A Study for Consecutive Precipitation Pattern Based on Stochastic Ordering

: Consecutive precipitation extremes may cause more catastrophes than occasional extreme 8 events. They may pose more serious threats to the safety of people's lives and property. They also can 9 cause great damage to the healthy development of social economy. It is of practical significance to explore 10 this issue. In this work, a nonparametric approach based on stochastic ordering combined with EI Barmi- 11 McKeague test was employed, which is more flexible if the trend is non-monotonic or more complex to 12 model. The average summer consecutive precipitation in 31 provinces of China were compared in three 13 periods, 960-1965, 1985-1990 and 2010-2015. Based on this approach, the results showed that, in 17 out of 14 the 31 provinces, the consecutive precipitation in summer increase stochastically from period 1 to period 15 2 or period 3, or increase stochastically from period 2 to period 3. These 17 provinces mainly located in 16 Northwest and Southeast China. Given the increases in the average summer consecutive precipitation 17 and the high single consecutive precipitation of provinces which located in the Southeast China and socio- 18 economic vulnerability to such extremes in China, the local government and relevant national 19 departments should adopt more strategies to alleviate and adapt to the increasing trend of consecutive 20 precipitation extremes.


Introduction 23
Under the background of global warming, the change of extreme weather and climate 24 events has attracted extensive attention of scholars worldwide. There have been a lot of studies 25 on extreme climate change in China. The precipitation indices include maximum 1-day 26 precipitation, annual total wet-day precipitation, the number of heavy precipitation days and 27 very wet days. Most of the studies are about the frequency or intensity of extreme weather 28 events, such as the frequency of extreme wet events [8], trends of maximum 1-day precipitation, 29 annual total wet-day precipitation, the number of heavy precipitation days and very wet days 30 [9], the tendency of annual mean and extreme precipitation [10]. 31 Much work on the trends of precipitation also has been conducted in different regions, 32 and has shown that the trends in precipitation were uneven in both space and time. Gong et al. 33 (2004)[11]studied the daily precipitation changes and found out that the precipitation amounts 34 in the semi-arid region over northern China show slightly decreasing trends. There are almost 35 no significant trends in annual mean and extreme precipitation in the Zhujiang River Basin [10]. 36 The regional maximum 1-day precipitation and annual total wet-day precipitation on average, 37 show insignificant increases in the arid area of northwestern China [9]. In Sichuan Province, the 38 characteristics of the total summer precipitation, extreme summer precipitation days, and 39 summer extreme precipitation intensity were inconspicuous, while the extreme precipitation 40 in the late-21st century exhibited a certain degree of increase [12]. During the summer monsoon 41 period, extreme wet events exhibit a slight decreasing trend with fluctuations in Southwestern 42 China [8]. Shi et al. (2018) [1] analyzed the temporal and spatial distributions and tendencies in 43 the consecutive temperature and precipitation extremes in China during 1961-2015, which 44 calculated linear trends of consecutive days of precipitation extremes. Insignificant decreasing 45 trends are also found for consecutive dry days in the arid area of northwestern China [9]. 46 Delta [13]. 48 Consecutive dry days and consecutive wet days are the two indices most frequently 49 involved in the studies for consecutive precipitation. Previous studies on extreme weather and 50 climate events have suggested that consecutive temperature or precipitation extremes may 51 cause more catastrophes than occasional extreme events, and will pose more serious threats to 52 the safety of people's lives and property and the healthy development of social economy [6,7]. 53 Here, we focus on the consecutive precipitation in summer over China in this work. We would 54 like to provide references for the scientific research on consecutive precipitation change and to 55 improve the risk prevention ability of regional disastrous weathers. 56 The outline of the current article is as follows. In Section 2, we introduce the sources where 57 the data comes from, the average consecutive precipitation, stochastic ordering of random 58 variables and empirical likelihood-based test for stochastic ordering. In Section 3, based on the 59 stochastic ordering and nonparametric test we compared the average summer consecutive 60 precipitation in 31 provinces of China in three periods, 960-1965, 1985-1990 and 2010-2015. 61 Finally, the discussion is presented in Section 4. 62

Study data 64
The observed daily total precipitation data covering 1960-2015 from 756 national key 65 meteorological stations were provided by the National Meteorological Information Center, 66 China Meteorological Administration. Based on a combination of criteria involved in the 67 spatial distribution of meteorological stations and the length, completeness and quality of data 68 series, data actually used in this study was further selected. Potential errors or outliers are taken 69 care of in the validation process. Preliminary quality controls were implemented to check the

Consecutive Precipitation 78
Wet days were defined as the precipitation less than 1 mm/day in many articles [1,2]. The 79 threshold also can be lowered in areas with relatively low precipitation. As considering the 80 vast regional differences in China, a threshold of 0.1mm/day was used in [3]. Since 31 provinces 81 (municipalities or autonomous regions) are involved in our research, we choose the latter 82 threshold, that is 0.1mm/day. There are many researches on the consecutive wet days[4,5], but 83 few researches involve the consecutive precipitation. Here, a consecutive precipitation refers 84 to the total amount of rainfall in the single consecutive wet days. For example, the rainfall in 85 some place starts on June 1 and ends on June 4, then the single consecutive precipitation here 86 is the total amount of rainfall from June 1 to June 4. Since in China, the summer rainfall in most 87 areas is the highest in the whole year, we only focus on the summer season (June to August) 88 here. We take the average value of the single summer consecutive precipitation over all stations 89 in each province, and explore their changes in three periods. 90

Stochastic Ordering of Random Variables 92
There have been many methodologies to study the spatial and temporal characteristics of 93 consecutive precipitation. article, we intend to explore the characteristics of consecutive precipitation for 31 provinces in 99 China. We take Neimenggu province for an example, Figure 2 shows the average consecutive

104
Obviously, the data are very skewed and have a heavy tail. Hence, linear regression model 105 is not a good choice for us. Although the Mann-Kendall (MK) test makes no assumption about 106 the probability distribution, it is only suitable to examine whether a time series has monotonic 107 trend. In the current article, we intend to explore the characteristics of summer consecutive 108 precipitation in 31 provinces, however, it doesn't show any obvious trend or change over time 109 in some provinces. We take Neimenggu as an example, as shown on  into stochastic ordering, a probability concept to sort random variables in an increasing or 123 decreasing order. Stochastic ordering is a powerful statistical procedure and would be helpful 124 that could detect changes, which has a higher power than the existing test procedure. less than or equal to that of another variable, then the random variable is called as stochastically 129 larger than another random variable. For example, the empirical cumulative distribution Surely, this need a formal statistical test to verify it，which will be introduced in the following 135 section. In the current article, we pick three periods of time, 1960-1965, 1985-1990, and 2010-136 2015, and take the summer consecutive precipitation as the random variables. Based on this 137 statistical method Stochastic Ordering, we do a pairwise comparison in these three periods.

Empirical Likelihood-Based Test for Stochastic Ordering 142
We are so lucky that we have resource for reference. Nan Ni  Given two random variables 1 and 2, with cumulative continuous distribution functions 151 1 and 2, if ( 1 > ) ≥ ( 2 > ) for all , or equivalently, 1( ) ≤ 2( ) for all , then the 152 ordering is denoted by 1 ≻ 2 or 1 ≻ 2. The main work to be done next is to test the 153 hypothesis 154 Suppose there are two random independent samples, whose sizes are and cumulative Here is a standard Brownian bridge. By simulations, they got the critical values 1.821 and 177 3.185, for the significance level = . and = .
respectively. If is greater than the 178 critical value, the null hypothesis will be rejected. 179

Examples 180
Let's go back to Figure 3, Figure 4 and Figure 5. We take the summer consecutive 181 precipitation in Neimenggu province as an example. As shown on Figure  test also confirms what be shown on Figure 4 and Figure 5. following three cases will be tested: 4 ≻ 3 , 5 ≻ 3 , 5 ≻ 4 . 203 The next work is divided into two steps.
Step one, for the sake of illustration, we divide 204 31 provinces into 4 different clusters. The basis of our classification is the hierarchical 205 clustering [27]. Provinces with similar average consecutive precipitation will fall into one 206 cluster. After applying the Euclidean distance to measure the similarity of different provinces, 207 we employ Ward's minimum variance method for clustering. We summarize the clustering 208 result in Table 1. The resulting clusters is shown on Figure 6, which is plotted in a dendogram.  Guangdong, Guangxi. The main reason for this difference is that the basis for clustering is 217 different. We divided the four clusters based on the average summer consecutive precipitation 218 in this work, while their basis is the summer daily precipitation. This also shows that our 219 research of this paper is of practical significance. Figure 7 better shows the geographical 220 characteristics of these four clusters than Figure Table 2 in [24]. But the regional 235 characteristics are more obvious. 236 We can see that there exist significant differences among different clusters. In Cluster 1, 239 all provinces have stochastically increasing for summer consecutive precipitation from period 240 1 to either period 2 or period 3, or increasing from period 2 to period 3. In Cluster 2, Only three 241 of the 11 provinces showed an increasing stochastically trend from period 1 to either period 242 2 or period 3, and there is no significant difference between period 2 and 3. In Cluster 3, the 243 summer consecutive precipitation increases stochastically from period 1 to either period 2 or 244 period 3, or increasing from period 2 to period 3 in 6 out of 10 provinces. In Cluster 4, the three 245 provinces Fujian, Guangdong, and Guangxi increase stochastically for summer consecutive 246 precipitation from period 2 to period 3, only Guangxi province have increasing stochastically 247 from period 1 to period 3. None of the five provinces show increasing stochastically from 248 period 1 to period 2. 249 In total, 9 out of the 31 provinces have summer consecutive precipitation increasing 250 stochastically from period 1 to period 2, 10 out of the 31 provinces have summer consecutive 251 precipitation increasing stochastically from period 1 to period 3, 7 out of the 31 provinces have 252 summer consecutive precipitation increasing stochastically from period 2 to period 3. 17 out of 253 the 31 provinces have summer consecutive precipitation increasing stochastically from period 254 1 to period 2 or period 3, or increasing stochastically from period 2 to period 3. We distinguish 255 the 17 provinces from the other 14 provinces by different colors in Figure 8, these 17 provinces 256 are marked green, and the other 14 provinces appear yellow. Furthermore, We note that the 257 summer consecutive precipitation have obviously region difference as shown in Figure 8.

Discussion 265
As a follow-up work of Nan Ni and Hao Zhang[24], we explore the spatial and temporal 266 changes in summer consecutive precipitation by using the method of stochastic ordering, and 267 applied the EI Barmi-McKeague test for stochastic ordering. In this work, we choose the same 268 three periods as in [24], which is 1960-1965, 1985-1990, and 2010-2015. The results show obvious 269 regional characteristics, 17 provinces which located in the North and South, especially in the 270 Northwest and Southeast China, show increasing stochastically from period 1 to period 2 or 271 period 3, or from period 2 to period 3, while the other 14 provinces have no significant 272 stochastically increasing. It is particularly noteworthy that the spatial characteristics are 273 significant for both the average summer consecutive precipitation and the results of EI Barmi-274 McKeague test for stochastic ordering, as shown in Figure 7 and Figure 8 respectively. For the 275 provinces which located in the Northwest China, such as Gansu, Ningxia, Neimenggu, Qinghai, 276 Xinjiang, the average single consecutive precipitation is small and the average total 277 precipitation of summer is only 5-25mm. As we can see in Table 1, the average single  278 consecutive precipitation is 4.5-13mm for provinces in Cluster1, and water resources are scarce 279 in these provinces. Hence the stochastically increasing should probably a good message for 280 them. However, for the provinces which located in the Southeast China, such as Fujian, 281 Guangdong, Guangxi, the average total precipitation of summer can be as high as 60-120mm, 282 the average single consecutive precipitation is 45-57mm. They are rainy provinces, water 283 resource is abundant. analysis for the maximum consecutive precipitation in each month [29]. The Barmi-McKeague 294 test employed in our current article is more flexible for the model, whose trend is non-295 monotonic or more complex, as we can see from the results above.   Figure 1 Please see the Manuscript PDF le for the complete gure caption Note: The designations employed and the presentation of the material on this map do not imply the expression of any opinion whatsoever on the part of Research Square concerning the legal status of any country, territory, city or area or of its authorities, or concerning the delimitation of its frontiers or boundaries. This map has been provided by the authors. Please see the Manuscript PDF le for the complete gure caption Figure 5 Please see the Manuscript PDF le for the complete gure caption Figure 6 Please see the Manuscript PDF le for the complete gure caption Figure 7 Please see the Manuscript PDF le for the complete gure caption Note: The designations employed and the presentation of the material on this map do not imply the expression of any opinion whatsoever on the part of Research Square concerning the legal status of any country, territory, city or area or of its authorities, or concerning the delimitation of its frontiers or boundaries. This map has been provided by the authors. Please see the Manuscript PDF le for the complete gure caption Note: The designations employed and the presentation of the material on this map do not imply the expression of any opinion whatsoever on the part of Research Square concerning the legal status of any country, territory, city or area or of its authorities, or concerning the delimitation of its frontiers or boundaries. This map has been provided by the authors.