Quantile-parameterized distribution: Difference between revisions

Revision as of 19:51, 12 February 2021

Revision as of 11:21, 18 February 2021

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=== Transformations ===

QPD transformations are governed by a general property of quantile functions: for any [[quantile function]] <math>x=Q(y)</math> and increasing function <math>t(x), x=t^{-1} (Q(y))</math> is a [[quantile function]].<ref>Gilchrist, W., 2000. Statistical modelling with quantile functions. CRC Press.</ref> For example, the [[quantile function]] of the [[normal distribution]], <math>x=\mu+\sigma \Phi^{-1} (y)</math>, is a QPD by the Keelin and Powley definition. The natural logarithm, <math>t(x)=\ln(x-b_l)</math>, is an increasing function, so <math>x=b_l+e^{\mu+\sigma \Phi^{-1} (y)}</math> is the [[quantile function]] of the [[Log-normal distribution|lognormal distribution]] with lower bound <math>b_l</math>. Importantly, this transformation converts an unbounded QPD into a semi-bounded QPD. Similarly, applying this log transformation to the [https://en.wikipedia.org/wiki/Metalog_distribution[Metalog distribution#Unbounded,_semibounded,_and_bounded_metalog_distributions |unbounded metalog distribution]]<ref name="UnboundedMetalog">[[doi:10.1287/deca.2016.0338|Keelin, T.W. (2016), Section 3, pp. 249–257.]]</ref> yields the [https://en.wikipedia.org/wiki/Metalog_distribution[Metalog distribution#Unbounded,_semibounded,_and_bounded_metalog_distributions |semi-bounded (log) metalog distribution]];<ref name="KeelinSec4">[[doi:10.1287/deca.2016.0338|Keelin, T.W. (2016), Section 4.]]</ref> likewise, applying the logit transformation, <math>t(x)=\ln((x-b_l)/(b_u-x))</math>, yields the [https://en.wikipedia.org/wiki/Metalog_distribution[Metalog distribution#Unbounded,_semibounded,_and_bounded_metalog_distributions |bounded (logit) metalog distribution]]<ref name="KeelinSec4" /> with lower and upper bounds <math>b_l</math> and <math>b_u</math>, respectively. Moreover, by considering <math>t(x)</math> to be <math>F^{-1} (y)</math> distributed, where <math>F^{-1} (y)</math> 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 <math>F^{-1} (y)</math>, which has <math>n-2</math> shape parameters; the log transformation has <math>n-1</math> shape parameters, and the logit transformation has <math>n</math> shape parameters. Moreover, such transformed QPDs share the same set of feasible coefficients as the underlying untransformed QPD.<ref>[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]</ref>

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* The quantile function of the [[logistic distribution]], <math>x=\mu+s \ln(y/(1-y) )</math>.

* The unbounded [[metalog distribution]], which is a power series expansion of the <math>\mu</math> and <math>s</math> parameters of the logistic quantile function.

* The [https://en.wikipedia.org/wiki/Metalog_distribution[Metalog distribution#Unbounded,_semibounded,_and_bounded_metalog_distributions |semi-bounded and bounded metalog distributions]], which are the log and logit transforms, respectively, of the unbounded metalog distribution.

* The [https://en.wikipedia.org/wiki/Metalog_distribution[Metalog distribution#SPT_metalog_distributions |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<ref>[[doi:10.1287/deca.1110.0213|Keelin, T.W., and Powley, B.W. (2011), pp. 208–210]]</ref>

* The metadistributions, including the meta-normal<ref>[[doi:10.1287/deca.2016.0338|Keelin, T.W. (2016), p. 253.]]</ref>

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== Applications ==

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.<ref>[[doi:10.1287/deca.2016.0338|Keelin, T.W. (2016), Section 6.2.2, pp. 271–274.]]</ref> 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 manner<ref>[[doi:10.1287/deca.2016.0338|Keelin, T.W. (2016), Section 6.1.1, Figure 10, pp 266–267.]]</ref>). Quantile parameterization enables a closed-form QPD representation of known distributions whose CDFs otherwise have no closed-form expression. Keelin et al. (2019)<ref>[https://dl.acm.org/doi/abs/10.5555/3400397.3400643 Keelin, T.W., Chrisman, L. and Savage, S.L. (2019). “The metalog distributions and extremely accurate sums of lognormals in closed form.” WSC '19: Proceedings of the Winter Simulation Conference. 3074–3085.]</ref> 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,<ref>[[doi:10.1111/risa.12453|Reinhardt, J.D., Chen, X., Liu, W., Manchev, P. and Pate-Cornell, M.E. (2016). “Asteroid risk assessment: A probabilistic approach.” Risk Analysis. 36 (2): 244–261]]</ref> cybersecurity,<ref name="Faber" /><ref>[https://www.sciencedirect.com/science/article/pii/S0167404819300604 Wang, J., Neil, M. and Fenton, N. (2020). “A Bayesian network approach for cybersecurity risk assessment implementing and extending the FAIR model.” Computers & Security. 89: 101659.]</ref> biases in projections of oil-field production when compared to observed production after the fact,<ref>[https://www.onepetro.org/journal-paper/SPE-195914-PA Bratvold, R.B., Mohus, E., Petutschnig, D. and Bickel, E. (2020). “Production forecasting: Optimistic and overconfident—Over and over again.” Society of Petroleum Engineers. doi:10.2118/195914-PA.]</ref> and future Canadian population projections based on combining the probabilistic views of multiple experts.<ref>[https://library.oapen.org/bitstream/handle/20.500.12657/42565/2020_Book_DevelopmentsInDemographicForec.pdf?sequence=1#page=51 Dion, P., Galbraith, N., Sirag, E. (2020). “Using expert elicitation to build long-term projection assumptions.” In Developments in Demographic Forecasting, Chapter 3, pp. 43–62. Springer]</ref> See [https://en.wikipedia.org/wiki/Metalog_distribution[Metalog distribution#Applications |metalog distributions]] and Keelin (2016)<ref name="Keelin2016" /> for additional applications of the metalog distribution.