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A set of probabilities along with corresponding quantiles are often used to define predictive distributions or probabilistic forecasts. These quantile predictions offer easily interpreted uncertainty of an event, and quantiles are generally straightforward to estimate using standard statistical and machine learning methods. However, compared to a distribution defined by a probability density or cumulative distribution function, a set of quantiles has less distributional information. When given estimated quantiles, it may be desirable to estimate a fully defined continuous distribution function. Many researchers do so to make evaluation or ensemble modeling simpler. Most existing methods for fitting a distribution to quantiles lack accurate representation of the inherent uncertainty from quantile estimation or are limited in their applications. In this manuscript, we present a Gaussian process model, the quantile Gaussian process –based on established asymptotic results of quantile functions and sample quantiles– to construct a probability distribution given estimated quantiles. In some applications, the form of an unknown distribution function from which sample quantiles are drawn must be estimated, for which case we propose the use of a latent truncated Dirichlet process mixture model for estimation. A Bayesian application of the quantile Gaussian process is evaluated for parameter inference and distribution approximation in simulation studies as well as in a real data analysis of quantile forecasts from the 2023-24 US Centers for Disease Control collaborative flu forecasting initiative. The simulation studies and data analysis show that compared to other existing methods, the quantile Gaussian process leads to accurate inference on model parameters, estimation of a continuous distribution, and uncertainty quantification of sample quantiles.
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Computing Accurate Probabilistic Estimates of One-D Entropy from Equiprobable Random Samples
We develop a simple Quantile Spacing (QS) method for accurate probabilistic estimation of one-dimensional entropy from equiprobable random samples, and compare it with the popular Bin-Counting (BC) and Kernel Density (KD) methods. In contrast to BC, which uses equal-width bins with varying probability mass, the QS method uses estimates of the quantiles that divide the support of the data generating probability density function (pdf) into equal-probability-mass intervals. And, whereas BC and KD each require optimal tuning of a hyper-parameter whose value varies with sample size and shape of the pdf, QS only requires specification of the number of quantiles to be used. Results indicate, for the class of distributions tested, that the optimal number of quantiles is a fixed fraction of the sample size (empirically determined to be ~0.25–0.35), and that this value is relatively insensitive to distributional form or sample size. This provides a clear advantage over BC and KD since hyper-parameter tuning is not required. Further, unlike KD, there is no need to select an appropriate kernel-type, and so QS is applicable to pdfs of arbitrary shape, including those with discontinuous slope and/or magnitude. Bootstrapping is used to approximate the sampling variability distribution of the resulting entropy estimate, and is shown to accurately reflect the true uncertainty. For the four distributional forms studied (Gaussian, Log-Normal, Exponential and Bimodal Gaussian Mixture), expected estimation bias is less than 1% and uncertainty is low even for samples of as few as 100 data points; in contrast, for KD the small sample bias can be as large as −10% and for BC as large as −50%. We speculate that estimating quantile locations, rather than bin-probabilities, results in more efficient use of the information in the data to approximate the underlying shape of an unknown data generating pdf.
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- Award ID(s):
- 1740858
- PAR ID:
- 10294762
- Date Published:
- Journal Name:
- Entropy
- Volume:
- 23
- Issue:
- 6
- ISSN:
- 1099-4300
- Page Range / eLocation ID:
- 740
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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- Free Publicly Accessible Full Text
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Accepted Manuscript
- Journal Article:
- https://doi.org/10.3390/e23060740
