Empirical Quantile Mapping

Authorship: Kunal Konar, Consultant (Business Development & Analytic Hydrology)

Note: This article was first appeared as a LinkedIn post about 4 years back.

Empirical Quantile Mapping (eQM) is a non-parametric statistical transformation technique whose origin may be (at that is as old as I have found through literature review) traced back to the work of Brier & Panofsky in 1968 (Brier & Panofsky, 1968). Brier & Panofsky argued that any cumulative distribution function (CDF) may be transformed into any other CDF, of a given form, by suitable transformation or functional relationships without knowing the functional form of the transformation. They achieved this transformation using empirical CDF of observed variables and defining the transformation or mapping as “the essential feature of the transformation of a variate from one distribution to a variate with a distribution of prescribed form is that the probability of being less than a given value of the variate shall be the same as the probability of being less than the corresponding value of the transformed variate.” Brier & Panofsky termed this transformation as “equiprobability transformation”.

In more recent time, Boe et al. (Boé et al., 2007) consolidated this concept of “equiprobability transformation” in their work and termed it as “quantile-quantile mapping transformation”. The functional form of eQM adopted by Boe et al. may be expressed as:

Figure-1: Empirical Quantile Mapping (Boé et al., 2007)

The authors applied the eQM at daily level, but independently for four seasons within a year. Themeβl et al. (Themeßl et al., 2012) adopted a slightly different procedure where eQM has been applied to each day of a typical year independently. Themeβl et al. accomplished this by applying a 31-day wide centered moving window for each day of year. In this formulation,

the functional form of eQM becomes,

Figure-2: Empirical Quantile Mapping (Themeßl et al., 2012)

That is, in this formulation, 366 functional relationships need to be estimated from the data in the calibration period.

This moving-window formulation of Themeβl et al. (2012) that I have found as one of most generic and useful algorithms for any type of bias correction for climatological or hydrometeorological application.

Computational Implementation

As I understand, probably there are number of implementations of the eQM algorithm. But, previously I have used qmap package (Gudmundsson, 2016) as implemented in R environment (R Core Team, 2017).

References

• Boé, J., Terray, L., Habets, F., & Martin, E. (2007). Statistical and dynamical downscaling of the Seine basin climate for hydro-meteorological studies. Int J Climatol, 27(12), 1643–1655.

• Brier, W., & Panofsky, H. (1968). Some Applications of Statistics to Meteorology; The Pennsylvania State University Press.

• Gudmundsson, L. (2016). qmap: Statistical transformations for post-processing climate model output. R package version 1.0-4. Vienna, Austria: Comprehensive R Archive Network.

• R Core Team. (2017). R: A language and environment for statistical computing. Vienna, Austria: R Foundation for Statistical Computing. Retrieved from https://www.R- project.org/

• Themeßl, M., Gobiet, A., & Heinrich, G. (2012). Empirical-statistical downscaling and error correction of regional climatemodels and its impact on the climate change signal. Clim. Change, 112(2), 449–468.