraints than traditional distributions, they have been used to fit a wide range of empirical data in order to represent those data sets as continuous distributions (e.g., reflecting bimodality that may exist in the data in a straightforward mannerKeelin, T.W. (2016), Section 6.1.1, Figure 10, pp 266–267.). Quantile parameterization enables a closed-form QPD representation of known distributions whose CDFs otherwise have no closed-form expression. Keelin et al. (2019){{cite book | url=https://dl.acm.org/doi/abs/10.5555/3400397.3400643 | isbn=9781728132839 | title=The metalog distributions and extremely accurate sums of lognormals in closed form | date=18 May 2020 | pages=3074–3085 | last1=Mustafee | first1=N. | publisher=Institute of Electrical and Electronics Engineers (IEEE) }} apply this to the sum of independent identically distributed lognormal distributions, where quantiles of the sum can be determined by a large number of simulations. Nine such quantiles are used to parameterize a semi-bounded metalog distribution that runs through each of these nine quantiles exactly. QPDs have also been applied to assess the risks of asteroid impact,{{cite journal | url=https://doi.org/10.1111/risa.12453 | doi=10.1111/risa.12453 | title=Asteroid Risk Assessment: A Probabilistic Approach | year=2016 | last1=Reinhardt | first1=Jason C. | last2=Chen | first2=Xi | last3=Liu | first3=Wenhao | last4=Manchev | first4=Petar | last5=Paté-Cornell | first5=M. Elisabeth | journal=Risk Analysis | volume=36 | issue=2 | pages=244–261 | pmid=26215051 | bibcode=2016RiskA..36..244R | s2cid=23308354 | url-access=subscription }} cybersecurity,{{cite journal | url=https://www.sciencedirect.com/science/article/pii/S0167404819300604 | doi=10.1016/j.cose.2019.101659 | title=A Bayesian network approach for cybersecurity risk assessment implementing and extending the FAIR model | year=2020 | last1=Wang | first1=Jiali | last2=Neil | first2=Martin | last3=Fenton | first3=Norman | journal=Computers & Security | volume=89 | page=101659 | s2cid=209099797 }} biases in projections of oil-field production when compared to observed production after the fact,{{Cite journal |url=https://www.onepetro.org/journal-paper/SPE-195914-PA |doi=10.2118/195914-PA |title=Production Forecasting: Optimistic and Overconfident—Over and over Again |year=2020 |last1=Bratvold |first1=Reidar B. |last2=Mohus |first2=Erlend |last3=Petutschnig |first3=David |last4=Bickel |first4=Eric |journal=Spe Reservoir Evaluation & Engineering |volume=23 |issue=3 |pages=0799–0810 |s2cid=219661316 |url-access=subscription }} and future Canadian population projections based on combining the probabilistic views of multiple experts.{{Cite book |url=https://library.oapen.org/bitstream/handle/20.500.12657/42565/2020_Book_DevelopmentsInDemographicForec.pdf?sequence=1#page=51 |title=Developments in Demographic Forecasting |year=2020 |isbn=978-3-030-42471-8 |series=The Springer Series on Demographic Methods and Population Analysis |volume=49 |pages=43–62 |doi=10.1007/978-3-030-42472-5 |hdl=20.500.12657/42565 |s2cid=226615299}} See metalog distributions and Keelin (2016) for additional applications of the metalog distribution.

• The Metalog Distributions, [http://www.metalogs.org www.metalogs.org]

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Category:Continuous distributions

Category:Systems of probability distributions