Evaluation of heterogeneity statistics as reasonable proxies of the error of precipitation quantile estimation in the Minneapolis-St. Paul region

Summary

Estimating precipitation frequency is important in engineering, agriculture, land use planning, and many other disciplines. The index flood method alleviates small sample size issues due to short record length by calculating normalized quantile estimates for averaged data from a “region” of gauges. For a perfectly homogeneous region this adds no error; heterogeneity statistics seek to quantify a real-world region’s deviation from this assumption. Hosking and Wallis (1997) introduced a Monte Carlo heterogeneity statistic called here $H_1$ and used a simulation study to assess its utility while rejecting two similar statistics called here $H_2$ and $H_3$. A nearly linear relationship was found between $H_1$ and the percentage root mean square error (RMSE) increase due to heterogeneity, establishing $H_1$ as a “reasonable proxy” of quantile error. The $H_1$-percent RMSE added relationship found in the simulation experiment was used to find equivalent RMSEs for heterogeneity thresholds against which all three H statistics were tested. In this study the “reasonable proxy” relationship is evaluated across a highly skewed daily precipitation dataset in Minnesota for $H_1\text{,}{H}_2$ and $H_3$. Simulated regions used in quantile error estimation are generated using at-site L-moment ratios scaled toward the regional mean with a shrinkage multiplier. A linear relationship is found between Monte Carlo estimates of quantile RMSE and both $H_1$ and $H_2$ across all possible regionalizations of twelve gauges. $H_2$’s relationship is less linear than $H_1$’s as quantified by Pearson’s r. A synthetic study is also undertaken using the same sample sizes, regional L-moment averages, and between-site variations as the Hosking and Wallis (1997) simulation. The $H_2$-percent RMSE added relationship is found to be nearly as linear as for $H_1$, complementing the enumeration study’s findings. Because $H_2$’s linear relationship with percent RMSE added has approximately one-fourth the slope of the$H_1$-RMSE relationship, heterogeneity thresholds calculated with reference to $H_1$ should not be applied to $H_2$. $H_2$ thresholds can be derived from the $H_2$-percent RMSE added relationship in analogous fashion to the method used in Hosking and Wallis (1997) for $H_1$. The resulting thresholds are one-fourth the magnitude of the $H_1$ thresholds.

Introduction

Rare or extreme precipitation events, which include events classified as natural disasters, have major ecological, economic, and public safety significance. Sample size is often a limiting factor in the estimation of extreme hydrological events; one rule of thumb for flood frequency estimation is that for reliable estimates of a return period T the record length in station-years must exceed $5T$ (Robson and Reed, 1999). Many statistical hydrologists have followed the index flood method of Dalrymple (1960), in which sample size is increased by grouping gauges, or“sites”, into regions, calculating a regional “growth curve” normalized by an index such as the mean or median of the at-gauge data, and estimating at-site quantiles by multiplying the index flood and the regional growth curve.

Linear moments, analogues of conventional central moments like skewness and kurtosis based on probability weighted moments (Greenwood et al., 1979), are often used in this context. L-moment estimators have lower bias than other common methods of estimation at small sample size (Hosking et al., 1985, Lettenmaier et al., 1987). They are less biased than conventional moment estimators, are not bounded by sample size, and are more robust to outliers. L-moment ratios can be more reliably predicted from a subsample than conventional moment ratios. L-moment ratios, the second through fourth of which are denoted the coefficient of L-variance (L-CV), L-skewness, and L-kurtosis (the first L-moment ratio does not exist), provide greater insight into the underlying distribution of high-skew data than conventional moment ratios. For example, L-moment ratio diagrams are used as decision aides for identifying the underlying distribution of regional data (Hosking, 1990, Vogel and Fennessey, 1993, Hosking and Wallis, 1997, Zafirakou-Koulouris et al., 1998).

L-moment analysis of regions formed according to hydrological characteristics has been conducted in recent decades on streamflow (Vogel et al., 1993, Ouarda et al., 2008, Noto and Loggia, 2009) and precipitation data (Guttman et al., 1993, Werick et al., 1994, Adamowski et al., 1996, Alila, 1999, Smithers and Schulze, 2001, Kyselý et al., 2007, Modarres and Sarhadi, 2011). L-moment ratios for daily data series using “wet-day” (non-zero only) and full datasets have been evaluated across the United States (Hanson and Vogel, 2008). Regional frequency analysis models using fuzzy regions (Jingyi and Hall, 2004, Rao and Srinivas, 2006) and fractional-membership regions of influence (Burn, 1990, Zrinji and Burn, 1994, Gaál et al., 2008) represent alternatives to the strict regional membership model.

The regional pooling mechanism of the index flood method involves the assumption of homogeneity across the sites in a candidate region - at-site differences in L-moment ratios are assumed to be due solely to sampling variability. The degree to which the homogeneity assumption is violated is therefore likely to be related to quantile error. Statistics quantifying the heterogeneity of a region based on Monte Carlo simulation have been proposed which sample from a Generalized extreme-value distribution (Lu and Stedinger, 1992, Alila, 1999).

Hosking and Wallis (1997) define three statistics based on the between-site variation of L-moment ratios, $H_1\text{,}{H}_2$, and $H_3$. All three H statistics fit the flexible four-parameter Kappa distribution with the average L-moment ratios of the region in question and use Monte Carlo simulation to generate simulated regions from the Kappa. For the real region and for each simulated region a statistic called $V_1\text{,}{V}_2$, or $V_3$ is calculated using the sum of the squared difference between each site’s L-moment ratio values and the regional average. $V_1$ uses only the L-CV, $V_2$ incorporates L-CV and L-skewness, and $V_3$ incorporates L-skewness and L-kurtosis. $H_1$ is calculated when $V_1$ for the real region minus the mean of $V_1$ for simulated regions is divided by the standard deviation of simulated regions’ $V_1\text{;}{H}_2$ and $H_3$ are calculated analogously (see Eqs. (6), (7), (8), (9)).

A simulation study is used in Hosking and Wallis (1997) to reject $H_2$ and $H_3$ and to accept $H_1$. Heterogeneous regions’ RMSEs are divided by their equivalent homogeneous region’s RMSE. This isolates the RMSE increase due to heterogeneity. $H_1$ is shown to have a linear relationship with percent RMSE added due to heterogeneity. Results for $H_2$ and $H_3$ are not reported.

Hosking and Wallis (1997) define thresholds below which regions can be considered “possibly” and “definitely” heterogeneous with reference to a range of percent RMSE added values implied by the $H_1$-percent RMSE added relationship. $H_1=1$ is found to indicate a 20–40% increase in RMSE, while $H_1=2$ is associated with 40–80% increases. These thresholds are also applied to $H_2$ and $H_3$, which are found to rarely exceed them.

Viglione et al. (2007) investigate $H_1$ and $H_2$ as well as two nonparametric heterogeneity statistics by measuring the fraction of simulated regions that are correctly and incorrectly identified as heterogeneous. The threshold of $H=2$ is used for both $H_1$ and $H_2$. They confirm the utility of $H_1$ for simulated data with L-skewness below 0.23 and reject $H_2$. The bootstrap Anderson–Darling test is found to be more powerful than either statistic for data with higher skewness.

Two approaches are used in this study to quantify the power of $H_1\text{,}{H}_2$, and $H_3$ as proxies of error due to heterogeneity. The original Hosking and Wallis (1997) simulation study is recapitulated and results for $H_2$ and $H_3$ are presented alongside those for $H_1$. Thresholds for $H_2$ are found using its linear relationship to quantile error, not $H_1$’s. An enumeration study is also conducted, estimating $H_1\text{,}{H}_2\text{,}{H}_3$, and quantile error for all possible regionalizations of a small daily precipitation gauge dataset. Components of error unrelated to heterogeneity are preserved in this study, allowing the heterogeneity statistics’ relationships with total estimated quantile error to be compared to the ideal case represented in the simulation experiment.

The remainder of this paper is structured as follows. Daily precipitation gauge data from which all possible regions are to be enumerated are presented in the following section, Section 2. Section 3 introduces the equations and methods used to calculate linear moments of the data, estimate regional heterogeneity, assign a regional distribution, estimate the RMSE of regional quantile estimates, and perform a simulation experiment analogous to that presented in Hosking and Wallis (1997). Section 4 presents the results first of the simulation experiment, then of the enumeration experiment, which compares heterogeneity and error estimates across all possible regions formed from the selected precipitation gauges. Section 5 discusses the results; Section 6 summarizes the paper and offers conclusions and potential avenues of future research.