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Due primarily to the devastating destructions of the extreme climate events, a number of studies have focused on identifying, validating and predicting extreme climate events and have found a number of indicators to analyze the extreme climate evolutions and variability. Some indicators called quantile indices, which count exceeded days above percentile-based threshold in a fixed period (i.e., TX10p, TX90p, TN10p, and TN90p), have been found an artificial inhomogeneity at the beginning and end of base period where their quantiles are calculated. Zhang et.al(2005) proposed a bootstrap method to remove it which has been highly recommended and its dataset have been released and available online. In this paper, a new systematic and physical modeling method is preferred, and it shows that the method we put forward is effective and robust both in the produced and climate data.

In Zhang's study, he produced 60-year daily data through lag 1-day auto regression AR (1) process: (1), with autocorrelation * *and white noise *Z*. In counting exceeding days over quantile-based (90^{th}-quantile for example) threshold in a year, computing daily threshold is the primary interest. The original method taking 90^{th} quantile of first 30-year daily observations (base period) as threshold will inevitably result in a sharp "jump" of exceedance rate in boundary of in-base and out-of-base period (Fig 1; black line). No doubt, no rapid change takes place because AR (1) process is smoothing, so the sharp "jump" completely results from threshold estimation method. Zhang et al. (2005) judged that biased threshold in base period is the key of inhomogeneity where sample in counting exceedance used repeatedly in computing its corresponding threshold. By extracting the data out of estimating its threshold in base period, the inhomogeneity of exceedance rate is successfully removed, that is bootstrap method.

Fig1. Average exceedance rate of daily values greater than 90^{th} quantile in 1000 simulations of lag 1-day auto regression process with autocorrelation 0.8. A sharp "jump" of the original method (black line) is shown in the boundary of in-base (the first 30 years) and out-of-base period (the second 30 years), while our quantile regression method (red line) remains stable around 10%.

However, Zhang's correcting only in base period is slightly subjective without powerful theoretical supporting. In base period, threshold on same day in the first 30 years differs, but out of base period, it remains same threshold on same day. Correction in base period with different threshold estimations across whole period may also introduce additional discontinuity between in-base and out-of-base period. In our opinion, quantile regression should be taken into consideration to capture distribution of quantile-based threshold rather than focus on the original. We assume that *q*^{th}-quantile of AR series *X _{q}|_{t }*has relationship with lag 1-3 day series

*X*|

*,*

_{t-1}*X*|

*, and*

_{t-2}*X*|

*, the quantile regression model should be built as: (2), where*

_{t-3}*b*is the regression coefficient. Fitting the model with base-period data and applying it to out-of-base period data, thresholds of the whole period could be output according to quantile regression. Then, the exceedance rate is easily obtained. Contrary to the original method, our quantile regression method indicates no visible inhomogeneity (Fig 1; red line). It is an exciting result because no correction is needed partly (but Zhang's bootstrap need), and the quantile regression model is consistently applied across the in-base and out-of-base period. Thus, our threshold estimation process is systematic without artificial systematic error and the threshold varies every day as indicated in Eq. (2). What is more, relating the quantile to the former behaviors (

*X*|

*,*

_{t-1}*X*|

*, and*

_{t-2}*X*|

*) is physically meaningful. For example, when comes a hot wave, a lower 90*

_{t-3}^{th}-quantile threshold is supposed to be due to the lower former records, and greater observation should be counted into the exceedance, when it wakens, higher threshold is supposed and it may below the threshold could not be counted in. The physical process is vividly reflected in the regressive process of quantile. Therefore, taking systematic error and physical meaning into account, our quantile-regression method makes great progress.

Fig2. The exceedance rate of the pencentile indices (a) warm days (TX90p), (b) warm nights (TN90p), (c) cold days (TX10p), (d) cold nights (TN10p). Differnces are displayed between the original method (black line), quantile regression method (red line) and bootstrap method (green line). Their base period is 1961-1990.

Different to produced data, annual cycle and daily variation is important in temperature data. It is flexible for our method to add a cubic spline function of days to the Eq. (2) describing seasonal cycle. In this part, we show our robustness of TX10p, TX90p, TN10p, and TN90p indices based on grid data located at 118.125°E, 32.1°N of BNU-ESM under CMIP5. Fig2 shows difference of original, bootstrap and quantile-regression method in exceedance rates. Shocked by little change between the original and bootstrap method, they both contain large random fluctuation. Though 5-day-window samples are used in Zhang's to improve stability, random effect still remains in quantile that may cover the real change. But our quantile-regression model is to estimate the average mode of quantile smoothing the random effect. As clearly shown in Fig 2(b), after 1990, Zhang's method containing more random effects with 15% mean exceedance rate of TN90p indicates increase warm-night events, while ours is still around 10% without obvious increase. The larger-biased of Zhang's method is a common phenomenon. Calculating spatial average values of TN90p from 2080 to 2100 in Chinese region based on BNU-ESM, we find that the exceedance rate based on Zhang's is 1.8 and 1.3 is larger than ours in the scenario of rcp45 and rcp85 respectively. Therefore, our modeling method is a more robust approach to reduce the random effects.

In conclusion, our quantile-regression method is a superior to remove inhomogeneity in quantile indices with the advantage of overall physical meaning and robustness. These released quantile indices are highly recommended to use our quantile-regression method to recalculate for accurate estimation of extreme climate change.